%------------------------------------------------------------------------------
% File     : ITP287_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_SuccPredImperative 00461_025495
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0094_VEBT_SuccPredImperative_00461_025495 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 13108 (5044 unt;2178 typ;   0 def)
%            Number of atoms       : 20216 (9163 equ)
%            Maximal formula atoms :   47 (   1 avg)
%            Number of connectives : 20039 (2130   ~; 331   |;2079   &)
%                                         (2118 <=>;13381  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   6 avg)
%            Maximal term depth    :   36 (   2 avg)
%            Number of FOOLs       : 2188 (1044 fml;1144 var)
%            Number of X terms     : 1317 (   0  [];1017 ite; 300 let)
%            Number of types       :   22 (  21 usr)
%            Number of type conns  : 1993 (1724   >; 269   *;   0   +;   0  <<)
%            Number of predicates  :  225 ( 222 usr;   2 prp; 0-13 aty)
%            Number of functors    : 1966 (1966 usr; 115 con; 0-7 aty)
%            Number of variables   : 37918 (33642   !; 814   ?;37918   :)
%                                         (3462  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TX1_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 21:01:28.601
%------------------------------------------------------------------------------
% Could-be-implicit typings (35)
tff(ty_t_VEBT__BuildupMemImp_OVEBTi,type,
    vEBT_VEBTi: $tType ).

tff(ty_t_VEBT__Definitions_OVEBT,type,
    vEBT_VEBT: $tType ).

tff(ty_t_Heap__Time__Monad_OHeap,type,
    heap_Time_Heap: $tType > $tType ).

tff(ty_t_Code__Numeral_Ointeger,type,
    code_integer: $tType ).

tff(ty_t_Heap_Oheap_Oheap__ext,type,
    heap_ext: $tType > $tType ).

tff(ty_t_Product__Type_Ounit,type,
    product_unit: $tType ).

tff(ty_t_Product__Type_Oprod,type,
    product_prod: ( $tType * $tType ) > $tType ).

tff(ty_t_Numeral__Type_Onum1,type,
    numeral_num1: $tType ).

tff(ty_t_Numeral__Type_Onum0,type,
    numeral_num0: $tType ).

tff(ty_t_Numeral__Type_Obit1,type,
    numeral_bit1: $tType > $tType ).

tff(ty_t_Numeral__Type_Obit0,type,
    numeral_bit0: $tType > $tType ).

tff(ty_t_Extended__Nat_Oenat,type,
    extended_enat: $tType ).

tff(ty_t_Multiset_Omultiset,type,
    multiset: $tType > $tType ).

tff(ty_t_Complex_Ocomplex,type,
    complex: $tType ).

tff(ty_t_Assertions_Oassn,type,
    assn: $tType ).

tff(ty_t_Enum_Ofinite__3,type,
    finite_3: $tType ).

tff(ty_t_Enum_Ofinite__2,type,
    finite_2: $tType ).

tff(ty_t_Enum_Ofinite__1,type,
    finite_1: $tType ).

tff(ty_t_Uint32_Ouint32,type,
    uint32: $tType ).

tff(ty_t_Option_Ooption,type,
    option: $tType > $tType ).

tff(ty_t_Filter_Ofilter,type,
    filter: $tType > $tType ).

tff(ty_t_String_Ochar,type,
    char: $tType ).

tff(ty_t_Heap_Oarray,type,
    array: $tType > $tType ).

tff(ty_t_Word_Oword,type,
    word: $tType > $tType ).

tff(ty_t_Real_Oreal,type,
    real: $tType ).

tff(ty_t_List_Olist,type,
    list: $tType > $tType ).

tff(ty_t_Set_Oset,type,
    set: $tType > $tType ).

tff(ty_t_Rat_Orat,type,
    rat: $tType ).

tff(ty_t_Num_Onum,type,
    num: $tType ).

tff(ty_t_Nat_Onat,type,
    nat: $tType ).

tff(ty_t_Int_Oint,type,
    int: $tType ).

tff(ty_t_itself,type,
    itself: $tType > $tType ).

tff(ty_t_fun,type,
    fun: ( $tType * $tType ) > $tType ).

tff(ty_tf_b,type,
    b: $tType ).

tff(ty_tf_a,type,
    a: $tType ).

% Explicit typings (2143)
tff(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
    comple5582772986160207858norder: 
      !>[A: $tType] : $o ).

tff(sy_cl_Enum_Oenum,type,
    enum: 
      !>[A: $tType] : $o ).

tff(sy_cl_Typerep_Otyperep,type,
    typerep: 
      !>[A: $tType] : $o ).

tff(sy_cl_HOL_Otype,type,
    type: 
      !>[A: $tType] : $o ).

tff(sy_cl_Nat_Osize,type,
    size: 
      !>[A: $tType] : $o ).

tff(sy_cl_HOL_Oequal,type,
    cl_HOL_Oequal: 
      !>[A: $tType] : $o ).

tff(sy_cl_Heap_Oheap,type,
    heap: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Odvd,type,
    dvd: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Oone,type,
    one: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Oidom,type,
    idom: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Oring,type,
    ring: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Oplus,type,
    plus: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Ozero,type,
    zero: 
      !>[A: $tType] : $o ).

tff(sy_cl_Num_Onumeral,type,
    numeral: 
      !>[A: $tType] : $o ).

tff(sy_cl_Power_Opower,type,
    power: 
      !>[A: $tType] : $o ).

tff(sy_cl_Fields_Ofield,type,
    field: 
      !>[A: $tType] : $o ).

tff(sy_cl_GCD_Oring__gcd,type,
    ring_gcd: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Oabs__if,type,
    abs_if: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Oring__1,type,
    ring_1: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Ouminus,type,
    uminus: 
      !>[A: $tType] : $o ).

tff(sy_cl_Orderings_Oord,type,
    ord: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Osemidom,type,
    semidom: 
      !>[A: $tType] : $o ).

tff(sy_cl_Fields_Oinverse,type,
    inverse: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Osemiring,type,
    semiring: 
      !>[A: $tType] : $o ).

tff(sy_cl_Nat_Oring__char__0,type,
    ring_char_0: 
      !>[A: $tType] : $o ).

tff(sy_cl_Num_Oneg__numeral,type,
    neg_numeral: 
      !>[A: $tType] : $o ).

tff(sy_cl_Orderings_Oorder,type,
    order: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Ocomm__ring,type,
    comm_ring: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Omult__zero,type,
    mult_zero: 
      !>[A: $tType] : $o ).

tff(sy_cl_Type__Length_Olen,type,
    type_len: 
      !>[A: $tType] : $o ).

tff(sy_cl_GCD_Osemiring__gcd,type,
    semiring_gcd: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Ogroup__add,type,
    group_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Orderings_Ono__bot,type,
    no_bot: 
      !>[A: $tType] : $o ).

tff(sy_cl_Orderings_Ono__top,type,
    no_top: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Osemiring__0,type,
    semiring_0: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Osemiring__1,type,
    semiring_1: 
      !>[A: $tType] : $o ).

tff(sy_cl_Type__Length_Olen0,type,
    type_len0: 
      !>[A: $tType] : $o ).

tff(sy_cl_Finite__Set_Ofinite,type,
    finite_finite: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Omonoid__add,type,
    monoid_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Ocomm__ring__1,type,
    comm_ring_1: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Oidom__divide,type,
    idom_divide: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Oidom__modulo,type,
    idom_modulo: 
      !>[A: $tType] : $o ).

tff(sy_cl_Transcendental_Oln,type,
    ln: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Omonoid__mult,type,
    monoid_mult: 
      !>[A: $tType] : $o ).

tff(sy_cl_Orderings_Olinorder,type,
    linorder: 
      !>[A: $tType] : $o ).

tff(sy_cl_Orderings_Opreorder,type,
    preorder: 
      !>[A: $tType] : $o ).

tff(sy_cl_Parity_Oring__parity,type,
    ring_parity: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Oidom__abs__sgn,type,
    idom_abs_sgn: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Oordered__ring,type,
    ordered_ring: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Ozero__neq__one,type,
    zero_neq_one: 
      !>[A: $tType] : $o ).

tff(sy_cl_Fields_Ofield__char__0,type,
    field_char_0: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Oab__group__add,type,
    ab_group_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Nat_Osemiring__char__0,type,
    semiring_char_0: 
      !>[A: $tType] : $o ).

tff(sy_cl_Orderings_Oorder__bot,type,
    order_bot: 
      !>[A: $tType] : $o ).

tff(sy_cl_Orderings_Oorder__top,type,
    order_top: 
      !>[A: $tType] : $o ).

tff(sy_cl_Orderings_Owellorder,type,
    wellorder: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Ocomm__semiring,type,
    comm_semiring: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Ozero__less__one,type,
    zero_less_one: 
      !>[A: $tType] : $o ).

tff(sy_cl_Fields_Odivision__ring,type,
    division_ring: 
      !>[A: $tType] : $o ).

tff(sy_cl_Fields_Ofield__abs__sgn,type,
    field_abs_sgn: 
      !>[A: $tType] : $o ).

tff(sy_cl_Num_Osemiring__numeral,type,
    semiring_numeral: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Osemidom__divide,type,
    semidom_divide: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Osemidom__modulo,type,
    semidom_modulo: 
      !>[A: $tType] : $o ).

tff(sy_cl_Orderings_Odense__order,type,
    dense_order: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Ocomm__semiring__0,type,
    comm_semiring_0: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Ocomm__semiring__1,type,
    comm_semiring_1: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Olinordered__idom,type,
    linordered_idom: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Olinordered__ring,type,
    linordered_ring: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Osemiring__modulo,type,
    semiring_modulo: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Ocomm__monoid__add,type,
    comm_monoid_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Parity_Osemiring__parity,type,
    semiring_parity: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Oordered__ring__abs,type,
    ordered_ring_abs: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Oordered__semiring,type,
    ordered_semiring: 
      !>[A: $tType] : $o ).

tff(sy_cl_Code__Evaluation_Oterm__of,type,
    code_term_of: 
      !>[A: $tType] : $o ).

tff(sy_cl_Fields_Olinordered__field,type,
    linordered_field: 
      !>[A: $tType] : $o ).

tff(sy_cl_Generic__set__bit_Oset__bit,type,
    generic_set_set_bit: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Ocomm__monoid__diff,type,
    comm_monoid_diff: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Ocomm__monoid__mult,type,
    comm_monoid_mult: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Oalgebraic__semidom,type,
    algebraic_semidom: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Osemiring__1__cancel,type,
    semiring_1_cancel: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Oab__semigroup__mult,type,
    ab_semigroup_mult: 
      !>[A: $tType] : $o ).

tff(sy_cl_Orderings_Odense__linorder,type,
    dense_linorder: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Olinordered__semidom,type,
    linordered_semidom: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Oordered__semiring__0,type,
    ordered_semiring_0: 
      !>[A: $tType] : $o ).

tff(sy_cl_Least__significant__bit_Olsb,type,
    least_6119777620449941438nt_lsb: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Obanach,type,
    real_Vector_banach: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Olinordered__semiring,type,
    linordered_semiring: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Ocancel__semigroup__add,type,
    cancel_semigroup_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Oordered__ab__group__add,type,
    ordered_ab_group_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Olinordered__semiring__1,type,
    linord6961819062388156250ring_1: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Oordered__comm__semiring,type,
    ordere2520102378445227354miring: 
      !>[A: $tType] : $o ).

tff(sy_cl_Topological__Spaces_Ot1__space,type,
    topological_t1_space: 
      !>[A: $tType] : $o ).

tff(sy_cl_Topological__Spaces_Ot2__space,type,
    topological_t2_space: 
      !>[A: $tType] : $o ).

tff(sy_cl_Bit__Operations_Osemiring__bits,type,
    bit_semiring_bits: 
      !>[A: $tType] : $o ).

tff(sy_cl_Limits_Otopological__group__add,type,
    topolo1633459387980952147up_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Ocomm__semiring__1__cancel,type,
    comm_s4317794764714335236cancel: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Olinordered__ring__strict,type,
    linord4710134922213307826strict: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Ocancel__comm__monoid__add,type,
    cancel1802427076303600483id_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Limits_Otopological__monoid__add,type,
    topolo6943815403480290642id_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Oreal__field,type,
    real_V7773925162809079976_field: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Oring__1__no__zero__divisors,type,
    ring_15535105094025558882visors: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Olinordered__ab__group__add,type,
    linord5086331880401160121up_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Oordered__comm__monoid__add,type,
    ordere6911136660526730532id_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Limits_Otopological__monoid__mult,type,
    topolo1898628316856586783d_mult: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
    real_V4867850818363320053vector: 
      !>[A: $tType] : $o ).

tff(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
    archim2362893244070406136eiling: 
      !>[A: $tType] : $o ).

tff(sy_cl_GCD_Osemiring__gcd__mult__normalize,type,
    semiri6843258321239162965malize: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Oordered__ab__group__add__abs,type,
    ordere166539214618696060dd_abs: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Oordered__ab__semigroup__add,type,
    ordere6658533253407199908up_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Limits_Otopological__ab__group__add,type,
    topolo1287966508704411220up_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Ometric__space,type,
    real_V7819770556892013058_space: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Osemiring__no__zero__divisors,type,
    semiri3467727345109120633visors: 
      !>[A: $tType] : $o ).

tff(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
    boolea8198339166811842893lgebra: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Olinordered__semiring__strict,type,
    linord8928482502909563296strict: 
      !>[A: $tType] : $o ).

tff(sy_cl_Topological__Spaces_Operfect__space,type,
    topolo8386298272705272623_space: 
      !>[A: $tType] : $o ).

tff(sy_cl_Euclidean__Division_Oeuclidean__ring,type,
    euclid5891614535332579305n_ring: 
      !>[A: $tType] : $o ).

tff(sy_cl_Limits_Otopological__semigroup__mult,type,
    topolo4211221413907600880p_mult: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Ocomplete__space,type,
    real_V8037385150606011577_space: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Oreal__algebra__1,type,
    real_V2191834092415804123ebra_1: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Olinordered__nonzero__semiring,type,
    linord181362715937106298miring: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
    semiri2026040879449505780visors: 
      !>[A: $tType] : $o ).

tff(sy_cl_Topological__Spaces_Oorder__topology,type,
    topolo2564578578187576103pology: 
      !>[A: $tType] : $o ).

tff(sy_cl_Bit__Operations_Oring__bit__operations,type,
    bit_ri3973907225187159222ations: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Olinordered__ab__semigroup__add,type,
    linord4140545234300271783up_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Limits_Otopological__comm__monoid__add,type,
    topolo5987344860129210374id_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Orderings_Ounbounded__dense__linorder,type,
    unboun7993243217541854897norder: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Olinordered__semiring__1__strict,type,
    linord715952674999750819strict: 
      !>[A: $tType] : $o ).

tff(sy_cl_Archimedean__Field_Oarchimedean__field,type,
    archim462609752435547400_field: 
      !>[A: $tType] : $o ).

tff(sy_cl_Bit__Comprehension_Obit__comprehension,type,
    bit_bi6583157726757044596ension: 
      !>[A: $tType] : $o ).

tff(sy_cl_Limits_Otopological__comm__monoid__mult,type,
    topolo4987421752381908075d_mult: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Oreal__div__algebra,type,
    real_V5047593784448816457lgebra: 
      !>[A: $tType] : $o ).

tff(sy_cl_Quickcheck__Narrowing_Opartial__term__of,type,
    quickc6926020345158392990erm_of: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Oreal__normed__field,type,
    real_V3459762299906320749_field: 
      !>[A: $tType] : $o ).

tff(sy_cl_Topological__Spaces_Odiscrete__topology,type,
    topolo8865339358273720382pology: 
      !>[A: $tType] : $o ).

tff(sy_cl_Topological__Spaces_Olinorder__topology,type,
    topolo1944317154257567458pology: 
      !>[A: $tType] : $o ).

tff(sy_cl_Topological__Spaces_Otopological__space,type,
    topolo4958980785337419405_space: 
      !>[A: $tType] : $o ).

tff(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
    euclid3725896446679973847miring: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
    canoni5634975068530333245id_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
    ordere8940638589300402666id_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
    strict7427464778891057005id_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
    real_V822414075346904944vector: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
    linord2810124833399127020strict: 
      !>[A: $tType] : $o ).

tff(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
    bit_se359711467146920520ations: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
    ordere2412721322843649153imp_le: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
    ordere580206878836729694up_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
    ordere1170586879665033532d_diff: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
    strict9044650504122735259up_add: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Oordered__real__vector,type,
    real_V5355595471888546746vector: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra,type,
    real_V4412858255891104859lgebra: 
      !>[A: $tType] : $o ).

tff(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
    semiri6575147826004484403cancel: 
      !>[A: $tType] : $o ).

tff(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
    euclid8851590272496341667cancel: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra__1,type,
    real_V2822296259951069270ebra_1: 
      !>[A: $tType] : $o ).

tff(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
    unique1627219031080169319umeral: 
      !>[A: $tType] : $o ).

tff(sy_cl_Real__Vector__Spaces_Oreal__normed__div__algebra,type,
    real_V8999393235501362500lgebra: 
      !>[A: $tType] : $o ).

tff(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
    topolo3112930676232923870pology: 
      !>[A: $tType] : $o ).

tff(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
    euclid4440199948858584721cancel: 
      !>[A: $tType] : $o ).

tff(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
    euclid3128863361964157862miring: 
      !>[A: $tType] : $o ).

tff(sy_cl_Topological__Spaces_Olinear__continuum__topology,type,
    topolo8458572112393995274pology: 
      !>[A: $tType] : $o ).

tff(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
    ordere1937475149494474687imp_le: 
      !>[A: $tType] : $o ).

tff(sy_cl_Euclidean__Division_Ounique__euclidean__ring__with__nat,type,
    euclid8789492081693882211th_nat: 
      !>[A: $tType] : $o ).

tff(sy_cl_Euclidean__Division_Ounique__euclidean__semiring__with__nat,type,
    euclid5411537665997757685th_nat: 
      !>[A: $tType] : $o ).

tff(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
    semiri1453513574482234551roduct: 
      !>[A: $tType] : $o ).

tff(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
    bit_un5681908812861735899ations: 
      !>[A: $tType] : $o ).

tff(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
    condit6923001295902523014norder: 
      !>[A: $tType] : $o ).

tff(sy_c_ATP_058Lamp__a____,type,
    aTP_Lamp_a: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aa____,type,
    aTP_Lamp_aa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aaa____,type,
    aTP_Lamp_aaa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aab____,type,
    aTP_Lamp_aab: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aac____,type,
    aTP_Lamp_aac: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aad____,type,
    aTP_Lamp_aad: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aae____,type,
    aTP_Lamp_aae: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aaf____,type,
    aTP_Lamp_aaf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aag____,type,
    aTP_Lamp_aag: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aah____,type,
    aTP_Lamp_aah: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aai____,type,
    aTP_Lamp_aai: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aaj____,type,
    aTP_Lamp_aaj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aak____,type,
    aTP_Lamp_aak: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aal____,type,
    aTP_Lamp_aal: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aam____,type,
    aTP_Lamp_aam: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aan____,type,
    aTP_Lamp_aan: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aao____,type,
    aTP_Lamp_aao: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aap____,type,
    aTP_Lamp_aap: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aaq____,type,
    aTP_Lamp_aaq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aar____,type,
    aTP_Lamp_aar: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aas____,type,
    aTP_Lamp_aas: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aat____,type,
    aTP_Lamp_aat: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aau____,type,
    aTP_Lamp_aau: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aav____,type,
    aTP_Lamp_aav: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aaw____,type,
    aTP_Lamp_aaw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aax____,type,
    aTP_Lamp_aax: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aay____,type,
    aTP_Lamp_aay: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aaz____,type,
    aTP_Lamp_aaz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ab____,type,
    aTP_Lamp_ab: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aba____,type,
    aTP_Lamp_aba: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abb____,type,
    aTP_Lamp_abb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abc____,type,
    aTP_Lamp_abc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abd____,type,
    aTP_Lamp_abd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abe____,type,
    aTP_Lamp_abe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abf____,type,
    aTP_Lamp_abf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abg____,type,
    aTP_Lamp_abg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abh____,type,
    aTP_Lamp_abh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abi____,type,
    aTP_Lamp_abi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abj____,type,
    aTP_Lamp_abj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abk____,type,
    aTP_Lamp_abk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abl____,type,
    aTP_Lamp_abl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abm____,type,
    aTP_Lamp_abm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abn____,type,
    aTP_Lamp_abn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abo____,type,
    aTP_Lamp_abo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abp____,type,
    aTP_Lamp_abp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abq____,type,
    aTP_Lamp_abq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abr____,type,
    aTP_Lamp_abr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abs____,type,
    aTP_Lamp_abs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abt____,type,
    aTP_Lamp_abt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abu____,type,
    aTP_Lamp_abu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abv____,type,
    aTP_Lamp_abv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abw____,type,
    aTP_Lamp_abw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abx____,type,
    aTP_Lamp_abx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aby____,type,
    aTP_Lamp_aby: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__abz____,type,
    aTP_Lamp_abz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ac____,type,
    aTP_Lamp_ac: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aca____,type,
    aTP_Lamp_aca: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acb____,type,
    aTP_Lamp_acb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acc____,type,
    aTP_Lamp_acc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acd____,type,
    aTP_Lamp_acd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ace____,type,
    aTP_Lamp_ace: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acf____,type,
    aTP_Lamp_acf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acg____,type,
    aTP_Lamp_acg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ach____,type,
    aTP_Lamp_ach: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aci____,type,
    aTP_Lamp_aci: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acj____,type,
    aTP_Lamp_acj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ack____,type,
    aTP_Lamp_ack: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acl____,type,
    aTP_Lamp_acl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acm____,type,
    aTP_Lamp_acm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acn____,type,
    aTP_Lamp_acn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aco____,type,
    aTP_Lamp_aco: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acp____,type,
    aTP_Lamp_acp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acq____,type,
    aTP_Lamp_acq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__acr____,type,
    aTP_Lamp_acr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acs____,type,
    aTP_Lamp_acs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__act____,type,
    aTP_Lamp_act: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__acu____,type,
    aTP_Lamp_acu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acv____,type,
    aTP_Lamp_acv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__acw____,type,
    aTP_Lamp_acw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acx____,type,
    aTP_Lamp_acx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__acy____,type,
    aTP_Lamp_acy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__acz____,type,
    aTP_Lamp_acz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ad____,type,
    aTP_Lamp_ad: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ada____,type,
    aTP_Lamp_ada: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adb____,type,
    aTP_Lamp_adb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adc____,type,
    aTP_Lamp_adc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__add____,type,
    aTP_Lamp_add: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ade____,type,
    aTP_Lamp_ade: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adf____,type,
    aTP_Lamp_adf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adg____,type,
    aTP_Lamp_adg: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__adh____,type,
    aTP_Lamp_adh: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__adi____,type,
    aTP_Lamp_adi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adj____,type,
    aTP_Lamp_adj: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__adk____,type,
    aTP_Lamp_adk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adl____,type,
    aTP_Lamp_adl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adm____,type,
    aTP_Lamp_adm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adn____,type,
    aTP_Lamp_adn: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ado____,type,
    aTP_Lamp_ado: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adp____,type,
    aTP_Lamp_adp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adq____,type,
    aTP_Lamp_adq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adr____,type,
    aTP_Lamp_adr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ads____,type,
    aTP_Lamp_ads: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adt____,type,
    aTP_Lamp_adt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adu____,type,
    aTP_Lamp_adu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adv____,type,
    aTP_Lamp_adv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adw____,type,
    aTP_Lamp_adw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adx____,type,
    aTP_Lamp_adx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ady____,type,
    aTP_Lamp_ady: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__adz____,type,
    aTP_Lamp_adz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ae____,type,
    aTP_Lamp_ae: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aea____,type,
    aTP_Lamp_aea: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aeb____,type,
    aTP_Lamp_aeb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aec____,type,
    aTP_Lamp_aec: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aed____,type,
    aTP_Lamp_aed: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aee____,type,
    aTP_Lamp_aee: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aef____,type,
    aTP_Lamp_aef: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aeg____,type,
    aTP_Lamp_aeg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aeh____,type,
    aTP_Lamp_aeh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aei____,type,
    aTP_Lamp_aei: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aej____,type,
    aTP_Lamp_aej: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aek____,type,
    aTP_Lamp_aek: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ael____,type,
    aTP_Lamp_ael: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aem____,type,
    aTP_Lamp_aem: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aen____,type,
    aTP_Lamp_aen: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aeo____,type,
    aTP_Lamp_aeo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aep____,type,
    aTP_Lamp_aep: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aeq____,type,
    aTP_Lamp_aeq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aer____,type,
    aTP_Lamp_aer: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aes____,type,
    aTP_Lamp_aes: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aet____,type,
    aTP_Lamp_aet: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aeu____,type,
    aTP_Lamp_aeu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aev____,type,
    aTP_Lamp_aev: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aew____,type,
    aTP_Lamp_aew: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aex____,type,
    aTP_Lamp_aex: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aey____,type,
    aTP_Lamp_aey: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aez____,type,
    aTP_Lamp_aez: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__af____,type,
    aTP_Lamp_af: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afa____,type,
    aTP_Lamp_afa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afb____,type,
    aTP_Lamp_afb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afc____,type,
    aTP_Lamp_afc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afd____,type,
    aTP_Lamp_afd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afe____,type,
    aTP_Lamp_afe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aff____,type,
    aTP_Lamp_aff: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afg____,type,
    aTP_Lamp_afg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afh____,type,
    aTP_Lamp_afh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afi____,type,
    aTP_Lamp_afi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afj____,type,
    aTP_Lamp_afj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afk____,type,
    aTP_Lamp_afk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afl____,type,
    aTP_Lamp_afl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afm____,type,
    aTP_Lamp_afm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afn____,type,
    aTP_Lamp_afn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afo____,type,
    aTP_Lamp_afo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afp____,type,
    aTP_Lamp_afp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afq____,type,
    aTP_Lamp_afq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afr____,type,
    aTP_Lamp_afr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afs____,type,
    aTP_Lamp_afs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aft____,type,
    aTP_Lamp_aft: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afu____,type,
    aTP_Lamp_afu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afv____,type,
    aTP_Lamp_afv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afw____,type,
    aTP_Lamp_afw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afx____,type,
    aTP_Lamp_afx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__afy____,type,
    aTP_Lamp_afy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__afz____,type,
    aTP_Lamp_afz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ag____,type,
    aTP_Lamp_ag: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aga____,type,
    aTP_Lamp_aga: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agb____,type,
    aTP_Lamp_agb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agc____,type,
    aTP_Lamp_agc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agd____,type,
    aTP_Lamp_agd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__age____,type,
    aTP_Lamp_age: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agf____,type,
    aTP_Lamp_agf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agg____,type,
    aTP_Lamp_agg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agh____,type,
    aTP_Lamp_agh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agi____,type,
    aTP_Lamp_agi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agj____,type,
    aTP_Lamp_agj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agk____,type,
    aTP_Lamp_agk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agl____,type,
    aTP_Lamp_agl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agm____,type,
    aTP_Lamp_agm: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__agn____,type,
    aTP_Lamp_agn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ago____,type,
    aTP_Lamp_ago: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__agp____,type,
    aTP_Lamp_agp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agq____,type,
    aTP_Lamp_agq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agr____,type,
    aTP_Lamp_agr: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ags____,type,
    aTP_Lamp_ags: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agt____,type,
    aTP_Lamp_agt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agu____,type,
    aTP_Lamp_agu: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__agv____,type,
    aTP_Lamp_agv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agw____,type,
    aTP_Lamp_agw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agx____,type,
    aTP_Lamp_agx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agy____,type,
    aTP_Lamp_agy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__agz____,type,
    aTP_Lamp_agz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ah____,type,
    aTP_Lamp_ah: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aha____,type,
    aTP_Lamp_aha: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahb____,type,
    aTP_Lamp_ahb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahc____,type,
    aTP_Lamp_ahc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahd____,type,
    aTP_Lamp_ahd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahe____,type,
    aTP_Lamp_ahe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahf____,type,
    aTP_Lamp_ahf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahg____,type,
    aTP_Lamp_ahg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahh____,type,
    aTP_Lamp_ahh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahi____,type,
    aTP_Lamp_ahi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahj____,type,
    aTP_Lamp_ahj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahk____,type,
    aTP_Lamp_ahk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahl____,type,
    aTP_Lamp_ahl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahm____,type,
    aTP_Lamp_ahm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahn____,type,
    aTP_Lamp_ahn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aho____,type,
    aTP_Lamp_aho: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahp____,type,
    aTP_Lamp_ahp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahq____,type,
    aTP_Lamp_ahq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahr____,type,
    aTP_Lamp_ahr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahs____,type,
    aTP_Lamp_ahs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aht____,type,
    aTP_Lamp_aht: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahu____,type,
    aTP_Lamp_ahu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahv____,type,
    aTP_Lamp_ahv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahw____,type,
    aTP_Lamp_ahw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahx____,type,
    aTP_Lamp_ahx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahy____,type,
    aTP_Lamp_ahy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ahz____,type,
    aTP_Lamp_ahz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ai____,type,
    aTP_Lamp_ai: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aia____,type,
    aTP_Lamp_aia: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aib____,type,
    aTP_Lamp_aib: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aic____,type,
    aTP_Lamp_aic: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aid____,type,
    aTP_Lamp_aid: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aie____,type,
    aTP_Lamp_aie: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aif____,type,
    aTP_Lamp_aif: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aig____,type,
    aTP_Lamp_aig: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aih____,type,
    aTP_Lamp_aih: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aii____,type,
    aTP_Lamp_aii: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aij____,type,
    aTP_Lamp_aij: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aik____,type,
    aTP_Lamp_aik: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ail____,type,
    aTP_Lamp_ail: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aim____,type,
    aTP_Lamp_aim: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ain____,type,
    aTP_Lamp_ain: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aio____,type,
    aTP_Lamp_aio: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aip____,type,
    aTP_Lamp_aip: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aiq____,type,
    aTP_Lamp_aiq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__air____,type,
    aTP_Lamp_air: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ais____,type,
    aTP_Lamp_ais: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ait____,type,
    aTP_Lamp_ait: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aiu____,type,
    aTP_Lamp_aiu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aiv____,type,
    aTP_Lamp_aiv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aiw____,type,
    aTP_Lamp_aiw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aix____,type,
    aTP_Lamp_aix: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aiy____,type,
    aTP_Lamp_aiy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aiz____,type,
    aTP_Lamp_aiz: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aj____,type,
    aTP_Lamp_aj: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aja____,type,
    aTP_Lamp_aja: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajb____,type,
    aTP_Lamp_ajb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajc____,type,
    aTP_Lamp_ajc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajd____,type,
    aTP_Lamp_ajd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aje____,type,
    aTP_Lamp_aje: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajf____,type,
    aTP_Lamp_ajf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajg____,type,
    aTP_Lamp_ajg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajh____,type,
    aTP_Lamp_ajh: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aji____,type,
    aTP_Lamp_aji: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajj____,type,
    aTP_Lamp_ajj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajk____,type,
    aTP_Lamp_ajk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajl____,type,
    aTP_Lamp_ajl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajm____,type,
    aTP_Lamp_ajm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajn____,type,
    aTP_Lamp_ajn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajo____,type,
    aTP_Lamp_ajo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajp____,type,
    aTP_Lamp_ajp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajq____,type,
    aTP_Lamp_ajq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajr____,type,
    aTP_Lamp_ajr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajs____,type,
    aTP_Lamp_ajs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajt____,type,
    aTP_Lamp_ajt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aju____,type,
    aTP_Lamp_aju: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajv____,type,
    aTP_Lamp_ajv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ajw____,type,
    aTP_Lamp_ajw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ajx____,type,
    aTP_Lamp_ajx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ajy____,type,
    aTP_Lamp_ajy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ajz____,type,
    aTP_Lamp_ajz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ak____,type,
    aTP_Lamp_ak: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aka____,type,
    aTP_Lamp_aka: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akb____,type,
    aTP_Lamp_akb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akc____,type,
    aTP_Lamp_akc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akd____,type,
    aTP_Lamp_akd: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ake____,type,
    aTP_Lamp_ake: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__akf____,type,
    aTP_Lamp_akf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akg____,type,
    aTP_Lamp_akg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akh____,type,
    aTP_Lamp_akh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aki____,type,
    aTP_Lamp_aki: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akj____,type,
    aTP_Lamp_akj: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__akk____,type,
    aTP_Lamp_akk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akl____,type,
    aTP_Lamp_akl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akm____,type,
    aTP_Lamp_akm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akn____,type,
    aTP_Lamp_akn: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ako____,type,
    aTP_Lamp_ako: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__akp____,type,
    aTP_Lamp_akp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akq____,type,
    aTP_Lamp_akq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akr____,type,
    aTP_Lamp_akr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aks____,type,
    aTP_Lamp_aks: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akt____,type,
    aTP_Lamp_akt: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aku____,type,
    aTP_Lamp_aku: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akv____,type,
    aTP_Lamp_akv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akw____,type,
    aTP_Lamp_akw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__akx____,type,
    aTP_Lamp_akx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aky____,type,
    aTP_Lamp_aky: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__akz____,type,
    aTP_Lamp_akz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__al____,type,
    aTP_Lamp_al: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ala____,type,
    aTP_Lamp_ala: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__alb____,type,
    aTP_Lamp_alb: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__alc____,type,
    aTP_Lamp_alc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ald____,type,
    aTP_Lamp_ald: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ale____,type,
    aTP_Lamp_ale: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alf____,type,
    aTP_Lamp_alf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alg____,type,
    aTP_Lamp_alg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alh____,type,
    aTP_Lamp_alh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ali____,type,
    aTP_Lamp_ali: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alj____,type,
    aTP_Lamp_alj: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__alk____,type,
    aTP_Lamp_alk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__all____,type,
    aTP_Lamp_all: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alm____,type,
    aTP_Lamp_alm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aln____,type,
    aTP_Lamp_aln: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alo____,type,
    aTP_Lamp_alo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alp____,type,
    aTP_Lamp_alp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alq____,type,
    aTP_Lamp_alq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alr____,type,
    aTP_Lamp_alr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__als____,type,
    aTP_Lamp_als: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alt____,type,
    aTP_Lamp_alt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alu____,type,
    aTP_Lamp_alu: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__alv____,type,
    aTP_Lamp_alv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__alw____,type,
    aTP_Lamp_alw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alx____,type,
    aTP_Lamp_alx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aly____,type,
    aTP_Lamp_aly: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__alz____,type,
    aTP_Lamp_alz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__am____,type,
    aTP_Lamp_am: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ama____,type,
    aTP_Lamp_ama: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__amb____,type,
    aTP_Lamp_amb: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__amc____,type,
    aTP_Lamp_amc: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__amd____,type,
    aTP_Lamp_amd: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ame____,type,
    aTP_Lamp_ame: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amf____,type,
    aTP_Lamp_amf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amg____,type,
    aTP_Lamp_amg: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__amh____,type,
    aTP_Lamp_amh: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ami____,type,
    aTP_Lamp_ami: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amj____,type,
    aTP_Lamp_amj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amk____,type,
    aTP_Lamp_amk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aml____,type,
    aTP_Lamp_aml: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amm____,type,
    aTP_Lamp_amm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amn____,type,
    aTP_Lamp_amn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amo____,type,
    aTP_Lamp_amo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amp____,type,
    aTP_Lamp_amp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amq____,type,
    aTP_Lamp_amq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amr____,type,
    aTP_Lamp_amr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ams____,type,
    aTP_Lamp_ams: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amt____,type,
    aTP_Lamp_amt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amu____,type,
    aTP_Lamp_amu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amv____,type,
    aTP_Lamp_amv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amw____,type,
    aTP_Lamp_amw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amx____,type,
    aTP_Lamp_amx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amy____,type,
    aTP_Lamp_amy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__amz____,type,
    aTP_Lamp_amz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__an____,type,
    aTP_Lamp_an: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ana____,type,
    aTP_Lamp_ana: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anb____,type,
    aTP_Lamp_anb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anc____,type,
    aTP_Lamp_anc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__and____,type,
    aTP_Lamp_and: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ane____,type,
    aTP_Lamp_ane: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anf____,type,
    aTP_Lamp_anf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ang____,type,
    aTP_Lamp_ang: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anh____,type,
    aTP_Lamp_anh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ani____,type,
    aTP_Lamp_ani: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anj____,type,
    aTP_Lamp_anj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ank____,type,
    aTP_Lamp_ank: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anl____,type,
    aTP_Lamp_anl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anm____,type,
    aTP_Lamp_anm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ann____,type,
    aTP_Lamp_ann: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ano____,type,
    aTP_Lamp_ano: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__anp____,type,
    aTP_Lamp_anp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anq____,type,
    aTP_Lamp_anq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__anr____,type,
    aTP_Lamp_anr: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ans____,type,
    aTP_Lamp_ans: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ant____,type,
    aTP_Lamp_ant: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anu____,type,
    aTP_Lamp_anu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anv____,type,
    aTP_Lamp_anv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__anw____,type,
    aTP_Lamp_anw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__anx____,type,
    aTP_Lamp_anx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__any____,type,
    aTP_Lamp_any: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__anz____,type,
    aTP_Lamp_anz: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ao____,type,
    aTP_Lamp_ao: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoa____,type,
    aTP_Lamp_aoa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aob____,type,
    aTP_Lamp_aob: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoc____,type,
    aTP_Lamp_aoc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aod____,type,
    aTP_Lamp_aod: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aoe____,type,
    aTP_Lamp_aoe: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aof____,type,
    aTP_Lamp_aof: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aog____,type,
    aTP_Lamp_aog: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoh____,type,
    aTP_Lamp_aoh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoi____,type,
    aTP_Lamp_aoi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoj____,type,
    aTP_Lamp_aoj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aok____,type,
    aTP_Lamp_aok: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aol____,type,
    aTP_Lamp_aol: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aom____,type,
    aTP_Lamp_aom: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aon____,type,
    aTP_Lamp_aon: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoo____,type,
    aTP_Lamp_aoo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aop____,type,
    aTP_Lamp_aop: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoq____,type,
    aTP_Lamp_aoq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aor____,type,
    aTP_Lamp_aor: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aos____,type,
    aTP_Lamp_aos: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aot____,type,
    aTP_Lamp_aot: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aou____,type,
    aTP_Lamp_aou: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aov____,type,
    aTP_Lamp_aov: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aow____,type,
    aTP_Lamp_aow: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aox____,type,
    aTP_Lamp_aox: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoy____,type,
    aTP_Lamp_aoy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aoz____,type,
    aTP_Lamp_aoz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ap____,type,
    aTP_Lamp_ap: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apa____,type,
    aTP_Lamp_apa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apb____,type,
    aTP_Lamp_apb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apc____,type,
    aTP_Lamp_apc: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__apd____,type,
    aTP_Lamp_apd: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ape____,type,
    aTP_Lamp_ape: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__apf____,type,
    aTP_Lamp_apf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apg____,type,
    aTP_Lamp_apg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aph____,type,
    aTP_Lamp_aph: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__api____,type,
    aTP_Lamp_api: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apj____,type,
    aTP_Lamp_apj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apk____,type,
    aTP_Lamp_apk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apl____,type,
    aTP_Lamp_apl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apm____,type,
    aTP_Lamp_apm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apn____,type,
    aTP_Lamp_apn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apo____,type,
    aTP_Lamp_apo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__app____,type,
    aTP_Lamp_app: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__apq____,type,
    aTP_Lamp_apq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apr____,type,
    aTP_Lamp_apr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aps____,type,
    aTP_Lamp_aps: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apt____,type,
    aTP_Lamp_apt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apu____,type,
    aTP_Lamp_apu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apv____,type,
    aTP_Lamp_apv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apw____,type,
    aTP_Lamp_apw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apx____,type,
    aTP_Lamp_apx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apy____,type,
    aTP_Lamp_apy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__apz____,type,
    aTP_Lamp_apz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aq____,type,
    aTP_Lamp_aq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqa____,type,
    aTP_Lamp_aqa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqb____,type,
    aTP_Lamp_aqb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqc____,type,
    aTP_Lamp_aqc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqd____,type,
    aTP_Lamp_aqd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqe____,type,
    aTP_Lamp_aqe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqf____,type,
    aTP_Lamp_aqf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqg____,type,
    aTP_Lamp_aqg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqh____,type,
    aTP_Lamp_aqh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqi____,type,
    aTP_Lamp_aqi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqj____,type,
    aTP_Lamp_aqj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqk____,type,
    aTP_Lamp_aqk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aql____,type,
    aTP_Lamp_aql: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqm____,type,
    aTP_Lamp_aqm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqn____,type,
    aTP_Lamp_aqn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqo____,type,
    aTP_Lamp_aqo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqp____,type,
    aTP_Lamp_aqp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqq____,type,
    aTP_Lamp_aqq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqr____,type,
    aTP_Lamp_aqr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqs____,type,
    aTP_Lamp_aqs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqt____,type,
    aTP_Lamp_aqt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqu____,type,
    aTP_Lamp_aqu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqv____,type,
    aTP_Lamp_aqv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqw____,type,
    aTP_Lamp_aqw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqx____,type,
    aTP_Lamp_aqx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aqy____,type,
    aTP_Lamp_aqy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aqz____,type,
    aTP_Lamp_aqz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ar____,type,
    aTP_Lamp_ar: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ara____,type,
    aTP_Lamp_ara: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arb____,type,
    aTP_Lamp_arb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arc____,type,
    aTP_Lamp_arc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ard____,type,
    aTP_Lamp_ard: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__are____,type,
    aTP_Lamp_are: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__arf____,type,
    aTP_Lamp_arf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arg____,type,
    aTP_Lamp_arg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arh____,type,
    aTP_Lamp_arh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ari____,type,
    aTP_Lamp_ari: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arj____,type,
    aTP_Lamp_arj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ark____,type,
    aTP_Lamp_ark: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arl____,type,
    aTP_Lamp_arl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arm____,type,
    aTP_Lamp_arm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arn____,type,
    aTP_Lamp_arn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aro____,type,
    aTP_Lamp_aro: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arp____,type,
    aTP_Lamp_arp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arq____,type,
    aTP_Lamp_arq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arr____,type,
    aTP_Lamp_arr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ars____,type,
    aTP_Lamp_ars: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__art____,type,
    aTP_Lamp_art: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aru____,type,
    aTP_Lamp_aru: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__arv____,type,
    aTP_Lamp_arv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arw____,type,
    aTP_Lamp_arw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arx____,type,
    aTP_Lamp_arx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ary____,type,
    aTP_Lamp_ary: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__arz____,type,
    aTP_Lamp_arz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__as____,type,
    aTP_Lamp_as: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asa____,type,
    aTP_Lamp_asa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asb____,type,
    aTP_Lamp_asb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asc____,type,
    aTP_Lamp_asc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asd____,type,
    aTP_Lamp_asd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ase____,type,
    aTP_Lamp_ase: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asf____,type,
    aTP_Lamp_asf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asg____,type,
    aTP_Lamp_asg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ash____,type,
    aTP_Lamp_ash: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asi____,type,
    aTP_Lamp_asi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asj____,type,
    aTP_Lamp_asj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ask____,type,
    aTP_Lamp_ask: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asl____,type,
    aTP_Lamp_asl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asm____,type,
    aTP_Lamp_asm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asn____,type,
    aTP_Lamp_asn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aso____,type,
    aTP_Lamp_aso: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asp____,type,
    aTP_Lamp_asp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asq____,type,
    aTP_Lamp_asq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asr____,type,
    aTP_Lamp_asr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ass____,type,
    aTP_Lamp_ass: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ast____,type,
    aTP_Lamp_ast: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asu____,type,
    aTP_Lamp_asu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asv____,type,
    aTP_Lamp_asv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asw____,type,
    aTP_Lamp_asw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asx____,type,
    aTP_Lamp_asx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asy____,type,
    aTP_Lamp_asy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__asz____,type,
    aTP_Lamp_asz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__at____,type,
    aTP_Lamp_at: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ata____,type,
    aTP_Lamp_ata: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atb____,type,
    aTP_Lamp_atb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atc____,type,
    aTP_Lamp_atc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atd____,type,
    aTP_Lamp_atd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ate____,type,
    aTP_Lamp_ate: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atf____,type,
    aTP_Lamp_atf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atg____,type,
    aTP_Lamp_atg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ath____,type,
    aTP_Lamp_ath: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ati____,type,
    aTP_Lamp_ati: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atj____,type,
    aTP_Lamp_atj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atk____,type,
    aTP_Lamp_atk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atl____,type,
    aTP_Lamp_atl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atm____,type,
    aTP_Lamp_atm: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__atn____,type,
    aTP_Lamp_atn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ato____,type,
    aTP_Lamp_ato: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atp____,type,
    aTP_Lamp_atp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atq____,type,
    aTP_Lamp_atq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atr____,type,
    aTP_Lamp_atr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ats____,type,
    aTP_Lamp_ats: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__att____,type,
    aTP_Lamp_att: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atu____,type,
    aTP_Lamp_atu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atv____,type,
    aTP_Lamp_atv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atw____,type,
    aTP_Lamp_atw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atx____,type,
    aTP_Lamp_atx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aty____,type,
    aTP_Lamp_aty: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__atz____,type,
    aTP_Lamp_atz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__au____,type,
    aTP_Lamp_au: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aua____,type,
    aTP_Lamp_aua: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aub____,type,
    aTP_Lamp_aub: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auc____,type,
    aTP_Lamp_auc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aud____,type,
    aTP_Lamp_aud: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aue____,type,
    aTP_Lamp_aue: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auf____,type,
    aTP_Lamp_auf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aug____,type,
    aTP_Lamp_aug: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auh____,type,
    aTP_Lamp_auh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aui____,type,
    aTP_Lamp_aui: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auj____,type,
    aTP_Lamp_auj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auk____,type,
    aTP_Lamp_auk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aul____,type,
    aTP_Lamp_aul: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aum____,type,
    aTP_Lamp_aum: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aun____,type,
    aTP_Lamp_aun: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auo____,type,
    aTP_Lamp_auo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aup____,type,
    aTP_Lamp_aup: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auq____,type,
    aTP_Lamp_auq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aur____,type,
    aTP_Lamp_aur: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aus____,type,
    aTP_Lamp_aus: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aut____,type,
    aTP_Lamp_aut: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__auu____,type,
    aTP_Lamp_auu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auv____,type,
    aTP_Lamp_auv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auw____,type,
    aTP_Lamp_auw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aux____,type,
    aTP_Lamp_aux: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auy____,type,
    aTP_Lamp_auy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__auz____,type,
    aTP_Lamp_auz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__av____,type,
    aTP_Lamp_av: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ava____,type,
    aTP_Lamp_ava: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avb____,type,
    aTP_Lamp_avb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avc____,type,
    aTP_Lamp_avc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avd____,type,
    aTP_Lamp_avd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ave____,type,
    aTP_Lamp_ave: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avf____,type,
    aTP_Lamp_avf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avg____,type,
    aTP_Lamp_avg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avh____,type,
    aTP_Lamp_avh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avi____,type,
    aTP_Lamp_avi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avj____,type,
    aTP_Lamp_avj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avk____,type,
    aTP_Lamp_avk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avl____,type,
    aTP_Lamp_avl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avm____,type,
    aTP_Lamp_avm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avn____,type,
    aTP_Lamp_avn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avo____,type,
    aTP_Lamp_avo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avp____,type,
    aTP_Lamp_avp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avq____,type,
    aTP_Lamp_avq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avr____,type,
    aTP_Lamp_avr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avs____,type,
    aTP_Lamp_avs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avt____,type,
    aTP_Lamp_avt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avu____,type,
    aTP_Lamp_avu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avv____,type,
    aTP_Lamp_avv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avw____,type,
    aTP_Lamp_avw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avx____,type,
    aTP_Lamp_avx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avy____,type,
    aTP_Lamp_avy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__avz____,type,
    aTP_Lamp_avz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aw____,type,
    aTP_Lamp_aw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awa____,type,
    aTP_Lamp_awa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awb____,type,
    aTP_Lamp_awb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awc____,type,
    aTP_Lamp_awc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awd____,type,
    aTP_Lamp_awd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awe____,type,
    aTP_Lamp_awe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awf____,type,
    aTP_Lamp_awf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awg____,type,
    aTP_Lamp_awg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awh____,type,
    aTP_Lamp_awh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awi____,type,
    aTP_Lamp_awi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awj____,type,
    aTP_Lamp_awj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awk____,type,
    aTP_Lamp_awk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awl____,type,
    aTP_Lamp_awl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awm____,type,
    aTP_Lamp_awm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awn____,type,
    aTP_Lamp_awn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awo____,type,
    aTP_Lamp_awo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awp____,type,
    aTP_Lamp_awp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awq____,type,
    aTP_Lamp_awq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awr____,type,
    aTP_Lamp_awr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aws____,type,
    aTP_Lamp_aws: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awt____,type,
    aTP_Lamp_awt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awu____,type,
    aTP_Lamp_awu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awv____,type,
    aTP_Lamp_awv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aww____,type,
    aTP_Lamp_aww: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awx____,type,
    aTP_Lamp_awx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awy____,type,
    aTP_Lamp_awy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__awz____,type,
    aTP_Lamp_awz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ax____,type,
    aTP_Lamp_ax: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axa____,type,
    aTP_Lamp_axa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axb____,type,
    aTP_Lamp_axb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axc____,type,
    aTP_Lamp_axc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axd____,type,
    aTP_Lamp_axd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axe____,type,
    aTP_Lamp_axe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axf____,type,
    aTP_Lamp_axf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axg____,type,
    aTP_Lamp_axg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axh____,type,
    aTP_Lamp_axh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axi____,type,
    aTP_Lamp_axi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axj____,type,
    aTP_Lamp_axj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axk____,type,
    aTP_Lamp_axk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axl____,type,
    aTP_Lamp_axl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axm____,type,
    aTP_Lamp_axm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axn____,type,
    aTP_Lamp_axn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axo____,type,
    aTP_Lamp_axo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axp____,type,
    aTP_Lamp_axp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axq____,type,
    aTP_Lamp_axq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axr____,type,
    aTP_Lamp_axr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axs____,type,
    aTP_Lamp_axs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axt____,type,
    aTP_Lamp_axt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axu____,type,
    aTP_Lamp_axu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axv____,type,
    aTP_Lamp_axv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axw____,type,
    aTP_Lamp_axw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axx____,type,
    aTP_Lamp_axx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__axy____,type,
    aTP_Lamp_axy: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__axz____,type,
    aTP_Lamp_axz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ay____,type,
    aTP_Lamp_ay: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aya____,type,
    aTP_Lamp_aya: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayb____,type,
    aTP_Lamp_ayb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayc____,type,
    aTP_Lamp_ayc: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ayd____,type,
    aTP_Lamp_ayd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aye____,type,
    aTP_Lamp_aye: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayf____,type,
    aTP_Lamp_ayf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayg____,type,
    aTP_Lamp_ayg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayh____,type,
    aTP_Lamp_ayh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayi____,type,
    aTP_Lamp_ayi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayj____,type,
    aTP_Lamp_ayj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayk____,type,
    aTP_Lamp_ayk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayl____,type,
    aTP_Lamp_ayl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__aym____,type,
    aTP_Lamp_aym: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayn____,type,
    aTP_Lamp_ayn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayo____,type,
    aTP_Lamp_ayo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayp____,type,
    aTP_Lamp_ayp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayq____,type,
    aTP_Lamp_ayq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayr____,type,
    aTP_Lamp_ayr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ays____,type,
    aTP_Lamp_ays: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayt____,type,
    aTP_Lamp_ayt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayu____,type,
    aTP_Lamp_ayu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayv____,type,
    aTP_Lamp_ayv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayw____,type,
    aTP_Lamp_ayw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayx____,type,
    aTP_Lamp_ayx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayy____,type,
    aTP_Lamp_ayy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ayz____,type,
    aTP_Lamp_ayz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__az____,type,
    aTP_Lamp_az: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aza____,type,
    aTP_Lamp_aza: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__azb____,type,
    aTP_Lamp_azb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azc____,type,
    aTP_Lamp_azc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azd____,type,
    aTP_Lamp_azd: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__aze____,type,
    aTP_Lamp_aze: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azf____,type,
    aTP_Lamp_azf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azg____,type,
    aTP_Lamp_azg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azh____,type,
    aTP_Lamp_azh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azi____,type,
    aTP_Lamp_azi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azj____,type,
    aTP_Lamp_azj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azk____,type,
    aTP_Lamp_azk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__azl____,type,
    aTP_Lamp_azl: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__azm____,type,
    aTP_Lamp_azm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azn____,type,
    aTP_Lamp_azn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azo____,type,
    aTP_Lamp_azo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azp____,type,
    aTP_Lamp_azp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azq____,type,
    aTP_Lamp_azq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azr____,type,
    aTP_Lamp_azr: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__azs____,type,
    aTP_Lamp_azs: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__azt____,type,
    aTP_Lamp_azt: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__azu____,type,
    aTP_Lamp_azu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azv____,type,
    aTP_Lamp_azv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azw____,type,
    aTP_Lamp_azw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azx____,type,
    aTP_Lamp_azx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azy____,type,
    aTP_Lamp_azy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__azz____,type,
    aTP_Lamp_azz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ba____,type,
    aTP_Lamp_ba: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__baa____,type,
    aTP_Lamp_baa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bab____,type,
    aTP_Lamp_bab: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bac____,type,
    aTP_Lamp_bac: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bad____,type,
    aTP_Lamp_bad: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bae____,type,
    aTP_Lamp_bae: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__baf____,type,
    aTP_Lamp_baf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bag____,type,
    aTP_Lamp_bag: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bah____,type,
    aTP_Lamp_bah: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bai____,type,
    aTP_Lamp_bai: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__baj____,type,
    aTP_Lamp_baj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bak____,type,
    aTP_Lamp_bak: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bal____,type,
    aTP_Lamp_bal: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bam____,type,
    aTP_Lamp_bam: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ban____,type,
    aTP_Lamp_ban: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bao____,type,
    aTP_Lamp_bao: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bap____,type,
    aTP_Lamp_bap: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__baq____,type,
    aTP_Lamp_baq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bar____,type,
    aTP_Lamp_bar: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bas____,type,
    aTP_Lamp_bas: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bat____,type,
    aTP_Lamp_bat: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bau____,type,
    aTP_Lamp_bau: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bav____,type,
    aTP_Lamp_bav: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__baw____,type,
    aTP_Lamp_baw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bax____,type,
    aTP_Lamp_bax: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bay____,type,
    aTP_Lamp_bay: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__baz____,type,
    aTP_Lamp_baz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bb____,type,
    aTP_Lamp_bb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bba____,type,
    aTP_Lamp_bba: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbb____,type,
    aTP_Lamp_bbb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbc____,type,
    aTP_Lamp_bbc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbd____,type,
    aTP_Lamp_bbd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbe____,type,
    aTP_Lamp_bbe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbf____,type,
    aTP_Lamp_bbf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbg____,type,
    aTP_Lamp_bbg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbh____,type,
    aTP_Lamp_bbh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbi____,type,
    aTP_Lamp_bbi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbj____,type,
    aTP_Lamp_bbj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbk____,type,
    aTP_Lamp_bbk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbl____,type,
    aTP_Lamp_bbl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbm____,type,
    aTP_Lamp_bbm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbn____,type,
    aTP_Lamp_bbn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbo____,type,
    aTP_Lamp_bbo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbp____,type,
    aTP_Lamp_bbp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbq____,type,
    aTP_Lamp_bbq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbr____,type,
    aTP_Lamp_bbr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbs____,type,
    aTP_Lamp_bbs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbt____,type,
    aTP_Lamp_bbt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbu____,type,
    aTP_Lamp_bbu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbv____,type,
    aTP_Lamp_bbv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbw____,type,
    aTP_Lamp_bbw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbx____,type,
    aTP_Lamp_bbx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bby____,type,
    aTP_Lamp_bby: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bbz____,type,
    aTP_Lamp_bbz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bc____,type,
    aTP_Lamp_bc: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__bca____,type,
    aTP_Lamp_bca: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcb____,type,
    aTP_Lamp_bcb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcc____,type,
    aTP_Lamp_bcc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcd____,type,
    aTP_Lamp_bcd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bce____,type,
    aTP_Lamp_bce: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcf____,type,
    aTP_Lamp_bcf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcg____,type,
    aTP_Lamp_bcg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bch____,type,
    aTP_Lamp_bch: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bci____,type,
    aTP_Lamp_bci: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcj____,type,
    aTP_Lamp_bcj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bck____,type,
    aTP_Lamp_bck: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcl____,type,
    aTP_Lamp_bcl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcm____,type,
    aTP_Lamp_bcm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcn____,type,
    aTP_Lamp_bcn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bco____,type,
    aTP_Lamp_bco: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcp____,type,
    aTP_Lamp_bcp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcq____,type,
    aTP_Lamp_bcq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcr____,type,
    aTP_Lamp_bcr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcs____,type,
    aTP_Lamp_bcs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bct____,type,
    aTP_Lamp_bct: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcu____,type,
    aTP_Lamp_bcu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcv____,type,
    aTP_Lamp_bcv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcw____,type,
    aTP_Lamp_bcw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcx____,type,
    aTP_Lamp_bcx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcy____,type,
    aTP_Lamp_bcy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bcz____,type,
    aTP_Lamp_bcz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bd____,type,
    aTP_Lamp_bd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bda____,type,
    aTP_Lamp_bda: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdb____,type,
    aTP_Lamp_bdb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdc____,type,
    aTP_Lamp_bdc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdd____,type,
    aTP_Lamp_bdd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bde____,type,
    aTP_Lamp_bde: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdf____,type,
    aTP_Lamp_bdf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdg____,type,
    aTP_Lamp_bdg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdh____,type,
    aTP_Lamp_bdh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdi____,type,
    aTP_Lamp_bdi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdj____,type,
    aTP_Lamp_bdj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdk____,type,
    aTP_Lamp_bdk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__bdl____,type,
    aTP_Lamp_bdl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdm____,type,
    aTP_Lamp_bdm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdn____,type,
    aTP_Lamp_bdn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdo____,type,
    aTP_Lamp_bdo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdp____,type,
    aTP_Lamp_bdp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdq____,type,
    aTP_Lamp_bdq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdr____,type,
    aTP_Lamp_bdr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bds____,type,
    aTP_Lamp_bds: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdt____,type,
    aTP_Lamp_bdt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdu____,type,
    aTP_Lamp_bdu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdv____,type,
    aTP_Lamp_bdv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdw____,type,
    aTP_Lamp_bdw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdx____,type,
    aTP_Lamp_bdx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdy____,type,
    aTP_Lamp_bdy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bdz____,type,
    aTP_Lamp_bdz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__be____,type,
    aTP_Lamp_be: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bea____,type,
    aTP_Lamp_bea: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__beb____,type,
    aTP_Lamp_beb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bec____,type,
    aTP_Lamp_bec: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bed____,type,
    aTP_Lamp_bed: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bee____,type,
    aTP_Lamp_bee: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bef____,type,
    aTP_Lamp_bef: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__beg____,type,
    aTP_Lamp_beg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__beh____,type,
    aTP_Lamp_beh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bei____,type,
    aTP_Lamp_bei: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bej____,type,
    aTP_Lamp_bej: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bek____,type,
    aTP_Lamp_bek: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__bel____,type,
    aTP_Lamp_bel: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bem____,type,
    aTP_Lamp_bem: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ben____,type,
    aTP_Lamp_ben: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__beo____,type,
    aTP_Lamp_beo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bep____,type,
    aTP_Lamp_bep: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__beq____,type,
    aTP_Lamp_beq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ber____,type,
    aTP_Lamp_ber: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bes____,type,
    aTP_Lamp_bes: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bf____,type,
    aTP_Lamp_bf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bg____,type,
    aTP_Lamp_bg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bh____,type,
    aTP_Lamp_bh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bi____,type,
    aTP_Lamp_bi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bj____,type,
    aTP_Lamp_bj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bk____,type,
    aTP_Lamp_bk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bl____,type,
    aTP_Lamp_bl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bm____,type,
    aTP_Lamp_bm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bn____,type,
    aTP_Lamp_bn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bo____,type,
    aTP_Lamp_bo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bp____,type,
    aTP_Lamp_bp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bq____,type,
    aTP_Lamp_bq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__br____,type,
    aTP_Lamp_br: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bs____,type,
    aTP_Lamp_bs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bt____,type,
    aTP_Lamp_bt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bu____,type,
    aTP_Lamp_bu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bv____,type,
    aTP_Lamp_bv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bw____,type,
    aTP_Lamp_bw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bx____,type,
    aTP_Lamp_bx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__by____,type,
    aTP_Lamp_by: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__bz____,type,
    aTP_Lamp_bz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ca____,type,
    aTP_Lamp_ca: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cb____,type,
    aTP_Lamp_cb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cc____,type,
    aTP_Lamp_cc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cd____,type,
    aTP_Lamp_cd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ce____,type,
    aTP_Lamp_ce: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cf____,type,
    aTP_Lamp_cf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cg____,type,
    aTP_Lamp_cg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ch____,type,
    aTP_Lamp_ch: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ci____,type,
    aTP_Lamp_ci: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cj____,type,
    aTP_Lamp_cj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ck____,type,
    aTP_Lamp_ck: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cl____,type,
    aTP_Lamp_cl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cm____,type,
    aTP_Lamp_cm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cn____,type,
    aTP_Lamp_cn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__co____,type,
    aTP_Lamp_co: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cp____,type,
    aTP_Lamp_cp: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__cq____,type,
    aTP_Lamp_cq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__cr____,type,
    aTP_Lamp_cr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cs____,type,
    aTP_Lamp_cs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ct____,type,
    aTP_Lamp_ct: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__cu____,type,
    aTP_Lamp_cu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cv____,type,
    aTP_Lamp_cv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cw____,type,
    aTP_Lamp_cw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cx____,type,
    aTP_Lamp_cx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cy____,type,
    aTP_Lamp_cy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cz____,type,
    aTP_Lamp_cz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__da____,type,
    aTP_Lamp_da: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__db____,type,
    aTP_Lamp_db: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dc____,type,
    aTP_Lamp_dc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dd____,type,
    aTP_Lamp_dd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__de____,type,
    aTP_Lamp_de: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__df____,type,
    aTP_Lamp_df: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dg____,type,
    aTP_Lamp_dg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dh____,type,
    aTP_Lamp_dh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__di____,type,
    aTP_Lamp_di: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dj____,type,
    aTP_Lamp_dj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dk____,type,
    aTP_Lamp_dk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dl____,type,
    aTP_Lamp_dl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dm____,type,
    aTP_Lamp_dm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dn____,type,
    aTP_Lamp_dn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__do____,type,
    aTP_Lamp_do: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__dp____,type,
    aTP_Lamp_dp: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__dq____,type,
    aTP_Lamp_dq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dr____,type,
    aTP_Lamp_dr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ds____,type,
    aTP_Lamp_ds: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dt____,type,
    aTP_Lamp_dt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__du____,type,
    aTP_Lamp_du: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dv____,type,
    aTP_Lamp_dv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dw____,type,
    aTP_Lamp_dw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dx____,type,
    aTP_Lamp_dx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dy____,type,
    aTP_Lamp_dy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dz____,type,
    aTP_Lamp_dz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ea____,type,
    aTP_Lamp_ea: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eb____,type,
    aTP_Lamp_eb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ec____,type,
    aTP_Lamp_ec: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ed____,type,
    aTP_Lamp_ed: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ee____,type,
    aTP_Lamp_ee: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ef____,type,
    aTP_Lamp_ef: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eg____,type,
    aTP_Lamp_eg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eh____,type,
    aTP_Lamp_eh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ei____,type,
    aTP_Lamp_ei: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ej____,type,
    aTP_Lamp_ej: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ek____,type,
    aTP_Lamp_ek: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__el____,type,
    aTP_Lamp_el: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__em____,type,
    aTP_Lamp_em: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__en____,type,
    aTP_Lamp_en: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eo____,type,
    aTP_Lamp_eo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ep____,type,
    aTP_Lamp_ep: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eq____,type,
    aTP_Lamp_eq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__er____,type,
    aTP_Lamp_er: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__es____,type,
    aTP_Lamp_es: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__et____,type,
    aTP_Lamp_et: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__eu____,type,
    aTP_Lamp_eu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ev____,type,
    aTP_Lamp_ev: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ew____,type,
    aTP_Lamp_ew: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ex____,type,
    aTP_Lamp_ex: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ey____,type,
    aTP_Lamp_ey: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ez____,type,
    aTP_Lamp_ez: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fa____,type,
    aTP_Lamp_fa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fb____,type,
    aTP_Lamp_fb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fc____,type,
    aTP_Lamp_fc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fd____,type,
    aTP_Lamp_fd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fe____,type,
    aTP_Lamp_fe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ff____,type,
    aTP_Lamp_ff: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fg____,type,
    aTP_Lamp_fg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fh____,type,
    aTP_Lamp_fh: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__fi____,type,
    aTP_Lamp_fi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fj____,type,
    aTP_Lamp_fj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fk____,type,
    aTP_Lamp_fk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fl____,type,
    aTP_Lamp_fl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fm____,type,
    aTP_Lamp_fm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fn____,type,
    aTP_Lamp_fn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fo____,type,
    aTP_Lamp_fo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fp____,type,
    aTP_Lamp_fp: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__fq____,type,
    aTP_Lamp_fq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__fr____,type,
    aTP_Lamp_fr: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__fs____,type,
    aTP_Lamp_fs: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ft____,type,
    aTP_Lamp_ft: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fu____,type,
    aTP_Lamp_fu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fv____,type,
    aTP_Lamp_fv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fw____,type,
    aTP_Lamp_fw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fx____,type,
    aTP_Lamp_fx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fy____,type,
    aTP_Lamp_fy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fz____,type,
    aTP_Lamp_fz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ga____,type,
    aTP_Lamp_ga: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gb____,type,
    aTP_Lamp_gb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gc____,type,
    aTP_Lamp_gc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gd____,type,
    aTP_Lamp_gd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ge____,type,
    aTP_Lamp_ge: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gf____,type,
    aTP_Lamp_gf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gg____,type,
    aTP_Lamp_gg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gh____,type,
    aTP_Lamp_gh: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__gi____,type,
    aTP_Lamp_gi: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__gj____,type,
    aTP_Lamp_gj: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__gk____,type,
    aTP_Lamp_gk: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__gl____,type,
    aTP_Lamp_gl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gm____,type,
    aTP_Lamp_gm: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__gn____,type,
    aTP_Lamp_gn: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__go____,type,
    aTP_Lamp_go: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gp____,type,
    aTP_Lamp_gp: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__gq____,type,
    aTP_Lamp_gq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gr____,type,
    aTP_Lamp_gr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gs____,type,
    aTP_Lamp_gs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gt____,type,
    aTP_Lamp_gt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gu____,type,
    aTP_Lamp_gu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gv____,type,
    aTP_Lamp_gv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gw____,type,
    aTP_Lamp_gw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gx____,type,
    aTP_Lamp_gx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gy____,type,
    aTP_Lamp_gy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gz____,type,
    aTP_Lamp_gz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ha____,type,
    aTP_Lamp_ha: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hb____,type,
    aTP_Lamp_hb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hc____,type,
    aTP_Lamp_hc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hd____,type,
    aTP_Lamp_hd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__he____,type,
    aTP_Lamp_he: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hf____,type,
    aTP_Lamp_hf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hg____,type,
    aTP_Lamp_hg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hh____,type,
    aTP_Lamp_hh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hi____,type,
    aTP_Lamp_hi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hj____,type,
    aTP_Lamp_hj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hk____,type,
    aTP_Lamp_hk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hl____,type,
    aTP_Lamp_hl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hm____,type,
    aTP_Lamp_hm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hn____,type,
    aTP_Lamp_hn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ho____,type,
    aTP_Lamp_ho: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hp____,type,
    aTP_Lamp_hp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hq____,type,
    aTP_Lamp_hq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hr____,type,
    aTP_Lamp_hr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hs____,type,
    aTP_Lamp_hs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ht____,type,
    aTP_Lamp_ht: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hu____,type,
    aTP_Lamp_hu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hv____,type,
    aTP_Lamp_hv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hw____,type,
    aTP_Lamp_hw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hx____,type,
    aTP_Lamp_hx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hy____,type,
    aTP_Lamp_hy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__hz____,type,
    aTP_Lamp_hz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ia____,type,
    aTP_Lamp_ia: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ib____,type,
    aTP_Lamp_ib: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ic____,type,
    aTP_Lamp_ic: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__id____,type,
    aTP_Lamp_id: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ie____,type,
    aTP_Lamp_ie: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__if____,type,
    aTP_Lamp_if: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ig____,type,
    aTP_Lamp_ig: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ih____,type,
    aTP_Lamp_ih: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ii____,type,
    aTP_Lamp_ii: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ij____,type,
    aTP_Lamp_ij: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ik____,type,
    aTP_Lamp_ik: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__il____,type,
    aTP_Lamp_il: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__im____,type,
    aTP_Lamp_im: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__in____,type,
    aTP_Lamp_in: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__io____,type,
    aTP_Lamp_io: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ip____,type,
    aTP_Lamp_ip: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iq____,type,
    aTP_Lamp_iq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ir____,type,
    aTP_Lamp_ir: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__is____,type,
    aTP_Lamp_is: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__it____,type,
    aTP_Lamp_it: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iu____,type,
    aTP_Lamp_iu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iv____,type,
    aTP_Lamp_iv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iw____,type,
    aTP_Lamp_iw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ix____,type,
    aTP_Lamp_ix: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iy____,type,
    aTP_Lamp_iy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__iz____,type,
    aTP_Lamp_iz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ja____,type,
    aTP_Lamp_ja: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jb____,type,
    aTP_Lamp_jb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jc____,type,
    aTP_Lamp_jc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jd____,type,
    aTP_Lamp_jd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__je____,type,
    aTP_Lamp_je: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jf____,type,
    aTP_Lamp_jf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jg____,type,
    aTP_Lamp_jg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jh____,type,
    aTP_Lamp_jh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ji____,type,
    aTP_Lamp_ji: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jj____,type,
    aTP_Lamp_jj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jk____,type,
    aTP_Lamp_jk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jl____,type,
    aTP_Lamp_jl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jm____,type,
    aTP_Lamp_jm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jn____,type,
    aTP_Lamp_jn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jo____,type,
    aTP_Lamp_jo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jp____,type,
    aTP_Lamp_jp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jq____,type,
    aTP_Lamp_jq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jr____,type,
    aTP_Lamp_jr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__js____,type,
    aTP_Lamp_js: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jt____,type,
    aTP_Lamp_jt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ju____,type,
    aTP_Lamp_ju: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jv____,type,
    aTP_Lamp_jv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__jw____,type,
    aTP_Lamp_jw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__jx____,type,
    aTP_Lamp_jx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jy____,type,
    aTP_Lamp_jy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jz____,type,
    aTP_Lamp_jz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ka____,type,
    aTP_Lamp_ka: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kb____,type,
    aTP_Lamp_kb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kc____,type,
    aTP_Lamp_kc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kd____,type,
    aTP_Lamp_kd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ke____,type,
    aTP_Lamp_ke: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kf____,type,
    aTP_Lamp_kf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kg____,type,
    aTP_Lamp_kg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kh____,type,
    aTP_Lamp_kh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ki____,type,
    aTP_Lamp_ki: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kj____,type,
    aTP_Lamp_kj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kk____,type,
    aTP_Lamp_kk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kl____,type,
    aTP_Lamp_kl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__km____,type,
    aTP_Lamp_km: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kn____,type,
    aTP_Lamp_kn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ko____,type,
    aTP_Lamp_ko: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kp____,type,
    aTP_Lamp_kp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kq____,type,
    aTP_Lamp_kq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__kr____,type,
    aTP_Lamp_kr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ks____,type,
    aTP_Lamp_ks: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kt____,type,
    aTP_Lamp_kt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ku____,type,
    aTP_Lamp_ku: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kv____,type,
    aTP_Lamp_kv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kw____,type,
    aTP_Lamp_kw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kx____,type,
    aTP_Lamp_kx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ky____,type,
    aTP_Lamp_ky: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__kz____,type,
    aTP_Lamp_kz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__la____,type,
    aTP_Lamp_la: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lb____,type,
    aTP_Lamp_lb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lc____,type,
    aTP_Lamp_lc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ld____,type,
    aTP_Lamp_ld: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__le____,type,
    aTP_Lamp_le: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lf____,type,
    aTP_Lamp_lf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lg____,type,
    aTP_Lamp_lg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lh____,type,
    aTP_Lamp_lh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__li____,type,
    aTP_Lamp_li: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lj____,type,
    aTP_Lamp_lj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lk____,type,
    aTP_Lamp_lk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ll____,type,
    aTP_Lamp_ll: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lm____,type,
    aTP_Lamp_lm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ln____,type,
    aTP_Lamp_ln: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lo____,type,
    aTP_Lamp_lo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lp____,type,
    aTP_Lamp_lp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lq____,type,
    aTP_Lamp_lq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__lr____,type,
    aTP_Lamp_lr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ls____,type,
    aTP_Lamp_ls: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lt____,type,
    aTP_Lamp_lt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lu____,type,
    aTP_Lamp_lu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lv____,type,
    aTP_Lamp_lv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lw____,type,
    aTP_Lamp_lw: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__lx____,type,
    aTP_Lamp_lx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ly____,type,
    aTP_Lamp_ly: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__lz____,type,
    aTP_Lamp_lz: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ma____,type,
    aTP_Lamp_ma: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mb____,type,
    aTP_Lamp_mb: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mc____,type,
    aTP_Lamp_mc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__md____,type,
    aTP_Lamp_md: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__me____,type,
    aTP_Lamp_me: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mf____,type,
    aTP_Lamp_mf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mg____,type,
    aTP_Lamp_mg: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mh____,type,
    aTP_Lamp_mh: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mi____,type,
    aTP_Lamp_mi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mj____,type,
    aTP_Lamp_mj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mk____,type,
    aTP_Lamp_mk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ml____,type,
    aTP_Lamp_ml: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mm____,type,
    aTP_Lamp_mm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mn____,type,
    aTP_Lamp_mn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mo____,type,
    aTP_Lamp_mo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mp____,type,
    aTP_Lamp_mp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mq____,type,
    aTP_Lamp_mq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mr____,type,
    aTP_Lamp_mr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ms____,type,
    aTP_Lamp_ms: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mt____,type,
    aTP_Lamp_mt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mu____,type,
    aTP_Lamp_mu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mv____,type,
    aTP_Lamp_mv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mw____,type,
    aTP_Lamp_mw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mx____,type,
    aTP_Lamp_mx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__my____,type,
    aTP_Lamp_my: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mz____,type,
    aTP_Lamp_mz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__na____,type,
    aTP_Lamp_na: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nb____,type,
    aTP_Lamp_nb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nc____,type,
    aTP_Lamp_nc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nd____,type,
    aTP_Lamp_nd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ne____,type,
    aTP_Lamp_ne: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nf____,type,
    aTP_Lamp_nf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ng____,type,
    aTP_Lamp_ng: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nh____,type,
    aTP_Lamp_nh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ni____,type,
    aTP_Lamp_ni: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nj____,type,
    aTP_Lamp_nj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nk____,type,
    aTP_Lamp_nk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nl____,type,
    aTP_Lamp_nl: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__nm____,type,
    aTP_Lamp_nm: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__nn____,type,
    aTP_Lamp_nn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__no____,type,
    aTP_Lamp_no: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__np____,type,
    aTP_Lamp_np: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nq____,type,
    aTP_Lamp_nq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nr____,type,
    aTP_Lamp_nr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ns____,type,
    aTP_Lamp_ns: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nt____,type,
    aTP_Lamp_nt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nu____,type,
    aTP_Lamp_nu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nv____,type,
    aTP_Lamp_nv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nw____,type,
    aTP_Lamp_nw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nx____,type,
    aTP_Lamp_nx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ny____,type,
    aTP_Lamp_ny: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nz____,type,
    aTP_Lamp_nz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oa____,type,
    aTP_Lamp_oa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ob____,type,
    aTP_Lamp_ob: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oc____,type,
    aTP_Lamp_oc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__od____,type,
    aTP_Lamp_od: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oe____,type,
    aTP_Lamp_oe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__of____,type,
    aTP_Lamp_of: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__og____,type,
    aTP_Lamp_og: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oh____,type,
    aTP_Lamp_oh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oi____,type,
    aTP_Lamp_oi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oj____,type,
    aTP_Lamp_oj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ok____,type,
    aTP_Lamp_ok: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ol____,type,
    aTP_Lamp_ol: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__om____,type,
    aTP_Lamp_om: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__on____,type,
    aTP_Lamp_on: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oo____,type,
    aTP_Lamp_oo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__op____,type,
    aTP_Lamp_op: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oq____,type,
    aTP_Lamp_oq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__or____,type,
    aTP_Lamp_or: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__os____,type,
    aTP_Lamp_os: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ot____,type,
    aTP_Lamp_ot: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ou____,type,
    aTP_Lamp_ou: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ov____,type,
    aTP_Lamp_ov: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ow____,type,
    aTP_Lamp_ow: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ox____,type,
    aTP_Lamp_ox: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oy____,type,
    aTP_Lamp_oy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__oz____,type,
    aTP_Lamp_oz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pa____,type,
    aTP_Lamp_pa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pb____,type,
    aTP_Lamp_pb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pc____,type,
    aTP_Lamp_pc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pd____,type,
    aTP_Lamp_pd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pe____,type,
    aTP_Lamp_pe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pf____,type,
    aTP_Lamp_pf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pg____,type,
    aTP_Lamp_pg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ph____,type,
    aTP_Lamp_ph: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pi____,type,
    aTP_Lamp_pi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pj____,type,
    aTP_Lamp_pj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pk____,type,
    aTP_Lamp_pk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pl____,type,
    aTP_Lamp_pl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pm____,type,
    aTP_Lamp_pm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pn____,type,
    aTP_Lamp_pn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__po____,type,
    aTP_Lamp_po: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pp____,type,
    aTP_Lamp_pp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pq____,type,
    aTP_Lamp_pq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pr____,type,
    aTP_Lamp_pr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ps____,type,
    aTP_Lamp_ps: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pt____,type,
    aTP_Lamp_pt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pu____,type,
    aTP_Lamp_pu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pv____,type,
    aTP_Lamp_pv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pw____,type,
    aTP_Lamp_pw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__px____,type,
    aTP_Lamp_px: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__py____,type,
    aTP_Lamp_py: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__pz____,type,
    aTP_Lamp_pz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qa____,type,
    aTP_Lamp_qa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qb____,type,
    aTP_Lamp_qb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qc____,type,
    aTP_Lamp_qc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qd____,type,
    aTP_Lamp_qd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qe____,type,
    aTP_Lamp_qe: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__qf____,type,
    aTP_Lamp_qf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qg____,type,
    aTP_Lamp_qg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qh____,type,
    aTP_Lamp_qh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qi____,type,
    aTP_Lamp_qi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qj____,type,
    aTP_Lamp_qj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qk____,type,
    aTP_Lamp_qk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ql____,type,
    aTP_Lamp_ql: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qm____,type,
    aTP_Lamp_qm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qn____,type,
    aTP_Lamp_qn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qo____,type,
    aTP_Lamp_qo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qp____,type,
    aTP_Lamp_qp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qq____,type,
    aTP_Lamp_qq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qr____,type,
    aTP_Lamp_qr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qs____,type,
    aTP_Lamp_qs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qt____,type,
    aTP_Lamp_qt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qu____,type,
    aTP_Lamp_qu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qv____,type,
    aTP_Lamp_qv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qw____,type,
    aTP_Lamp_qw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qx____,type,
    aTP_Lamp_qx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qy____,type,
    aTP_Lamp_qy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__qz____,type,
    aTP_Lamp_qz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ra____,type,
    aTP_Lamp_ra: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rb____,type,
    aTP_Lamp_rb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rc____,type,
    aTP_Lamp_rc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rd____,type,
    aTP_Lamp_rd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__re____,type,
    aTP_Lamp_re: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rf____,type,
    aTP_Lamp_rf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rg____,type,
    aTP_Lamp_rg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rh____,type,
    aTP_Lamp_rh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ri____,type,
    aTP_Lamp_ri: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rj____,type,
    aTP_Lamp_rj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rk____,type,
    aTP_Lamp_rk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rl____,type,
    aTP_Lamp_rl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rm____,type,
    aTP_Lamp_rm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rn____,type,
    aTP_Lamp_rn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ro____,type,
    aTP_Lamp_ro: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rp____,type,
    aTP_Lamp_rp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rq____,type,
    aTP_Lamp_rq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rr____,type,
    aTP_Lamp_rr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rs____,type,
    aTP_Lamp_rs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rt____,type,
    aTP_Lamp_rt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ru____,type,
    aTP_Lamp_ru: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rv____,type,
    aTP_Lamp_rv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rw____,type,
    aTP_Lamp_rw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rx____,type,
    aTP_Lamp_rx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ry____,type,
    aTP_Lamp_ry: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__rz____,type,
    aTP_Lamp_rz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sa____,type,
    aTP_Lamp_sa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sb____,type,
    aTP_Lamp_sb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sc____,type,
    aTP_Lamp_sc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sd____,type,
    aTP_Lamp_sd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__se____,type,
    aTP_Lamp_se: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sf____,type,
    aTP_Lamp_sf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sg____,type,
    aTP_Lamp_sg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sh____,type,
    aTP_Lamp_sh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__si____,type,
    aTP_Lamp_si: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sj____,type,
    aTP_Lamp_sj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sk____,type,
    aTP_Lamp_sk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sl____,type,
    aTP_Lamp_sl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sm____,type,
    aTP_Lamp_sm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sn____,type,
    aTP_Lamp_sn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__so____,type,
    aTP_Lamp_so: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sp____,type,
    aTP_Lamp_sp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sq____,type,
    aTP_Lamp_sq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sr____,type,
    aTP_Lamp_sr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ss____,type,
    aTP_Lamp_ss: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__st____,type,
    aTP_Lamp_st: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__su____,type,
    aTP_Lamp_su: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sv____,type,
    aTP_Lamp_sv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sw____,type,
    aTP_Lamp_sw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sx____,type,
    aTP_Lamp_sx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sy____,type,
    aTP_Lamp_sy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__sz____,type,
    aTP_Lamp_sz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ta____,type,
    aTP_Lamp_ta: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tb____,type,
    aTP_Lamp_tb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tc____,type,
    aTP_Lamp_tc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__td____,type,
    aTP_Lamp_td: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__te____,type,
    aTP_Lamp_te: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tf____,type,
    aTP_Lamp_tf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tg____,type,
    aTP_Lamp_tg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__th____,type,
    aTP_Lamp_th: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ti____,type,
    aTP_Lamp_ti: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tj____,type,
    aTP_Lamp_tj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tk____,type,
    aTP_Lamp_tk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tl____,type,
    aTP_Lamp_tl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tm____,type,
    aTP_Lamp_tm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tn____,type,
    aTP_Lamp_tn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__to____,type,
    aTP_Lamp_to: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tp____,type,
    aTP_Lamp_tp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tq____,type,
    aTP_Lamp_tq: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__tr____,type,
    aTP_Lamp_tr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ts____,type,
    aTP_Lamp_ts: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__tt____,type,
    aTP_Lamp_tt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tu____,type,
    aTP_Lamp_tu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tv____,type,
    aTP_Lamp_tv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tw____,type,
    aTP_Lamp_tw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tx____,type,
    aTP_Lamp_tx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ty____,type,
    aTP_Lamp_ty: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__tz____,type,
    aTP_Lamp_tz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ua____,type,
    aTP_Lamp_ua: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ub____,type,
    aTP_Lamp_ub: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uc____,type,
    aTP_Lamp_uc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ud____,type,
    aTP_Lamp_ud: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ue____,type,
    aTP_Lamp_ue: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uf____,type,
    aTP_Lamp_uf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ug____,type,
    aTP_Lamp_ug: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uh____,type,
    aTP_Lamp_uh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ui____,type,
    aTP_Lamp_ui: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uj____,type,
    aTP_Lamp_uj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uk____,type,
    aTP_Lamp_uk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ul____,type,
    aTP_Lamp_ul: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__um____,type,
    aTP_Lamp_um: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__un____,type,
    aTP_Lamp_un: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uo____,type,
    aTP_Lamp_uo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__up____,type,
    aTP_Lamp_up: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uq____,type,
    aTP_Lamp_uq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ur____,type,
    aTP_Lamp_ur: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__us____,type,
    aTP_Lamp_us: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ut____,type,
    aTP_Lamp_ut: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uu____,type,
    aTP_Lamp_uu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uv____,type,
    aTP_Lamp_uv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uw____,type,
    aTP_Lamp_uw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ux____,type,
    aTP_Lamp_ux: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uy____,type,
    aTP_Lamp_uy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__uz____,type,
    aTP_Lamp_uz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__va____,type,
    aTP_Lamp_va: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vb____,type,
    aTP_Lamp_vb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vc____,type,
    aTP_Lamp_vc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vd____,type,
    aTP_Lamp_vd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ve____,type,
    aTP_Lamp_ve: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vf____,type,
    aTP_Lamp_vf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vg____,type,
    aTP_Lamp_vg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vh____,type,
    aTP_Lamp_vh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vi____,type,
    aTP_Lamp_vi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vj____,type,
    aTP_Lamp_vj: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__vk____,type,
    aTP_Lamp_vk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vl____,type,
    aTP_Lamp_vl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vm____,type,
    aTP_Lamp_vm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vn____,type,
    aTP_Lamp_vn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vo____,type,
    aTP_Lamp_vo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vp____,type,
    aTP_Lamp_vp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vq____,type,
    aTP_Lamp_vq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vr____,type,
    aTP_Lamp_vr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vs____,type,
    aTP_Lamp_vs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vt____,type,
    aTP_Lamp_vt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vu____,type,
    aTP_Lamp_vu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vv____,type,
    aTP_Lamp_vv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vw____,type,
    aTP_Lamp_vw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vx____,type,
    aTP_Lamp_vx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vy____,type,
    aTP_Lamp_vy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__vz____,type,
    aTP_Lamp_vz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wa____,type,
    aTP_Lamp_wa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wb____,type,
    aTP_Lamp_wb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wc____,type,
    aTP_Lamp_wc: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wd____,type,
    aTP_Lamp_wd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__we____,type,
    aTP_Lamp_we: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wf____,type,
    aTP_Lamp_wf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wg____,type,
    aTP_Lamp_wg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wh____,type,
    aTP_Lamp_wh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wi____,type,
    aTP_Lamp_wi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wj____,type,
    aTP_Lamp_wj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wk____,type,
    aTP_Lamp_wk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wl____,type,
    aTP_Lamp_wl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wm____,type,
    aTP_Lamp_wm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wn____,type,
    aTP_Lamp_wn: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wo____,type,
    aTP_Lamp_wo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wp____,type,
    aTP_Lamp_wp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wq____,type,
    aTP_Lamp_wq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wr____,type,
    aTP_Lamp_wr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ws____,type,
    aTP_Lamp_ws: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wt____,type,
    aTP_Lamp_wt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wu____,type,
    aTP_Lamp_wu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wv____,type,
    aTP_Lamp_wv: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ww____,type,
    aTP_Lamp_ww: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__wx____,type,
    aTP_Lamp_wx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wy____,type,
    aTP_Lamp_wy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__wz____,type,
    aTP_Lamp_wz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xa____,type,
    aTP_Lamp_xa: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xb____,type,
    aTP_Lamp_xb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xc____,type,
    aTP_Lamp_xc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xd____,type,
    aTP_Lamp_xd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xe____,type,
    aTP_Lamp_xe: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xf____,type,
    aTP_Lamp_xf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xg____,type,
    aTP_Lamp_xg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xh____,type,
    aTP_Lamp_xh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xi____,type,
    aTP_Lamp_xi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xj____,type,
    aTP_Lamp_xj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xk____,type,
    aTP_Lamp_xk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xl____,type,
    aTP_Lamp_xl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xm____,type,
    aTP_Lamp_xm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xn____,type,
    aTP_Lamp_xn: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xo____,type,
    aTP_Lamp_xo: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xp____,type,
    aTP_Lamp_xp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xq____,type,
    aTP_Lamp_xq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xr____,type,
    aTP_Lamp_xr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xs____,type,
    aTP_Lamp_xs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xt____,type,
    aTP_Lamp_xt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xu____,type,
    aTP_Lamp_xu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xv____,type,
    aTP_Lamp_xv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xw____,type,
    aTP_Lamp_xw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xx____,type,
    aTP_Lamp_xx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xy____,type,
    aTP_Lamp_xy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xz____,type,
    aTP_Lamp_xz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ya____,type,
    aTP_Lamp_ya: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yb____,type,
    aTP_Lamp_yb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yc____,type,
    aTP_Lamp_yc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yd____,type,
    aTP_Lamp_yd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ye____,type,
    aTP_Lamp_ye: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yf____,type,
    aTP_Lamp_yf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yg____,type,
    aTP_Lamp_yg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yh____,type,
    aTP_Lamp_yh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yi____,type,
    aTP_Lamp_yi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yj____,type,
    aTP_Lamp_yj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yk____,type,
    aTP_Lamp_yk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yl____,type,
    aTP_Lamp_yl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ym____,type,
    aTP_Lamp_ym: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yn____,type,
    aTP_Lamp_yn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yo____,type,
    aTP_Lamp_yo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yp____,type,
    aTP_Lamp_yp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yq____,type,
    aTP_Lamp_yq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yr____,type,
    aTP_Lamp_yr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ys____,type,
    aTP_Lamp_ys: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yt____,type,
    aTP_Lamp_yt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yu____,type,
    aTP_Lamp_yu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yv____,type,
    aTP_Lamp_yv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yw____,type,
    aTP_Lamp_yw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yx____,type,
    aTP_Lamp_yx: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__yy____,type,
    aTP_Lamp_yy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__yz____,type,
    aTP_Lamp_yz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__za____,type,
    aTP_Lamp_za: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__zb____,type,
    aTP_Lamp_zb: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zc____,type,
    aTP_Lamp_zc: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zd____,type,
    aTP_Lamp_zd: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ze____,type,
    aTP_Lamp_ze: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zf____,type,
    aTP_Lamp_zf: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zg____,type,
    aTP_Lamp_zg: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zh____,type,
    aTP_Lamp_zh: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zi____,type,
    aTP_Lamp_zi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zj____,type,
    aTP_Lamp_zj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zk____,type,
    aTP_Lamp_zk: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zl____,type,
    aTP_Lamp_zl: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zm____,type,
    aTP_Lamp_zm: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zn____,type,
    aTP_Lamp_zn: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zo____,type,
    aTP_Lamp_zo: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zp____,type,
    aTP_Lamp_zp: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zq____,type,
    aTP_Lamp_zq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zr____,type,
    aTP_Lamp_zr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zs____,type,
    aTP_Lamp_zs: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zt____,type,
    aTP_Lamp_zt: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zu____,type,
    aTP_Lamp_zu: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zv____,type,
    aTP_Lamp_zv: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zw____,type,
    aTP_Lamp_zw: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zx____,type,
    aTP_Lamp_zx: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zy____,type,
    aTP_Lamp_zy: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__zz____,type,
    aTP_Lamp_zz: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_Archimedean__Field_Oceiling,type,
    archimedean_ceiling: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
    archim6421214686448440834_floor: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Archimedean__Field_Ofrac,type,
    archimedean_frac: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Archimedean__Field_Oround,type,
    archimedean_round: 
      !>[A: $tType] : ( A > int ) ).

tff(sy_c_Array__Time_Oalloc,type,
    array_alloc: 
      !>[A: $tType] : ( ( list(A) * heap_ext(product_unit) ) > product_prod(array(A),heap_ext(product_unit)) ) ).

tff(sy_c_Array__Time_Ofreeze,type,
    array_freeze: 
      !>[A: $tType] : ( array(A) > heap_Time_Heap(list(A)) ) ).

tff(sy_c_Array__Time_Olen,type,
    array_len: 
      !>[A: $tType] : ( array(A) > heap_Time_Heap(nat) ) ).

tff(sy_c_Array__Time_Olen_H,type,
    array_len2: 
      !>[B: $tType,A: $tType] : ( array(B) > heap_Time_Heap(A) ) ).

tff(sy_c_Array__Time_Omake,type,
    array_make: 
      !>[A: $tType] : ( ( nat * fun(nat,A) ) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Array__Time_Omap__entry,type,
    array_map_entry: 
      !>[A: $tType] : ( ( nat * fun(A,A) * array(A) ) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Array__Time_Onew,type,
    array_new: 
      !>[A: $tType] : ( ( nat * A ) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Array__Time_Onth,type,
    array_nth: 
      !>[A: $tType] : ( array(A) > fun(nat,heap_Time_Heap(A)) ) ).

tff(sy_c_Array__Time_Onth_H,type,
    array_nth2: 
      !>[A: $tType] : ( array(A) > fun(code_integer,heap_Time_Heap(A)) ) ).

tff(sy_c_Array__Time_Oof__list,type,
    array_of_list: 
      !>[A: $tType] : ( list(A) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Array__Time_Oswap,type,
    array_swap: 
      !>[A: $tType] : ( ( nat * A * array(A) ) > heap_Time_Heap(A) ) ).

tff(sy_c_Array__Time_Oupd,type,
    array_upd: 
      !>[A: $tType] : ( ( nat * A * array(A) ) > heap_Time_Heap(array(A)) ) ).

tff(sy_c_Assertions_Oassn_ORep__assn,type,
    rep_assn: ( assn * product_prod(heap_ext(product_unit),set(nat)) ) > $o ).

tff(sy_c_Assertions_Oentails,type,
    entails: fun(assn,fun(assn,$o)) ).

tff(sy_c_Assertions_Oex__assn,type,
    ex_assn: 
      !>[A: $tType] : ( fun(A,assn) > assn ) ).

tff(sy_c_Assertions_Opure__assn,type,
    pure_assn: $o > assn ).

tff(sy_c_Assertions_Osnga__assn,type,
    snga_assn: 
      !>[A: $tType] : ( array(A) > fun(list(A),assn) ) ).

tff(sy_c_Automation_OFI,type,
    fi: ( list(product_prod(assn,assn)) * assn * assn * assn * assn * assn ) > $o ).

tff(sy_c_Automation_OFI__RESULT,type,
    fI_RESULT: ( list(product_prod(assn,assn)) * assn * assn * assn ) > $o ).

tff(sy_c_Binomial_Obinomial,type,
    binomial: nat > fun(nat,nat) ).

tff(sy_c_Binomial_Ogbinomial,type,
    gbinomial: 
      !>[A: $tType] : ( A > fun(nat,A) ) ).

tff(sy_c_Bit__Comprehension_Obit__comprehension__class_Oset__bits,type,
    bit_bi4170147762399595738t_bits: 
      !>[A: $tType] : ( fun(nat,$o) > A ) ).

tff(sy_c_Bit__Comprehension_Owf__set__bits__int,type,
    bit_wf_set_bits_int: fun(nat,$o) > $o ).

tff(sy_c_Bit__Operations_Oand__int__rel,type,
    bit_and_int_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_Bit__Operations_Oand__not__num,type,
    bit_and_not_num: ( num * num ) > option(num) ).

tff(sy_c_Bit__Operations_Oconcat__bit,type,
    bit_concat_bit: ( nat * int ) > fun(int,int) ).

tff(sy_c_Bit__Operations_Oor__not__num__neg,type,
    bit_or_not_num_neg: ( num * num ) > num ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
    bit_ri4277139882892585799ns_not: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
    bit_ri4674362597316999326ke_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
    bit_se5824344872417868541ns_and: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
    bit_se4197421643247451524op_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
    bit_se8732182000553998342ip_bit: 
      !>[A: $tType] : ( ( nat * A ) > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
    bit_se2239418461657761734s_mask: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
    bit_se1065995026697491101ons_or: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
    bit_se4730199178511100633sh_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
    bit_se5668285175392031749et_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
    bit_se2584673776208193580ke_bit: 
      !>[A: $tType] : ( nat > fun(A,A) ) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
    bit_se2638667681897837118et_bit: 
      !>[A: $tType] : fun(nat,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
    bit_se5824344971392196577ns_xor: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
    bit_se5641148757651400278ts_bit: 
      !>[A: $tType] : fun(A,fun(nat,$o)) ).

tff(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
    bit_se6407376104438227557le_bit: 
      !>[A: $tType] : ( ( itself(A) * nat ) > $o ) ).

tff(sy_c_Bit__Operations_Otake__bit__num,type,
    bit_take_bit_num: ( nat * num ) > option(num) ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftl,type,
    bit_Sh4282982442137083160shiftl: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osemiring__bit__operations__class_Oshiftr,type,
    bit_Sh4282982442137083166shiftr: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Bit__Shifts__Infix__Syntax_Osshiftr,type,
    bit_Sh8784991116023147202shiftr: 
      !>[A: $tType] : ( ( word(A) * nat ) > word(A) ) ).

tff(sy_c_Bits__Integer_OBit__integer,type,
    bits_Bit_integer: ( code_integer * $o ) > code_integer ).

tff(sy_c_Bits__Integer_Obin__last__integer,type,
    bits_b8758750999018896077nteger: code_integer > $o ).

tff(sy_c_Bits__Integer_Obin__rest__integer,type,
    bits_b2549910563261871055nteger: code_integer > code_integer ).

tff(sy_c_Bits__Integer_Ointeger__set__bit,type,
    bits_integer_set_bit: ( code_integer * code_integer * $o ) > code_integer ).

tff(sy_c_Bits__Integer_Ointeger__shiftl,type,
    bits_integer_shiftl: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Bits__Integer_Ointeger__shiftr,type,
    bits_integer_shiftr: ( code_integer * code_integer ) > code_integer ).

tff(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
    boolea3799213064322606851m_diff: 
      !>[A: $tType] : ( ( fun(A,fun(A,A)) * fun(A,fun(A,A)) * fun(A,A) * A * A * fun(A,fun(A,A)) ) > $o ) ).

tff(sy_c_Code__Int__Integer__Conversion_Oint__of__integer__symbolic,type,
    code_I935103866777955880mbolic: code_integer > int ).

tff(sy_c_Code__Numeral_Obit__cut__integer,type,
    code_bit_cut_integer: code_integer > product_prod(code_integer,$o) ).

tff(sy_c_Code__Numeral_Odivmod__abs,type,
    code_divmod_abs: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Odivmod__integer,type,
    code_divmod_integer: ( code_integer * code_integer ) > product_prod(code_integer,code_integer) ).

tff(sy_c_Code__Numeral_Odup,type,
    code_dup: code_integer > code_integer ).

tff(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
    code_int_of_integer: code_integer > int ).

tff(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
    code_integer_of_int: int > code_integer ).

tff(sy_c_Code__Numeral_Ointeger__of__nat,type,
    code_integer_of_nat: nat > code_integer ).

tff(sy_c_Code__Numeral_Onat__of__integer,type,
    code_nat_of_integer: fun(code_integer,nat) ).

tff(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: code_integer > num ).

tff(sy_c_Code__Target__Int_Onegative,type,
    code_Target_negative: fun(num,int) ).

tff(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Onarrowing__samples,type,
    code_T4080844693773952564amples: 
      !>[A: $tType] : ( ( fun(code_integer,product_prod(A,A)) * A * code_integer ) > list(A) ) ).

tff(sy_c_Code__Target__Word__Base_Oquickcheck__narrowing__samples_Opartial__term__of__sample,type,
    code_T4081349890594273596sample: 
      !>[A: $tType] : ( ( fun(code_integer,product_prod(A,A)) * A * code_integer ) > A ) ).

tff(sy_c_Code__Target__Word__Base_Oset__bits__aux,type,
    code_T2661198915054445665ts_aux: 
      !>[A: $tType] : ( ( fun(nat,$o) * nat * word(A) ) > word(A) ) ).

tff(sy_c_Complete__Partial__Order_Occpo_Oadmissible,type,
    comple1908693960933563346ssible: 
      !>[A: $tType] : ( ( fun(set(A),A) * fun(A,fun(A,$o)) * fun(A,$o) ) > $o ) ).

tff(sy_c_Complete__Partial__Order_Occpo_Ofixp,type,
    comple187402453842119260l_fixp: 
      !>[A: $tType] : ( ( fun(set(A),A) * fun(A,fun(A,$o)) * fun(A,A) ) > A ) ).

tff(sy_c_Complete__Partial__Order_Omonotone,type,
    comple7038119648293358887notone: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(A,$o)) * fun(B,fun(B,$o)) * fun(A,B) ) > $o ) ).

tff(sy_c_Complex_OArg,type,
    arg: complex > real ).

tff(sy_c_Complex_Ocis,type,
    cis: real > complex ).

tff(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: ( real * real ) > complex ).

tff(sy_c_Complex_Ocsqrt,type,
    csqrt: complex > complex ).

tff(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

tff(sy_c_Deriv_Ohas__derivative,type,
    has_derivative: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * fun(A,B) * filter(A) ) > $o ) ).

tff(sy_c_Deriv_Ohas__field__derivative,type,
    has_field_derivative: 
      !>[A: $tType] : ( ( fun(A,A) * A * filter(A) ) > $o ) ).

tff(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod(int,int) > int ).

tff(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: ( int * int ) > int ).

tff(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: ( nat * nat ) > product_prod(nat,nat) ).

tff(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: ( int * int * product_prod(int,int) ) > $o ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
    unique5940410009612947441es_aux: 
      !>[A: $tType] : ( product_prod(A,A) > $o ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
    unique8689654367752047608divmod: 
      !>[A: $tType] : ( ( num * num ) > product_prod(A,A) ) ).

tff(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
    unique1321980374590559556d_step: 
      !>[A: $tType] : ( ( num * product_prod(A,A) ) > product_prod(A,A) ) ).

tff(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
    comm_s3205402744901411588hammer: 
      !>[A: $tType] : ( ( A * nat ) > A ) ).

tff(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
    semiring_char_0_fact: 
      !>[A: $tType] : ( nat > A ) ).

tff(sy_c_Fields_Oinverse__class_Oinverse,type,
    inverse_inverse: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Filter_Oat__top,type,
    at_top: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_Filter_Oeventually,type,
    eventually: 
      !>[A: $tType] : ( ( fun(A,$o) * filter(A) ) > $o ) ).

tff(sy_c_Filter_Ofilterlim,type,
    filterlim: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * filter(B) * filter(A) ) > $o ) ).

tff(sy_c_Finite__Set_Ofinite,type,
    finite_finite2: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Finite__Set_Ofold,type,
    finite_fold: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * B * set(A) ) > B ) ).

tff(sy_c_Fun_Obij__betw,type,
    bij_betw: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) * set(B) ) > $o ) ).

tff(sy_c_Fun_Ocomp,type,
    comp: 
      !>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).

tff(sy_c_Fun_Oinj__on,type,
    inj_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > $o ) ).

tff(sy_c_Fun_Othe__inv__into,type,
    the_inv_into: 
      !>[A: $tType,B: $tType] : ( ( set(A) * fun(A,B) * B ) > A ) ).

tff(sy_c_GCD_Obezw,type,
    bezw: ( nat * nat ) > product_prod(int,int) ).

tff(sy_c_GCD_Obezw__rel,type,
    bezw_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_GCD_Ogcd__class_Ogcd,type,
    gcd_gcd: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_GCD_Ogcd__nat__rel,type,
    gcd_nat_rel: fun(product_prod(nat,nat),fun(product_prod(nat,nat),$o)) ).

tff(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
    semiri4206861660011772517g_char: 
      !>[A: $tType] : ( itself(A) > nat ) ).

tff(sy_c_Generic__set__bit_Oset__bit__class_Oset__bit,type,
    generi7602027413899671122et_bit: 
      !>[A: $tType] : fun(A,fun(nat,fun($o,A))) ).

tff(sy_c_Groups_Oabs__class_Oabs,type,
    abs_abs: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Groups_Ominus__class_Ominus,type,
    minus_minus: 
      !>[A: $tType] : ( A > fun(A,A) ) ).

tff(sy_c_Groups_Oone__class_Oone,type,
    one_one: 
      !>[A: $tType] : A ).

tff(sy_c_Groups_Oplus__class_Oplus,type,
    plus_plus: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Osgn__class_Osgn,type,
    sgn_sgn: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Otimes__class_Otimes,type,
    times_times: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Groups_Ouminus__class_Ouminus,type,
    uminus_uminus: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Groups_Ozero__class_Ozero,type,
    zero_zero: 
      !>[A: $tType] : A ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
    groups7311177749621191930dd_sum: 
      !>[B: $tType,A: $tType] : ( fun(B,A) > fun(set(B),A) ) ).

tff(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
    groups1027152243600224163dd_sum: 
      !>[C: $tType,A: $tType] : ( ( fun(C,A) * set(C) ) > A ) ).

tff(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
    groups7121269368397514597t_prod: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * set(B) ) > A ) ).

tff(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
    groups4207007520872428315er_sum: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * A * list(B) ) > A ) ).

tff(sy_c_HOL_OThe,type,
    the: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_HOL_Oundefined,type,
    undefined: 
      !>[A: $tType] : A ).

tff(sy_c_Hash__Instances_Ohash__code__list,type,
    hash_hash_code_list: 
      !>[A: $tType] : ( ( fun(A,uint32) * list(A) ) > uint32 ) ).

tff(sy_c_Hash__Instances_Ohash__code__option,type,
    hash_h1887023736457453652option: 
      !>[A: $tType] : ( ( fun(A,uint32) * option(A) ) > uint32 ) ).

tff(sy_c_Hash__Instances_Ohash__code__prod,type,
    hash_hash_code_prod: 
      !>[A: $tType,B: $tType] : ( ( fun(A,uint32) * fun(B,uint32) * product_prod(A,B) ) > uint32 ) ).

tff(sy_c_Heap_Oarray_Osize__array,type,
    size_array: 
      !>[A: $tType] : ( ( fun(A,nat) * array(A) ) > nat ) ).

tff(sy_c_Heap__Time__Monad_OHeap_OHeap,type,
    heap_Time_Heap2: 
      !>[A: $tType] : ( fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_OHeap_Osize__Heap,type,
    heap_Time_size_Heap: 
      !>[A: $tType] : ( ( fun(A,nat) * heap_Time_Heap(A) ) > nat ) ).

tff(sy_c_Heap__Time__Monad_OHeap__lub,type,
    heap_Time_Heap_lub: 
      !>[A: $tType] : fun(set(heap_Time_Heap(A)),heap_Time_Heap(A)) ).

tff(sy_c_Heap__Time__Monad_OHeap__ord,type,
    heap_Time_Heap_ord: 
      !>[A: $tType] : fun(heap_Time_Heap(A),fun(heap_Time_Heap(A),$o)) ).

tff(sy_c_Heap__Time__Monad_Oassert,type,
    heap_Time_assert: 
      !>[A: $tType] : ( ( fun(A,$o) * A ) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Obind,type,
    heap_Time_bind: 
      !>[A: $tType,B: $tType] : ( ( heap_Time_Heap(A) * fun(A,heap_Time_Heap(B)) ) > heap_Time_Heap(B) ) ).

tff(sy_c_Heap__Time__Monad_Oeffect,type,
    heap_Time_effect: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) * heap_ext(product_unit) * A * nat ) > $o ) ).

tff(sy_c_Heap__Time__Monad_Oexecute,type,
    heap_Time_execute: 
      !>[A: $tType] : ( heap_Time_Heap(A) > fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))) ) ).

tff(sy_c_Heap__Time__Monad_Oguard,type,
    heap_Time_guard: 
      !>[A: $tType] : ( ( fun(heap_ext(product_unit),$o) * fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))) ) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Oheap,type,
    heap_Time_heap: 
      !>[A: $tType] : ( fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_Olift,type,
    heap_Time_lift: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(A,heap_Time_Heap(B)) ) ).

tff(sy_c_Heap__Time__Monad_Oreturn,type,
    heap_Time_return: 
      !>[A: $tType] : fun(A,heap_Time_Heap(A)) ).

tff(sy_c_Heap__Time__Monad_Osuccess,type,
    heap_Time_success: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > $o ) ).

tff(sy_c_Heap__Time__Monad_Otap,type,
    heap_Time_tap: 
      !>[A: $tType] : ( fun(heap_ext(product_unit),A) > heap_Time_Heap(A) ) ).

tff(sy_c_Heap__Time__Monad_OtimeFrame,type,
    heap_Time_timeFrame: 
      !>[A: $tType] : ( ( nat * option(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) > option(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ).

tff(sy_c_Heap__Time__Monad_Oureturn,type,
    heap_Time_ureturn: 
      !>[A: $tType] : fun(A,heap_Time_Heap(A)) ).

tff(sy_c_Heap__Time__Monad_Owait,type,
    heap_Time_wait: nat > heap_Time_Heap(product_unit) ).

tff(sy_c_Hoare__Triple_Ohoare__triple,type,
    hoare_hoare_triple: 
      !>[A: $tType] : ( ( assn * heap_Time_Heap(A) * fun(A,assn) ) > $o ) ).

tff(sy_c_Hoare__Triple_Onew__addrs,type,
    hoare_new_addrs: ( heap_ext(product_unit) * set(nat) * heap_ext(product_unit) ) > set(nat) ).

tff(sy_c_Int_Oint__ge__less__than,type,
    int_ge_less_than: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Oint__ge__less__than2,type,
    int_ge_less_than2: int > set(product_prod(int,int)) ).

tff(sy_c_Int_Onat,type,
    nat2: int > nat ).

tff(sy_c_Int_Oring__1__class_OInts,type,
    ring_1_Ints: 
      !>[A: $tType] : set(A) ).

tff(sy_c_Int_Oring__1__class_Oof__int,type,
    ring_1_of_int: 
      !>[A: $tType] : fun(int,A) ).

tff(sy_c_Lattices__Big_Olinorder__class_OMax,type,
    lattic643756798349783984er_Max: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Least__significant__bit_Olsb__class_Olsb,type,
    least_8051144512741203767sb_lsb: 
      !>[A: $tType] : fun(A,$o) ).

tff(sy_c_Limits_Oat__infinity,type,
    at_infinity: 
      !>[A: $tType] : filter(A) ).

tff(sy_c_List_OBleast,type,
    bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,$o) ) > A ) ).

tff(sy_c_List_Oabort__Bleast,type,
    abort_Bleast: 
      !>[A: $tType] : ( ( set(A) * fun(A,$o) ) > A ) ).

tff(sy_c_List_Oappend,type,
    append: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_List_Oconcat,type,
    concat: 
      !>[A: $tType] : ( list(list(A)) > list(A) ) ).

tff(sy_c_List_Ocount__list,type,
    count_list: 
      !>[A: $tType] : ( list(A) > fun(A,nat) ) ).

tff(sy_c_List_Oenumerate,type,
    enumerate: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(product_prod(nat,A)) ) ).

tff(sy_c_List_Ofilter,type,
    filter2: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > list(A) ) ).

tff(sy_c_List_Ofoldr,type,
    foldr: 
      !>[A: $tType,B: $tType] : ( ( fun(A,fun(B,B)) * list(A) * B ) > B ) ).

tff(sy_c_List_Olenlex,type,
    lenlex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olex,type,
    lex: 
      !>[A: $tType] : ( set(product_prod(A,A)) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olexn,type,
    lexn: 
      !>[A: $tType] : ( ( set(product_prod(A,A)) * nat ) > set(product_prod(list(A),list(A))) ) ).

tff(sy_c_List_Olinorder_Ostable__sort__key,type,
    stable_sort_key: 
      !>[B: $tType,A: $tType] : ( fun(fun(B,A),fun(list(B),list(B))) > $o ) ).

tff(sy_c_List_Olinorder__class_Osort__key,type,
    linorder_sort_key: 
      !>[B: $tType,A: $tType] : ( fun(B,A) > fun(list(B),list(B)) ) ).

tff(sy_c_List_Olist_OCons,type,
    cons: 
      !>[A: $tType] : ( A > fun(list(A),list(A)) ) ).

tff(sy_c_List_Olist_ONil,type,
    nil: 
      !>[A: $tType] : list(A) ).

tff(sy_c_List_Olist_Ocase__list,type,
    case_list: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,fun(list(A),B)) ) > fun(list(A),B) ) ).

tff(sy_c_List_Olist_Ohd,type,
    hd: 
      !>[A: $tType] : fun(list(A),A) ).

tff(sy_c_List_Olist_Omap,type,
    map: 
      !>[A: $tType,Aa: $tType] : ( ( fun(A,Aa) * list(A) ) > list(Aa) ) ).

tff(sy_c_List_Olist_Oset,type,
    set2: 
      !>[A: $tType] : ( list(A) > set(A) ) ).

tff(sy_c_List_Olist_Osize__list,type,
    size_list: 
      !>[A: $tType] : ( ( fun(A,nat) * list(A) ) > nat ) ).

tff(sy_c_List_Olist_Otl,type,
    tl: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_List_Olist__update,type,
    list_update: 
      !>[A: $tType] : ( ( list(A) * nat * A ) > list(A) ) ).

tff(sy_c_List_Omap__filter,type,
    map_filter: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * list(A) ) > list(B) ) ).

tff(sy_c_List_Omin__list,type,
    min_list: 
      !>[A: $tType] : ( list(A) > A ) ).

tff(sy_c_List_Omin__list__rel,type,
    min_list_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),$o)) ).

tff(sy_c_List_On__lists,type,
    n_lists: 
      !>[A: $tType] : ( ( nat * list(A) ) > list(list(A)) ) ).

tff(sy_c_List_Onth,type,
    nth: 
      !>[A: $tType] : ( list(A) > fun(nat,A) ) ).

tff(sy_c_List_Oproduct,type,
    product: 
      !>[A: $tType,B: $tType] : ( ( list(A) * list(B) ) > list(product_prod(A,B)) ) ).

tff(sy_c_List_Oproduct__lists,type,
    product_lists: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Oreplicate,type,
    replicate: 
      !>[A: $tType] : ( ( nat * A ) > list(A) ) ).

tff(sy_c_List_Oset__Cons,type,
    set_Cons: 
      !>[A: $tType] : ( ( set(A) * set(list(A)) ) > set(list(A)) ) ).

tff(sy_c_List_Osubseqs,type,
    subseqs: 
      !>[A: $tType] : ( list(A) > list(list(A)) ) ).

tff(sy_c_List_Otranspose,type,
    transpose: 
      !>[A: $tType] : ( list(list(A)) > list(list(A)) ) ).

tff(sy_c_List_Otranspose__rel,type,
    transpose_rel: 
      !>[A: $tType] : fun(list(list(A)),fun(list(list(A)),$o)) ).

tff(sy_c_List_Oupt,type,
    upt: ( nat * nat ) > list(nat) ).

tff(sy_c_List_Oupto,type,
    upto: ( int * int ) > list(int) ).

tff(sy_c_List_Oupto__aux,type,
    upto_aux: ( int * int * list(int) ) > list(int) ).

tff(sy_c_List_Oupto__rel,type,
    upto_rel: fun(product_prod(int,int),fun(product_prod(int,int),$o)) ).

tff(sy_c_Map_Oran,type,
    ran: 
      !>[A: $tType,B: $tType] : ( fun(A,option(B)) > set(B) ) ).

tff(sy_c_Misc_OEps__Opt,type,
    eps_Opt: 
      !>[A: $tType] : ( fun(A,$o) > option(A) ) ).

tff(sy_c_Misc_Ofun__of__rel,type,
    fun_of_rel: 
      !>[B: $tType,A: $tType] : ( ( set(product_prod(B,A)) * B ) > A ) ).

tff(sy_c_Misc_Omergesort__by__rel__split,type,
    merges295452479951948502_split: 
      !>[A: $tType] : ( ( product_prod(list(A),list(A)) * list(A) ) > product_prod(list(A),list(A)) ) ).

tff(sy_c_Misc_Opairself,type,
    pairself: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(product_prod(A,A),product_prod(B,B)) ) ).

tff(sy_c_Misc_Orel__of,type,
    rel_of: 
      !>[A: $tType,B: $tType] : ( ( fun(A,option(B)) * fun(product_prod(A,B),$o) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Misc_Oslice,type,
    slice: 
      !>[A: $tType] : ( ( nat * nat * list(A) ) > list(A) ) ).

tff(sy_c_Most__significant__bit_Omsb__class_Omsb,type,
    most_s684356279273892711sb_msb: 
      !>[A: $tType] : ( A > $o ) ).

tff(sy_c_Multiset_Ois__empty,type,
    is_empty: 
      !>[A: $tType] : ( multiset(A) > $o ) ).

tff(sy_c_Multiset_Olinorder__class_Opart,type,
    linorder_part: 
      !>[B: $tType,A: $tType] : ( ( fun(B,A) * A * list(B) ) > product_prod(list(B),product_prod(list(B),list(B))) ) ).

tff(sy_c_Nat_OSuc,type,
    suc: fun(nat,nat) ).

tff(sy_c_Nat_Onat_Ocase__nat,type,
    case_nat: 
      !>[A: $tType] : ( ( A * fun(nat,A) * nat ) > A ) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
    semiring_1_of_nat: 
      !>[A: $tType] : fun(nat,A) ).

tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
    semiri8178284476397505188at_aux: 
      !>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).

tff(sy_c_Nat_Osize__class_Osize,type,
    size_size: 
      !>[A: $tType] : fun(A,nat) ).

tff(sy_c_Nat__Bijection_Oset__decode,type,
    nat_set_decode: nat > set(nat) ).

tff(sy_c_Nat__Bijection_Oset__encode,type,
    nat_set_encode: fun(set(nat),nat) ).

tff(sy_c_Nat__Bijection_Otriangle,type,
    nat_triangle: nat > nat ).

tff(sy_c_NthRoot_Oroot,type,
    root: nat > fun(real,real) ).

tff(sy_c_NthRoot_Osqrt,type,
    sqrt: fun(real,real) ).

tff(sy_c_Num_OBitM,type,
    bitM: num > num ).

tff(sy_c_Num_Oinc,type,
    inc: num > num ).

tff(sy_c_Num_Oneg__numeral__class_Odbl,type,
    neg_numeral_dbl: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
    neg_numeral_dbl_dec: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
    neg_numeral_dbl_inc: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Num_Onum_OBit0,type,
    bit0: num > num ).

tff(sy_c_Num_Onum_OBit1,type,
    bit1: fun(num,num) ).

tff(sy_c_Num_Onum_OOne,type,
    one2: num ).

tff(sy_c_Num_Onum_Ocase__num,type,
    case_num: 
      !>[A: $tType] : ( ( A * fun(num,A) * fun(num,A) * num ) > A ) ).

tff(sy_c_Num_Onum_Osize__num,type,
    size_num: num > nat ).

tff(sy_c_Num_Onum__of__nat,type,
    num_of_nat: nat > num ).

tff(sy_c_Num_Onumeral__class_Onumeral,type,
    numeral_numeral: 
      !>[A: $tType] : fun(num,A) ).

tff(sy_c_Num_Opow,type,
    pow: ( num * num ) > num ).

tff(sy_c_Num_Opred__numeral,type,
    pred_numeral: num > nat ).

tff(sy_c_Option_Ooption_ONone,type,
    none: 
      !>[A: $tType] : option(A) ).

tff(sy_c_Option_Ooption_OSome,type,
    some: 
      !>[A: $tType] : fun(A,option(A)) ).

tff(sy_c_Option_Ooption_Ocase__option,type,
    case_option: 
      !>[B: $tType,A: $tType] : ( ( B * fun(A,B) * option(A) ) > B ) ).

tff(sy_c_Option_Ooption_Osize__option,type,
    size_option: 
      !>[A: $tType] : ( ( fun(A,nat) * option(A) ) > nat ) ).

tff(sy_c_Option_Ooption_Othe,type,
    the2: 
      !>[A: $tType] : fun(option(A),A) ).

tff(sy_c_Orderings_Obot__class_Obot,type,
    bot_bot: 
      !>[A: $tType] : A ).

tff(sy_c_Orderings_Oord__class_OLeast,type,
    ord_Least: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_Orderings_Oord__class_Oless,type,
    ord_less: 
      !>[A: $tType] : ( A > fun(A,$o) ) ).

tff(sy_c_Orderings_Oord__class_Oless__eq,type,
    ord_less_eq: 
      !>[A: $tType] : ( A > fun(A,$o) ) ).

tff(sy_c_Orderings_Oord__class_Omax,type,
    ord_max: 
      !>[A: $tType] : ( A > fun(A,A) ) ).

tff(sy_c_Orderings_Oord__class_Omin,type,
    ord_min: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Orderings_Oorder__class_OGreatest,type,
    order_Greatest: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_Orderings_Otop__class_Otop,type,
    top_top: 
      !>[A: $tType] : A ).

tff(sy_c_Partial__Function_Oflat__lub,type,
    partial_flat_lub: 
      !>[A: $tType] : ( A > fun(set(A),A) ) ).

tff(sy_c_Partial__Function_Oflat__ord,type,
    partial_flat_ord: 
      !>[A: $tType] : ( A > fun(A,fun(A,$o)) ) ).

tff(sy_c_Partial__Function_Ofun__lub,type,
    partial_fun_lub: 
      !>[C: $tType,B: $tType,A: $tType] : ( fun(set(C),B) > fun(set(fun(A,C)),fun(A,B)) ) ).

tff(sy_c_Partial__Function_Ofun__ord,type,
    partial_fun_ord: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,$o)) > fun(fun(C,A),fun(fun(C,B),$o)) ) ).

tff(sy_c_Partial__Function_Opartial__function__definitions,type,
    partia7178651479351089652itions: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * fun(set(A),A) ) > $o ) ).

tff(sy_c_Power_Opower__class_Opower,type,
    power_power: 
      !>[A: $tType] : fun(A,fun(nat,A)) ).

tff(sy_c_Product__Type_OPair,type,
    product_Pair: 
      !>[A: $tType,B: $tType] : fun(A,fun(B,product_prod(A,B))) ).

tff(sy_c_Product__Type_Oapsnd,type,
    product_apsnd: 
      !>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(product_prod(A,B),product_prod(A,C))) ).

tff(sy_c_Product__Type_Ocurry,type,
    product_curry: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(product_prod(A,B),C) > fun(A,fun(B,C)) ) ).

tff(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
    product_rec_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > T ) ).

tff(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
    product_rec_set_prod: 
      !>[A: $tType,B: $tType,T: $tType] : ( ( fun(A,fun(B,T)) * product_prod(A,B) ) > fun(T,$o) ) ).

tff(sy_c_Product__Type_Oprod_Ocase__prod,type,
    product_case_prod: 
      !>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(product_prod(A,B),C) ) ).

tff(sy_c_Product__Type_Oprod_Ofst,type,
    product_fst: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),A) ).

tff(sy_c_Product__Type_Oprod_Osnd,type,
    product_snd: 
      !>[A: $tType,B: $tType] : fun(product_prod(A,B),B) ).

tff(sy_c_Product__Type_Oproduct,type,
    product_product: 
      !>[A: $tType,B: $tType] : ( ( set(A) * set(B) ) > set(product_prod(A,B)) ) ).

tff(sy_c_Pure_Otype,type,
    type2: 
      !>[A: $tType] : itself(A) ).

tff(sy_c_Quicksort_Olinorder__class_Oquicksort,type,
    linorder_quicksort: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_Quicksort_Olinorder__class_Oquicksort__rel,type,
    linord6200660962353139674rt_rel: 
      !>[A: $tType] : fun(list(A),fun(list(A),$o)) ).

tff(sy_c_Rat_OFrct,type,
    frct: product_prod(int,int) > rat ).

tff(sy_c_Rat_Onormalize,type,
    normalize: product_prod(int,int) > product_prod(int,int) ).

tff(sy_c_Rat_Oof__int,type,
    of_int: int > rat ).

tff(sy_c_Rat_Oquotient__of,type,
    quotient_of: rat > product_prod(int,int) ).

tff(sy_c_Real__Vector__Spaces_Obounded__linear,type,
    real_V3181309239436604168linear: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > $o ) ).

tff(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
    real_V557655796197034286t_dist: 
      !>[A: $tType] : ( ( A * A ) > real ) ).

tff(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
    real_V7770717601297561774m_norm: 
      !>[A: $tType] : ( A > real ) ).

tff(sy_c_Real__Vector__Spaces_Oof__real,type,
    real_Vector_of_real: 
      !>[A: $tType] : fun(real,A) ).

tff(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
    real_V8093663219630862766scaleR: 
      !>[A: $tType] : ( real > fun(A,A) ) ).

tff(sy_c_Refine__Imp__Hol_Oassert_H,type,
    refine_Imp_assert: $o > heap_Time_Heap(product_unit) ).

tff(sy_c_Refine__Imp__Hol_Orefines,type,
    refine_Imp_refines: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_Time_Heap(A) ) > $o ) ).

tff(sy_c_Rings_Odivide__class_Odivide,type,
    divide_divide: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Rings_Odvd__class_Odvd,type,
    dvd_dvd: 
      !>[A: $tType] : ( A > fun(A,$o) ) ).

tff(sy_c_Rings_Omodulo__class_Omodulo,type,
    modulo_modulo: 
      !>[A: $tType] : fun(A,fun(A,A)) ).

tff(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
    zero_neq_one_of_bool: 
      !>[A: $tType] : fun($o,A) ).

tff(sy_c_Series_Osuminf,type,
    suminf: 
      !>[A: $tType] : ( fun(nat,A) > A ) ).

tff(sy_c_Series_Osummable,type,
    summable: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Series_Osums,type,
    sums: 
      !>[A: $tType] : ( fun(nat,A) > fun(A,$o) ) ).

tff(sy_c_Set_OCollect,type,
    collect: 
      !>[A: $tType] : ( fun(A,$o) > set(A) ) ).

tff(sy_c_Set_Ofilter,type,
    filter3: 
      !>[A: $tType] : ( ( fun(A,$o) * set(A) ) > set(A) ) ).

tff(sy_c_Set_Oimage,type,
    image: 
      !>[A: $tType,B: $tType] : ( fun(A,B) > fun(set(A),set(B)) ) ).

tff(sy_c_Set_Oinsert,type,
    insert: 
      !>[A: $tType] : ( A > fun(set(A),set(A)) ) ).

tff(sy_c_Set_Othe__elem,type,
    the_elem: 
      !>[A: $tType] : ( set(A) > A ) ).

tff(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
    set_fo6178422350223883121st_nat: 
      !>[A: $tType] : ( ( fun(nat,fun(A,A)) * nat * nat * A ) > A ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
    set_or1337092689740270186AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
    set_or7035219750837199246ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OatMost,type,
    set_ord_atMost: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
    set_or3652927894154168847AtMost: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
    set_or5935395276787703475ssThan: 
      !>[A: $tType] : ( ( A * A ) > set(A) ) ).

tff(sy_c_Set__Interval_Oord__class_OlessThan,type,
    set_ord_lessThan: 
      !>[A: $tType] : ( A > set(A) ) ).

tff(sy_c_Signed__Division_Osigned__division__class_Osigned__divide,type,
    signed7115095781618012415divide: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Signed__Division_Osigned__division__class_Osigned__modulo,type,
    signed6721504322012087516modulo: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_String_Oascii__of,type,
    ascii_of: char > char ).

tff(sy_c_String_Ochar_OChar,type,
    char2: ( $o * $o * $o * $o * $o * $o * $o ) > fun($o,char) ).

tff(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

tff(sy_c_String_Ochar__of__integer,type,
    char_of_integer: code_integer > char ).

tff(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
    comm_s6883823935334413003f_char: 
      !>[A: $tType] : fun(char,A) ).

tff(sy_c_String_Ointeger__of__char,type,
    integer_of_char: char > code_integer ).

tff(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
    unique5772411509450598832har_of: 
      !>[A: $tType] : fun(A,char) ).

tff(sy_c_Syntax__Match_Osyntax__fo__nomatch,type,
    syntax7388354845996824322omatch: 
      !>[A: $tType,B: $tType] : ( ( A * B ) > $o ) ).

tff(sy_c_Time__Reasoning_OEQ,type,
    time_EQ: 
      !>[A: $tType] : ( ( A * A ) > $o ) ).

tff(sy_c_Time__Reasoning_OTBOUND,type,
    time_TBOUND: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * nat ) > $o ) ).

tff(sy_c_Time__Reasoning_Ofails,type,
    time_fails: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > $o ) ).

tff(sy_c_Time__Reasoning_Ohtt,type,
    time_htt: 
      !>[A: $tType] : ( ( assn * heap_Time_Heap(A) * fun(A,assn) * nat ) > $o ) ).

tff(sy_c_Time__Reasoning_Othe__heap,type,
    time_the_heap: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > heap_ext(product_unit) ) ).

tff(sy_c_Time__Reasoning_Othe__res,type,
    time_the_res: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > A ) ).

tff(sy_c_Time__Reasoning_Otime,type,
    time_time: 
      !>[A: $tType] : ( ( heap_Time_Heap(A) * heap_ext(product_unit) ) > nat ) ).

tff(sy_c_Topological__Spaces_Ocontinuous,type,
    topolo3448309680560233919inuous: 
      !>[A: $tType,B: $tType] : ( ( filter(A) * fun(A,B) ) > $o ) ).

tff(sy_c_Topological__Spaces_Omonoseq,type,
    topological_monoseq: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
    topolo1002775350975398744n_open: 
      !>[A: $tType] : ( set(A) > $o ) ).

tff(sy_c_Topological__Spaces_Ot2__space__class_OLim,type,
    topolo3827282254853284352ce_Lim: 
      !>[F: $tType,A: $tType] : ( ( filter(F) * fun(F,A) ) > A ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
    topolo174197925503356063within: 
      !>[A: $tType] : ( ( A * set(A) ) > filter(A) ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
    topolo7230453075368039082e_nhds: 
      !>[A: $tType] : ( A > filter(A) ) ).

tff(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy,type,
    topolo3814608138187158403Cauchy: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Transcendental_Oarccos,type,
    arccos: fun(real,real) ).

tff(sy_c_Transcendental_Oarcosh,type,
    arcosh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: fun(real,real) ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: fun(real,real) ).

tff(sy_c_Transcendental_Oarsinh,type,
    arsinh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Oartanh,type,
    artanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Ocos,type,
    cos: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocos__coeff,type,
    cos_coeff: nat > real ).

tff(sy_c_Transcendental_Ocosh,type,
    cosh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Ocot,type,
    cot: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Odiffs,type,
    diffs: 
      !>[A: $tType] : ( fun(nat,A) > fun(nat,A) ) ).

tff(sy_c_Transcendental_Oexp,type,
    exp: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Oln__class_Oln,type,
    ln_ln: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Olog,type,
    log: real > fun(real,real) ).

tff(sy_c_Transcendental_Opi,type,
    pi: real ).

tff(sy_c_Transcendental_Opowr,type,
    powr: 
      !>[A: $tType] : ( ( A * A ) > A ) ).

tff(sy_c_Transcendental_Osin,type,
    sin: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Osin__coeff,type,
    sin_coeff: nat > real ).

tff(sy_c_Transcendental_Osinh,type,
    sinh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Otan,type,
    tan: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Transcendental_Otanh,type,
    tanh: 
      !>[A: $tType] : fun(A,A) ).

tff(sy_c_Type__Length_Olen0__class_Olen__of,type,
    type_len0_len_of: 
      !>[A: $tType] : ( itself(A) > nat ) ).

tff(sy_c_Uint32_ORep__uint32_H,type,
    rep_uint32: uint32 > word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))) ).

tff(sy_c_Uint32_OUint32,type,
    uint322: fun(code_integer,uint32) ).

tff(sy_c_Uint32_OUint32__signed,type,
    uint32_signed: code_integer > uint32 ).

tff(sy_c_Uint32_Odiv0__uint32,type,
    div0_uint32: uint32 > uint32 ).

tff(sy_c_Uint32_Ointeger__of__uint32,type,
    integer_of_uint32: fun(uint32,code_integer) ).

tff(sy_c_Uint32_Ointeger__of__uint32__signed,type,
    intege5370686899274169573signed: uint32 > code_integer ).

tff(sy_c_Uint32_Omod0__uint32,type,
    mod0_uint32: uint32 > uint32 ).

tff(sy_c_Uint32_Oset__bits__aux__uint32,type,
    set_bits_aux_uint32: ( fun(nat,$o) * nat * uint32 ) > uint32 ).

tff(sy_c_Uint32_Osigned__drop__bit__uint32,type,
    signed489701013188660438uint32: fun(nat,fun(uint32,uint32)) ).

tff(sy_c_Uint32_Ouint32_OAbs__uint32,type,
    abs_uint32: word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))) > uint32 ).

tff(sy_c_Uint32_Ouint32_ORep__uint32,type,
    rep_uint322: uint32 > word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))) ).

tff(sy_c_Uint32_Ouint32__div,type,
    uint32_div: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Uint32_Ouint32__divmod,type,
    uint32_divmod: ( uint32 * uint32 ) > product_prod(uint32,uint32) ).

tff(sy_c_Uint32_Ouint32__mod,type,
    uint32_mod: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Uint32_Ouint32__sdiv,type,
    uint32_sdiv: ( uint32 * uint32 ) > uint32 ).

tff(sy_c_Uint32_Ouint32__set__bit,type,
    uint32_set_bit: ( uint32 * code_integer * $o ) > uint32 ).

tff(sy_c_Uint32_Ouint32__shiftl,type,
    uint32_shiftl: ( uint32 * code_integer ) > uint32 ).

tff(sy_c_Uint32_Ouint32__shiftr,type,
    uint32_shiftr: ( uint32 * code_integer ) > uint32 ).

tff(sy_c_Uint32_Ouint32__sshiftr,type,
    uint32_sshiftr: ( uint32 * code_integer ) > uint32 ).

tff(sy_c_Uint32_Ouint32__test__bit,type,
    uint32_test_bit: ( uint32 * code_integer ) > $o ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t,type,
    vEBT_T_i_n_s_e_r_t: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H,type,
    vEBT_T_i_n_s_e_r_t2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H__rel,type,
    vEBT_T5076183648494686801_t_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t__rel,type,
    vEBT_T9217963907923527482_t_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t,type,
    vEBT_T_m_a_x_t: vEBT_VEBT > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t__rel,type,
    vEBT_T_m_a_x_t_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r,type,
    vEBT_T_m_e_m_b_e_r: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H,type,
    vEBT_T_m_e_m_b_e_r2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H__rel,type,
    vEBT_T8099345112685741742_r_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r__rel,type,
    vEBT_T5837161174952499735_r_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l,type,
    vEBT_T_m_i_n_N_u_l_l: vEBT_VEBT > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l__rel,type,
    vEBT_T5462971552011256508_l_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t,type,
    vEBT_T_m_i_n_t: vEBT_VEBT > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t__rel,type,
    vEBT_T_m_i_n_t_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d,type,
    vEBT_T_p_r_e_d: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H,type,
    vEBT_T_p_r_e_d2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H__rel,type,
    vEBT_T_p_r_e_d_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d__rel,type,
    vEBT_T_p_r_e_d_rel2: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c,type,
    vEBT_T_s_u_c_c: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H,type,
    vEBT_T_s_u_c_c2: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H__rel,type,
    vEBT_T_s_u_c_c_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Bounds_OT_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c__rel,type,
    vEBT_T_s_u_c_c_rel2: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi,type,
    vEBT_V441764108873111860ildupi: nat > nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H,type,
    vEBT_V9176841429113362141ildupi: nat > int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi_H__rel,type,
    vEBT_V3352910403632780892pi_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OT__vebt__buildupi__rel,type,
    vEBT_V2957053500504383685pi_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb,type,
    vEBT_VEBT_Tb: nat > int ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H,type,
    vEBT_VEBT_Tb2: nat > nat ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb_H__rel,type,
    vEBT_VEBT_Tb_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OTb__rel,type,
    vEBT_VEBT_Tb_rel2: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ohighi,type,
    vEBT_VEBT_highi: ( nat * nat ) > heap_Time_Heap(nat) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Olowi,type,
    vEBT_VEBT_lowi: ( nat * nat ) > heap_Time_Heap(nat) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli,type,
    vEBT_VEBT_minNulli: vEBT_VEBTi > heap_Time_Heap($o) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_OminNulli__rel,type,
    vEBT_V5740978063120863272li_rel: fun(vEBT_VEBTi,fun(vEBT_VEBTi,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei,type,
    vEBT_VEBT_replicatei: 
      !>[A: $tType] : ( ( nat * heap_Time_Heap(A) ) > heap_Time_Heap(list(A)) ) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Oreplicatei__rel,type,
    vEBT_V8535202610096940344ei_rel: 
      !>[A: $tType] : fun(product_prod(nat,heap_Time_Heap(A)),fun(product_prod(nat,heap_Time_Heap(A)),$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H,type,
    vEBT_V739175172307565963ildupi: nat > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__buildupi_H__rel,type,
    vEBT_V254170901696579886pi_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__inserti_H,type,
    vEBT_V3964819847710782039nserti: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBT__internal_Ovebt__memberi_H,type,
    vEBT_V854960066525838166emberi: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_OLeafi,type,
    vEBT_Leafi: ( $o * $o ) > vEBT_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_ONodei,type,
    vEBT_Nodei: ( option(product_prod(nat,nat)) * nat * array(vEBT_VEBTi) * vEBT_VEBTi ) > vEBT_VEBTi ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Ocase__VEBTi,type,
    vEBT_case_VEBTi: 
      !>[A: $tType] : ( ( fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))) * fun($o,fun($o,A)) * vEBT_VEBTi ) > A ) ).

tff(sy_c_VEBT__BuildupMemImp_OVEBTi_Osize__VEBTi,type,
    vEBT_size_VEBTi: fun(vEBT_VEBTi,nat) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw,type,
    vEBT_vebt_assn_raw: fun(vEBT_VEBT,fun(vEBT_VEBTi,assn)) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__assn__raw__rel,type,
    vEBT_v8524038756793281170aw_rel: fun(product_prod(vEBT_VEBT,vEBT_VEBTi),fun(product_prod(vEBT_VEBT,vEBT_VEBTi),$o)) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi,type,
    vEBT_vebt_buildupi: nat > heap_Time_Heap(vEBT_VEBTi) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__buildupi__rel,type,
    vEBT_v1230518104690509829pi_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__inserti,type,
    vEBT_vebt_inserti: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti,type,
    vEBT_vebt_maxti: vEBT_VEBTi > heap_Time_Heap(option(nat)) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__maxti__rel,type,
    vEBT_vebt_maxti_rel: fun(vEBT_VEBTi,fun(vEBT_VEBTi,$o)) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__memberi,type,
    vEBT_vebt_memberi: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti,type,
    vEBT_vebt_minti: vEBT_VEBTi > heap_Time_Heap(option(nat)) ).

tff(sy_c_VEBT__BuildupMemImp_Ovebt__minti__rel,type,
    vEBT_vebt_minti_rel: fun(vEBT_VEBTi,fun(vEBT_VEBTi,$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
    vEBT_Leaf: ( $o * $o ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_ONode,type,
    vEBT_Node: ( option(product_prod(nat,nat)) * nat * list(vEBT_VEBT) * vEBT_VEBT ) > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_OVEBT_Ocase__VEBT,type,
    vEBT_case_VEBT: 
      !>[A: $tType] : ( ( fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))) * fun($o,fun($o,A)) * vEBT_VEBT ) > A ) ).

tff(sy_c_VEBT__Definitions_OVEBT_Ois__Node,type,
    vEBT_is_Node: vEBT_VEBT > $o ).

tff(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
    vEBT_size_VEBT: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > fun(nat,$o) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
    vEBT_VEBT_high: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
    vEBT_V5917875025757280293ildren: ( nat * list(vEBT_VEBT) * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
    vEBT_VEBT_low: ( nat * nat ) > nat ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
    vEBT_VEBT_membermima: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
    vEBT_V4351362008482014158ma_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
    vEBT_V5719532721284313246member: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
    vEBT_V5765760719290551771er_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
    vEBT_VEBT_valid: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
    vEBT_VEBT_valid_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Definitions_Oinvar__vebt,type,
    vEBT_invar_vebt: ( vEBT_VEBT * nat ) > $o ).

tff(sy_c_VEBT__Definitions_Oset__vebt,type,
    vEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

tff(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
    vEBT_v4011308405150292612up_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e,type,
    vEBT_T_d_e_l_e_t_e: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__DeleteBounds_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e__rel,type,
    vEBT_T8441311223069195367_e_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H,type,
    vEBT_V1232361888498592333_e_t_e: ( vEBT_VEBT * nat ) > nat ).

tff(sy_c_VEBT__DeleteBounds_OVEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H__rel,type,
    vEBT_V6368547301243506412_e_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Delete_Ovebt__delete,type,
    vEBT_vebt_delete: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Delete_Ovebt__delete__rel,type,
    vEBT_vebt_delete_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight,type,
    vEBT_VEBT_height: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Height_OVEBT__internal_Oheight__rel,type,
    vEBT_VEBT_height_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Insert_Ovebt__insert,type,
    vEBT_vebt_insert: ( vEBT_VEBT * nat ) > vEBT_VEBT ).

tff(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
    vEBT_vebt_insert_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__List__Assn_OlistI__assn,type,
    vEBT_List_listI_assn: 
      !>[A: $tType,B: $tType] : ( ( set(nat) * fun(A,fun(B,assn)) * list(A) * list(B) ) > assn ) ).

tff(sy_c_VEBT__List__Assn_Olist__assn,type,
    vEBT_List_list_assn: 
      !>[A: $tType,C: $tType] : ( ( fun(A,fun(C,assn)) * list(A) ) > fun(list(C),assn) ) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: ( nat * nat * nat ) > nat ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
    vEBT_VEBT_minNull: vEBT_VEBT > $o ).

tff(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
    vEBT_V6963167321098673237ll_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set(nat) ).

tff(sy_c_VEBT__Member_Ovebt__member,type,
    vEBT_vebt_member: vEBT_VEBT > fun(nat,$o) ).

tff(sy_c_VEBT__Member_Ovebt__member__rel,type,
    vEBT_vebt_member_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
    vEBT_VEBT_add: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
    vEBT_VEBT_greater: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
    vEBT_VEBT_less: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
    vEBT_VEBT_lesseq: ( option(nat) * option(nat) ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
    vEBT_VEBT_max_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
    vEBT_VEBT_min_in_set: ( set(nat) * nat ) > $o ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
    vEBT_VEBT_mul: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift,type,
    vEBT_V2048590022279873568_shift: 
      !>[A: $tType] : ( fun(A,fun(A,A)) > fun(option(A),fun(option(A),option(A))) ) ).

tff(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
    vEBT_VEBT_power: fun(option(nat),fun(option(nat),option(nat))) ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt,type,
    vEBT_vebt_maxt: vEBT_VEBT > option(nat) ).

tff(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
    vEBT_vebt_maxt_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__MinMax_Ovebt__mint,type,
    vEBT_vebt_mint: vEBT_VEBT > option(nat) ).

tff(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
    vEBT_vebt_mint_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Pred_Ois__pred__in__set,type,
    vEBT_is_pred_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Pred_Ovebt__pred,type,
    vEBT_vebt_pred: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Pred_Ovebt__pred__rel,type,
    vEBT_vebt_pred_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d,type,
    vEBT_V8646137997579335489_i_l_d: nat > nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p,type,
    vEBT_V8346862874174094_d_u_p: nat > nat ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p__rel,type,
    vEBT_V1247956027447740395_p_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d__rel,type,
    vEBT_V5144397997797733112_d_rel: fun(nat,fun(nat,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt,type,
    vEBT_VEBT_cnt: fun(vEBT_VEBT,real) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H,type,
    vEBT_VEBT_cnt2: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt_H__rel,type,
    vEBT_VEBT_cnt_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ocnt__rel,type,
    vEBT_VEBT_cnt_rel2: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace,type,
    vEBT_VEBT_space: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H,type,
    vEBT_VEBT_space2: fun(vEBT_VEBT,nat) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace_H__rel,type,
    vEBT_VEBT_space_rel: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__Space_OVEBT__internal_Ospace__rel,type,
    vEBT_VEBT_space_rel2: fun(vEBT_VEBT,fun(vEBT_VEBT,$o)) ).

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__predi_H,type,
    vEBT_VEBT_vebt_predi: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))) ).

tff(sy_c_VEBT__SuccPredImperative_OVEBT__internal_Ovebt__succi_H,type,
    vEBT_VEBT_vebt_succi: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))) ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__predi,type,
    vEBT_vebt_predi: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))) ).

tff(sy_c_VEBT__SuccPredImperative_Ovebt__succi,type,
    vEBT_vebt_succi: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))) ).

tff(sy_c_VEBT__Succ_Ois__succ__in__set,type,
    vEBT_is_succ_in_set: ( set(nat) * nat * nat ) > $o ).

tff(sy_c_VEBT__Succ_Ovebt__succ,type,
    vEBT_vebt_succ: ( vEBT_VEBT * nat ) > option(nat) ).

tff(sy_c_VEBT__Succ_Ovebt__succ__rel,type,
    vEBT_vebt_succ_rel: fun(product_prod(vEBT_VEBT,nat),fun(product_prod(vEBT_VEBT,nat),$o)) ).

tff(sy_c_Wellfounded_Oaccp,type,
    accp: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * A ) > $o ) ).

tff(sy_c_Word_Oeven__word,type,
    even_word: 
      !>[A: $tType] : fun(word(A),$o) ).

tff(sy_c_Word_Orevcast,type,
    revcast: 
      !>[A: $tType,B: $tType] : ( word(A) > word(B) ) ).

tff(sy_c_Word_Oring__1__class_Osigned,type,
    ring_1_signed: 
      !>[B: $tType,A: $tType] : ( word(B) > A ) ).

tff(sy_c_Word_Osemiring__1__class_Ounsigned,type,
    semiring_1_unsigned: 
      !>[B: $tType,A: $tType] : fun(word(B),A) ).

tff(sy_c_Word_Osigned__drop__bit,type,
    signed_drop_bit: 
      !>[A: $tType] : ( ( nat * word(A) ) > word(A) ) ).

tff(sy_c_Word_Oslice,type,
    slice2: 
      !>[A: $tType,B: $tType] : ( ( nat * word(A) ) > word(B) ) ).

tff(sy_c_Word_Oslice1,type,
    slice1: 
      !>[A: $tType,B: $tType] : ( ( nat * word(A) ) > word(B) ) ).

tff(sy_c_Word_Oword__cat,type,
    word_cat: 
      !>[A: $tType,B: $tType,C: $tType] : ( ( word(A) * word(B) ) > word(C) ) ).

tff(sy_c_Word_Oword__int__case,type,
    word_int_case: 
      !>[B: $tType,A: $tType] : ( ( fun(int,B) * word(A) ) > B ) ).

tff(sy_c_Word_Oword__pred,type,
    word_pred: 
      !>[A: $tType] : ( word(A) > word(A) ) ).

tff(sy_c_Word_Oword__roti,type,
    word_roti: 
      !>[A: $tType] : ( ( int * word(A) ) > word(A) ) ).

tff(sy_c_Word_Oword__rotl,type,
    word_rotl: 
      !>[A: $tType] : ( ( nat * word(A) ) > word(A) ) ).

tff(sy_c_Word_Oword__rotr,type,
    word_rotr: 
      !>[A: $tType] : ( ( nat * word(A) ) > word(A) ) ).

tff(sy_c_Word_Oword__sle,type,
    word_sle: 
      !>[A: $tType] : ( ( word(A) * word(A) ) > $o ) ).

tff(sy_c_Word_Oword__sless,type,
    word_sless: 
      !>[A: $tType] : ( ( word(A) * word(A) ) > $o ) ).

tff(sy_c_Word_Oword__split,type,
    word_split: 
      !>[A: $tType,B: $tType,C: $tType] : ( word(A) > product_prod(word(B),word(C)) ) ).

tff(sy_c_Word_Oword__succ,type,
    word_succ: 
      !>[A: $tType] : ( word(A) > word(A) ) ).

tff(sy_c_Word__Type__Copies_Oword__type__copy__misc,type,
    word_T1870391057261780392y_misc: 
      !>[B: $tType,A: $tType] : ( ( fun(word(B),A) * fun(A,word(B)) * fun(nat,fun(A,A)) * fun(nat,A) * fun(A,nat) * fun(int,A) * fun(A,int) * fun(code_integer,A) * fun(A,code_integer) * nat * fun(fun(nat,$o),fun(nat,fun(A,A))) ) > $o ) ).

tff(sy_c_Word__Type__Copies_Oword__type__copy__misc__axioms,type,
    word_T8964210463127689547axioms: 
      !>[A: $tType,B: $tType] : ( ( fun(A,word(B)) * nat * fun(fun(nat,$o),fun(nat,fun(A,A))) ) > $o ) ).

tff(sy_c_aa,type,
    aa: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).

tff(sy_c_fChoice,type,
    fChoice: 
      !>[A: $tType] : ( fun(A,$o) > A ) ).

tff(sy_c_fequal,type,
    fequal: 
      !>[A: $tType] : ( A > fun(A,$o) ) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : ( ( A * set(A) ) > $o ) ).

tff(sy_v_f____,type,
    f: ( vEBT_VEBT * vEBT_VEBTi * nat ) > heap_Time_Heap(option(nat)) ).

tff(sy_v_ta____,type,
    ta: vEBT_VEBT ).

tff(sy_v_tia____,type,
    tia: vEBT_VEBTi ).

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (9649)
tff(fact_0_refines__case__VEBTi,axiom,
    ! [A: $tType,Tib: vEBT_VEBTi,Ti: vEBT_VEBTi,F1: fun($o,fun($o,heap_Time_Heap(A))),F12: fun($o,fun($o,heap_Time_Heap(A))),F2: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))))),F22: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)))))] :
      ( ( Tib = Ti )
     => ( ! [A2: $o,B2: $o] : refine_Imp_refines(A,aa($o,heap_Time_Heap(A),aa($o,fun($o,heap_Time_Heap(A)),F1,(A2)),(B2)),aa($o,heap_Time_Heap(A),aa($o,fun($o,heap_Time_Heap(A)),F12,(A2)),(B2)))
       => ( ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeArray: array(vEBT_VEBTi),Summary: vEBT_VEBTi] : refine_Imp_refines(A,aa(vEBT_VEBTi,heap_Time_Heap(A),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)))),F2,Info),Deg),TreeArray),Summary),aa(vEBT_VEBTi,heap_Time_Heap(A),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)))),F22,Info),Deg),TreeArray),Summary))
         => refine_Imp_refines(A,vEBT_case_VEBTi(heap_Time_Heap(A),F2,F1,Tib),vEBT_case_VEBTi(heap_Time_Heap(A),F22,F12,Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_1_greater__shift,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa(nat,$o,ord_less(nat,Y),Xb)
    <=> vEBT_VEBT_greater(aa(nat,option(nat),some(nat),Xb),aa(nat,option(nat),some(nat),Y)) ) ).

% greater_shift
tff(fact_2_max__in__set__def,axiom,
    ! [Xs: set(nat),Xb: nat] :
      ( vEBT_VEBT_max_in_set(Xs,Xb)
    <=> ( member(nat,Xb,Xs)
        & ! [X: nat] :
            ( member(nat,X,Xs)
           => aa(nat,$o,ord_less_eq(nat,X),Xb) ) ) ) ).

% max_in_set_def
tff(fact_3_min__in__set__def,axiom,
    ! [Xs: set(nat),Xb: nat] :
      ( vEBT_VEBT_min_in_set(Xs,Xb)
    <=> ( member(nat,Xb,Xs)
        & ! [X: nat] :
            ( member(nat,X,Xs)
           => aa(nat,$o,ord_less_eq(nat,Xb),X) ) ) ) ).

% min_in_set_def
tff(fact_4_power__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xb),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_power,aa(nat,option(nat),some(nat),Xb)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% power_shift
tff(fact_5_bit__split__inv,axiom,
    ! [Xb: nat,D: nat] : vEBT_VEBT_bit_concat(vEBT_VEBT_high(Xb,D),vEBT_VEBT_low(Xb,D),D) = Xb ).

% bit_split_inv
tff(fact_6_high__def,axiom,
    ! [Xb: nat,N: nat] : vEBT_VEBT_high(Xb,N) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% high_def
tff(fact_7_lesseq__shift,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa(nat,$o,ord_less_eq(nat,Xb),Y)
    <=> vEBT_VEBT_lesseq(aa(nat,option(nat),some(nat),Xb),aa(nat,option(nat),some(nat),Y)) ) ).

% lesseq_shift
tff(fact_8_vebt__predi_Osimps,axiom,
    ! [Tb: vEBT_VEBTi,Xb: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_vebt_predi,Tb),Xb) = vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aTP_Lamp_bj(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),Xb),aTP_Lamp_bk(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Xb),Tb) ).

% vebt_predi.simps
tff(fact_9_VEBT__internal_Ovebt__predi_H_Osimps,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),vEBT_VEBT_vebt_predi,Tb),Tib),Xb) = vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_cb(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Tb),Xb),aTP_Lamp_bk(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Xb),Tib) ).

% VEBT_internal.vebt_predi'.simps
tff(fact_10_refines__replicate,axiom,
    ! [A: $tType,Fa: heap_Time_Heap(A),F3: heap_Time_Heap(A),N: nat] :
      ( refine_Imp_refines(A,Fa,F3)
     => refine_Imp_refines(list(A),vEBT_VEBT_replicatei(A,N,Fa),vEBT_VEBT_replicatei(A,N,F3)) ) ).

% refines_replicate
tff(fact_11_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,M: nat,N: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
         => ( aa(A,$o,ord_less(A,B3),one_one(A))
           => ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N))
            <=> aa(nat,$o,ord_less_eq(nat,N),M) ) ) ) ) ).

% power_decreasing_iff
tff(fact_12_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_13_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_14_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
            <=> ( Xb = Y ) ) ) ) ) ).

% power2_eq_iff_nonneg
tff(fact_15_bits__1__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% bits_1_div_2
tff(fact_16_one__div__two__eq__zero,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ).

% one_div_two_eq_zero
tff(fact_17_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,Xb: nat,Y: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),B3)
         => ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),Y))
          <=> aa(nat,$o,ord_less_eq(nat,Xb),Y) ) ) ) ).

% power_increasing_iff
tff(fact_18_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B3: A,N: nat] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
             => ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N))
              <=> aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ) ) ).

% power_mono_iff
tff(fact_19_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,M: nat,N: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
         => ( aa(A,$o,ord_less(A,B3),one_one(A))
           => ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N))
            <=> aa(nat,$o,ord_less(nat,N),M) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_20_zero__eq__power2,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% zero_eq_power2
tff(fact_21_half__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),zero_zero(int))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% half_negative_int_iff
tff(fact_22_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% half_nonnegative_int_iff
tff(fact_23_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_24_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% bits_div_0
tff(fact_25_bits__div__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),one_one(A)) = A3 ) ).

% bits_div_by_1
tff(fact_26_power__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),N) = one_one(A) ) ).

% power_one
tff(fact_27_power__one__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),one_one(nat)) = A3 ) ).

% power_one_right
tff(fact_28_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,N: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A3)
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) )
          <=> ( M = N ) ) ) ) ).

% power_inject_exp
tff(fact_29_power__zero__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(num,nat,numeral_numeral(nat),K)) = zero_zero(A) ) ).

% power_zero_numeral
tff(fact_30_div__less,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) ) ) ).

% div_less
tff(fact_31_nat__zero__less__power__iff,axiom,
    ! [Xb: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xb),N))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Xb)
        | ( N = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_32_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,Xb: nat,Y: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),B3)
         => ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),Y))
          <=> aa(nat,$o,ord_less(nat,Xb),Y) ) ) ) ).

% power_strict_increasing_iff
tff(fact_33_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ) ) ).

% power_eq_0_iff
tff(fact_34_div__eq__dividend__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = M )
      <=> ( N = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_35_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V2048590022279873568_shift(nat,power_power(nat)) ).

% local.power_def
tff(fact_36_not__exp__less__eq__0__int,axiom,
    ! [N: nat] : ~ aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),zero_zero(int)) ).

% not_exp_less_eq_0_int
tff(fact_37_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A3: A,N: nat] :
          ( ( A3 != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) != zero_zero(A) ) ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_38_power__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N)) ) ).

% power_divide
tff(fact_39_div__le__dividend,axiom,
    ! [M: nat,N: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),M) ).

% div_le_dividend
tff(fact_40_div__le__mono,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),K)) ) ).

% div_le_mono
tff(fact_41_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ).

% zero_le_power
tff(fact_42_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B3: A,N: nat] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
           => aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N)) ) ) ) ).

% power_mono
tff(fact_43_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ).

% zero_less_power
tff(fact_44_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less_eq(A,one_one(A)),A3)
         => aa(A,$o,ord_less_eq(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ).

% one_le_power
tff(fact_45_mem__Collect__eq,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] :
      ( member(A,A3,collect(A,P))
    <=> aa(A,$o,P,A3) ) ).

% mem_Collect_eq
tff(fact_46_Collect__mem__eq,axiom,
    ! [A: $tType,A4: set(A)] : collect(A,aTP_Lamp_cc(set(A),fun(A,$o),A4)) = A4 ).

% Collect_mem_eq
tff(fact_47_Collect__cong,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X2: A] :
          ( aa(A,$o,P,X2)
        <=> aa(A,$o,Q,X2) )
     => ( collect(A,P) = collect(A,Q) ) ) ).

% Collect_cong
tff(fact_48_HOL_Oext,axiom,
    ! [B: $tType,A: $tType,Fa: fun(A,B),G: fun(A,B)] :
      ( ! [X2: A] : aa(A,B,Fa,X2) = aa(A,B,G,X2)
     => ( Fa = G ) ) ).

% HOL.ext
tff(fact_49_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_one_over
tff(fact_50_power__0,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),zero_zero(nat)) = one_one(A) ) ).

% power_0
tff(fact_51_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = zero_zero(nat) )
    <=> ( aa(nat,$o,ord_less(nat,M),N)
        | ( N = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_52_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),I)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N))
       => aa(nat,$o,ord_less(nat,M),N) ) ) ).

% nat_power_less_imp_less
tff(fact_53_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B3: A] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less(A,A3),B3) ) ) ) ).

% power_less_imp_less_base
tff(fact_54_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,A3),one_one(A))
           => aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),one_one(A)) ) ) ) ).

% power_le_one
tff(fact_55_numeral__Bit0__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),bit0(N))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(num,A,numeral_numeral(A),N) ) ).

% numeral_Bit0_div_2
tff(fact_56_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = $ite(N = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% power_0_left
tff(fact_57_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N2: nat,A3: A] :
          ( aa(nat,$o,ord_less(nat,N),N2)
         => ( aa(A,$o,ord_less(A,one_one(A)),A3)
           => aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)) ) ) ) ).

% power_strict_increasing
tff(fact_58_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,N: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A3)
         => ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))
           => aa(nat,$o,ord_less(nat,M),N) ) ) ) ).

% power_less_imp_less_exp
tff(fact_59_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),N) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_60_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N2: nat,A3: A] :
          ( aa(nat,$o,ord_less_eq(nat,N),N2)
         => ( aa(A,$o,ord_less_eq(A,one_one(A)),A3)
           => aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)) ) ) ) ).

% power_increasing
tff(fact_61_div__greater__zero__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N))
    <=> ( aa(nat,$o,ord_less_eq(nat,N),M)
        & aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ) ).

% div_greater_zero_iff
tff(fact_62_div__le__mono2,axiom,
    ! [M: nat,N: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(nat,$o,ord_less_eq(nat,M),N)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),M)) ) ) ).

% div_le_mono2
tff(fact_63_div__less__dividend,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,one_one(nat)),N)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),M) ) ) ).

% div_less_dividend
tff(fact_64_zero__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) ) ) ).

% zero_power2
tff(fact_65_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N2: nat,A3: A] :
          ( aa(nat,$o,ord_less(nat,N),N2)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
           => ( aa(A,$o,ord_less(A,A3),one_one(A))
             => aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_66_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,N2: nat,A3: A] :
          ( aa(nat,$o,ord_less_eq(nat,N),N2)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
           => ( aa(A,$o,ord_less_eq(A,A3),one_one(A))
             => aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ) ).

% power_decreasing
tff(fact_67_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B3: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N) )
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
             => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
               => ( A3 = B3 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_68_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,A3: A,B3: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
             => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N) )
              <=> ( A3 = B3 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_69_one__power2,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) ) ) ).

% one_power2
tff(fact_70_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,M: nat,N: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))
           => aa(nat,$o,ord_less_eq(nat,M),N) ) ) ) ).

% power_le_imp_le_exp
tff(fact_71_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less_eq(A,one_one(A)),A3)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
           => aa(A,$o,ord_less_eq(A,A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ).

% self_le_power
tff(fact_72_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A3)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
           => aa(A,$o,ord_less(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ).

% one_less_power
tff(fact_73_power2__nat__le__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),N)
     => aa(nat,$o,ord_less_eq(nat,M),N) ) ).

% power2_nat_le_imp_le
tff(fact_74_power2__nat__le__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> aa(nat,$o,ord_less_eq(nat,M),N) ) ).

% power2_nat_le_eq_le
tff(fact_75_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)) ) ).

% self_le_ge2_pow
tff(fact_76_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less_eq(A,Xb),Y) ) ) ) ).

% power2_le_imp_le
tff(fact_77_power2__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
             => ( Xb = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_78_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% zero_le_power2
tff(fact_79_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : ~ aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A)) ) ).

% power2_less_0
tff(fact_80_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B3: A,N: nat] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
           => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
             => aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N)) ) ) ) ) ).

% power_strict_mono
tff(fact_81_power2__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less(A,Xb),Y) ) ) ) ).

% power2_less_imp_less
tff(fact_82_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3))
          <=> aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_83_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3))
          <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_84_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),one_one(A))
          <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_85_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),one_one(A))
          <=> aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_86_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ).

% zero_less_divide_1_iff
tff(fact_87_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3))
          <=> aa(A,$o,ord_less(A,A3),B3) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_88_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3))
          <=> aa(A,$o,ord_less(A,B3),A3) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_89_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),one_one(A))
          <=> aa(A,$o,ord_less(A,B3),A3) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_90_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),one_one(A))
          <=> aa(A,$o,ord_less(A,A3),B3) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_91_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% divide_less_0_1_iff
tff(fact_92_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ).

% zero_le_divide_1_iff
tff(fact_93_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) ) ) ).

% divide_le_0_1_iff
tff(fact_94_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B3 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_95_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B3) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B3 ) ) ) ) ).

% divide_cancel_left
tff(fact_96_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,C2: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B3 ) ) ) ) ).

% divide_cancel_right
tff(fact_97_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% division_ring_divide_zero
tff(fact_98_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = one_one(A) )
        <=> ( ( B3 != zero_zero(A) )
            & ( A3 = B3 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_99_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) )
        <=> ( ( B3 != zero_zero(A) )
            & ( A3 = B3 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_100_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = one_one(A) ) ) ) ).

% divide_self
tff(fact_101_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = $ite(A3 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% divide_self_if
tff(fact_102_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3) = one_one(A) )
        <=> ( ( A3 != zero_zero(A) )
            & ( A3 = B3 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_103_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3) )
        <=> ( ( A3 != zero_zero(A) )
            & ( A3 = B3 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_104_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_105_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_106_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,K),zero_zero(int))
     => ( aa(int,$o,ord_less(int,L),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_neg_neg_trivial
tff(fact_107_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less(int,K),L)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = zero_zero(int) ) ) ) ).

% div_pos_pos_trivial
tff(fact_108_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X3: A] :
        ? [Y2: A] : aa(A,$o,ord_less(A,Y2),X3) ) ).

% linordered_field_no_lb
tff(fact_109_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X3: A] :
        ? [X_1: A] : aa(A,$o,ord_less(A,X3),X_1) ) ).

% linordered_field_no_ub
tff(fact_110_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),zero_zero(A))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) )
            | ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) ) ) ) ) ).

% divide_le_0_iff
tff(fact_111_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) ) ) ) ).

% divide_right_mono
tff(fact_112_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) )
            | ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_113_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_nonneg_nonneg
tff(fact_114_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
         => ( aa(A,$o,ord_less_eq(A,Y),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_nonpos
tff(fact_115_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_nonneg
tff(fact_116_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,Y),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_nonpos_nonpos
tff(fact_117_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)) ) ) ) ).

% divide_right_mono_neg
tff(fact_118_divide__neg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),zero_zero(A))
         => ( aa(A,$o,ord_less(A,Y),zero_zero(A))
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_neg_neg
tff(fact_119_divide__neg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),zero_zero(A))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_neg_pos
tff(fact_120_divide__pos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xb)
         => ( aa(A,$o,ord_less(A,Y),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_pos_neg
tff(fact_121_divide__pos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xb)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_pos_pos
tff(fact_122_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),zero_zero(A))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less(A,B3),zero_zero(A)) )
            | ( aa(A,$o,ord_less(A,A3),zero_zero(A))
              & aa(A,$o,ord_less(A,zero_zero(A)),B3) ) ) ) ) ).

% divide_less_0_iff
tff(fact_123_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A3),B3) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B3),A3) )
            & ( C2 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_124_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less(A,zero_zero(A)),B3) )
            | ( aa(A,$o,ord_less(A,A3),zero_zero(A))
              & aa(A,$o,ord_less(A,B3),zero_zero(A)) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_125_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) ) ) ) ).

% divide_strict_right_mono
tff(fact_126_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( aa(A,$o,ord_less(A,C2),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_127_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = one_one(A) )
          <=> ( A3 = B3 ) ) ) ) ).

% right_inverse_eq
tff(fact_128_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A,W: A,Z: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less_eq(A,Xb),Y)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),W)
             => ( aa(A,$o,ord_less_eq(A,W),Z)
               => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_le
tff(fact_129_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
         => ( aa(A,$o,ord_less(A,Xb),Y)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),W)
             => ( aa(A,$o,ord_less_eq(A,W),Z)
               => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_less
tff(fact_130_frac__less2,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xb)
         => ( aa(A,$o,ord_less_eq(A,Xb),Y)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),W)
             => ( aa(A,$o,ord_less(A,W),Z)
               => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),W)) ) ) ) ) ) ).

% frac_less2
tff(fact_131_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less_eq(A,A3),B3) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ) ).

% divide_le_cancel
tff(fact_132_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
         => ( aa(A,$o,ord_less(A,Y),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_nonneg_neg
tff(fact_133_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_nonneg_pos
tff(fact_134_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),zero_zero(A))
         => ( aa(A,$o,ord_less(A,Y),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% divide_nonpos_neg
tff(fact_135_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),zero_zero(A))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),zero_zero(A)) ) ) ) ).

% divide_nonpos_pos
tff(fact_136_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),one_one(A))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less(A,B3),A3) )
            | ( aa(A,$o,ord_less(A,A3),zero_zero(A))
              & aa(A,$o,ord_less(A,A3),B3) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_137_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less(A,A3),B3) )
            | ( aa(A,$o,ord_less(A,A3),zero_zero(A))
              & aa(A,$o,ord_less(A,B3),A3) ) ) ) ) ).

% less_divide_eq_1
tff(fact_138_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),one_one(A))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less_eq(A,B3),A3) )
            | ( aa(A,$o,ord_less(A,A3),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,A3),B3) )
            | ( A3 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_139_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less_eq(A,A3),B3) )
            | ( aa(A,$o,ord_less(A,A3),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ) ).

% le_divide_eq_1
tff(fact_140_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( aa(A,$o,ord_less(A,one_one(A)),aa(num,A,numeral_numeral(A),N))
        <=> aa(num,$o,ord_less(num,one2),N) ) ) ).

% one_less_numeral_iff
tff(fact_141_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] :
          ( aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),N)),one_one(A))
        <=> aa(num,$o,ord_less_eq(num,N),one2) ) ) ).

% numeral_le_one_iff
tff(fact_142_int__div__same__is__1,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),A3)
     => ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3) = A3 )
      <=> ( B3 = one_one(int) ) ) ) ).

% int_div_same_is_1
tff(fact_143_option_Ocollapse,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
     => ( aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) = Option ) ) ).

% option.collapse
tff(fact_144_prod_Ocollapse,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) = Prod ).

% prod.collapse
tff(fact_145_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),bit0(V))),aa(num,int,numeral_numeral(int),bit0(W))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit0
tff(fact_146_less__one,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,N),one_one(nat))
    <=> ( N = zero_zero(nat) ) ) ).

% less_one
tff(fact_147_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),A3) = one_one(A) ) ) ) ).

% div_self
tff(fact_148_numeral__eq__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] :
          ( ( aa(num,A,numeral_numeral(A),N) = one_one(A) )
        <=> ( N = one2 ) ) ) ).

% numeral_eq_one_iff
tff(fact_149_one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),N) )
        <=> ( one2 = N ) ) ) ).

% one_eq_numeral_iff
tff(fact_150_two__pow__div__gt__le,axiom,
    ! [V: nat,N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,V),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
     => aa(nat,$o,ord_less_eq(nat,M),N) ) ).

% two_pow_div_gt_le
tff(fact_151_vebt__succi_Osimps,axiom,
    ! [Tb: vEBT_VEBTi,Xb: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_vebt_succi,Tb),Xb) = vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aTP_Lamp_ck(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),Xb),aTP_Lamp_cl(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Xb),Tb) ).

% vebt_succi.simps
tff(fact_152_less__shift,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa(nat,$o,ord_less(nat,Xb),Y)
    <=> vEBT_VEBT_less(aa(nat,option(nat),some(nat),Xb),aa(nat,option(nat),some(nat),Y)) ) ).

% less_shift
tff(fact_153_numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: num,N: num] :
          ( ( aa(num,A,numeral_numeral(A),M) = aa(num,A,numeral_numeral(A),N) )
        <=> ( M = N ) ) ) ).

% numeral_eq_iff
tff(fact_154_old_Oprod_Oinject,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B4: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) )
    <=> ( ( A3 = A5 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
tff(fact_155_prod_Oinject,axiom,
    ! [A: $tType,B: $tType,X1: A,X22: B,Y1: A,Y22: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y22) )
    <=> ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
tff(fact_156_option_Oinject,axiom,
    ! [A: $tType,X22: A,Y22: A] :
      ( ( aa(A,option(A),some(A),X22) = aa(A,option(A),some(A),Y22) )
    <=> ( X22 = Y22 ) ) ).

% option.inject
tff(fact_157_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_158_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% div_0
tff(fact_159_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),one_one(A)) = A3 ) ).

% div_by_1
tff(fact_160_less__nat__zero__code,axiom,
    ! [N: nat] : ~ aa(nat,$o,ord_less(nat,N),zero_zero(nat)) ).

% less_nat_zero_code
tff(fact_161_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ).

% neq0_conv
tff(fact_162_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero(nat) )
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),A3) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_163_le0,axiom,
    ! [N: nat] : aa(nat,$o,ord_less_eq(nat,zero_zero(nat)),N) ).

% le0
tff(fact_164_bot__nat__0_Oextremum,axiom,
    ! [A3: nat] : aa(nat,$o,ord_less_eq(nat,zero_zero(nat)),A3) ).

% bot_nat_0.extremum
tff(fact_165_not__Some__eq,axiom,
    ! [A: $tType,Xb: option(A)] :
      ( ! [Y3: A] : Xb != aa(A,option(A),some(A),Y3)
    <=> ( Xb = none(A) ) ) ).

% not_Some_eq
tff(fact_166_not__None__eq,axiom,
    ! [A: $tType,Xb: option(A)] :
      ( ( Xb != none(A) )
    <=> ? [Y3: A] : Xb = aa(A,option(A),some(A),Y3) ) ).

% not_None_eq
tff(fact_167_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,Fa: fun(B,fun(C,A)),A3: B,B3: C] : aa(product_prod(B,C),A,product_case_prod(B,C,A,Fa),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B3)) = aa(C,A,aa(B,fun(C,A),Fa,A3),B3) ).

% case_prod_conv
tff(fact_168_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,N: num] :
          ( aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))
        <=> aa(num,$o,ord_less_eq(num,M),N) ) ) ).

% numeral_le_iff
tff(fact_169_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,N: num] :
          ( aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))
        <=> aa(num,$o,ord_less(num,M),N) ) ) ).

% numeral_less_iff
tff(fact_170_vebt__succi__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xb: nat,Tb: vEBT_VEBT] : refine_Imp_refines(option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_vebt_succi,Tib),Xb),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),vEBT_VEBT_vebt_succi,Tb),Tib),Xb)) ).

% vebt_succi_refines
tff(fact_171_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [Fa: fun(A,B),P: fun(A,$o),A3: A] :
          ( ! [X2: A] :
              ( ! [Y4: A] :
                  ( aa(B,$o,ord_less(B,aa(A,B,Fa,Y4)),aa(A,B,Fa,X2))
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X2) )
         => aa(A,$o,P,A3) ) ) ).

% measure_induct_rule
tff(fact_172_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [Fa: fun(A,B),P: fun(A,$o),A3: A] :
          ( ! [X2: A] :
              ( ! [Y4: A] :
                  ( aa(B,$o,ord_less(B,aa(A,B,Fa,Y4)),aa(A,B,Fa,X2))
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X2) )
         => aa(A,$o,P,A3) ) ) ).

% measure_induct
tff(fact_173_linorder__neqE__linordered__idom,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
         => ( ~ aa(A,$o,ord_less(A,Xb),Y)
           => aa(A,$o,ord_less(A,Y),Xb) ) ) ) ).

% linorder_neqE_linordered_idom
tff(fact_174_le__num__One__iff,axiom,
    ! [Xb: num] :
      ( aa(num,$o,ord_less_eq(num,Xb),one2)
    <=> ( Xb = one2 ) ) ).

% le_num_One_iff
tff(fact_175_Pair__inject,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: B,A5: A,B4: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A5),B4) )
     => ~ ( ( A3 = A5 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
tff(fact_176_prod__cases,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o),P2: product_prod(A,B)] :
      ( ! [A2: A,B2: B] : aa(product_prod(A,B),$o,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2))
     => aa(product_prod(A,B),$o,P,P2) ) ).

% prod_cases
tff(fact_177_surj__pair,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(A,B)] :
    ? [X2: A,Y2: B] : P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) ).

% surj_pair
tff(fact_178_old_Oprod_Oexhaust,axiom,
    ! [A: $tType,B: $tType,Y: product_prod(A,B)] :
      ~ ! [A2: A,B2: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) ).

% old.prod.exhaust
tff(fact_179_infinite__descent__measure,axiom,
    ! [A: $tType,P: fun(A,$o),V2: fun(A,nat),Xb: A] :
      ( ! [X2: A] :
          ( ~ aa(A,$o,P,X2)
         => ? [Y4: A] :
              ( aa(nat,$o,ord_less(nat,aa(A,nat,V2,Y4)),aa(A,nat,V2,X2))
              & ~ aa(A,$o,P,Y4) ) )
     => aa(A,$o,P,Xb) ) ).

% infinite_descent_measure
tff(fact_180_linorder__neqE__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( Xb != Y )
     => ( ~ aa(nat,$o,ord_less(nat,Xb),Y)
       => aa(nat,$o,ord_less(nat,Y),Xb) ) ) ).

% linorder_neqE_nat
tff(fact_181_infinite__descent,axiom,
    ! [P: fun(nat,$o),N: nat] :
      ( ! [N3: nat] :
          ( ~ aa(nat,$o,P,N3)
         => ? [M2: nat] :
              ( aa(nat,$o,ord_less(nat,M2),N3)
              & ~ aa(nat,$o,P,M2) ) )
     => aa(nat,$o,P,N) ) ).

% infinite_descent
tff(fact_182_nat__less__induct,axiom,
    ! [P: fun(nat,$o),N: nat] :
      ( ! [N3: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,ord_less(nat,M2),N3)
             => aa(nat,$o,P,M2) )
         => aa(nat,$o,P,N3) )
     => aa(nat,$o,P,N) ) ).

% nat_less_induct
tff(fact_183_less__irrefl__nat,axiom,
    ! [N: nat] : ~ aa(nat,$o,ord_less(nat,N),N) ).

% less_irrefl_nat
tff(fact_184_less__not__refl3,axiom,
    ! [S: nat,Tb: nat] :
      ( aa(nat,$o,ord_less(nat,S),Tb)
     => ( S != Tb ) ) ).

% less_not_refl3
tff(fact_185_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,N),M)
     => ( M != N ) ) ).

% less_not_refl2
tff(fact_186_less__not__refl,axiom,
    ! [N: nat] : ~ aa(nat,$o,ord_less(nat,N),N) ).

% less_not_refl
tff(fact_187_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
    <=> ( aa(nat,$o,ord_less(nat,M),N)
        | aa(nat,$o,ord_less(nat,N),M) ) ) ).

% nat_neq_iff
tff(fact_188_size__neq__size__imp__neq,axiom,
    ! [A: $tType] :
      ( size(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,nat,size_size(A),Xb) != aa(A,nat,size_size(A),Y) )
         => ( Xb != Y ) ) ) ).

% size_neq_size_imp_neq
tff(fact_189_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B3: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y2: nat] :
            ( aa(nat,$o,P,Y2)
           => aa(nat,$o,ord_less_eq(nat,Y2),B3) )
       => ? [X2: nat] :
            ( aa(nat,$o,P,X2)
            & ! [Y4: nat] :
                ( aa(nat,$o,P,Y4)
               => aa(nat,$o,ord_less_eq(nat,Y4),X2) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_190_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
      | aa(nat,$o,ord_less_eq(nat,N),M) ) ).

% nat_le_linear
tff(fact_191_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => ( aa(nat,$o,ord_less_eq(nat,N),M)
       => ( M = N ) ) ) ).

% le_antisym
tff(fact_192_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => aa(nat,$o,ord_less_eq(nat,M),N) ) ).

% eq_imp_le
tff(fact_193_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,ord_less_eq(nat,J),K)
       => aa(nat,$o,ord_less_eq(nat,I),K) ) ) ).

% le_trans
tff(fact_194_le__refl,axiom,
    ! [N: nat] : aa(nat,$o,ord_less_eq(nat,N),N) ).

% le_refl
tff(fact_195_prod_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,D2: $tType,C: $tType,H: fun(B,A),Fa: fun(C,fun(D2,B)),Prod: product_prod(C,D2)] : aa(B,A,H,aa(product_prod(C,D2),B,product_case_prod(C,D2,B,Fa),Prod)) = aa(product_prod(C,D2),A,product_case_prod(C,D2,A,aa(fun(C,fun(D2,B)),fun(C,fun(D2,A)),aTP_Lamp_cm(fun(B,A),fun(fun(C,fun(D2,B)),fun(C,fun(D2,A))),H),Fa)),Prod) ).

% prod.case_distrib
tff(fact_196_option_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType,H: fun(B,A),F1: B,F2: fun(C,B),Option: option(C)] : aa(B,A,H,case_option(B,C,F1,F2,Option)) = case_option(A,C,aa(B,A,H,F1),aa(fun(C,B),fun(C,A),aTP_Lamp_cn(fun(B,A),fun(fun(C,B),fun(C,A)),H),F2),Option) ).

% option.case_distrib
tff(fact_197_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,ord_less_eq(A,zero_zero(A)),zero_zero(A)) ) ).

% le_numeral_extra(3)
tff(fact_198_less__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,ord_less(A,zero_zero(A)),zero_zero(A)) ) ).

% less_numeral_extra(3)
tff(fact_199_zero__neq__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: num] : zero_zero(A) != aa(num,A,numeral_numeral(A),N) ) ).

% zero_neq_numeral
tff(fact_200_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_201_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,ord_less_eq(A,one_one(A)),one_one(A)) ) ).

% le_numeral_extra(4)
tff(fact_202_less__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,ord_less(A,one_one(A)),one_one(A)) ) ).

% less_numeral_extra(4)
tff(fact_203_infinite__descent0__measure,axiom,
    ! [A: $tType,V2: fun(A,nat),P: fun(A,$o),Xb: A] :
      ( ! [X2: A] :
          ( ( aa(A,nat,V2,X2) = zero_zero(nat) )
         => aa(A,$o,P,X2) )
     => ( ! [X2: A] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(A,nat,V2,X2))
           => ( ~ aa(A,$o,P,X2)
             => ? [Y4: A] :
                  ( aa(nat,$o,ord_less(nat,aa(A,nat,V2,Y4)),aa(A,nat,V2,X2))
                  & ~ aa(A,$o,P,Y4) ) ) )
       => aa(A,$o,P,Xb) ) ) ).

% infinite_descent0_measure
tff(fact_204_infinite__descent0,axiom,
    ! [P: fun(nat,$o),N: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N3: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N3)
           => ( ~ aa(nat,$o,P,N3)
             => ? [M2: nat] :
                  ( aa(nat,$o,ord_less(nat,M2),N3)
                  & ~ aa(nat,$o,P,M2) ) ) )
       => aa(nat,$o,P,N) ) ) ).

% infinite_descent0
tff(fact_205_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),N)
     => ( N != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_206_less__zeroE,axiom,
    ! [N: nat] : ~ aa(nat,$o,ord_less(nat,N),zero_zero(nat)) ).

% less_zeroE
tff(fact_207_not__less0,axiom,
    ! [N: nat] : ~ aa(nat,$o,ord_less(nat,N),zero_zero(nat)) ).

% not_less0
tff(fact_208_not__gr0,axiom,
    ! [N: nat] :
      ( ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
    <=> ( N = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_209_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ).

% gr0I
tff(fact_210_bot__nat__0_Oextremum__strict,axiom,
    ! [A3: nat] : ~ aa(nat,$o,ord_less(nat,A3),zero_zero(nat)) ).

% bot_nat_0.extremum_strict
tff(fact_211_le__0__eq,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,N),zero_zero(nat))
    <=> ( N = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_212_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( aa(nat,$o,ord_less_eq(nat,A3),zero_zero(nat))
     => ( A3 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_213_bot__nat__0_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( aa(nat,$o,ord_less_eq(nat,A3),zero_zero(nat))
    <=> ( A3 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_214_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : aa(nat,$o,ord_less_eq(nat,zero_zero(nat)),N) ).

% less_eq_nat.simps(1)
tff(fact_215_less__mono__imp__le__mono,axiom,
    ! [Fa: fun(nat,nat),I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),J2)
         => aa(nat,$o,ord_less(nat,aa(nat,nat,Fa,I2)),aa(nat,nat,Fa,J2)) )
     => ( aa(nat,$o,ord_less_eq(nat,I),J)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,Fa,I)),aa(nat,nat,Fa,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_216_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => ( ( M != N )
       => aa(nat,$o,ord_less(nat,M),N) ) ) ).

% le_neq_implies_less
tff(fact_217_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,$o,ord_less(nat,M),N)
        | ( M = N ) )
     => aa(nat,$o,ord_less_eq(nat,M),N) ) ).

% less_or_eq_imp_le
tff(fact_218_le__eq__less__or__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
    <=> ( aa(nat,$o,ord_less(nat,M),N)
        | ( M = N ) ) ) ).

% le_eq_less_or_eq
tff(fact_219_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),N)
     => aa(nat,$o,ord_less_eq(nat,M),N) ) ).

% less_imp_le_nat
tff(fact_220_nat__less__le,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),N)
    <=> ( aa(nat,$o,ord_less_eq(nat,M),N)
        & ( M != N ) ) ) ).

% nat_less_le
tff(fact_221_old_Oprod_Ocase,axiom,
    ! [B: $tType,A: $tType,C: $tType,Fa: fun(B,fun(C,A)),X1: B,X22: C] : aa(product_prod(B,C),A,product_case_prod(B,C,A,Fa),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X1),X22)) = aa(C,A,aa(B,fun(C,A),Fa,X1),X22) ).

% old.prod.case
tff(fact_222_fst__conv,axiom,
    ! [B: $tType,A: $tType,X1: A,X22: B] : aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X22)) = X1 ).

% fst_conv
tff(fact_223_fst__eqD,axiom,
    ! [B: $tType,A: $tType,Xb: A,Y: B,A3: A] :
      ( ( aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y)) = A3 )
     => ( Xb = A3 ) ) ).

% fst_eqD
tff(fact_224_combine__options__cases,axiom,
    ! [A: $tType,B: $tType,Xb: option(A),P: fun(option(A),fun(option(B),$o)),Y: option(B)] :
      ( ( ( Xb = none(A) )
       => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xb),Y) )
     => ( ( ( Y = none(B) )
         => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xb),Y) )
       => ( ! [A2: A,B2: B] :
              ( ( Xb = aa(A,option(A),some(A),A2) )
             => ( ( Y = aa(B,option(B),some(B),B2) )
               => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xb),Y) ) )
         => aa(option(B),$o,aa(option(A),fun(option(B),$o),P,Xb),Y) ) ) ) ).

% combine_options_cases
tff(fact_225_split__option__all,axiom,
    ! [A: $tType,P: fun(option(A),$o)] :
      ( ! [X_12: option(A)] : aa(option(A),$o,P,X_12)
    <=> ( aa(option(A),$o,P,none(A))
        & ! [X: A] : aa(option(A),$o,P,aa(A,option(A),some(A),X)) ) ) ).

% split_option_all
tff(fact_226_split__option__ex,axiom,
    ! [A: $tType,P: fun(option(A),$o)] :
      ( ? [X_12: option(A)] : aa(option(A),$o,P,X_12)
    <=> ( aa(option(A),$o,P,none(A))
        | ? [X: A] : aa(option(A),$o,P,aa(A,option(A),some(A),X)) ) ) ).

% split_option_ex
tff(fact_227_option_Oexhaust,axiom,
    ! [A: $tType,Y: option(A)] :
      ( ( Y != none(A) )
     => ~ ! [X23: A] : Y != aa(A,option(A),some(A),X23) ) ).

% option.exhaust
tff(fact_228_option_OdiscI,axiom,
    ! [A: $tType,Option: option(A),X22: A] :
      ( ( Option = aa(A,option(A),some(A),X22) )
     => ( Option != none(A) ) ) ).

% option.discI
tff(fact_229_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X22: A] : none(A) != aa(A,option(A),some(A),X22) ).

% option.distinct(1)
tff(fact_230_snd__conv,axiom,
    ! [B: $tType,A: $tType,X1: B,X22: A] : aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X1),X22)) = X22 ).

% snd_conv
tff(fact_231_snd__eqD,axiom,
    ! [B: $tType,A: $tType,Xb: B,Y: A,A3: A] :
      ( ( aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),Y)) = A3 )
     => ( Y = A3 ) ) ).

% snd_eqD
tff(fact_232_prod__induct3,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,C)),$o),Xb: product_prod(A,product_prod(B,C))] :
      ( ! [A2: A,B2: B,C3: C] : aa(product_prod(A,product_prod(B,C)),$o,P,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C3)))
     => aa(product_prod(A,product_prod(B,C)),$o,P,Xb) ) ).

% prod_induct3
tff(fact_233_prod__cases3,axiom,
    ! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
      ~ ! [A2: A,B2: B,C3: C] : Y != aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B2),C3)) ).

% prod_cases3
tff(fact_234_option_Osel,axiom,
    ! [A: $tType,X22: A] : aa(option(A),A,the2(A),aa(A,option(A),some(A),X22)) = X22 ).

% option.sel
tff(fact_235_prod__eq__iff,axiom,
    ! [B: $tType,A: $tType,S: product_prod(A,B),Tb: product_prod(A,B)] :
      ( ( S = Tb )
    <=> ( ( aa(product_prod(A,B),A,product_fst(A,B),S) = aa(product_prod(A,B),A,product_fst(A,B),Tb) )
        & ( aa(product_prod(A,B),B,product_snd(A,B),S) = aa(product_prod(A,B),B,product_snd(A,B),Tb) ) ) ) ).

% prod_eq_iff
tff(fact_236_prod__eqI,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),Q2: product_prod(A,B)] :
      ( ( aa(product_prod(A,B),A,product_fst(A,B),P2) = aa(product_prod(A,B),A,product_fst(A,B),Q2) )
     => ( ( aa(product_prod(A,B),B,product_snd(A,B),P2) = aa(product_prod(A,B),B,product_snd(A,B),Q2) )
       => ( P2 = Q2 ) ) ) ).

% prod_eqI
tff(fact_237_prod_Oexpand,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B),Prod2: product_prod(A,B)] :
      ( ( ( aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,product_fst(A,B),Prod2) )
        & ( aa(product_prod(A,B),B,product_snd(A,B),Prod) = aa(product_prod(A,B),B,product_snd(A,B),Prod2) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
tff(fact_238_option_Oexpand,axiom,
    ! [A: $tType,Option: option(A),Option2: option(A)] :
      ( ( ( Option = none(A) )
      <=> ( Option2 = none(A) ) )
     => ( ( ( Option != none(A) )
         => ( ( Option2 != none(A) )
           => ( aa(option(A),A,the2(A),Option) = aa(option(A),A,the2(A),Option2) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
tff(fact_239_option_Osimps_I5_J,axiom,
    ! [A: $tType,B: $tType,F1: A,F2: fun(B,A),X22: B] : case_option(A,B,F1,F2,aa(B,option(B),some(B),X22)) = aa(B,A,F2,X22) ).

% option.simps(5)
tff(fact_240_option_Osimps_I4_J,axiom,
    ! [B: $tType,A: $tType,F1: A,F2: fun(B,A)] : case_option(A,B,F1,F2,none(B)) = F1 ).

% option.simps(4)
tff(fact_241_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,Fa: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
      ( ! [X2: A,Y2: B] : aa(B,C,aa(A,fun(B,C),Fa,X2),Y2) = aa(product_prod(A,B),C,G,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2))
     => ( product_case_prod(A,B,C,Fa) = G ) ) ).

% cond_case_prod_eta
tff(fact_242_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,Fa: fun(product_prod(A,B),C)] : product_case_prod(A,B,C,aTP_Lamp_co(fun(product_prod(A,B),C),fun(A,fun(B,C)),Fa)) = Fa ).

% case_prod_eta
tff(fact_243_case__prodE2,axiom,
    ! [B: $tType,A: $tType,C: $tType,Q: fun(A,$o),P: fun(B,fun(C,A)),Z: product_prod(B,C)] :
      ( aa(A,$o,Q,aa(product_prod(B,C),A,product_case_prod(B,C,A,P),Z))
     => ~ ! [X2: B,Y2: C] :
            ( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X2),Y2) )
           => ~ aa(A,$o,Q,aa(C,A,aa(B,fun(C,A),P,X2),Y2)) ) ) ).

% case_prodE2
tff(fact_244_fst__def,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Prod) = aa(product_prod(A,B),A,product_case_prod(A,B,A,aTP_Lamp_cp(A,fun(B,A))),Prod) ).

% fst_def
tff(fact_245_snd__def,axiom,
    ! [A: $tType,B: $tType,Prod: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Prod) = aa(product_prod(B,A),A,product_case_prod(B,A,A,aTP_Lamp_cq(B,fun(A,A))),Prod) ).

% snd_def
tff(fact_246_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),N)),zero_zero(A)) ) ).

% not_numeral_le_zero
tff(fact_247_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(num,A,numeral_numeral(A),N)) ) ).

% zero_le_numeral
tff(fact_248_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),N)),zero_zero(A)) ) ).

% not_numeral_less_zero
tff(fact_249_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : aa(A,$o,ord_less(A,zero_zero(A)),aa(num,A,numeral_numeral(A),N)) ) ).

% zero_less_numeral
tff(fact_250_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,ord_less_eq(A,one_one(A)),zero_zero(A)) ) ).

% not_one_le_zero
tff(fact_251_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,ord_less_eq(A,zero_zero(A)),one_one(A)) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_252_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,ord_less_eq(A,zero_zero(A)),one_one(A)) ) ).

% zero_less_one_class.zero_le_one
tff(fact_253_not__one__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,ord_less(A,one_one(A)),zero_zero(A)) ) ).

% not_one_less_zero
tff(fact_254_zero__less__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,ord_less(A,zero_zero(A)),one_one(A)) ) ).

% zero_less_one
tff(fact_255_less__numeral__extra_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,ord_less(A,zero_zero(A)),one_one(A)) ) ).

% less_numeral_extra(1)
tff(fact_256_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : aa(A,$o,ord_less_eq(A,one_one(A)),aa(num,A,numeral_numeral(A),N)) ) ).

% one_le_numeral
tff(fact_257_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: num] : ~ aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),N)),one_one(A)) ) ).

% not_numeral_less_one
tff(fact_258_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_259_divide__numeral__1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),one2)) = A3 ) ).

% divide_numeral_1
tff(fact_260_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_261_ex__least__nat__le,axiom,
    ! [P: fun(nat,$o),N: nat] :
      ( aa(nat,$o,P,N)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,ord_less_eq(nat,K2),N)
            & ! [I3: nat] :
                ( aa(nat,$o,ord_less(nat,I3),K2)
               => ~ aa(nat,$o,P,I3) )
            & aa(nat,$o,P,K2) ) ) ) ).

% ex_least_nat_le
tff(fact_262_surjective__pairing,axiom,
    ! [B: $tType,A: $tType,Tb: product_prod(A,B)] : Tb = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Tb)),aa(product_prod(A,B),B,product_snd(A,B),Tb)) ).

% surjective_pairing
tff(fact_263_prod_Oexhaust__sel,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : Prod = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(product_prod(A,B),A,product_fst(A,B),Prod)),aa(product_prod(A,B),B,product_snd(A,B),Prod)) ).

% prod.exhaust_sel
tff(fact_264_option_Oexhaust__sel,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
     => ( Option = aa(A,option(A),some(A),aa(option(A),A,the2(A),Option)) ) ) ).

% option.exhaust_sel
tff(fact_265_case__prod__beta,axiom,
    ! [A: $tType,C: $tType,B: $tType,Fa: fun(B,fun(C,A)),P2: product_prod(B,C)] : aa(product_prod(B,C),A,product_case_prod(B,C,A,Fa),P2) = aa(C,A,aa(B,fun(C,A),Fa,aa(product_prod(B,C),B,product_fst(B,C),P2)),aa(product_prod(B,C),C,product_snd(B,C),P2)) ).

% case_prod_beta
tff(fact_266_split__beta,axiom,
    ! [A: $tType,C: $tType,B: $tType,Fa: fun(B,fun(C,A)),Prod: product_prod(B,C)] : aa(product_prod(B,C),A,product_case_prod(B,C,A,Fa),Prod) = aa(C,A,aa(B,fun(C,A),Fa,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) ).

% split_beta
tff(fact_267_pos__imp__zdiv__neg__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
     => ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),zero_zero(int))
      <=> aa(int,$o,ord_less(int,A3),zero_zero(int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_268_neg__imp__zdiv__neg__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,ord_less(int,B3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),zero_zero(int))
      <=> aa(int,$o,ord_less(int,zero_zero(int)),A3) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_269_int__div__less__self,axiom,
    ! [Xb: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),Xb)
     => ( aa(int,$o,ord_less(int,one_one(int)),K)
       => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),K)),Xb) ) ) ).

% int_div_less_self
tff(fact_270_div__neg__pos__less0,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less(int,A3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
       => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),zero_zero(int)) ) ) ).

% div_neg_pos_less0
tff(fact_271_prod__induct7,axiom,
    ! [G2: $tType,F: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2)))))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2))))))] :
      ( ! [A2: A,B2: B,C3: C,D3: D2,E2: E,F4: F,G3: G2] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2)))))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2)))))),A2),aa(product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2)))),product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2))))),aa(B,fun(product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2)))),product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2)))))),product_Pair(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2))))),B2),aa(product_prod(D2,product_prod(E,product_prod(F,G2))),product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2)))),aa(C,fun(product_prod(D2,product_prod(E,product_prod(F,G2))),product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2))))),product_Pair(C,product_prod(D2,product_prod(E,product_prod(F,G2)))),C3),aa(product_prod(E,product_prod(F,G2)),product_prod(D2,product_prod(E,product_prod(F,G2))),aa(D2,fun(product_prod(E,product_prod(F,G2)),product_prod(D2,product_prod(E,product_prod(F,G2)))),product_Pair(D2,product_prod(E,product_prod(F,G2))),D3),aa(product_prod(F,G2),product_prod(E,product_prod(F,G2)),aa(E,fun(product_prod(F,G2),product_prod(E,product_prod(F,G2))),product_Pair(E,product_prod(F,G2)),E2),aa(G2,product_prod(F,G2),aa(F,fun(G2,product_prod(F,G2)),product_Pair(F,G2),F4),G3)))))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2)))))),$o,P,Xb) ) ).

% prod_induct7
tff(fact_272_prod__induct6,axiom,
    ! [F: $tType,E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F))))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F)))))] :
      ( ! [A2: A,B2: B,C3: C,D3: D2,E2: E,F4: F] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F))))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F))))),A2),aa(product_prod(C,product_prod(D2,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F)))),aa(B,fun(product_prod(C,product_prod(D2,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F))))),product_Pair(B,product_prod(C,product_prod(D2,product_prod(E,F)))),B2),aa(product_prod(D2,product_prod(E,F)),product_prod(C,product_prod(D2,product_prod(E,F))),aa(C,fun(product_prod(D2,product_prod(E,F)),product_prod(C,product_prod(D2,product_prod(E,F)))),product_Pair(C,product_prod(D2,product_prod(E,F))),C3),aa(product_prod(E,F),product_prod(D2,product_prod(E,F)),aa(D2,fun(product_prod(E,F),product_prod(D2,product_prod(E,F))),product_Pair(D2,product_prod(E,F)),D3),aa(F,product_prod(E,F),aa(E,fun(F,product_prod(E,F)),product_Pair(E,F),E2),F4))))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F))))),$o,P,Xb) ) ).

% prod_induct6
tff(fact_273_prod__induct5,axiom,
    ! [E: $tType,D2: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D2,E)))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D2,E))))] :
      ( ! [A2: A,B2: B,C3: C,D3: D2,E2: E] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D2,E)))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D2,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D2,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D2,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D2,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D2,E)))),A2),aa(product_prod(C,product_prod(D2,E)),product_prod(B,product_prod(C,product_prod(D2,E))),aa(B,fun(product_prod(C,product_prod(D2,E)),product_prod(B,product_prod(C,product_prod(D2,E)))),product_Pair(B,product_prod(C,product_prod(D2,E))),B2),aa(product_prod(D2,E),product_prod(C,product_prod(D2,E)),aa(C,fun(product_prod(D2,E),product_prod(C,product_prod(D2,E))),product_Pair(C,product_prod(D2,E)),C3),aa(E,product_prod(D2,E),aa(D2,fun(E,product_prod(D2,E)),product_Pair(D2,E),D3),E2)))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D2,E)))),$o,P,Xb) ) ).

% prod_induct5
tff(fact_274_prod__induct4,axiom,
    ! [D2: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,D2))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,D2)))] :
      ( ! [A2: A,B2: B,C3: C,D3: D2] : aa(product_prod(A,product_prod(B,product_prod(C,D2))),$o,P,aa(product_prod(B,product_prod(C,D2)),product_prod(A,product_prod(B,product_prod(C,D2))),aa(A,fun(product_prod(B,product_prod(C,D2)),product_prod(A,product_prod(B,product_prod(C,D2)))),product_Pair(A,product_prod(B,product_prod(C,D2))),A2),aa(product_prod(C,D2),product_prod(B,product_prod(C,D2)),aa(B,fun(product_prod(C,D2),product_prod(B,product_prod(C,D2))),product_Pair(B,product_prod(C,D2)),B2),aa(D2,product_prod(C,D2),aa(C,fun(D2,product_prod(C,D2)),product_Pair(C,D2),C3),D3))))
     => aa(product_prod(A,product_prod(B,product_prod(C,D2))),$o,P,Xb) ) ).

% prod_induct4
tff(fact_275_prod__cases7,axiom,
    ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F: $tType,G2: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2))))))] :
      ~ ! [A2: A,B2: B,C3: C,D3: D2,E2: E,F4: F,G3: G2] : Y != aa(product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2)))))),A2),aa(product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2)))),product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2))))),aa(B,fun(product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2)))),product_prod(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2)))))),product_Pair(B,product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2))))),B2),aa(product_prod(D2,product_prod(E,product_prod(F,G2))),product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2)))),aa(C,fun(product_prod(D2,product_prod(E,product_prod(F,G2))),product_prod(C,product_prod(D2,product_prod(E,product_prod(F,G2))))),product_Pair(C,product_prod(D2,product_prod(E,product_prod(F,G2)))),C3),aa(product_prod(E,product_prod(F,G2)),product_prod(D2,product_prod(E,product_prod(F,G2))),aa(D2,fun(product_prod(E,product_prod(F,G2)),product_prod(D2,product_prod(E,product_prod(F,G2)))),product_Pair(D2,product_prod(E,product_prod(F,G2))),D3),aa(product_prod(F,G2),product_prod(E,product_prod(F,G2)),aa(E,fun(product_prod(F,G2),product_prod(E,product_prod(F,G2))),product_Pair(E,product_prod(F,G2)),E2),aa(G2,product_prod(F,G2),aa(F,fun(G2,product_prod(F,G2)),product_Pair(F,G2),F4),G3)))))) ).

% prod_cases7
tff(fact_276_prod__cases6,axiom,
    ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,F: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F)))))] :
      ~ ! [A2: A,B2: B,C3: C,D3: D2,E2: E,F4: F] : Y != aa(product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F))))),A2),aa(product_prod(C,product_prod(D2,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F)))),aa(B,fun(product_prod(C,product_prod(D2,product_prod(E,F))),product_prod(B,product_prod(C,product_prod(D2,product_prod(E,F))))),product_Pair(B,product_prod(C,product_prod(D2,product_prod(E,F)))),B2),aa(product_prod(D2,product_prod(E,F)),product_prod(C,product_prod(D2,product_prod(E,F))),aa(C,fun(product_prod(D2,product_prod(E,F)),product_prod(C,product_prod(D2,product_prod(E,F)))),product_Pair(C,product_prod(D2,product_prod(E,F))),C3),aa(product_prod(E,F),product_prod(D2,product_prod(E,F)),aa(D2,fun(product_prod(E,F),product_prod(D2,product_prod(E,F))),product_Pair(D2,product_prod(E,F)),D3),aa(F,product_prod(E,F),aa(E,fun(F,product_prod(E,F)),product_Pair(E,F),E2),F4))))) ).

% prod_cases6
tff(fact_277_prod__cases5,axiom,
    ! [A: $tType,B: $tType,C: $tType,D2: $tType,E: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D2,E))))] :
      ~ ! [A2: A,B2: B,C3: C,D3: D2,E2: E] : Y != aa(product_prod(B,product_prod(C,product_prod(D2,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D2,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D2,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D2,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D2,E)))),A2),aa(product_prod(C,product_prod(D2,E)),product_prod(B,product_prod(C,product_prod(D2,E))),aa(B,fun(product_prod(C,product_prod(D2,E)),product_prod(B,product_prod(C,product_prod(D2,E)))),product_Pair(B,product_prod(C,product_prod(D2,E))),B2),aa(product_prod(D2,E),product_prod(C,product_prod(D2,E)),aa(C,fun(product_prod(D2,E),product_prod(C,product_prod(D2,E))),product_Pair(C,product_prod(D2,E)),C3),aa(E,product_prod(D2,E),aa(D2,fun(E,product_prod(D2,E)),product_Pair(D2,E),D3),E2)))) ).

% prod_cases5
tff(fact_278_prod__cases4,axiom,
    ! [A: $tType,B: $tType,C: $tType,D2: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D2)))] :
      ~ ! [A2: A,B2: B,C3: C,D3: D2] : Y != aa(product_prod(B,product_prod(C,D2)),product_prod(A,product_prod(B,product_prod(C,D2))),aa(A,fun(product_prod(B,product_prod(C,D2)),product_prod(A,product_prod(B,product_prod(C,D2)))),product_Pair(A,product_prod(B,product_prod(C,D2))),A2),aa(product_prod(C,D2),product_prod(B,product_prod(C,D2)),aa(B,fun(product_prod(C,D2),product_prod(B,product_prod(C,D2))),product_Pair(B,product_prod(C,D2)),B2),aa(D2,product_prod(C,D2),aa(C,fun(D2,product_prod(C,D2)),product_Pair(C,D2),C3),D3))) ).

% prod_cases4
tff(fact_279_option_Ocase__eq__if,axiom,
    ! [A: $tType,B: $tType,F1: A,F2: fun(B,A),Option: option(B)] :
      case_option(A,B,F1,F2,Option) = $ite(Option = none(B),F1,aa(B,A,F2,aa(option(B),B,the2(B),Option))) ).

% option.case_eq_if
tff(fact_280_case__prod__unfold,axiom,
    ! [C: $tType,B: $tType,A: $tType,X3: fun(A,fun(B,C)),Xa: product_prod(A,B)] : aa(product_prod(A,B),C,product_case_prod(A,B,C,X3),Xa) = aa(B,C,aa(A,fun(B,C),X3,aa(product_prod(A,B),A,product_fst(A,B),Xa)),aa(product_prod(A,B),B,product_snd(A,B),Xa)) ).

% case_prod_unfold
tff(fact_281_case__prod__beta_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,Fa: fun(A,fun(B,C)),X3: product_prod(A,B)] : aa(product_prod(A,B),C,product_case_prod(A,B,C,Fa),X3) = aa(B,C,aa(A,fun(B,C),Fa,aa(product_prod(A,B),A,product_fst(A,B),X3)),aa(product_prod(A,B),B,product_snd(A,B),X3)) ).

% case_prod_beta'
tff(fact_282_split__comp__eq,axiom,
    ! [B: $tType,C: $tType,D2: $tType,A: $tType,Fa: fun(D2,fun(B,C)),G: fun(A,D2)] : aa(fun(A,D2),fun(product_prod(A,B),C),aTP_Lamp_cr(fun(D2,fun(B,C)),fun(fun(A,D2),fun(product_prod(A,B),C)),Fa),G) = product_case_prod(A,B,C,aa(fun(A,D2),fun(A,fun(B,C)),aTP_Lamp_cs(fun(D2,fun(B,C)),fun(fun(A,D2),fun(A,fun(B,C))),Fa),G)) ).

% split_comp_eq
tff(fact_283_option_Othe__def,axiom,
    ! [A: $tType,Option: option(A)] : aa(option(A),A,the2(A),Option) = case_option(A,A,undefined(A),aTP_Lamp_ct(A,A),Option) ).

% option.the_def
tff(fact_284_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
         => ( aa(A,$o,ord_less_eq(A,B3),A3)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ) ) ).

% div_positive
tff(fact_285_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_286_prod_Osplit__sel__asm,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,$o),Fa: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
      ( aa(A,$o,P,aa(product_prod(B,C),A,product_case_prod(B,C,A,Fa),Prod))
    <=> ~ ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
          & ~ aa(A,$o,P,aa(C,A,aa(B,fun(C,A),Fa,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod))) ) ) ).

% prod.split_sel_asm
tff(fact_287_prod_Osplit__sel,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,$o),Fa: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
      ( aa(A,$o,P,aa(product_prod(B,C),A,product_case_prod(B,C,A,Fa),Prod))
    <=> ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
       => aa(A,$o,P,aa(C,A,aa(B,fun(C,A),Fa,aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod))) ) ) ).

% prod.split_sel
tff(fact_288_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3))
      <=> ( aa(int,$o,ord_less_eq(int,B3),A3)
          & aa(int,$o,ord_less(int,zero_zero(int)),B3) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_289_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3))
      <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),A3) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_290_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,ord_less(int,B3),zero_zero(int))
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3))
      <=> aa(int,$o,ord_less_eq(int,A3),zero_zero(int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_291_zdiv__le__dividend,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),A3) ) ) ).

% zdiv_le_dividend
tff(fact_292_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K))
      <=> aa(int,$o,ord_less_eq(int,K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_293_div__nonpos__pos__le0,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),zero_zero(int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_294_div__nonneg__neg__le0,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less(int,B3),zero_zero(int))
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),zero_zero(int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_295_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,L),K)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),L)
       => aa(int,$o,ord_less(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)) ) ) ).

% div_positive_int
tff(fact_296_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L))
    <=> ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) )
        | ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
          & aa(int,$o,ord_less_eq(int,zero_zero(int)),L) )
        | ( aa(int,$o,ord_less(int,K),zero_zero(int))
          & aa(int,$o,ord_less(int,L),zero_zero(int)) ) ) ) ).

% div_int_pos_iff
tff(fact_297_zdiv__mono2__neg,axiom,
    ! [A3: int,B4: int,B3: int] :
      ( aa(int,$o,ord_less(int,A3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B4)
       => ( aa(int,$o,ord_less_eq(int,B4),B3)
         => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B4)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_298_zdiv__mono1__neg,axiom,
    ! [A3: int,A5: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),A5)
     => ( aa(int,$o,ord_less(int,B3),zero_zero(int))
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),B3)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)) ) ) ).

% zdiv_mono1_neg
tff(fact_299_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),I),K) = zero_zero(int) )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,ord_less_eq(int,zero_zero(int)),I)
          & aa(int,$o,ord_less(int,I),K) )
        | ( aa(int,$o,ord_less_eq(int,I),zero_zero(int))
          & aa(int,$o,ord_less(int,K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_300_zdiv__mono2,axiom,
    ! [A3: int,B4: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B4)
       => ( aa(int,$o,ord_less_eq(int,B4),B3)
         => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B4)) ) ) ) ).

% zdiv_mono2
tff(fact_301_zdiv__mono1,axiom,
    ! [A3: int,A5: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),A5)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A5),B3)) ) ) ).

% zdiv_mono1
tff(fact_302_option_Osplit__sel__asm,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F1: A,F2: fun(B,A),Option: option(B)] :
      ( aa(A,$o,P,case_option(A,B,F1,F2,Option))
    <=> ~ ( ( ( Option = none(B) )
            & ~ aa(A,$o,P,F1) )
          | ( ( Option = aa(B,option(B),some(B),aa(option(B),B,the2(B),Option)) )
            & ~ aa(A,$o,P,aa(B,A,F2,aa(option(B),B,the2(B),Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_303_option_Osplit__sel,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F1: A,F2: fun(B,A),Option: option(B)] :
      ( aa(A,$o,P,case_option(A,B,F1,F2,Option))
    <=> ( ( ( Option = none(B) )
         => aa(A,$o,P,F1) )
        & ( ( Option = aa(B,option(B),some(B),aa(option(B),B,the2(B),Option)) )
         => aa(A,$o,P,aa(B,A,F2,aa(option(B),B,the2(B),Option))) ) ) ) ).

% option.split_sel
tff(fact_304_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ).

% half_gt_zero_iff
tff(fact_305_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% half_gt_zero
tff(fact_306_semiring__norm_I76_J,axiom,
    ! [N: num] : aa(num,$o,ord_less(num,one2),bit0(N)) ).

% semiring_norm(76)
tff(fact_307_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa(num,$o,ord_less_eq(num,bit0(M)),one2) ).

% semiring_norm(69)
tff(fact_308_pred__list__to__short,axiom,
    ! [Deg2: nat,Xb: nat,Ma: nat,TreeList: list(vEBT_VEBT),Mi: nat,Summary2: vEBT_VEBT] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
     => ( aa(nat,$o,ord_less_eq(nat,Xb),Ma)
       => ( aa(nat,$o,ord_less_eq(nat,aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = none(nat) ) ) ) ) ).

% pred_list_to_short
tff(fact_309_VEBT__internal_Ovebt__succi_H_Osimps,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),vEBT_VEBT_vebt_succi,Tb),Tib),Xb) = vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_dk(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Tb),Xb),aTP_Lamp_cl(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Xb),Tib) ).

% VEBT_internal.vebt_succi'.simps
tff(fact_310_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa(num,$o,ord_less(num,M),one2) ).

% semiring_norm(75)
tff(fact_311_semiring__norm_I68_J,axiom,
    ! [N: num] : aa(num,$o,ord_less_eq(num,one2),N) ).

% semiring_norm(68)
tff(fact_312_semiring__norm_I78_J,axiom,
    ! [M: num,N: num] :
      ( aa(num,$o,ord_less(num,bit0(M)),bit0(N))
    <=> aa(num,$o,ord_less(num,M),N) ) ).

% semiring_norm(78)
tff(fact_313_semiring__norm_I71_J,axiom,
    ! [M: num,N: num] :
      ( aa(num,$o,ord_less_eq(num,bit0(M)),bit0(N))
    <=> aa(num,$o,ord_less_eq(num,M),N) ) ).

% semiring_norm(71)
tff(fact_314_pred__max,axiom,
    ! [Deg2: nat,Ma: nat,Xb: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
     => ( aa(nat,$o,ord_less(nat,Ma),Xb)
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = aa(nat,option(nat),some(nat),Ma) ) ) ) ).

% pred_max
tff(fact_315_less__option__None__Some__code,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A] : aa(option(A),$o,ord_less(option(A),none(A)),aa(A,option(A),some(A),Xb)) ) ).

% less_option_None_Some_code
tff(fact_316_less__eq__option__Some__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A] : ~ aa(option(A),$o,ord_less_eq(option(A),aa(A,option(A),some(A),Xb)),none(A)) ) ).

% less_eq_option_Some_None
tff(fact_317_enat__ord__number_I1_J,axiom,
    ! [M: num,N: num] :
      ( aa(extended_enat,$o,ord_less_eq(extended_enat,aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),N))
    <=> aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N)) ) ).

% enat_ord_number(1)
tff(fact_318_semiring__norm_I87_J,axiom,
    ! [M: num,N: num] :
      ( ( bit0(M) = bit0(N) )
    <=> ( M = N ) ) ).

% semiring_norm(87)
tff(fact_319_case__prodI,axiom,
    ! [A: $tType,B: $tType,Fa: fun(A,fun(B,$o)),A3: A,B3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),Fa,A3),B3)
     => aa(product_prod(A,B),$o,product_case_prod(A,B,$o,Fa),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3)) ) ).

% case_prodI
tff(fact_320_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,$o))] :
      ( ! [A2: A,B2: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) )
         => aa(B,$o,aa(A,fun(B,$o),C2,A2),B2) )
     => aa(product_prod(A,B),$o,product_case_prod(A,B,$o,C2),P2) ) ).

% case_prodI2
tff(fact_321_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z: A,C2: fun(B,fun(C,set(A))),A3: B,B3: C] :
      ( member(A,Z,aa(C,set(A),aa(B,fun(C,set(A)),C2,A3),B3))
     => member(A,Z,aa(product_prod(B,C),set(A),product_case_prod(B,C,set(A),C2),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B3))) ) ).

% mem_case_prodI
tff(fact_322_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: product_prod(A,B),Z: C,C2: fun(A,fun(B,set(C)))] :
      ( ! [A2: A,B2: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) )
         => member(C,Z,aa(B,set(C),aa(A,fun(B,set(C)),C2,A2),B2)) )
     => member(C,Z,aa(product_prod(A,B),set(C),product_case_prod(A,B,set(C),C2),P2)) ) ).

% mem_case_prodI2
tff(fact_323_case__prodI2_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,fun(C,$o))),Xb: C] :
      ( ! [A2: A,B2: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) = P2 )
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,A2),B2),Xb) )
     => aa(C,$o,aa(product_prod(A,B),fun(C,$o),product_case_prod(A,B,fun(C,$o),C2),P2),Xb) ) ).

% case_prodI2'
tff(fact_324_semiring__norm_I85_J,axiom,
    ! [M: num] : bit0(M) != one2 ).

% semiring_norm(85)
tff(fact_325_semiring__norm_I83_J,axiom,
    ! [N: num] : one2 != bit0(N) ).

% semiring_norm(83)
tff(fact_326_less__eq__option__None__code,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: option(A)] : aa(option(A),$o,ord_less_eq(option(A),none(A)),Xb) ) ).

% less_eq_option_None_code
tff(fact_327_succ__min,axiom,
    ! [Deg2: nat,Xb: nat,Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
     => ( aa(nat,$o,ord_less(nat,Xb),Mi)
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = aa(nat,option(nat),some(nat),Mi) ) ) ) ).

% succ_min
tff(fact_328_less__option__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: option(A)] : ~ aa(option(A),$o,ord_less(option(A),Xb),none(A)) ) ).

% less_option_None
tff(fact_329_succ__list__to__short,axiom,
    ! [Deg2: nat,Mi: nat,Xb: nat,TreeList: list(vEBT_VEBT),Ma: nat,Summary2: vEBT_VEBT] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
     => ( aa(nat,$o,ord_less_eq(nat,Mi),Xb)
       => ( aa(nat,$o,ord_less_eq(nat,aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = none(nat) ) ) ) ) ).

% succ_list_to_short
tff(fact_330_enat__ord__number_I2_J,axiom,
    ! [M: num,N: num] :
      ( aa(extended_enat,$o,ord_less(extended_enat,aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),N))
    <=> aa(nat,$o,ord_less(nat,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N)) ) ).

% enat_ord_number(2)
tff(fact_331_less__eq__option__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(option(A),$o,ord_less_eq(option(A),aa(A,option(A),some(A),Xb)),aa(A,option(A),some(A),Y))
        <=> aa(A,$o,ord_less_eq(A,Xb),Y) ) ) ).

% less_eq_option_Some
tff(fact_332_less__option__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(option(A),$o,ord_less(option(A),aa(A,option(A),some(A),Xb)),aa(A,option(A),some(A),Y))
        <=> aa(A,$o,ord_less(A,Xb),Y) ) ) ).

% less_option_Some
tff(fact_333_succ__greatereq__min,axiom,
    ! [Deg2: nat,Mi: nat,Xb: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
     => ( aa(nat,$o,ord_less_eq(nat,Mi),Xb)
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = $let(
              l: nat,
              l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  $let(
                    maxlow: option(nat),
                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( maxlow != none(nat) )
                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                      $let(
                        sc: option(nat),
                        sc:= vEBT_vebt_succ(Summary2,h),
                        $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_334_succ__less__length__list,axiom,
    ! [Deg2: nat,Mi: nat,Xb: nat,TreeList: list(vEBT_VEBT),Ma: nat,Summary2: vEBT_VEBT] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
     => ( aa(nat,$o,ord_less_eq(nat,Mi),Xb)
       => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
         => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = $let(
                l: nat,
                l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $let(
                    maxlow: option(nat),
                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( maxlow != none(nat) )
                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                      $let(
                        sc: option(nat),
                        sc:= vEBT_vebt_succ(Summary2,h),
                        $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_335_pred__less__length__list,axiom,
    ! [Deg2: nat,Xb: nat,Ma: nat,TreeList: list(vEBT_VEBT),Mi: nat,Summary2: vEBT_VEBT] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
     => ( aa(nat,$o,ord_less_eq(nat,Xb),Ma)
       => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
         => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = $let(
                l: nat,
                l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( minlow != none(nat) )
                      & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                      $let(
                        pr: option(nat),
                        pr:= vEBT_vebt_pred(Summary2,h),
                        $ite(
                          pr = none(nat),
                          $ite(aa(nat,$o,ord_less(nat,Mi),Xb),aa(nat,option(nat),some(nat),Mi),none(nat)),
                          aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_336_pred__lesseq__max,axiom,
    ! [Deg2: nat,Xb: nat,Ma: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
     => ( aa(nat,$o,ord_less_eq(nat,Xb),Ma)
       => ( vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = $let(
              l: nat,
              l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    $ite(
                      ( ( minlow != none(nat) )
                      & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                      $let(
                        pr: option(nat),
                        pr:= vEBT_vebt_pred(Summary2,h),
                        $ite(
                          pr = none(nat),
                          $ite(aa(nat,$o,ord_less(nat,Mi),Xb),aa(nat,option(nat),some(nat),Mi),none(nat)),
                          aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_337_i0__lb,axiom,
    ! [N: extended_enat] : aa(extended_enat,$o,ord_less_eq(extended_enat,zero_zero(extended_enat)),N) ).

% i0_lb
tff(fact_338_ile0__eq,axiom,
    ! [N: extended_enat] :
      ( aa(extended_enat,$o,ord_less_eq(extended_enat,N),zero_zero(extended_enat))
    <=> ( N = zero_zero(extended_enat) ) ) ).

% ile0_eq
tff(fact_339_less__eq__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: option(A),Y: option(A)] :
          ( aa(option(A),$o,ord_less_eq(option(A),Xb),Y)
        <=> case_option($o,A,$true,aTP_Lamp_dl(option(A),fun(A,$o),Y),Xb) ) ) ).

% less_eq_option_def
tff(fact_340_less__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: option(A),Y: option(A)] :
          ( aa(option(A),$o,ord_less(option(A),Xb),Y)
        <=> case_option($o,A,$false,aTP_Lamp_dn(option(A),fun(A,$o),Xb),Y) ) ) ).

% less_option_def
tff(fact_341_mem__case__prodE,axiom,
    ! [B: $tType,A: $tType,C: $tType,Z: A,C2: fun(B,fun(C,set(A))),P2: product_prod(B,C)] :
      ( member(A,Z,aa(product_prod(B,C),set(A),product_case_prod(B,C,set(A),C2),P2))
     => ~ ! [X2: B,Y2: C] :
            ( ( P2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X2),Y2) )
           => ~ member(A,Z,aa(C,set(A),aa(B,fun(C,set(A)),C2,X2),Y2)) ) ) ).

% mem_case_prodE
tff(fact_342_case__prodD,axiom,
    ! [A: $tType,B: $tType,Fa: fun(A,fun(B,$o)),A3: A,B3: B] :
      ( aa(product_prod(A,B),$o,product_case_prod(A,B,$o,Fa),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3))
     => aa(B,$o,aa(A,fun(B,$o),Fa,A3),B3) ) ).

% case_prodD
tff(fact_343_case__prodE,axiom,
    ! [A: $tType,B: $tType,C2: fun(A,fun(B,$o)),P2: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,product_case_prod(A,B,$o,C2),P2)
     => ~ ! [X2: A,Y2: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) )
           => ~ aa(B,$o,aa(A,fun(B,$o),C2,X2),Y2) ) ) ).

% case_prodE
tff(fact_344_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: fun(A,fun(B,fun(C,$o))),A3: A,B3: B,C2: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),product_case_prod(A,B,fun(C,$o),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3)),C2)
     => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),R,A3),B3),C2) ) ).

% case_prodD'
tff(fact_345_case__prodE_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: fun(A,fun(B,fun(C,$o))),P2: product_prod(A,B),Z: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),product_case_prod(A,B,fun(C,$o),C2),P2),Z)
     => ~ ! [X2: A,Y2: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2) )
           => ~ aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,X2),Y2),Z) ) ) ).

% case_prodE'
tff(fact_346_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> case_option($o,A,$false,aTP_Lamp_do(A,$o),Option) ) ).

% option.disc_eq_case(2)
tff(fact_347_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> case_option($o,A,$true,aTP_Lamp_dp(A,$o),Option) ) ).

% option.disc_eq_case(1)
tff(fact_348_Product__Type_OCollect__case__prodD,axiom,
    ! [B: $tType,A: $tType,Xb: product_prod(A,B),A4: fun(A,fun(B,$o))] :
      ( member(product_prod(A,B),Xb,collect(product_prod(A,B),product_case_prod(A,B,$o,A4)))
     => aa(B,$o,aa(A,fun(B,$o),A4,aa(product_prod(A,B),A,product_fst(A,B),Xb)),aa(product_prod(A,B),B,product_snd(A,B),Xb)) ) ).

% Product_Type.Collect_case_prodD
tff(fact_349_option_Osize__neq,axiom,
    ! [A: $tType,Xb: option(A)] : aa(option(A),nat,size_size(option(A)),Xb) != zero_zero(nat) ).

% option.size_neq
tff(fact_350_case__optionE,axiom,
    ! [A: $tType,P: $o,Q: fun(A,$o),Xb: option(A)] :
      ( case_option($o,A,(P),Q,Xb)
     => ( ( ( Xb = none(A) )
         => ~ (P) )
       => ~ ! [Y2: A] :
              ( ( Xb = aa(A,option(A),some(A),Y2) )
             => ~ aa(A,$o,Q,Y2) ) ) ) ).

% case_optionE
tff(fact_351_less__eq__option__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: option(A)] : aa(option(A),$o,ord_less_eq(option(A),none(A)),Xb) ) ).

% less_eq_option_None
tff(fact_352_less__eq__option__None__is__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: option(A)] :
          ( aa(option(A),$o,ord_less_eq(option(A),Xb),none(A))
         => ( Xb = none(A) ) ) ) ).

% less_eq_option_None_is_None
tff(fact_353_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_354_less__option__None__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A] : aa(option(A),$o,ord_less(option(A),none(A)),aa(A,option(A),some(A),Xb)) ) ).

% less_option_None_Some
tff(fact_355_less__option__None__is__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: option(A)] :
          ( aa(option(A),$o,ord_less(option(A),none(A)),Xb)
         => ? [Z2: A] : Xb = aa(A,option(A),some(A),Z2) ) ) ).

% less_option_None_is_Some
tff(fact_356_tdeletemimi_H,axiom,
    ! [Deg2: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
     => aa(nat,$o,ord_less_eq(nat,vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Mi)),Deg2,TreeList,Summary2),Xb)),one_one(nat)) ) ).

% tdeletemimi'
tff(fact_357_VEBT__internal_Ovebt__predi_H_Oraw__induct,axiom,
    ! [P: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))))))),Xaa: product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: option(nat),N: nat] :
      ( ! [Vebt_predi: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))] :
          ( ! [A6: vEBT_VEBT,B5: vEBT_VEBTi,Ba: nat,H3: heap_ext(product_unit),H4: heap_ext(product_unit),R3: option(nat),N4: nat] :
              ( heap_Time_effect(option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Vebt_predi,A6),B5),Ba),H3,H4,R3,N4)
             => aa(nat,$o,aa(option(nat),fun(nat,$o),aa(heap_ext(product_unit),fun(option(nat),fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))),aa(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))),aa(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))))),P,A6),B5),Ba),H3),H4),R3),N4) )
         => ! [T2: vEBT_VEBT,Ti2: vEBT_VEBTi,X2: nat,Ta: heap_ext(product_unit),Tia: heap_ext(product_unit),Xa2: option(nat),N3: nat] :
              ( heap_Time_effect(option(nat),vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aTP_Lamp_ee(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),Vebt_predi),T2),X2),aTP_Lamp_bk(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),X2),Ti2),Ta,Tia,Xa2,N3)
             => aa(nat,$o,aa(option(nat),fun(nat,$o),aa(heap_ext(product_unit),fun(option(nat),fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))),aa(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))),aa(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))))),P,T2),Ti2),X2),Ta),Tia),Xa2),N3) ) )
     => ( heap_Time_effect(option(nat),aa(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat)),product_case_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(option(nat)),product_case_prod(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_VEBT_vebt_predi)),Xaa),H,H2,R2,N)
       => aa(nat,$o,aa(option(nat),fun(nat,$o),aa(heap_ext(product_unit),fun(option(nat),fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))),aa(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))),product_case_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))),product_case_prod(vEBT_VEBT,vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))))),P)),Xaa),H),H2),R2),N) ) ) ).

% VEBT_internal.vebt_predi'.raw_induct
tff(fact_358_VEBT__internal_Ovebt__succi_H_Oraw__induct,axiom,
    ! [P: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))))))),Xaa: product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: option(nat),N: nat] :
      ( ! [Vebt_succi: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))] :
          ( ! [A6: vEBT_VEBT,B5: vEBT_VEBTi,Ba: nat,H3: heap_ext(product_unit),H4: heap_ext(product_unit),R3: option(nat),N4: nat] :
              ( heap_Time_effect(option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Vebt_succi,A6),B5),Ba),H3,H4,R3,N4)
             => aa(nat,$o,aa(option(nat),fun(nat,$o),aa(heap_ext(product_unit),fun(option(nat),fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))),aa(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))),aa(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))))),P,A6),B5),Ba),H3),H4),R3),N4) )
         => ! [T2: vEBT_VEBT,Ti2: vEBT_VEBTi,X2: nat,Ta: heap_ext(product_unit),Tia: heap_ext(product_unit),Xa2: option(nat),N3: nat] :
              ( heap_Time_effect(option(nat),vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aTP_Lamp_et(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),Vebt_succi),T2),X2),aTP_Lamp_cl(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),X2),Ti2),Ta,Tia,Xa2,N3)
             => aa(nat,$o,aa(option(nat),fun(nat,$o),aa(heap_ext(product_unit),fun(option(nat),fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))),aa(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))),aa(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))))),P,T2),Ti2),X2),Ta),Tia),Xa2),N3) ) )
     => ( heap_Time_effect(option(nat),aa(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat)),product_case_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(option(nat)),product_case_prod(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_VEBT_vebt_succi)),Xaa),H,H2,R2,N)
       => aa(nat,$o,aa(option(nat),fun(nat,$o),aa(heap_ext(product_unit),fun(option(nat),fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))),aa(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))),product_case_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))),product_case_prod(vEBT_VEBT,vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))))),P)),Xaa),H),H2),R2),N) ) ) ).

% VEBT_internal.vebt_succi'.raw_induct
tff(fact_359_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2)),Xb)
     => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
        & ( ( Xb = Mi )
          | ( Xb = Ma )
          | ( aa(nat,$o,ord_less(nat,Xb),Ma)
            & aa(nat,$o,ord_less(nat,Mi),Xb)
            & aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
            & aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% member_inv
tff(fact_360_insert__simp__mima,axiom,
    ! [Xb: nat,Mi: nat,Ma: nat,Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( ( ( Xb = Mi )
        | ( Xb = Ma ) )
     => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
       => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2) ) ) ) ).

% insert_simp_mima
tff(fact_361_vebt__predi_Oraw__induct,axiom,
    ! [P: fun(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))))),Xaa: product_prod(vEBT_VEBTi,nat),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: option(nat),N: nat] :
      ( ! [Vebt_predi2: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))] :
          ( ! [A6: vEBT_VEBTi,B5: nat,H3: heap_ext(product_unit),H4: heap_ext(product_unit),R3: option(nat),N4: nat] :
              ( heap_Time_effect(option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),Vebt_predi2,A6),B5),H3,H4,R3,N4)
             => aa(nat,$o,aa(option(nat),fun(nat,$o),aa(heap_ext(product_unit),fun(option(nat),fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))),aa(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))),aa(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))))),P,A6),B5),H3),H4),R3),N4) )
         => ! [T2: vEBT_VEBTi,X2: nat,Ta: heap_ext(product_unit),Xa2: heap_ext(product_unit),R4: option(nat),N3: nat] :
              ( heap_Time_effect(option(nat),vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_ez(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Vebt_predi2),X2),aTP_Lamp_bk(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),X2),T2),Ta,Xa2,R4,N3)
             => aa(nat,$o,aa(option(nat),fun(nat,$o),aa(heap_ext(product_unit),fun(option(nat),fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))),aa(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))),aa(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))))),P,T2),X2),Ta),Xa2),R4),N3) ) )
     => ( heap_Time_effect(option(nat),aa(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat)),product_case_prod(vEBT_VEBTi,nat,heap_Time_Heap(option(nat)),vEBT_vebt_predi),Xaa),H,H2,R2,N)
       => aa(nat,$o,aa(option(nat),fun(nat,$o),aa(heap_ext(product_unit),fun(option(nat),fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))),aa(product_prod(vEBT_VEBTi,nat),fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))),product_case_prod(vEBT_VEBTi,nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))),P),Xaa),H),H2),R2),N) ) ) ).

% vebt_predi.raw_induct
tff(fact_362_vebt__succi_Oraw__induct,axiom,
    ! [P: fun(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))))),Xaa: product_prod(vEBT_VEBTi,nat),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: option(nat),N: nat] :
      ( ! [Vebt_succi2: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))] :
          ( ! [A6: vEBT_VEBTi,B5: nat,H3: heap_ext(product_unit),H4: heap_ext(product_unit),R3: option(nat),N4: nat] :
              ( heap_Time_effect(option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),Vebt_succi2,A6),B5),H3,H4,R3,N4)
             => aa(nat,$o,aa(option(nat),fun(nat,$o),aa(heap_ext(product_unit),fun(option(nat),fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))),aa(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))),aa(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))))),P,A6),B5),H3),H4),R3),N4) )
         => ! [T2: vEBT_VEBTi,X2: nat,Ta: heap_ext(product_unit),Xa2: heap_ext(product_unit),R4: option(nat),N3: nat] :
              ( heap_Time_effect(option(nat),vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_ff(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Vebt_succi2),X2),aTP_Lamp_cl(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),X2),T2),Ta,Xa2,R4,N3)
             => aa(nat,$o,aa(option(nat),fun(nat,$o),aa(heap_ext(product_unit),fun(option(nat),fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))),aa(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))),aa(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))))),P,T2),X2),Ta),Xa2),R4),N3) ) )
     => ( heap_Time_effect(option(nat),aa(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat)),product_case_prod(vEBT_VEBTi,nat,heap_Time_Heap(option(nat)),vEBT_vebt_succi),Xaa),H,H2,R2,N)
       => aa(nat,$o,aa(option(nat),fun(nat,$o),aa(heap_ext(product_unit),fun(option(nat),fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o))),aa(product_prod(vEBT_VEBTi,nat),fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))),product_case_prod(vEBT_VEBTi,nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(option(nat),fun(nat,$o)))),P),Xaa),H),H2),R2),N) ) ) ).

% vebt_succi.raw_induct
tff(fact_363_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xb: nat,Deg2: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary2: vEBT_VEBT] :
      ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),Deg2)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2)),Xb) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_364_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg2: nat,Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),Deg2)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2)),Xb)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
          | ( Xb = Mi )
          | ( Xb = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_365_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Deg2)
     => ( ( Mi != Ma )
       => ( aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Summary2)) = vEBT_VEBT_high(Ma,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% summaxma
tff(fact_366_delt__out__of__range,axiom,
    ! [Xb: nat,Mi: nat,Ma: nat,Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( ( aa(nat,$o,ord_less(nat,Xb),Mi)
        | aa(nat,$o,ord_less(nat,Ma),Xb) )
     => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2) ) ) ) ).

% delt_out_of_range
tff(fact_367_valid__tree__deg__neq__0,axiom,
    ! [Tb: vEBT_VEBT] : ~ vEBT_invar_vebt(Tb,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_368_valid__0__not,axiom,
    ! [Tb: vEBT_VEBT] : ~ vEBT_invar_vebt(Tb,zero_zero(nat)) ).

% valid_0_not
tff(fact_369_deg__deg__n,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info2,Deg2,TreeList,Summary2),N)
     => ( Deg2 = N ) ) ).

% deg_deg_n
tff(fact_370_delete__pres__valid,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => vEBT_invar_vebt(vEBT_vebt_delete(Tb,Xb),N) ) ).

% delete_pres_valid
tff(fact_371_deg__not__0,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ).

% deg_not_0
tff(fact_372_maxbmo,axiom,
    ! [Tb: vEBT_VEBT,Xb: nat] :
      ( ( vEBT_vebt_maxt(Tb) = aa(nat,option(nat),some(nat),Xb) )
     => aa(nat,$o,vEBT_V8194947554948674370ptions(Tb),Xb) ) ).

% maxbmo
tff(fact_373_dele__bmo__cont__corr,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_delete(Tb,Xb)),Y)
      <=> ( ( Xb != Y )
          & aa(nat,$o,vEBT_V8194947554948674370ptions(Tb),Y) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_374_both__member__options__equiv__member,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Tb),Xb)
      <=> aa(nat,$o,vEBT_vebt_member(Tb),Xb) ) ) ).

% both_member_options_equiv_member
tff(fact_375_valid__member__both__member__options,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Tb),Xb)
       => aa(nat,$o,vEBT_vebt_member(Tb),Xb) ) ) ).

% valid_member_both_member_options
tff(fact_376_dele__member__cont__corr,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( aa(nat,$o,vEBT_vebt_member(vEBT_vebt_delete(Tb,Xb)),Y)
      <=> ( ( Xb != Y )
          & aa(nat,$o,vEBT_vebt_member(Tb),Y) ) ) ) ).

% dele_member_cont_corr
tff(fact_377_mint__member,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Maxi: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( vEBT_vebt_mint(Tb) = aa(nat,option(nat),some(nat),Maxi) )
       => aa(nat,$o,vEBT_vebt_member(Tb),Maxi) ) ) ).

% mint_member
tff(fact_378_maxt__member,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Maxi: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( vEBT_vebt_maxt(Tb) = aa(nat,option(nat),some(nat),Maxi) )
       => aa(nat,$o,vEBT_vebt_member(Tb),Maxi) ) ) ).

% maxt_member
tff(fact_379_mint__corr__help,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Mini: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( vEBT_vebt_mint(Tb) = aa(nat,option(nat),some(nat),Mini) )
       => ( aa(nat,$o,vEBT_vebt_member(Tb),Xb)
         => aa(nat,$o,ord_less_eq(nat,Mini),Xb) ) ) ) ).

% mint_corr_help
tff(fact_380_maxt__corr__help,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Maxi: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( vEBT_vebt_maxt(Tb) = aa(nat,option(nat),some(nat),Maxi) )
       => ( aa(nat,$o,vEBT_vebt_member(Tb),Xb)
         => aa(nat,$o,ord_less_eq(nat,Xb),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_381_split__part,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: fun(A,fun(B,$o)),X3: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,product_case_prod(A,B,$o,aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_fg($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(P)),Q)),X3)
    <=> ( (P)
        & aa(product_prod(A,B),$o,product_case_prod(A,B,$o,Q),X3) ) ) ).

% split_part
tff(fact_382_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xb: nat,N: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Tree),Xb)
     => ( vEBT_invar_vebt(Tree,N)
       => aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ) ).

% member_bound
tff(fact_383_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),N)
     => ( aa(nat,$o,ord_less_eq(nat,Ma),Xb)
       => ( vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = none(nat) ) ) ) ).

% geqmaxNone
tff(fact_384_del__single__cont,axiom,
    ! [Xb: nat,Mi: nat,Ma: nat,Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( ( ( Xb = Mi )
        & ( Xb = Ma ) )
     => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary2) ) ) ) ).

% del_single_cont
tff(fact_385_misiz,axiom,
    ! [Tb: vEBT_VEBT,N: nat,M: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( aa(nat,option(nat),some(nat),M) = vEBT_vebt_mint(Tb) )
       => aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ) ).

% misiz
tff(fact_386_helpyd,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( vEBT_vebt_succ(Tb,Xb) = aa(nat,option(nat),some(nat),Y) )
       => aa(nat,$o,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ) ).

% helpyd
tff(fact_387_helpypredd,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( vEBT_vebt_pred(Tb,Xb) = aa(nat,option(nat),some(nat),Y) )
       => aa(nat,$o,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ) ).

% helpypredd
tff(fact_388_valid__insert__both__member__options__pres,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
       => ( aa(nat,$o,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Tb),Xb)
           => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Tb,Y)),Xb) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_389_valid__insert__both__member__options__add,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
       => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Tb,Xb)),Xb) ) ) ).

% valid_insert_both_member_options_add
tff(fact_390_post__member__pre__member,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
       => ( aa(nat,$o,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
         => ( aa(nat,$o,vEBT_vebt_member(vEBT_vebt_insert(Tb,Xb)),Y)
           => ( aa(nat,$o,vEBT_vebt_member(Tb),Y)
              | ( Xb = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_391_i0__less,axiom,
    ! [N: extended_enat] :
      ( aa(extended_enat,$o,ord_less(extended_enat,zero_zero(extended_enat)),N)
    <=> ( N != zero_zero(extended_enat) ) ) ).

% i0_less
tff(fact_392_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),N)
     => ( aa(nat,$o,ord_less_eq(nat,Mi),Ma)
        & aa(nat,$o,ord_less(nat,Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)) ) ) ).

% mi_ma_2_deg
tff(fact_393_member__correct,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( aa(nat,$o,vEBT_vebt_member(Tb),Xb)
      <=> member(nat,Xb,vEBT_set_vebt(Tb)) ) ) ).

% member_correct
tff(fact_394_both__member__options__ding,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info2,Deg2,TreeList,Summary2),N)
     => ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2))
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(Info2,Deg2,TreeList,Summary2)),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_395_zero__one__enat__neq_I1_J,axiom,
    zero_zero(extended_enat) != one_one(extended_enat) ).

% zero_one_enat_neq(1)
tff(fact_396_not__iless0,axiom,
    ! [N: extended_enat] : ~ aa(extended_enat,$o,ord_less(extended_enat,N),zero_zero(extended_enat)) ).

% not_iless0
tff(fact_397_enat__less__induct,axiom,
    ! [P: fun(extended_enat,$o),N: extended_enat] :
      ( ! [N3: extended_enat] :
          ( ! [M2: extended_enat] :
              ( aa(extended_enat,$o,ord_less(extended_enat,M2),N3)
             => aa(extended_enat,$o,P,M2) )
         => aa(extended_enat,$o,P,N3) )
     => aa(extended_enat,$o,P,N) ) ).

% enat_less_induct
tff(fact_398_Collect__case__prod__mono,axiom,
    ! [B: $tType,A: $tType,A4: fun(A,fun(B,$o)),B6: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,ord_less_eq(fun(A,fun(B,$o)),A4),B6)
     => aa(set(product_prod(A,B)),$o,ord_less_eq(set(product_prod(A,B)),collect(product_prod(A,B),product_case_prod(A,B,$o,A4))),collect(product_prod(A,B),product_case_prod(A,B,$o,B6))) ) ).

% Collect_case_prod_mono
tff(fact_399_prod_Odisc__eq__case,axiom,
    ! [B: $tType,A: $tType,Prod: product_prod(A,B)] : aa(product_prod(A,B),$o,product_case_prod(A,B,$o,aTP_Lamp_fh(A,fun(B,$o))),Prod) ).

% prod.disc_eq_case
tff(fact_400_in__children__def,axiom,
    ! [N: nat,TreeList: list(vEBT_VEBT),Xb: nat] :
      ( vEBT_V5917875025757280293ildren(N,TreeList,Xb)
    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xb,N))),vEBT_VEBT_low(Xb,N)) ) ).

% in_children_def
tff(fact_401_maxt__corr,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( vEBT_vebt_maxt(Tb) = aa(nat,option(nat),some(nat),Xb) )
       => vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Tb),Xb) ) ) ).

% maxt_corr
tff(fact_402_maxt__sound,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Tb),Xb)
       => ( vEBT_vebt_maxt(Tb) = aa(nat,option(nat),some(nat),Xb) ) ) ) ).

% maxt_sound
tff(fact_403_mint__sound,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Tb),Xb)
       => ( vEBT_vebt_mint(Tb) = aa(nat,option(nat),some(nat),Xb) ) ) ) ).

% mint_sound
tff(fact_404_mint__corr,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( vEBT_vebt_mint(Tb) = aa(nat,option(nat),some(nat),Xb) )
       => vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Tb),Xb) ) ) ).

% mint_corr
tff(fact_405_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),N)
     => ( ( Mi != Ma )
       => ( aa(nat,$o,ord_less(nat,Mi),Ma)
          & ? [M3: nat] :
              ( ( aa(nat,option(nat),some(nat),M3) = vEBT_vebt_mint(Summary2) )
              & aa(nat,$o,ord_less(nat,M3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ).

% mintlistlength
tff(fact_406_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TreeList,Summary2),Xb) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
tff(fact_407_set__n__deg__not__0,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,M: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member(vEBT_VEBT,X2,set2(vEBT_VEBT,TreeList))
         => vEBT_invar_vebt(X2,N) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
       => aa(nat,$o,ord_less_eq(nat,one_one(nat)),N) ) ) ).

% set_n_deg_not_0
tff(fact_408_set__vebt_H__def,axiom,
    ! [Tb: vEBT_VEBT] : vEBT_VEBT_set_vebt(Tb) = collect(nat,vEBT_vebt_member(Tb)) ).

% set_vebt'_def
tff(fact_409_vebt__pred_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xb) = $ite(
        aa(nat,$o,ord_less(nat,Ma),Xb),
        aa(nat,option(nat),some(nat),Ma),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
              $let(
                minlow: option(nat),
                minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( minlow != none(nat) )
                  & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                  aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  $let(
                    pr: option(nat),
                    pr:= vEBT_vebt_pred(Summary2,h),
                    $ite(
                      pr = none(nat),
                      $ite(aa(nat,$o,ord_less(nat,Mi),Xb),aa(nat,option(nat),some(nat),Mi),none(nat)),
                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
              none(nat) ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_410_vebt__succ_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xb) = $ite(
        aa(nat,$o,ord_less(nat,Xb),Mi),
        aa(nat,option(nat),some(nat),Mi),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( maxlow != none(nat) )
                  & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                  aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  $let(
                    sc: option(nat),
                    sc:= vEBT_vebt_succ(Summary2,h),
                    $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
              none(nat) ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_411_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tree,aa(nat,nat,suc,aa(nat,nat,suc,N)))
     => ? [Info: option(product_prod(nat,nat)),TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] : Tree = vEBT_Node(Info,aa(nat,nat,suc,aa(nat,nat,suc,N)),TreeList2,S2) ) ).

% deg_SUcn_Node
tff(fact_412_set__vebt__set__vebt_H__valid,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( vEBT_set_vebt(Tb) = vEBT_VEBT_set_vebt(Tb) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_413_inthall,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),N: nat] :
      ( ! [X2: A] :
          ( member(A,X2,set2(A,Xs))
         => aa(A,$o,P,X2) )
     => ( aa(nat,$o,ord_less(nat,N),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),N)) ) ) ).

% inthall
tff(fact_414_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( aa(nat,nat,suc,X22) = aa(nat,nat,suc,Y22) )
    <=> ( X22 = Y22 ) ) ).

% nat.inject
tff(fact_415_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( aa(nat,nat,suc,Nat) = aa(nat,nat,suc,Nat2) )
    <=> ( Nat = Nat2 ) ) ).

% old.nat.inject
tff(fact_416_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Deg2)
     => ( ( Mi = Ma )
       => ( ! [X3: vEBT_VEBT] :
              ( member(vEBT_VEBT,X3,set2(vEBT_VEBT,TreeList))
             => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_13) )
          & ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_13) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_417_lessI,axiom,
    ! [N: nat] : aa(nat,$o,ord_less(nat,N),aa(nat,nat,suc,N)) ).

% lessI
tff(fact_418_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),N)
     => aa(nat,$o,ord_less(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) ) ).

% Suc_mono
tff(fact_419_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,N))
    <=> aa(nat,$o,ord_less(nat,M),N) ) ).

% Suc_less_eq
tff(fact_420_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,N)),aa(nat,nat,suc,M))
    <=> aa(nat,$o,ord_less_eq(nat,N),M) ) ).

% Suc_le_mono
tff(fact_421_diff__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,minus_minus(nat,M),N) ).

% diff_Suc_Suc
tff(fact_422_Suc__diff__diff,axiom,
    ! [M: nat,N: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),N)),aa(nat,nat,suc,K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),N)),K) ).

% Suc_diff_diff
tff(fact_423_diff__0__eq__0,axiom,
    ! [N: nat] : aa(nat,nat,minus_minus(nat,zero_zero(nat)),N) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_424_diff__self__eq__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),M) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_425_diff__diff__cancel,axiom,
    ! [I: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),N)
     => ( aa(nat,nat,minus_minus(nat,N),aa(nat,nat,minus_minus(nat,N),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_426_diff__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,one_one(A)),one_one(A)) = zero_zero(A) ) ) ).

% diff_numeral_special(9)
tff(fact_427_power__0__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(nat,nat,suc,N)) = zero_zero(A) ) ).

% power_0_Suc
tff(fact_428_power__Suc0__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% power_Suc0_right
tff(fact_429_less__Suc0,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,N),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( N = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_430_zero__less__Suc,axiom,
    ! [N: nat] : aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,suc,N)) ).

% zero_less_Suc
tff(fact_431_div__by__Suc__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,zero_zero(nat))) = M ).

% div_by_Suc_0
tff(fact_432_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,minus_minus(nat,N),M))
    <=> aa(nat,$o,ord_less(nat,M),N) ) ).

% zero_less_diff
tff(fact_433_power__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,suc,zero_zero(nat)) ).

% power_Suc_0
tff(fact_434_nat__power__eq__Suc__0__iff,axiom,
    ! [Xb: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xb),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = zero_zero(nat) )
        | ( Xb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% nat_power_eq_Suc_0_iff
tff(fact_435_diff__Suc__1,axiom,
    ! [N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,N)),one_one(nat)) = N ).

% diff_Suc_1
tff(fact_436_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),N) = zero_zero(nat) )
    <=> aa(nat,$o,ord_less_eq(nat,M),N) ) ).

% diff_is_0_eq
tff(fact_437_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => ( aa(nat,nat,minus_minus(nat,M),N) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_438_Suc__pred,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).

% Suc_pred
tff(fact_439_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% Suc_1
tff(fact_440_div2__Suc__Suc,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% div2_Suc_Suc
tff(fact_441_Suc__diff__1,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,N),one_one(nat))) = N ) ) ).

% Suc_diff_1
tff(fact_442_pred__member,axiom,
    ! [Tb: vEBT_VEBT,Xb: nat,Y: nat] :
      ( vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Tb),Xb,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(Tb),Y)
        & aa(nat,$o,ord_less(nat,Y),Xb)
        & ! [Z3: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(Tb),Z3)
              & aa(nat,$o,ord_less(nat,Z3),Xb) )
           => aa(nat,$o,ord_less_eq(nat,Z3),Y) ) ) ) ).

% pred_member
tff(fact_443_succ__member,axiom,
    ! [Tb: vEBT_VEBT,Xb: nat,Y: nat] :
      ( vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Tb),Xb,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(Tb),Y)
        & aa(nat,$o,ord_less(nat,Xb),Y)
        & ! [Z3: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(Tb),Z3)
              & aa(nat,$o,ord_less(nat,Xb),Z3) )
           => aa(nat,$o,ord_less_eq(nat,Y),Z3) ) ) ) ).

% succ_member
tff(fact_444_succ__corr,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Sx: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( vEBT_vebt_succ(Tb,Xb) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Tb),Xb,Sx) ) ) ).

% succ_corr
tff(fact_445_pred__corr,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Px: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( vEBT_vebt_pred(Tb,Xb) = aa(nat,option(nat),some(nat),Px) )
      <=> vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Tb),Xb,Px) ) ) ).

% pred_corr
tff(fact_446_succ__correct,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Sx: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( vEBT_vebt_succ(Tb,Xb) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_set_vebt(Tb),Xb,Sx) ) ) ).

% succ_correct
tff(fact_447_pred__correct,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat,Sx: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( vEBT_vebt_pred(Tb,Xb) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_pred_in_set(vEBT_set_vebt(Tb),Xb,Sx) ) ) ).

% pred_correct
tff(fact_448_Suc__inject,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa(nat,nat,suc,Xb) = aa(nat,nat,suc,Y) )
     => ( Xb = Y ) ) ).

% Suc_inject
tff(fact_449_n__not__Suc__n,axiom,
    ! [N: nat] : N != aa(nat,nat,suc,N) ).

% n_not_Suc_n
tff(fact_450_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I),J)),K) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I),K)),J) ).

% diff_commute
tff(fact_451_zero__induct__lemma,axiom,
    ! [P: fun(nat,$o),K: nat,I: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [N3: nat] :
            ( aa(nat,$o,P,aa(nat,nat,suc,N3))
           => aa(nat,$o,P,N3) )
       => aa(nat,$o,P,aa(nat,nat,minus_minus(nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_452_diff__less__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,M),N)),aa(nat,nat,suc,M)) ).

% diff_less_Suc
tff(fact_453_Suc__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,N),M)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,N))) = aa(nat,nat,minus_minus(nat,M),N) ) ) ).

% Suc_diff_Suc
tff(fact_454_Suc__diff__le,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,N),M)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,M),N)) ) ) ).

% Suc_diff_le
tff(fact_455_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,N)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),one_one(nat))),N) ).

% diff_Suc_eq_diff_pred
tff(fact_456_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,I))),N) ) ).

% diff_Suc_less
tff(fact_457_diff__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,A3),B3)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) ) ).

% diff_divide_distrib
tff(fact_458_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),zero_zero(nat)) = M ).

% minus_nat.diff_0
tff(fact_459_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),N) = zero_zero(nat) )
     => ( ( aa(nat,nat,minus_minus(nat,N),M) = zero_zero(nat) )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
tff(fact_460_diff__less__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( aa(nat,$o,ord_less(nat,M),N)
     => ( aa(nat,$o,ord_less(nat,M),L)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,L),N)),aa(nat,nat,minus_minus(nat,L),M)) ) ) ).

% diff_less_mono2
tff(fact_461_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,J),K)
     => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,J),N)),K) ) ).

% less_imp_diff_less
tff(fact_462_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),M)
     => ( aa(nat,$o,ord_less_eq(nat,K),N)
       => ( ( aa(nat,nat,minus_minus(nat,M),K) = aa(nat,nat,minus_minus(nat,N),K) )
        <=> ( M = N ) ) ) ) ).

% eq_diff_iff
tff(fact_463_le__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),M)
     => ( aa(nat,$o,ord_less_eq(nat,K),N)
       => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,N),K))
        <=> aa(nat,$o,ord_less_eq(nat,M),N) ) ) ) ).

% le_diff_iff
tff(fact_464_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),M)
     => ( aa(nat,$o,ord_less_eq(nat,K),N)
       => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,N),K)) = aa(nat,nat,minus_minus(nat,M),N) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_465_diff__le__mono,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),L)),aa(nat,nat,minus_minus(nat,N),L)) ) ).

% diff_le_mono
tff(fact_466_diff__le__self,axiom,
    ! [M: nat,N: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),N)),M) ).

% diff_le_self
tff(fact_467_le__diff__iff_H,axiom,
    ! [A3: nat,C2: nat,B3: nat] :
      ( aa(nat,$o,ord_less_eq(nat,A3),C2)
     => ( aa(nat,$o,ord_less_eq(nat,B3),C2)
       => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,C2),A3)),aa(nat,nat,minus_minus(nat,C2),B3))
        <=> aa(nat,$o,ord_less_eq(nat,B3),A3) ) ) ) ).

% le_diff_iff'
tff(fact_468_diff__le__mono2,axiom,
    ! [M: nat,N: nat,L: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,L),N)),aa(nat,nat,minus_minus(nat,L),M)) ) ).

% diff_le_mono2
tff(fact_469_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] : zero_zero(nat) != aa(nat,nat,suc,X22) ).

% nat.distinct(1)
tff(fact_470_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_471_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_472_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat = aa(nat,nat,suc,X22) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_473_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_474_nat__induct,axiom,
    ! [P: fun(nat,$o),N: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N3: nat] :
            ( aa(nat,$o,P,N3)
           => aa(nat,$o,P,aa(nat,nat,suc,N3)) )
       => aa(nat,$o,P,N) ) ) ).

% nat_induct
tff(fact_475_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),M: nat,N: nat] :
      ( ! [X2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,X2),zero_zero(nat))
     => ( ! [Y2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,zero_zero(nat)),aa(nat,nat,suc,Y2))
       => ( ! [X2: nat,Y2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,X2),Y2)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,aa(nat,nat,suc,X2)),aa(nat,nat,suc,Y2)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,M),N) ) ) ) ).

% diff_induct
tff(fact_476_zero__induct,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [N3: nat] :
            ( aa(nat,$o,P,aa(nat,nat,suc,N3))
           => aa(nat,$o,P,N3) )
       => aa(nat,$o,P,zero_zero(nat)) ) ) ).

% zero_induct
tff(fact_477_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_478_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_479_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_480_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ? [M3: nat] : N = aa(nat,nat,suc,M3) ) ).

% not0_implies_Suc
tff(fact_481_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,I),K)
     => ( ( K != aa(nat,nat,suc,I) )
       => ~ ! [J2: nat] :
              ( aa(nat,$o,ord_less(nat,I),J2)
             => ( K != aa(nat,nat,suc,J2) ) ) ) ) ).

% Nat.lessE
tff(fact_482_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,M)),N)
     => aa(nat,$o,ord_less(nat,M),N) ) ).

% Suc_lessD
tff(fact_483_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,I)),K)
     => ~ ! [J2: nat] :
            ( aa(nat,$o,ord_less(nat,I),J2)
           => ( K != aa(nat,nat,suc,J2) ) ) ) ).

% Suc_lessE
tff(fact_484_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),N)
     => ( ( aa(nat,nat,suc,M) != N )
       => aa(nat,$o,ord_less(nat,aa(nat,nat,suc,M)),N) ) ) ).

% Suc_lessI
tff(fact_485_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,N))
     => ( ~ aa(nat,$o,ord_less(nat,M),N)
       => ( M = N ) ) ) ).

% less_SucE
tff(fact_486_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),N)
     => aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,N)) ) ).

% less_SucI
tff(fact_487_Ex__less__Suc,axiom,
    ! [N: nat,P: fun(nat,$o)] :
      ( ? [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,suc,N))
          & aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,N)
        | ? [I4: nat] :
            ( aa(nat,$o,ord_less(nat,I4),N)
            & aa(nat,$o,P,I4) ) ) ) ).

% Ex_less_Suc
tff(fact_488_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,N))
    <=> ( aa(nat,$o,ord_less(nat,M),N)
        | ( M = N ) ) ) ).

% less_Suc_eq
tff(fact_489_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ aa(nat,$o,ord_less(nat,M),N)
    <=> aa(nat,$o,ord_less(nat,N),aa(nat,nat,suc,M)) ) ).

% not_less_eq
tff(fact_490_Nat_OAll__less__Suc,axiom,
    ! [N: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,suc,N))
         => aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,N)
        & ! [I4: nat] :
            ( aa(nat,$o,ord_less(nat,I4),N)
           => aa(nat,$o,P,I4) ) ) ) ).

% Nat.All_less_Suc
tff(fact_491_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,N)),M)
    <=> ? [M4: nat] :
          ( ( M = aa(nat,nat,suc,M4) )
          & aa(nat,$o,ord_less(nat,N),M4) ) ) ).

% Suc_less_eq2
tff(fact_492_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ aa(nat,$o,ord_less(nat,N),M)
     => ( aa(nat,$o,ord_less(nat,N),aa(nat,nat,suc,M))
       => ( M = N ) ) ) ).

% less_antisym
tff(fact_493_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,N))
     => aa(nat,$o,ord_less(nat,M),N) ) ).

% Suc_less_SucD
tff(fact_494_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( aa(nat,$o,ord_less(nat,J),K)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,suc,I)),K) ) ) ).

% less_trans_Suc
tff(fact_495_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( ! [I2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,I2),aa(nat,nat,suc,I2))
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( aa(nat,$o,ord_less(nat,I2),J2)
             => ( aa(nat,$o,ord_less(nat,J2),K2)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),P,I2),J2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),P,J2),K2)
                   => aa(nat,$o,aa(nat,fun(nat,$o),P,I2),K2) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_496_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( ! [I2: nat] :
            ( ( J = aa(nat,nat,suc,I2) )
           => aa(nat,$o,P,I2) )
       => ( ! [I2: nat] :
              ( aa(nat,$o,ord_less(nat,I2),J)
             => ( aa(nat,$o,P,aa(nat,nat,suc,I2))
               => aa(nat,$o,P,I2) ) )
         => aa(nat,$o,P,I) ) ) ) ).

% strict_inc_induct
tff(fact_497_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ aa(nat,$o,ord_less(nat,N),M)
     => ( aa(nat,$o,ord_less(nat,N),aa(nat,nat,suc,M))
      <=> ( N = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_498_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,M)),N)
     => aa(nat,$o,ord_less_eq(nat,M),N) ) ).

% Suc_leD
tff(fact_499_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,N))
     => ( ~ aa(nat,$o,ord_less_eq(nat,M),N)
       => ( M = aa(nat,nat,suc,N) ) ) ) ).

% le_SucE
tff(fact_500_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,N)) ) ).

% le_SucI
tff(fact_501_Suc__le__D,axiom,
    ! [N: nat,M5: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,N)),M5)
     => ? [M3: nat] : M5 = aa(nat,nat,suc,M3) ) ).

% Suc_le_D
tff(fact_502_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,N))
    <=> ( aa(nat,$o,ord_less_eq(nat,M),N)
        | ( M = aa(nat,nat,suc,N) ) ) ) ).

% le_Suc_eq
tff(fact_503_Suc__n__not__le__n,axiom,
    ! [N: nat] : ~ aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,N)),N) ).

% Suc_n_not_le_n
tff(fact_504_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ~ aa(nat,$o,ord_less_eq(nat,M),N)
    <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,N)),M) ) ).

% not_less_eq_eq
tff(fact_505_full__nat__induct,axiom,
    ! [P: fun(nat,$o),N: nat] :
      ( ! [N3: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,M2)),N3)
             => aa(nat,$o,P,M2) )
         => aa(nat,$o,P,N3) )
     => aa(nat,$o,P,N) ) ).

% full_nat_induct
tff(fact_506_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => ( aa(nat,$o,P,M)
       => ( ! [N3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,M),N3)
             => ( aa(nat,$o,P,N3)
               => aa(nat,$o,P,aa(nat,nat,suc,N3)) ) )
         => aa(nat,$o,P,N) ) ) ) ).

% nat_induct_at_least
tff(fact_507_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => ( ! [X2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,X2),X2)
       => ( ! [X2: nat,Y2: nat,Z2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R,X2),Y2)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R,Y2),Z2)
               => aa(nat,$o,aa(nat,fun(nat,$o),R,X2),Z2) ) )
         => ( ! [N3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,N3),aa(nat,nat,suc,N3))
           => aa(nat,$o,aa(nat,fun(nat,$o),R,M),N) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_508_Suc__pred_H,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( N = aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_509_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),N) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_510_div__if,axiom,
    ! [M: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = $ite(
        ( aa(nat,$o,ord_less(nat,M),N)
        | ( N = zero_zero(nat) ) ),
        zero_zero(nat),
        aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,M),N)),N)) ) ).

% div_if
tff(fact_511_div__geq,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( ~ aa(nat,$o,ord_less(nat,M),N)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,M),N)),N)) ) ) ) ).

% div_geq
tff(fact_512_le__div__geq,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,$o,ord_less_eq(nat,N),M)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,M),N)),N)) ) ) ) ).

% le_div_geq
tff(fact_513_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,M),N)),M) ) ) ).

% diff_less
tff(fact_514_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Fa: fun(nat,A),N: nat,N5: nat] :
          ( ! [N3: nat] : aa(A,$o,ord_less(A,aa(nat,A,Fa,N3)),aa(nat,A,Fa,aa(nat,nat,suc,N3)))
         => ( aa(nat,$o,ord_less(nat,N),N5)
           => aa(A,$o,ord_less(A,aa(nat,A,Fa,N)),aa(nat,A,Fa,N5)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_515_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Fa: fun(nat,A),N: nat,M: nat] :
          ( ! [N3: nat] : aa(A,$o,ord_less(A,aa(nat,A,Fa,N3)),aa(nat,A,Fa,aa(nat,nat,suc,N3)))
         => ( aa(A,$o,ord_less(A,aa(nat,A,Fa,N)),aa(nat,A,Fa,M))
          <=> aa(nat,$o,ord_less(nat,N),M) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_516_less__diff__iff,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),M)
     => ( aa(nat,$o,ord_less_eq(nat,K),N)
       => ( aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,N),K))
        <=> aa(nat,$o,ord_less(nat,M),N) ) ) ) ).

% less_diff_iff
tff(fact_517_diff__less__mono,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( aa(nat,$o,ord_less(nat,A3),B3)
     => ( aa(nat,$o,ord_less_eq(nat,C2),A3)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,A3),C2)),aa(nat,nat,minus_minus(nat,B3),C2)) ) ) ).

% diff_less_mono
tff(fact_518_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Fa: fun(nat,A),N: nat,N5: nat] :
          ( ! [N3: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,Fa,N3)),aa(nat,A,Fa,aa(nat,nat,suc,N3)))
         => ( aa(nat,$o,ord_less_eq(nat,N),N5)
           => aa(A,$o,ord_less_eq(A,aa(nat,A,Fa,N)),aa(nat,A,Fa,N5)) ) ) ) ).

% lift_Suc_mono_le
tff(fact_519_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Fa: fun(nat,A),N: nat,N5: nat] :
          ( ! [N3: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,Fa,aa(nat,nat,suc,N3))),aa(nat,A,Fa,N3))
         => ( aa(nat,$o,ord_less_eq(nat,N),N5)
           => aa(A,$o,ord_less_eq(A,aa(nat,A,Fa,N5)),aa(nat,A,Fa,N)) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_520_Ex__less__Suc2,axiom,
    ! [N: nat,P: fun(nat,$o)] :
      ( ? [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,suc,N))
          & aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        | ? [I4: nat] :
            ( aa(nat,$o,ord_less(nat,I4),N)
            & aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% Ex_less_Suc2
tff(fact_521_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
    <=> ? [M6: nat] : N = aa(nat,nat,suc,M6) ) ).

% gr0_conv_Suc
tff(fact_522_All__less__Suc2,axiom,
    ! [N: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,suc,N))
         => aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I4: nat] :
            ( aa(nat,$o,ord_less(nat,I4),N)
           => aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% All_less_Suc2
tff(fact_523_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ? [M3: nat] : N = aa(nat,nat,suc,M3) ) ).

% gr0_implies_Suc
tff(fact_524_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,N))
    <=> ( ( M = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( M = aa(nat,nat,suc,J3) )
            & aa(nat,$o,ord_less(nat,J3),N) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_525_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_526_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),N)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,M)),N) ) ).

% Suc_leI
tff(fact_527_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,M)),N)
    <=> aa(nat,$o,ord_less(nat,M),N) ) ).

% Suc_le_eq
tff(fact_528_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,P,I)
       => ( ! [N3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I),N3)
             => ( aa(nat,$o,ord_less(nat,N3),J)
               => ( aa(nat,$o,P,N3)
                 => aa(nat,$o,P,aa(nat,nat,suc,N3)) ) ) )
         => aa(nat,$o,P,J) ) ) ) ).

% dec_induct
tff(fact_529_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,P,J)
       => ( ! [N3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I),N3)
             => ( aa(nat,$o,ord_less(nat,N3),J)
               => ( aa(nat,$o,P,aa(nat,nat,suc,N3))
                 => aa(nat,$o,P,N3) ) ) )
         => aa(nat,$o,P,I) ) ) ) ).

% inc_induct
tff(fact_530_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,M)),N)
     => aa(nat,$o,ord_less(nat,M),N) ) ).

% Suc_le_lessD
tff(fact_531_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => ( aa(nat,$o,ord_less(nat,N),aa(nat,nat,suc,M))
      <=> ( N = M ) ) ) ).

% le_less_Suc_eq
tff(fact_532_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,N))
    <=> aa(nat,$o,ord_less_eq(nat,M),N) ) ).

% less_Suc_eq_le
tff(fact_533_less__eq__Suc__le,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,N),M)
    <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,N)),M) ) ).

% less_eq_Suc_le
tff(fact_534_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,N)) ) ).

% le_imp_less_Suc
tff(fact_535_Suc__div__le__mono,axiom,
    ! [M: nat,N: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,M)),N)) ).

% Suc_div_le_mono
tff(fact_536_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),K),M)),K) = aa(nat,int,aa(int,fun(nat,int),power_power(int),K),aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% int_power_div_base
tff(fact_537_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B3: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),aa(nat,nat,suc,N)) )
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
             => ( A3 = B3 ) ) ) ) ) ).

% power_inject_base
tff(fact_538_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),aa(nat,nat,suc,N)))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% power_le_imp_le_base
tff(fact_539_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A3)
         => aa(A,$o,ord_less(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))) ) ) ).

% power_gt1
tff(fact_540_numeral__1__eq__Suc__0,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = aa(nat,nat,suc,zero_zero(nat)) ).

% numeral_1_eq_Suc_0
tff(fact_541_ex__least__nat__less,axiom,
    ! [P: fun(nat,$o),N: nat] :
      ( aa(nat,$o,P,N)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K2: nat] :
            ( aa(nat,$o,ord_less(nat,K2),N)
            & ! [I3: nat] :
                ( aa(nat,$o,ord_less_eq(nat,I3),K2)
               => ~ aa(nat,$o,P,I3) )
            & aa(nat,$o,P,aa(nat,nat,suc,K2)) ) ) ) ).

% ex_least_nat_less
tff(fact_542_nat__induct__non__zero,axiom,
    ! [N: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N3: nat] :
              ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N3)
             => ( aa(nat,$o,P,N3)
               => aa(nat,$o,P,aa(nat,nat,suc,N3)) ) )
         => aa(nat,$o,P,N) ) ) ) ).

% nat_induct_non_zero
tff(fact_543_power__gt__expt,axiom,
    ! [N: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),N)
     => aa(nat,$o,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),K)) ) ).

% power_gt_expt
tff(fact_544_nat__one__le__power,axiom,
    ! [I: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),I)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N)) ) ).

% nat_one_le_power
tff(fact_545_power2__commute,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,minus_minus(A,Xb),Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,minus_minus(A,Y),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_commute
tff(fact_546_vebt__succ_Osimps_I5_J,axiom,
    ! [V: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT,Vi: nat] : vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh),Vi) = none(nat) ).

% vebt_succ.simps(5)
tff(fact_547_vebt__pred_Osimps_I6_J,axiom,
    ! [V: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT,Vj: nat] : vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi),Vj) = none(nat) ).

% vebt_pred.simps(6)
tff(fact_548_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,N: nat,M: nat] :
          ( ( A3 != zero_zero(A) )
         => ( aa(nat,$o,ord_less_eq(nat,N),M)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,minus_minus(nat,M),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ) ).

% power_diff
tff(fact_549_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,A3),one_one(A))
           => aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),A3) ) ) ) ).

% power_Suc_le_self
tff(fact_550_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,A3),one_one(A))
           => aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N))),one_one(A)) ) ) ) ).

% power_Suc_less_one
tff(fact_551_option_Osize_I4_J,axiom,
    ! [A: $tType,X22: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X22)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_552_option_Osize_I3_J,axiom,
    ! [A: $tType] : aa(option(A),nat,size_size(option(A)),none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(3)
tff(fact_553_numeral__2__eq__2,axiom,
    aa(num,nat,numeral_numeral(nat),bit0(one2)) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% numeral_2_eq_2
tff(fact_554_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary2),Xb) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
tff(fact_555_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,M: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_556_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,M: nat,N: nat] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) = $ite(aa(nat,$o,ord_less_eq(nat,N),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,minus_minus(nat,M),N)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,minus_minus(nat,N),M)))) ) ) ) ).

% power_diff_power_eq
tff(fact_557_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),N)),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),N))) ) ).

% diff_le_diff_pow
tff(fact_558_less__2__cases,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,N),aa(num,nat,numeral_numeral(nat),bit0(one2)))
     => ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_559_less__2__cases__iff,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,N),aa(num,nat,numeral_numeral(nat),bit0(one2)))
    <=> ( ( N = zero_zero(nat) )
        | ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_560_power__sub,axiom,
    ! [N: nat,M: nat,A3: nat] :
      ( aa(nat,$o,ord_less_eq(nat,N),M)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),A3)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A3),aa(nat,nat,minus_minus(nat,M),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A3),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A3),N)) ) ) ) ).

% power_sub
tff(fact_561_power__minus__is__div,axiom,
    ! [B3: nat,A3: nat] :
      ( aa(nat,$o,ord_less_eq(nat,B3),A3)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A3)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)) ) ) ).

% power_minus_is_div
tff(fact_562_div__2__gt__zero,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),N)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% div_2_gt_zero
tff(fact_563_Suc__n__div__2__gt__zero,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_564_less__two__pow__divD,axiom,
    ! [Xb: nat,N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
     => ( aa(nat,$o,ord_less_eq(nat,M),N)
        & aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M))) ) ) ).

% less_two_pow_divD
tff(fact_565_less__two__pow__divI,axiom,
    ! [Xb: nat,N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M)))
     => ( aa(nat,$o,ord_less_eq(nat,M),N)
       => aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))) ) ) ).

% less_two_pow_divI
tff(fact_566_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Va: nat] : vEBT_vebt_succ(vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz),Va) = none(nat) ).

% vebt_succ.simps(3)
tff(fact_567_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list(vEBT_VEBT),Va: vEBT_VEBT,Vb: nat] : vEBT_vebt_pred(vEBT_Node(none(product_prod(nat,nat)),Uy,Uz,Va),Vb) = none(nat) ).

% vebt_pred.simps(4)
tff(fact_568_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I4_J,axiom,
    ! [Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Uu: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary2),Uu) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
tff(fact_569_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve: nat] : vEBT_vebt_succ(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc,Vd),Ve) = none(nat) ).

% vebt_succ.simps(4)
tff(fact_570_vebt__pred_Osimps_I5_J,axiom,
    ! [V: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT,Vf: nat] : vEBT_vebt_pred(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vd,Ve),Vf) = none(nat) ).

% vebt_pred.simps(5)
tff(fact_571_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2)),Xb)
    <=> $ite(
          Xb = Mi,
          $true,
          $ite(
            Xb = Ma,
            $true,
            $ite(
              aa(nat,$o,ord_less(nat,Xb),Mi),
              $false,
              $ite(
                aa(nat,$o,ord_less(nat,Ma),Xb),
                $false,
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $ite(aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_572_Suc__diff,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),M)
       => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,N),M)) = aa(nat,nat,minus_minus(nat,N),aa(nat,nat,minus_minus(nat,M),one_one(nat))) ) ) ) ).

% Suc_diff
tff(fact_573_not__Some__eq2,axiom,
    ! [B: $tType,A: $tType,V: option(product_prod(A,B))] :
      ( ! [X: A,Y3: B] : V != aa(product_prod(A,B),option(product_prod(A,B)),some(product_prod(A,B)),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Y3))
    <=> ( V = none(product_prod(A,B)) ) ) ).

% not_Some_eq2
tff(fact_574_vebt__pred_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(Xb,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv: $o] : Xb = vEBT_Leaf((Uu2),(Uv))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != none(nat) ) ) )
       => ( ! [A2: $o] :
              ( ? [Uw: $o] : Xb = vEBT_Leaf((A2),(Uw))
             => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) )
         => ( ! [A2: $o,B2: $o] :
                ( ( Xb = vEBT_Leaf((A2),(B2)) )
               => ( ? [Va2: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va2))
                 => ( Y != $ite(
                        (B2),
                        aa(nat,option(nat),some(nat),one_one(nat)),
                        $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)
               => ( Y != none(nat) ) )
             => ( ( ? [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2)
                 => ( Y != none(nat) ) )
               => ( ( ? [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)
                   => ( Y != none(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( Y != $ite(
                              aa(nat,$o,ord_less(nat,Ma2),Xaa),
                              aa(nat,option(nat),some(nat),Ma2),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $ite(
                                    aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                    $let(
                                      minlow: option(nat),
                                      minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                      $ite(
                                        ( ( minlow != none(nat) )
                                        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                        aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                        $let(
                                          pr: option(nat),
                                          pr:= vEBT_vebt_pred(Summary,h),
                                          $ite(
                                            pr = none(nat),
                                            $ite(aa(nat,$o,ord_less(nat,Mi2),Xaa),aa(nat,option(nat),some(nat),Mi2),none(nat)),
                                            aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                                    none(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
tff(fact_575_vebt__succ_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = Y )
     => ( ! [Uu2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((Uu2),(B2)) )
           => ( ( Xaa = zero_zero(nat) )
             => ( Y != $ite((B2),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : Xb = vEBT_Leaf((Uv),(Uw))
           => ( ? [N3: nat] : Xaa = aa(nat,nat,suc,N3)
             => ( Y != none(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != none(nat) ) )
           => ( ( ? [V3: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc2,Vd2)
               => ( Y != none(nat) ) )
             => ( ( ? [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)
                 => ( Y != none(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( Y != $ite(
                            aa(nat,$o,ord_less(nat,Xaa),Mi2),
                            aa(nat,option(nat),some(nat),Mi2),
                            $let(
                              l: nat,
                              l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                  $let(
                                    maxlow: option(nat),
                                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                    $ite(
                                      ( ( maxlow != none(nat) )
                                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                      aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                      $let(
                                        sc: option(nat),
                                        sc:= vEBT_vebt_succ(Summary,h),
                                        $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                                  none(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_576_zero__comp__diff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,minus_minus(A,A3),B3))
        <=> aa(A,$o,ord_less(A,B3),A3) ) ) ).

% zero_comp_diff_simps(2)
tff(fact_577_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,minus_minus(A,A3),B3))
        <=> aa(A,$o,ord_less(A,B3),A3) ) ) ).

% diff_gt_0_iff_gt
tff(fact_578_zero__comp__diff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,minus_minus(A,A3),B3))
        <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ).

% zero_comp_diff_simps(1)
tff(fact_579_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,minus_minus(A,A3),B3))
        <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ).

% diff_ge_0_iff_ge
tff(fact_580_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList,Summary2),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xb)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList,Summary2) ).

% vebt_insert.simps(4)
tff(fact_581_inrange,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => aa(set(nat),$o,ord_less_eq(set(nat),vEBT_VEBT_set_vebt(Tb)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),one_one(nat)))) ) ).

% inrange
tff(fact_582_Leaf__0__not,axiom,
    ! [A3: $o,B3: $o] : ~ vEBT_invar_vebt(vEBT_Leaf((A3),(B3)),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_583_deg__1__Leafy,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( N = one_one(nat) )
       => ? [A2: $o,B2: $o] : Tb = vEBT_Leaf((A2),(B2)) ) ) ).

% deg_1_Leafy
tff(fact_584_deg__1__Leaf,axiom,
    ! [Tb: vEBT_VEBT] :
      ( vEBT_invar_vebt(Tb,one_one(nat))
     => ? [A2: $o,B2: $o] : Tb = vEBT_Leaf((A2),(B2)) ) ).

% deg_1_Leaf
tff(fact_585_deg1Leaf,axiom,
    ! [Tb: vEBT_VEBT] :
      ( vEBT_invar_vebt(Tb,one_one(nat))
    <=> ? [A7: $o,B7: $o] : Tb = vEBT_Leaf((A7),(B7)) ) ).

% deg1Leaf
tff(fact_586_idiff__0__right,axiom,
    ! [N: extended_enat] : aa(extended_enat,extended_enat,minus_minus(extended_enat,N),zero_zero(extended_enat)) = N ).

% idiff_0_right
tff(fact_587_idiff__0,axiom,
    ! [N: extended_enat] : aa(extended_enat,extended_enat,minus_minus(extended_enat,zero_zero(extended_enat)),N) = zero_zero(extended_enat) ).

% idiff_0
tff(fact_588_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( aa(A,$o,ord_less_eq(A,N),zero_zero(A))
        <=> ( N = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_589_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ~ aa(A,$o,ord_less(A,zero_zero(A)),N)
        <=> ( N = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_590_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,A3),A3) = zero_zero(A) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_591_diff__zero,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,A3),zero_zero(A)) = A3 ) ).

% diff_zero
tff(fact_592_zero__diff,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,zero_zero(A)),A3) = zero_zero(A) ) ).

% zero_diff
tff(fact_593_diff__0__right,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,A3),zero_zero(A)) = A3 ) ).

% diff_0_right
tff(fact_594_diff__self,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,A3),A3) = zero_zero(A) ) ).

% diff_self
tff(fact_595_vebt__member_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Leaf((A3),(B3))),Xb)
    <=> $ite(
          Xb = zero_zero(nat),
          (A3),
          $ite(Xb = one_one(nat),(B3),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_596_vebt__insert_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,Xb: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A3),(B3)),Xb) = $ite(
        Xb = zero_zero(nat),
        vEBT_Leaf($true,(B3)),
        $ite(Xb = one_one(nat),vEBT_Leaf((A3),$true),vEBT_Leaf((A3),(B3))) ) ).

% vebt_insert.simps(1)
tff(fact_597_zero__reorient,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xb: A] :
          ( ( zero_zero(A) = Xb )
        <=> ( Xb = zero_zero(A) ) ) ) ).

% zero_reorient
tff(fact_598_ord__eq__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( ( A3 = B3 )
         => ( aa(A,$o,ord_less_eq(A,B3),C2)
           => ( ( C2 = D )
             => aa(A,$o,ord_less_eq(A,A3),D) ) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_599_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,N: nat] : vEBT_vebt_succ(vEBT_Leaf((Uv2),(Uw2)),aa(nat,nat,suc,N)) = none(nat) ).

% vebt_succ.simps(2)
tff(fact_600_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o] : vEBT_vebt_pred(vEBT_Leaf((Uu),(Uv2)),zero_zero(nat)) = none(nat) ).

% vebt_pred.simps(1)
tff(fact_601_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [Xb: A] :
          ( ( one_one(A) = Xb )
        <=> ( Xb = one_one(A) ) ) ) ).

% one_reorient
tff(fact_602_bex2I,axiom,
    ! [A: $tType,B: $tType,A3: A,B3: B,S3: set(product_prod(A,B)),P: fun(A,fun(B,$o))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),S3)
     => ( ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),S3)
         => aa(B,$o,aa(A,fun(B,$o),P,A3),B3) )
       => ? [A2: A,B2: B] :
            ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),S3)
            & aa(B,$o,aa(A,fun(B,$o),P,A2),B2) ) ) ) ).

% bex2I
tff(fact_603_pairself_Ocases,axiom,
    ! [B: $tType,A: $tType,Xb: product_prod(fun(A,B),product_prod(A,A))] :
      ~ ! [F4: fun(A,B),A2: A,B2: A] : Xb != aa(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A)),aa(fun(A,B),fun(product_prod(A,A),product_prod(fun(A,B),product_prod(A,A))),product_Pair(fun(A,B),product_prod(A,A)),F4),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A2),B2)) ).

% pairself.cases
tff(fact_604_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I3_J,axiom,
    ! [A3: $o,B3: $o,N: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Leaf((A3),(B3)),aa(nat,nat,suc,aa(nat,nat,suc,N))) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
tff(fact_605_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] : vEBT_V1232361888498592333_e_t_e(vEBT_Leaf((A3),(B3)),zero_zero(nat)) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(1)
tff(fact_606_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I2_J,axiom,
    ! [A3: $o,B3: $o] : vEBT_V1232361888498592333_e_t_e(vEBT_Leaf((A3),(B3)),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(2)
tff(fact_607_is__succ__in__set__def,axiom,
    ! [Xs: set(nat),Xb: nat,Y: nat] :
      ( vEBT_is_succ_in_set(Xs,Xb,Y)
    <=> ( member(nat,Y,Xs)
        & aa(nat,$o,ord_less(nat,Xb),Y)
        & ! [X: nat] :
            ( member(nat,X,Xs)
           => ( aa(nat,$o,ord_less(nat,Xb),X)
             => aa(nat,$o,ord_less_eq(nat,Y),X) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_608_is__pred__in__set__def,axiom,
    ! [Xs: set(nat),Xb: nat,Y: nat] :
      ( vEBT_is_pred_in_set(Xs,Xb,Y)
    <=> ( member(nat,Y,Xs)
        & aa(nat,$o,ord_less(nat,Y),Xb)
        & ! [X: nat] :
            ( member(nat,X,Xs)
           => ( aa(nat,$o,ord_less(nat,X),Xb)
             => aa(nat,$o,ord_less_eq(nat,X),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_609_vebt__pred_Osimps_I2_J,axiom,
    ! [A3: $o,Uw2: $o] :
      vEBT_vebt_pred(vEBT_Leaf((A3),(Uw2)),aa(nat,nat,suc,zero_zero(nat))) = $ite((A3),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ).

% vebt_pred.simps(2)
tff(fact_610_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu: $o,B3: $o] :
      vEBT_vebt_succ(vEBT_Leaf((Uu),(B3)),zero_zero(nat)) = $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ).

% vebt_succ.simps(1)
tff(fact_611_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A2: $o,B2: $o] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),zero_zero(nat))
     => ( ! [A2: $o,B2: $o] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A2: $o,B2: $o,N3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),aa(nat,nat,suc,aa(nat,nat,suc,N3)))
         => ( ! [Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,Uu2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary)),Uu2)
           => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary)),X2)
             => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary)),X2)
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),X2) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
tff(fact_612_vebt__pred_Osimps_I3_J,axiom,
    ! [A3: $o,B3: $o,Va: nat] :
      vEBT_vebt_pred(vEBT_Leaf((A3),(B3)),aa(nat,nat,suc,aa(nat,nat,suc,Va))) = $ite(
        (B3),
        aa(nat,option(nat),some(nat),one_one(nat)),
        $ite((A3),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ).

% vebt_pred.simps(3)
tff(fact_613_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb) ) ).

% zero_le
tff(fact_614_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),N)
        <=> ( N != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_615_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M: A,N: A] :
          ( aa(A,$o,ord_less(A,M),N)
         => ( N != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_616_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] : ~ aa(A,$o,ord_less(A,N),zero_zero(A)) ) ).

% not_less_zero
tff(fact_617_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [N: A] :
          ( ( N != zero_zero(A) )
         => aa(A,$o,ord_less(A,zero_zero(A)),N) ) ) ).

% gr_zeroI
tff(fact_618_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
        <=> ( aa(A,A,minus_minus(A,A3),B3) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_619_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( ( aa(A,A,minus_minus(A,A3),B3) = aa(A,A,minus_minus(A,C2),D) )
         => ( aa(A,$o,ord_less_eq(A,A3),B3)
          <=> aa(A,$o,ord_less_eq(A,C2),D) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_620_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,A3),C2)),aa(A,A,minus_minus(A,B3),C2)) ) ) ).

% diff_right_mono
tff(fact_621_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,C2),A3)),aa(A,A,minus_minus(A,C2),B3)) ) ) ).

% diff_left_mono
tff(fact_622_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,D: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,D),C2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,A3),C2)),aa(A,A,minus_minus(A,B3),D)) ) ) ) ).

% diff_mono
tff(fact_623_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A3),C2)),aa(A,A,minus_minus(A,B3),C2)) ) ) ).

% diff_strict_right_mono
tff(fact_624_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,C2),A3)),aa(A,A,minus_minus(A,C2),B3)) ) ) ).

% diff_strict_left_mono
tff(fact_625_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( ( aa(A,A,minus_minus(A,A3),B3) = aa(A,A,minus_minus(A,C2),D) )
         => ( aa(A,$o,ord_less(A,A3),B3)
          <=> aa(A,$o,ord_less(A,C2),D) ) ) ) ).

% diff_eq_diff_less
tff(fact_626_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,D: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,D),C2)
           => aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A3),C2)),aa(A,A,minus_minus(A,B3),D)) ) ) ) ).

% diff_strict_mono
tff(fact_627_exists__leI,axiom,
    ! [N: nat,P: fun(nat,$o)] :
      ( ( ! [N6: nat] :
            ( aa(nat,$o,ord_less(nat,N6),N)
           => ~ aa(nat,$o,P,N6) )
       => aa(nat,$o,P,N) )
     => ? [N7: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N7),N)
          & aa(nat,$o,P,N7) ) ) ).

% exists_leI
tff(fact_628_fstE,axiom,
    ! [B: $tType,A: $tType,Xb: product_prod(A,B),A3: A,B3: B,P: fun(A,$o)] :
      ( ( Xb = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3) )
     => ( aa(A,$o,P,aa(product_prod(A,B),A,product_fst(A,B),Xb))
       => aa(A,$o,P,A3) ) ) ).

% fstE
tff(fact_629_sndE,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(A,B),A3: A,B3: B,P: fun(B,$o)] :
      ( ( Xb = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3) )
     => ( aa(B,$o,P,aa(product_prod(A,B),B,product_snd(A,B),Xb))
       => aa(B,$o,P,B3) ) ) ).

% sndE
tff(fact_630_All__prod__contract,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] :
      ( ! [A7: A,X_12: B] : aa(B,$o,aa(A,fun(B,$o),P,A7),X_12)
    <=> ! [Z3: product_prod(A,B)] : aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(A,B),A,product_fst(A,B),Z3)),aa(product_prod(A,B),B,product_snd(A,B),Z3)) ) ).

% All_prod_contract
tff(fact_631_Ex__prod__contract,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] :
      ( ? [A7: A,X_12: B] : aa(B,$o,aa(A,fun(B,$o),P,A7),X_12)
    <=> ? [Z3: product_prod(A,B)] : aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(A,B),A,product_fst(A,B),Z3)),aa(product_prod(A,B),B,product_snd(A,B),Z3)) ) ).

% Ex_prod_contract
tff(fact_632_fn__fst__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,Fa: fun(A,C)] : aTP_Lamp_fi(fun(A,C),fun(product_prod(A,B),C),Fa) = product_case_prod(A,B,C,aTP_Lamp_fj(fun(A,C),fun(A,fun(B,C)),Fa)) ).

% fn_fst_conv
tff(fact_633_fn__snd__conv,axiom,
    ! [A: $tType,C: $tType,B: $tType,Fa: fun(B,C)] : aTP_Lamp_fk(fun(B,C),fun(product_prod(A,B),C),Fa) = product_case_prod(A,B,C,aTP_Lamp_fl(fun(B,C),fun(A,fun(B,C)),Fa)) ).

% fn_snd_conv
tff(fact_634_nested__case__prod__simp,axiom,
    ! [D2: $tType,A: $tType,C: $tType,B: $tType,Fa: fun(B,fun(C,fun(D2,A))),Xb: product_prod(B,C),Y: D2] : aa(D2,A,aa(product_prod(B,C),fun(D2,A),product_case_prod(B,C,fun(D2,A),Fa),Xb),Y) = aa(product_prod(B,C),A,product_case_prod(B,C,A,aa(D2,fun(B,fun(C,A)),aTP_Lamp_fm(fun(B,fun(C,fun(D2,A))),fun(D2,fun(B,fun(C,A))),Fa),Y)),Xb) ).

% nested_case_prod_simp
tff(fact_635_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,A3),B3)),zero_zero(A)) ) ) ).

% le_iff_diff_le_0
tff(fact_636_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
        <=> aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A3),B3)),zero_zero(A)) ) ) ).

% less_iff_diff_less_0
tff(fact_637_nat__compl__induct_H,axiom,
    ! [P: fun(nat,$o),N: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N3: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,ord_less_eq(nat,Nn),N3)
               => aa(nat,$o,P,Nn) )
           => aa(nat,$o,P,aa(nat,nat,suc,N3)) )
       => aa(nat,$o,P,N) ) ) ).

% nat_compl_induct'
tff(fact_638_nat__compl__induct,axiom,
    ! [P: fun(nat,$o),N: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N3: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,ord_less_eq(nat,Nn),N3)
               => aa(nat,$o,P,Nn) )
           => aa(nat,$o,P,aa(nat,nat,suc,N3)) )
       => aa(nat,$o,P,N) ) ) ).

% nat_compl_induct
tff(fact_639_nat__in__between__eq_I1_J,axiom,
    ! [A3: nat,B3: nat] :
      ( ( aa(nat,$o,ord_less(nat,A3),B3)
        & aa(nat,$o,ord_less_eq(nat,B3),aa(nat,nat,suc,A3)) )
    <=> ( B3 = aa(nat,nat,suc,A3) ) ) ).

% nat_in_between_eq(1)
tff(fact_640_nat__in__between__eq_I2_J,axiom,
    ! [A3: nat,B3: nat] :
      ( ( aa(nat,$o,ord_less_eq(nat,A3),B3)
        & aa(nat,$o,ord_less(nat,B3),aa(nat,nat,suc,A3)) )
    <=> ( B3 = A3 ) ) ).

% nat_in_between_eq(2)
tff(fact_641_Suc__to__right,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(nat,nat,suc,N) = M )
     => ( N = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,zero_zero(nat))) ) ) ).

% Suc_to_right
tff(fact_642_nat__geq__1__eq__neqz,axiom,
    ! [Xb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),Xb)
    <=> ( Xb != zero_zero(nat) ) ) ).

% nat_geq_1_eq_neqz
tff(fact_643_obtain__list__from__elements,axiom,
    ! [A: $tType,N: nat,P: fun(A,fun(nat,$o))] :
      ( ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),N)
         => ? [Li: A] : aa(nat,$o,aa(A,fun(nat,$o),P,Li),I2) )
     => ~ ! [L2: list(A)] :
            ( ( aa(list(A),nat,size_size(list(A)),L2) = N )
           => ~ ! [I3: nat] :
                  ( aa(nat,$o,ord_less(nat,I3),N)
                 => aa(nat,$o,aa(A,fun(nat,$o),P,aa(nat,A,nth(A,L2),I3)),I3) ) ) ) ).

% obtain_list_from_elements
tff(fact_644_vebt__insert_Osimps_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info2,zero_zero(nat),Ts,S),Xb) = vEBT_Node(Info2,zero_zero(nat),Ts,S) ).

% vebt_insert.simps(2)
tff(fact_645_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Xb: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv2,Uw2)),Xb) ).

% vebt_member.simps(2)
tff(fact_646_le__some__optE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M: A,Xb: option(A)] :
          ( aa(option(A),$o,ord_less_eq(option(A),aa(A,option(A),some(A),M)),Xb)
         => ~ ! [M7: A] :
                ( ( Xb = aa(A,option(A),some(A),M7) )
               => ~ aa(A,$o,ord_less_eq(A,M),M7) ) ) ) ).

% le_some_optE
tff(fact_647_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A2),
                  $ite(Xaa = one_one(nat),(B2),$false) ) )
       => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)
             => ~ $ite(
                    Xaa = Mi2,
                    $true,
                    $ite(
                      Xaa = Ma2,
                      $true,
                      $ite(
                        aa(nat,$o,ord_less(nat,Xaa),Mi2),
                        $false,
                        $ite(
                          aa(nat,$o,ord_less(nat,Ma2),Xaa),
                          $false,
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_648_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A2),
                    $ite(Xaa = one_one(nat),(B2),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)
           => (Y) )
         => ( ( ? [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)
             => (Y) )
           => ( ( ? [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => (Y) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi2,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa(nat,$o,ord_less(nat,Xaa),Mi2),
                                $false,
                                $ite(
                                  aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_649_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A2),
                $ite(Xaa = one_one(nat),(B2),$false) ) )
       => ( ! [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)
         => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)
           => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)
                   => $ite(
                        Xaa = Mi2,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa(nat,$o,ord_less(nat,Xaa),Mi2),
                            $false,
                            $ite(
                              aa(nat,$o,ord_less(nat,Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_650_all__set__conv__nth,axiom,
    ! [A: $tType,L: list(A),P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,set2(A,L))
         => aa(A,$o,P,X) )
    <=> ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(list(A),nat,size_size(list(A)),L))
         => aa(A,$o,P,aa(nat,A,nth(A,L),I4)) ) ) ).

% all_set_conv_nth
tff(fact_651_vebt__insert_Osimps_I3_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S),Xb) = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S) ).

% vebt_insert.simps(3)
tff(fact_652_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Xb: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz)),Xb) ).

% vebt_member.simps(3)
tff(fact_653_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( aa(nat,$o,ord_less_eq(nat,K),M)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,K),aa(nat,nat,suc,zero_zero(nat)))),M) ) ) ).

% nz_le_conv_less
tff(fact_654_Suc__n__minus__m__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => ( aa(nat,$o,ord_less(nat,one_one(nat)),M)
       => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,N),M)) = aa(nat,nat,minus_minus(nat,N),aa(nat,nat,minus_minus(nat,M),one_one(nat))) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_655_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Xb: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc)),Xb) ).

% vebt_member.simps(4)
tff(fact_656_vebt__maxt_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ( Y != $ite(
                  (B2),
                  aa(nat,option(nat),some(nat),one_one(nat)),
                  $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)
           => ( Y != none(nat) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Ma2) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_657_vebt__mint_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ( Y != $ite(
                  (A2),
                  aa(nat,option(nat),some(nat),zero_zero(nat)),
                  $ite((B2),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)
           => ( Y != none(nat) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Mi2) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_658_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less_eq(int,W),aa(int,int,minus_minus(int,Z),one_one(int)))
    <=> aa(int,$o,ord_less(int,W),Z) ) ).

% zle_diff1_eq
tff(fact_659_VEBT_Oinject_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,Y11: option(product_prod(nat,nat)),Y12: nat,Y13: list(vEBT_VEBT),Y14: vEBT_VEBT] :
      ( ( vEBT_Node(X11,X12,X13,X14) = vEBT_Node(Y11,Y12,Y13,Y14) )
    <=> ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_660_minus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,minus_minus(int,K),zero_zero(int)) = K ).

% minus_int_code(1)
tff(fact_661_VEBT_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),F2: fun($o,fun($o,B)),VEBT: vEBT_VEBT] : aa(B,A,H,vEBT_case_VEBT(B,F1,F2,VEBT)) = vEBT_case_VEBT(A,aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),aTP_Lamp_fn(fun(B,A),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A))))),H),F1),aa(fun($o,fun($o,B)),fun($o,fun($o,A)),aTP_Lamp_fo(fun(B,A),fun(fun($o,fun($o,B)),fun($o,fun($o,A))),H),F2),VEBT) ).

% VEBT.case_distrib
tff(fact_662_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))] :
      ( ! [Uu2: fun(A,fun(A,$o)),Uv: option(A)] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A))),aa(fun(A,fun(A,$o)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(option(A),option(A))),Uu2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Uv))
     => ( ! [Uw: fun(A,fun(A,$o)),V3: A] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A))),aa(fun(A,fun(A,$o)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(option(A),option(A))),Uw),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V3)),none(A)))
       => ~ ! [F4: fun(A,fun(A,$o)),X2: A,Y2: A] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A))),aa(fun(A,fun(A,$o)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$o)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(option(A),option(A))),F4),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),X2)),aa(A,option(A),some(A),Y2))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_663_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))] :
      ( ! [Uu2: fun(A,fun(A,A)),Uv: option(A)] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),Uu2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Uv))
     => ( ! [Uw: fun(A,fun(A,A)),V3: A] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),Uw),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),V3)),none(A)))
       => ~ ! [F4: fun(A,fun(A,A)),A2: A,B2: A] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),F4),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),aa(A,option(A),some(A),A2)),aa(A,option(A),some(A),B2))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_664_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Leaf((X21),(X222))) = zero_zero(nat) ).

% VEBT.size(4)
tff(fact_665_less__eq__int__code_I1_J,axiom,
    aa(int,$o,ord_less_eq(int,zero_zero(int)),zero_zero(int)) ).

% less_eq_int_code(1)
tff(fact_666_less__int__code_I1_J,axiom,
    ~ aa(int,$o,ord_less(int,zero_zero(int)),zero_zero(int)) ).

% less_int_code(1)
tff(fact_667_int__le__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,ord_less_eq(int,I),K)
     => ( aa(int,$o,P,K)
       => ( ! [I2: int] :
              ( aa(int,$o,ord_less_eq(int,I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_le_induct
tff(fact_668_int__less__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,I),K)
     => ( aa(int,$o,P,aa(int,int,minus_minus(int,K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,ord_less(int,I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_less_induct
tff(fact_669_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv: $o,D3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv))),D3)
     => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,Deg3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg,TreeList2,Summary)),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_670_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option(product_prod(nat,nat)),X122: nat,X132: list(vEBT_VEBT),X142: vEBT_VEBT] : Y != vEBT_Node(X112,X122,X132,X142)
     => ~ ! [X212: $o,X223: $o] : Y != vEBT_Leaf((X212),(X223)) ) ).

% VEBT.exhaust
tff(fact_671_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT,X21: $o,X222: $o] : vEBT_Node(X11,X12,X13,X14) != vEBT_Leaf((X21),(X222)) ).

% VEBT.distinct(1)
tff(fact_672_VEBT_Odisc_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : vEBT_is_Node(vEBT_Node(X11,X12,X13,X14)) ).

% VEBT.disc(1)
tff(fact_673_VEBT_OdiscI_I1_J,axiom,
    ! [VEBT: vEBT_VEBT,X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] :
      ( ( VEBT = vEBT_Node(X11,X12,X13,X14) )
     => vEBT_is_Node(VEBT) ) ).

% VEBT.discI(1)
tff(fact_674_is__Node__def,axiom,
    ! [VEBT: vEBT_VEBT] :
      ( vEBT_is_Node(VEBT)
    <=> ? [X113: option(product_prod(nat,nat)),X123: nat,X133: list(vEBT_VEBT),X143: vEBT_VEBT] : VEBT = vEBT_Node(X113,X123,X133,X143) ) ).

% is_Node_def
tff(fact_675_VEBT_Osimps_I5_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),F2: fun($o,fun($o,A)),X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : vEBT_case_VEBT(A,F1,F2,vEBT_Node(X11,X12,X13,X14)) = aa(vEBT_VEBT,A,aa(list(vEBT_VEBT),fun(vEBT_VEBT,A),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A))),F1,X11),X12),X13),X14) ).

% VEBT.simps(5)
tff(fact_676_VEBT_Odisc_I2_J,axiom,
    ! [X21: $o,X222: $o] : ~ vEBT_is_Node(vEBT_Leaf((X21),(X222))) ).

% VEBT.disc(2)
tff(fact_677_VEBT_Osimps_I6_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),F2: fun($o,fun($o,A)),X21: $o,X222: $o] : vEBT_case_VEBT(A,F1,F2,vEBT_Leaf((X21),(X222))) = aa($o,A,aa($o,fun($o,A),F2,(X21)),(X222)) ).

% VEBT.simps(6)
tff(fact_678_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [A: $tType,Fa: fun(A,fun(A,A)),A3: A,B3: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Fa),aa(A,option(A),some(A),A3)),aa(A,option(A),some(A),B3)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),Fa,A3),B3)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_679_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uv2: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Uu),none(A)),Uv2) = none(A) ).

% VEBT_internal.option_shift.simps(1)
tff(fact_680_set__vebt__def,axiom,
    ! [Tb: vEBT_VEBT] : vEBT_set_vebt(Tb) = collect(nat,vEBT_V8194947554948674370ptions(Tb)) ).

% set_vebt_def
tff(fact_681_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [A: $tType,Uw2: fun(A,fun(A,A)),V: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Uw2),aa(A,option(A),some(A),V)),none(A)) = none(A) ).

% VEBT_internal.option_shift.simps(2)
tff(fact_682_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,A)),Xaa: option(A),Xba: option(A),Y: option(A)] :
      ( ( aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Xb),Xaa),Xba) = Y )
     => ( ( ( Xaa = none(A) )
         => ( Y != none(A) ) )
       => ( ( ? [V3: A] : Xaa = aa(A,option(A),some(A),V3)
           => ( ( Xba = none(A) )
             => ( Y != none(A) ) ) )
         => ~ ! [A2: A] :
                ( ( Xaa = aa(A,option(A),some(A),A2) )
               => ! [B2: A] :
                    ( ( Xba = aa(A,option(A),some(A),B2) )
                   => ( Y != aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),Xb,A2),B2)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_683_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A2: $o,B2: $o,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),X2)
     => ( ! [Uu2: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Ux2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv,Uw)),Ux2)
       => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2)),X2) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_684_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less_eq(int,one_one(int)),Z)
    <=> aa(int,$o,ord_less(int,zero_zero(int)),Z) ) ).

% int_one_le_iff_zero_less
tff(fact_685_invar__vebt_Ointros_I1_J,axiom,
    ! [A3: $o,B3: $o] : vEBT_invar_vebt(vEBT_Leaf((A3),(B3)),aa(nat,nat,suc,zero_zero(nat))) ).

% invar_vebt.intros(1)
tff(fact_686_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ( Xb != vEBT_Leaf($false,$false) )
     => ( ! [Uv: $o] : Xb != vEBT_Leaf($true,(Uv))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uw,Ux2,Uy2)
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_687_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv2,Uw2)) = none(nat) ).

% vebt_mint.simps(2)
tff(fact_688_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv2,Uw2)) = none(nat) ).

% vebt_maxt.simps(2)
tff(fact_689_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A2: $o,B2: $o] : Xb != vEBT_Leaf((A2),(B2))
     => ( ! [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_690_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux,Uy,Uz)) = aa(nat,option(nat),some(nat),Mi) ).

% vebt_mint.simps(3)
tff(fact_691_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux,Uy,Uz)) = aa(nat,option(nat),some(nat),Ma) ).

% vebt_maxt.simps(3)
tff(fact_692_vebt__mint_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      vEBT_vebt_mint(vEBT_Leaf((A3),(B3))) = $ite(
        (A3),
        aa(nat,option(nat),some(nat),zero_zero(nat)),
        $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ).

% vebt_mint.simps(1)
tff(fact_693_vebt__maxt_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      vEBT_vebt_maxt(vEBT_Leaf((A3),(B3))) = $ite(
        (B3),
        aa(nat,option(nat),some(nat),one_one(nat)),
        $ite((A3),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ).

% vebt_maxt.simps(1)
tff(fact_694_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv: $o,Uw: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv))),Uw)
     => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT,Uz2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Uz2)
       => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),X2)
         => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc2)),X2)
           => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)),X2) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_695_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv: $o] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv))),zero_zero(nat))
     => ( ! [A2: $o,Uw: $o] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(Uw))),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A2: $o,B2: $o,Va2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))
         => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT,Vb2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Vb2)
           => ( ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT,Vf2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2)),Vf2)
             => ( ! [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT,Vj2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Vj2)
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),X2) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
tff(fact_696_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,B2: $o] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(B2))),zero_zero(nat))
     => ( ! [Uv: $o,Uw: $o,N3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv),(Uw))),aa(nat,nat,suc,N3))
       => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,Va3: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Va3)
         => ( ! [V3: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT,Ve2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc2,Vd2)),Ve2)
           => ( ! [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT,Vi2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Vi2)
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),X2) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_697_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A2: $o,B2: $o,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),X2)
     => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info,zero_zero(nat),Ts2,S2)),X2)
       => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),X2)
         => ( ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary)),X2)
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),X2) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_698_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A2: $o,B2: $o,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),X2)
     => ( ! [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)),X2)
       => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),X2)
         => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),X2)
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT,X2: nat] : Xb != aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),X2) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_699_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or1337092689740270186AtMost(A,A3,B3)),set_or1337092689740270186AtMost(A,C2,D))
        <=> ( ~ aa(A,$o,ord_less_eq(A,A3),B3)
            | ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less_eq(A,B3),D) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_700_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list(vEBT_VEBT),Sm: vEBT_VEBT,Xb: nat] : vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) ).

% vebt_delete.simps(6)
tff(fact_701_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
     => ( ( ? [A2: $o,B2: $o] : A1 = vEBT_Leaf((A2),(B2))
         => ( A22 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList2: list(vEBT_VEBT),N3: nat,Summary: vEBT_VEBT,M3: nat,Deg: nat] :
              ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary) )
             => ( ( A22 = Deg )
               => ( ! [X3: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X3,set2(vEBT_VEBT,TreeList2))
                     => vEBT_invar_vebt(X3,N3) )
                 => ( vEBT_invar_vebt(Summary,M3)
                   => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3) )
                     => ( ( M3 = N3 )
                       => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),M3) )
                         => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_13)
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X3,set2(vEBT_VEBT,TreeList2))
                                 => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_13) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list(vEBT_VEBT),N3: nat,Summary: vEBT_VEBT,M3: nat,Deg: nat] :
                ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary) )
               => ( ( A22 = Deg )
                 => ( ! [X3: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X3,set2(vEBT_VEBT,TreeList2))
                       => vEBT_invar_vebt(X3,N3) )
                   => ( vEBT_invar_vebt(Summary,M3)
                     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3) )
                       => ( ( M3 = aa(nat,nat,suc,N3) )
                         => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),M3) )
                           => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_13)
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X3,set2(vEBT_VEBT,TreeList2))
                                   => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_13) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list(vEBT_VEBT),N3: nat,Summary: vEBT_VEBT,M3: nat,Deg: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Deg,TreeList2,Summary) )
                 => ( ( A22 = Deg )
                   => ( ! [X3: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X3,set2(vEBT_VEBT,TreeList2))
                         => vEBT_invar_vebt(X3,N3) )
                     => ( vEBT_invar_vebt(Summary,M3)
                       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3) )
                         => ( ( M3 = N3 )
                           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),M3) )
                             => ( ! [I3: nat] :
                                    ( aa(nat,$o,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3))
                                   => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),X_12)
                                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I3) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X3: vEBT_VEBT] :
                                        ( member(vEBT_VEBT,X3,set2(vEBT_VEBT,TreeList2))
                                       => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_13) ) )
                                 => ( aa(nat,$o,ord_less_eq(nat,Mi2),Ma2)
                                   => ( aa(nat,$o,ord_less(nat,Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I3: nat] :
                                              ( aa(nat,$o,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3))
                                             => ( ( ( vEBT_VEBT_high(Ma2,N3) = I3 )
                                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(Ma2,N3)) )
                                                & ! [X3: nat] :
                                                    ( ( ( vEBT_VEBT_high(X3,N3) = I3 )
                                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(X3,N3)) )
                                                   => ( aa(nat,$o,ord_less(nat,Mi2),X3)
                                                      & aa(nat,$o,ord_less_eq(nat,X3),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list(vEBT_VEBT),N3: nat,Summary: vEBT_VEBT,M3: nat,Deg: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Deg,TreeList2,Summary) )
                   => ( ( A22 = Deg )
                     => ( ! [X3: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X3,set2(vEBT_VEBT,TreeList2))
                           => vEBT_invar_vebt(X3,N3) )
                       => ( vEBT_invar_vebt(Summary,M3)
                         => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3) )
                           => ( ( M3 = aa(nat,nat,suc,N3) )
                             => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),M3) )
                               => ( ! [I3: nat] :
                                      ( aa(nat,$o,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3))
                                     => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),X_12)
                                      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I3) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X3: vEBT_VEBT] :
                                          ( member(vEBT_VEBT,X3,set2(vEBT_VEBT,TreeList2))
                                         => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_13) ) )
                                   => ( aa(nat,$o,ord_less_eq(nat,Mi2),Ma2)
                                     => ( aa(nat,$o,ord_less(nat,Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I3: nat] :
                                                ( aa(nat,$o,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M3))
                                               => ( ( ( vEBT_VEBT_high(Ma2,N3) = I3 )
                                                   => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(Ma2,N3)) )
                                                  & ! [X3: nat] :
                                                      ( ( ( vEBT_VEBT_high(X3,N3) = I3 )
                                                        & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I3)),vEBT_VEBT_low(X3,N3)) )
                                                     => ( aa(nat,$o,ord_less(nat,Mi2),X3)
                                                        & aa(nat,$o,ord_less_eq(nat,X3),Ma2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_702_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
    <=> ( ( ? [A7: $o,B7: $o] : A1 = vEBT_Leaf((A7),(B7))
          & ( A22 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList3: list(vEBT_VEBT),N8: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList3,Summary3) )
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList3))
               => vEBT_invar_vebt(X,N8) )
            & vEBT_invar_vebt(Summary3,N8)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N8) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N8),N8) )
            & ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X_12)
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList3))
               => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_12) ) )
        | ? [TreeList3: list(vEBT_VEBT),N8: nat,Summary3: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList3,Summary3) )
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList3))
               => vEBT_invar_vebt(X,N8) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N8))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N8)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N8),aa(nat,nat,suc,N8)) )
            & ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),X_12)
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList3))
               => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_12) ) )
        | ? [TreeList3: list(vEBT_VEBT),N8: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList3))
               => vEBT_invar_vebt(X,N8) )
            & vEBT_invar_vebt(Summary3,N8)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N8) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N8),N8) )
            & ! [I4: nat] :
                ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N8))
               => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),X_12)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),I4) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList3))
                 => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_12) ) )
            & aa(nat,$o,ord_less_eq(nat,Mi3),Ma3)
            & aa(nat,$o,ord_less(nat,Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A22))
            & ( ( Mi3 != Ma3 )
             => ! [I4: nat] :
                  ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N8))
                 => ( ( ( vEBT_VEBT_high(Ma3,N8) = I4 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),vEBT_VEBT_low(Ma3,N8)) )
                    & ! [X: nat] :
                        ( ( ( vEBT_VEBT_high(X,N8) = I4 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),vEBT_VEBT_low(X,N8)) )
                       => ( aa(nat,$o,ord_less(nat,Mi3),X)
                          & aa(nat,$o,ord_less_eq(nat,X),Ma3) ) ) ) ) ) )
        | ? [TreeList3: list(vEBT_VEBT),N8: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi3),Ma3)),A22,TreeList3,Summary3) )
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList3))
               => vEBT_invar_vebt(X,N8) )
            & vEBT_invar_vebt(Summary3,aa(nat,nat,suc,N8))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList3) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N8)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N8),aa(nat,nat,suc,N8)) )
            & ! [I4: nat] :
                ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N8)))
               => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),X_12)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary3),I4) ) )
            & ( ( Mi3 = Ma3 )
             => ! [X: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList3))
                 => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_12) ) )
            & aa(nat,$o,ord_less_eq(nat,Mi3),Ma3)
            & aa(nat,$o,ord_less(nat,Ma3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A22))
            & ( ( Mi3 != Ma3 )
             => ! [I4: nat] :
                  ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N8)))
                 => ( ( ( vEBT_VEBT_high(Ma3,N8) = I4 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),vEBT_VEBT_low(Ma3,N8)) )
                    & ! [X: nat] :
                        ( ( ( vEBT_VEBT_high(X,N8) = I4 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList3),I4)),vEBT_VEBT_low(X,N8)) )
                       => ( aa(nat,$o,ord_less(nat,Mi3),X)
                          & aa(nat,$o,ord_less_eq(nat,X),Ma3) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_703_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xb,Xaa)
     => ( ! [Uu2: $o,Uv: $o] : Xb != vEBT_Leaf((Uu2),(Uv))
       => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( ( Xaa = Mi2 )
                  | ( Xaa = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc2)
                 => ( ( Xaa = Mi2 )
                    | ( Xaa = Ma2 )
                    | $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
             => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
tff(fact_704_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xb,Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv: $o] : Xb = vEBT_Leaf((Uu2),(Uv))
         => (Y) )
       => ( ( ? [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
           => (Y) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list(vEBT_VEBT)] :
                  ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc2)
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
             => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)
                   => ( (Y)
                    <=> ~ $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
tff(fact_705_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xb,Xaa)
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A2),
                $ite(Xaa = one_one(nat),(B2),$false) ) )
       => ( ! [Uu2: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb != vEBT_Node(Uu2,zero_zero(nat),Uv,Uw)
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S2: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2)
               => $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
tff(fact_706_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xb,Xaa)
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A2),
                  $ite(Xaa = one_one(nat),(B2),$false) ) )
       => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [S2: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2)
             => ~ $let(
                    pos: nat,
                    pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
tff(fact_707_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xb,Xaa)
      <=> (Y) )
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A2),
                    $ite(Xaa = one_one(nat),(B2),$false) ) ) )
       => ( ( ? [Uu2: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb = vEBT_Node(Uu2,zero_zero(nat),Uv,Uw)
           => (Y) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [S2: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2)
               => ( (Y)
                <=> ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
tff(fact_708_even__odd__cases,axiom,
    ! [Xb: nat] :
      ( ! [N3: nat] : Xb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),N3)
     => ~ ! [N3: nat] : Xb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),aa(nat,nat,suc,N3)) ) ).

% even_odd_cases
tff(fact_709_add__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xb),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(nat,option(nat),some(nat),Xb)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% add_shift
tff(fact_710_both__member__options__def,axiom,
    ! [Tb: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,vEBT_V8194947554948674370ptions(Tb),Xb)
    <=> ( vEBT_V5719532721284313246member(Tb,Xb)
        | vEBT_VEBT_membermima(Tb,Xb) ) ) ).

% both_member_options_def
tff(fact_711_add__def,axiom,
    vEBT_VEBT_add = vEBT_V2048590022279873568_shift(nat,plus_plus(nat)) ).

% add_def
tff(fact_712_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tree,N)
     => ( aa(nat,$o,vEBT_vebt_member(Tree),Xb)
       => ( vEBT_V5719532721284313246member(Tree,Xb)
          | vEBT_VEBT_membermima(Tree,Xb) ) ) ) ).

% member_valid_both_member_options
tff(fact_713_pow__sum,axiom,
    ! [A3: nat,B3: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B3))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A3)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B3) ).

% pow_sum
tff(fact_714_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)))
     => aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Ma,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ).

% high_bound_aux
tff(fact_715_add__0,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% add_0
tff(fact_716_zero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A,Y: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% zero_eq_add_iff_both_eq_0
tff(fact_717_add__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y) = zero_zero(A) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% add_eq_0_iff_both_eq_0
tff(fact_718_add__cancel__right__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% add_cancel_right_right
tff(fact_719_add__cancel__right__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3) )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% add_cancel_right_left
tff(fact_720_add__cancel__left__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = A3 )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% add_cancel_left_right
tff(fact_721_add__cancel__left__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [B3: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3) = A3 )
        <=> ( B3 = zero_zero(A) ) ) ) ).

% add_cancel_left_left
tff(fact_722_double__zero__sym,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% double_zero_sym
tff(fact_723_add_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% add.right_neutral
tff(fact_724_double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% double_eq_0_iff
tff(fact_725_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2))
        <=> aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% add_le_cancel_right
tff(fact_726_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3))
        <=> aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% add_le_cancel_left
tff(fact_727_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2))
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% add_less_cancel_right
tff(fact_728_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3))
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% add_less_cancel_left
tff(fact_729_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ) ).

% numeral_plus_numeral
tff(fact_730_add__numeral__left,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [V: num,W: num,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),W)),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W))),Z) ) ).

% add_numeral_left
tff(fact_731_add__Suc__right,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% add_Suc_right
tff(fact_732_Nat_Oadd__0__right,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),zero_zero(nat)) = M ).

% Nat.add_0_right
tff(fact_733_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        & ( N = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_734_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N))
    <=> aa(nat,$o,ord_less(nat,M),N) ) ).

% nat_add_left_cancel_less
tff(fact_735_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or1337092689740270186AtMost(A,L,U))
        <=> ( aa(A,$o,ord_less_eq(A,L),I)
            & aa(A,$o,ord_less_eq(A,I),U) ) ) ) ).

% atLeastAtMost_iff
tff(fact_736_Icc__eq__Icc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,H: A,L3: A,H2: A] :
          ( ( set_or1337092689740270186AtMost(A,L,H) = set_or1337092689740270186AtMost(A,L3,H2) )
        <=> ( ( ( L = L3 )
              & ( H = H2 ) )
            | ( ~ aa(A,$o,ord_less_eq(A,L),H)
              & ~ aa(A,$o,ord_less_eq(A,L3),H2) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_737_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N))
    <=> aa(nat,$o,ord_less_eq(nat,M),N) ) ).

% nat_add_left_cancel_le
tff(fact_738_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,I),J)),K) = aa(nat,nat,minus_minus(nat,I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ).

% diff_diff_left
tff(fact_739_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_740_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_741_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) ) ) ).

% le_add_same_cancel2
tff(fact_742_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) ) ) ).

% le_add_same_cancel1
tff(fact_743_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),B3)
        <=> aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) ) ) ).

% add_le_same_cancel2
tff(fact_744_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3)),B3)
        <=> aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) ) ) ).

% add_le_same_cancel1
tff(fact_745_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_746_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_747_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),B3) ) ) ).

% less_add_same_cancel2
tff(fact_748_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),B3) ) ) ).

% less_add_same_cancel1
tff(fact_749_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),B3)
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% add_less_same_cancel2
tff(fact_750_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3)),B3)
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% add_less_same_cancel1
tff(fact_751_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = zero_zero(A) ) ).

% diff_add_zero
tff(fact_752_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),aa(A,A,minus_minus(A,A3),B3)) = A3 ) ) ) ).

% le_add_diff_inverse
tff(fact_753_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A3),B3)),B3) = A3 ) ) ) ).

% le_add_diff_inverse2
tff(fact_754_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
        | aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ) ).

% add_gr_0
tff(fact_755_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,minus_minus(nat,J),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) ) ) ).

% Nat.add_diff_assoc
tff(fact_756_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,J),K)),I) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) ) ) ).

% Nat.add_diff_assoc2
tff(fact_757_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,minus_minus(nat,I),aa(nat,nat,minus_minus(nat,J),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.diff_diff_right
tff(fact_758_one__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).

% one_plus_numeral
tff(fact_759_numeral__plus__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ) ).

% numeral_plus_one
tff(fact_760_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,minus_minus(nat,I),aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,J),K))) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,suc,J)) ) ) ).

% diff_Suc_diff_eq1
tff(fact_761_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,J),K))),I) = aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,J)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I)) ) ) ).

% diff_Suc_diff_eq2
tff(fact_762_one__add__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% one_add_one
tff(fact_763_add__2__eq__Suc_H,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).

% add_2_eq_Suc'
tff(fact_764_add__2__eq__Suc,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).

% add_2_eq_Suc
tff(fact_765_add__self__div__2,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = M ).

% add_self_div_2
tff(fact_766_sum__power2__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = zero_zero(A) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_eq_zero_iff
tff(fact_767_is__num__normalize_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) ) ).

% is_num_normalize(1)
tff(fact_768_add_Ogroup__left__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% add.group_left_neutral
tff(fact_769_add_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% add.comm_neutral
tff(fact_770_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A3) = A3 ) ).

% comm_monoid_add_class.add_0
tff(fact_771_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2))
         => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% add_le_imp_le_right
tff(fact_772_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3))
         => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% add_le_imp_le_left
tff(fact_773_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
        <=> ? [C4: A] : B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C4) ) ) ).

% le_iff_add
tff(fact_774_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) ) ) ).

% add_right_mono
tff(fact_775_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ~ ! [C3: A] : B3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) ) ) ).

% less_eqE
tff(fact_776_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3)) ) ) ).

% add_left_mono
tff(fact_777_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),D)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),D)) ) ) ) ).

% add_mono
tff(fact_778_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,ord_less_eq(A,I),J)
            & aa(A,$o,ord_less_eq(A,K),L) )
         => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(1)
tff(fact_779_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,ord_less_eq(A,K),L) )
         => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(2)
tff(fact_780_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,ord_less_eq(A,I),J)
            & ( K = L ) )
         => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_semiring(3)
tff(fact_781_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2))
         => aa(A,$o,ord_less(A,A3),B3) ) ) ).

% add_less_imp_less_right
tff(fact_782_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3))
         => aa(A,$o,ord_less(A,A3),B3) ) ) ).

% add_less_imp_less_left
tff(fact_783_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) ) ) ).

% add_strict_right_mono
tff(fact_784_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3)) ) ) ).

% add_strict_left_mono
tff(fact_785_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,C2),D)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),D)) ) ) ) ).

% add_strict_mono
tff(fact_786_add__mono__thms__linordered__field_I1_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,ord_less(A,I),J)
            & ( K = L ) )
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(1)
tff(fact_787_add__mono__thms__linordered__field_I2_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & aa(A,$o,ord_less(A,K),L) )
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(2)
tff(fact_788_add__mono__thms__linordered__field_I5_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,ord_less(A,I),J)
            & aa(A,$o,ord_less(A,K),L) )
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(5)
tff(fact_789_add__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) ) ).

% add_divide_distrib
tff(fact_790_add__Suc__shift,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,N)) ).

% add_Suc_shift
tff(fact_791_add__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% add_Suc
tff(fact_792_nat__arith_Osuc1,axiom,
    ! [A4: nat,K: nat,A3: nat] :
      ( ( A4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),A3) )
     => ( aa(nat,nat,suc,A4) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,suc,A3)) ) ) ).

% nat_arith.suc1
tff(fact_793_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = M )
     => ( N = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_794_plus__nat_Oadd__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),N) = N ).

% plus_nat.add_0
tff(fact_795_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,K),L)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N) )
       => aa(nat,$o,ord_less(nat,M),N) ) ) ).

% less_add_eq_less
tff(fact_796_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => aa(nat,$o,ord_less(nat,I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J)) ) ).

% trans_less_add2
tff(fact_797_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => aa(nat,$o,ord_less(nat,I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M)) ) ).

% trans_less_add1
tff(fact_798_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_less_mono1
tff(fact_799_not__add__less2,axiom,
    ! [J: nat,I: nat] : ~ aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),I) ).

% not_add_less2
tff(fact_800_not__add__less1,axiom,
    ! [I: nat,J: nat] : ~ aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),I) ).

% not_add_less1
tff(fact_801_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( aa(nat,$o,ord_less(nat,K),L)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_less_mono
tff(fact_802_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K)
     => aa(nat,$o,ord_less(nat,I),K) ) ).

% add_lessD1
tff(fact_803_nat__le__iff__add,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
    <=> ? [K3: nat] : N = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3) ) ).

% nat_le_iff_add
tff(fact_804_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => aa(nat,$o,ord_less_eq(nat,I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J)) ) ).

% trans_le_add2
tff(fact_805_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => aa(nat,$o,ord_less_eq(nat,I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M)) ) ).

% trans_le_add1
tff(fact_806_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) ) ).

% add_le_mono1
tff(fact_807_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,ord_less_eq(nat,K),L)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),L)) ) ) ).

% add_le_mono
tff(fact_808_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),L)
     => ? [N3: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N3) ) ).

% le_Suc_ex
tff(fact_809_add__leD2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N)
     => aa(nat,$o,ord_less_eq(nat,K),N) ) ).

% add_leD2
tff(fact_810_add__leD1,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N)
     => aa(nat,$o,ord_less_eq(nat,M),N) ) ).

% add_leD1
tff(fact_811_le__add2,axiom,
    ! [N: nat,M: nat] : aa(nat,$o,ord_less_eq(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% le_add2
tff(fact_812_le__add1,axiom,
    ! [N: nat,M: nat] : aa(nat,$o,ord_less_eq(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) ).

% le_add1
tff(fact_813_add__leE,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),N)
     => ~ ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ~ aa(nat,$o,ord_less_eq(nat,K),N) ) ) ).

% add_leE
tff(fact_814_diff__add__inverse2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),N) = M ).

% diff_add_inverse2
tff(fact_815_diff__add__inverse,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),N) = M ).

% diff_add_inverse
tff(fact_816_diff__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,nat,minus_minus(nat,M),N) ).

% diff_cancel2
tff(fact_817_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N)) = aa(nat,nat,minus_minus(nat,M),N) ).

% Nat.diff_cancel
tff(fact_818_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,Y),zero_zero(A))
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y) = zero_zero(A) )
            <=> ( ( Xb = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonpos_eq_0_iff
tff(fact_819_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y) = zero_zero(A) )
            <=> ( ( Xb = zero_zero(A) )
                & ( Y = zero_zero(A) ) ) ) ) ) ) ).

% add_nonneg_eq_0_iff
tff(fact_820_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,B3),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),zero_zero(A)) ) ) ) ).

% add_nonpos_nonpos
tff(fact_821_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ) ).

% add_nonneg_nonneg
tff(fact_822_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less_eq(A,B3),A3)
           => aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% add_increasing2
tff(fact_823_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,A3),B3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B3) ) ) ) ).

% add_decreasing2
tff(fact_824_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,B3),C2)
           => aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% add_increasing
tff(fact_825_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,C2),B3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),B3) ) ) ) ).

% add_decreasing
tff(fact_826_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,B3),C2)
           => aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% pos_add_strict
tff(fact_827_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ~ ! [C3: A] :
                ( ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C3) )
               => ( C3 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_828_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ) ).

% add_pos_pos
tff(fact_829_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),zero_zero(A)) ) ) ) ).

% add_neg_neg
tff(fact_830_add__less__zeroD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),zero_zero(A))
         => ( aa(A,$o,ord_less(A,Xb),zero_zero(A))
            | aa(A,$o,ord_less(A,Y),zero_zero(A)) ) ) ) ).

% add_less_zeroD
tff(fact_831_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),D)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),D)) ) ) ) ).

% add_less_le_mono
tff(fact_832_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less(A,C2),D)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),D)) ) ) ) ).

% add_le_less_mono
tff(fact_833_add__mono__thms__linordered__field_I3_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,ord_less(A,I),J)
            & aa(A,$o,ord_less_eq(A,K),L) )
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(3)
tff(fact_834_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,ord_less_eq(A,I),J)
            & aa(A,$o,ord_less(A,K),L) )
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),L)) ) ) ).

% add_mono_thms_linordered_field(4)
tff(fact_835_numeral__Bit0,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),bit0(N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_Bit0
tff(fact_836_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),one_one(A))) ) ) ).

% add_mono1
tff(fact_837_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A] : aa(A,$o,ord_less(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))) ) ).

% less_add_one
tff(fact_838_one__plus__numeral__commute,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),Xb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Xb)),one_one(A)) ) ).

% one_plus_numeral_commute
tff(fact_839_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,A3),B3)),C2)
        <=> aa(A,$o,ord_less_eq(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3)) ) ) ).

% diff_le_eq
tff(fact_840_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,minus_minus(A,C2),B3))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) ) ) ).

% le_diff_eq
tff(fact_841_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B3),A3)),A3) = B3 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_842_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => aa(A,$o,ord_less_eq(A,C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)),A3)) ) ) ).

% le_add_diff
tff(fact_843_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),aa(A,A,minus_minus(A,B3),A3))
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),B3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_844_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,minus_minus(A,B3),A3)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_845_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,minus_minus(A,B3),A3)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_846_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B3),A3)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)),A3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_847_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B3),A3)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_848_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,minus_minus(A,C2),aa(A,A,minus_minus(A,B3),A3)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3)),B3) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_849_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,minus_minus(A,B3),A3)) = B3 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_850_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,A3),B3)
           => ( ( aa(A,A,minus_minus(A,B3),A3) = C2 )
            <=> ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A3) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_851_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,N: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),N)
         => aa(A,$o,ord_less_eq(A,I),aa(A,A,minus_minus(A,N),K)) ) ) ).

% add_le_imp_le_diff
tff(fact_852_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,N: A,J: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),N)
         => ( aa(A,$o,ord_less_eq(A,N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
           => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),N)
             => ( aa(A,$o,ord_less_eq(A,N),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
               => aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,N),K)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_853_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,minus_minus(A,C2),B3))
        <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) ) ) ).

% less_diff_eq
tff(fact_854_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A3),B3)),C2)
        <=> aa(A,$o,ord_less(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B3)) ) ) ).

% diff_less_eq
tff(fact_855_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),aa(A,A,minus_minus(A,A3),B3)) = A3 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_856_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_857_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( N = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_858_less__natE,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),N)
     => ~ ! [Q3: nat] : N != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)) ) ).

% less_natE
tff(fact_859_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : aa(nat,$o,ord_less(nat,I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),M))) ).

% less_add_Suc1
tff(fact_860_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : aa(nat,$o,ord_less(nat,I),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I))) ).

% less_add_Suc2
tff(fact_861_less__iff__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),N)
    <=> ? [K3: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3)) ) ).

% less_iff_Suc_add
tff(fact_862_less__imp__Suc__add,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),N)
     => ? [K2: nat] : N = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K2)) ) ).

% less_imp_Suc_add
tff(fact_863_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ? [K2: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K2)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K2) = J ) ) ) ).

% less_imp_add_positive
tff(fact_864_Suc__eq__plus1__left,axiom,
    ! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N) ).

% Suc_eq_plus1_left
tff(fact_865_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_866_Suc__eq__plus1,axiom,
    ! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_867_mono__nat__linear__lb,axiom,
    ! [Fa: fun(nat,nat),M: nat,K: nat] :
      ( ! [M3: nat,N3: nat] :
          ( aa(nat,$o,ord_less(nat,M3),N3)
         => aa(nat,$o,ord_less(nat,aa(nat,nat,Fa,M3)),aa(nat,nat,Fa,N3)) )
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,Fa,M)),K)),aa(nat,nat,Fa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K))) ) ).

% mono_nat_linear_lb
tff(fact_868_diff__add__0,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,minus_minus(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)) = zero_zero(nat) ).

% diff_add_0
tff(fact_869_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,I),aa(nat,nat,minus_minus(nat,J),K))
    <=> aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ).

% less_diff_conv
tff(fact_870_add__diff__inverse__nat,axiom,
    ! [M: nat,N: nat] :
      ( ~ aa(nat,$o,ord_less(nat,M),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,minus_minus(nat,M),N)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_871_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,J),K)),I)
    <=> aa(nat,$o,ord_less_eq(nat,J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ).

% le_diff_conv
tff(fact_872_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,$o,ord_less_eq(nat,I),aa(nat,nat,minus_minus(nat,J),K))
      <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J) ) ) ).

% Nat.le_diff_conv2
tff(fact_873_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),aa(nat,nat,minus_minus(nat,J),K)) ) ) ).

% Nat.diff_add_assoc
tff(fact_874_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),I)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,J),K)),I) ) ) ).

% Nat.diff_add_assoc2
tff(fact_875_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( ( aa(nat,nat,minus_minus(nat,J),I) = K )
      <=> ( J = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),I) ) ) ) ).

% Nat.le_imp_diff_is_add
tff(fact_876_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] :
          aa(num,A,numeral_numeral(A),bit0(N)) = $let(
            m: A,
            m:= aa(num,A,numeral_numeral(A),N),
            aa(A,A,aa(A,fun(A,A),plus_plus(A),m),m) ) ) ).

% numeral_code(2)
tff(fact_877_VEBT_Odisc__eq__case_I1_J,axiom,
    ! [VEBT: vEBT_VEBT] :
      ( vEBT_is_Node(VEBT)
    <=> vEBT_case_VEBT($o,aTP_Lamp_fp(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,$o)))),aTP_Lamp_fq($o,fun($o,$o)),VEBT) ) ).

% VEBT.disc_eq_case(1)
tff(fact_878_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,B3),C2)
           => aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% add_strict_increasing2
tff(fact_879_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,B3),C2)
           => aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2)) ) ) ) ).

% add_strict_increasing
tff(fact_880_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ) ).

% add_pos_nonneg
tff(fact_881_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),zero_zero(A)) ) ) ) ).

% add_nonpos_neg
tff(fact_882_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ) ).

% add_nonneg_pos
tff(fact_883_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,B3),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),zero_zero(A)) ) ) ) ).

% add_neg_nonpos
tff(fact_884_field__le__epsilon,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( ! [E2: A] :
              ( aa(A,$o,ord_less(A,zero_zero(A)),E2)
             => aa(A,$o,ord_less_eq(A,Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E2)) )
         => aa(A,$o,ord_less_eq(A,Xb),Y) ) ) ).

% field_le_epsilon
tff(fact_885_zero__less__two,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A))) ) ).

% zero_less_two
tff(fact_886_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),B3) ) ) ).

% discrete
tff(fact_887_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_888_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3)),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_889_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Ux: nat] : ~ vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero(nat),Uv2,Uw2),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_890_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => aa(A,$o,ord_less(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ) ).

% less_half_sum
tff(fact_891_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B3) ) ) ).

% gt_half_sum
tff(fact_892_nat__diff__split__asm,axiom,
    ! [P: fun(nat,$o),A3: nat,B3: nat] :
      ( aa(nat,$o,P,aa(nat,nat,minus_minus(nat,A3),B3))
    <=> ~ ( ( aa(nat,$o,ord_less(nat,A3),B3)
            & ~ aa(nat,$o,P,zero_zero(nat)) )
          | ? [D4: nat] :
              ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B3),D4) )
              & ~ aa(nat,$o,P,D4) ) ) ) ).

% nat_diff_split_asm
tff(fact_893_nat__diff__split,axiom,
    ! [P: fun(nat,$o),A3: nat,B3: nat] :
      ( aa(nat,$o,P,aa(nat,nat,minus_minus(nat,A3),B3))
    <=> ( ( aa(nat,$o,ord_less(nat,A3),B3)
         => aa(nat,$o,P,zero_zero(nat)) )
        & ! [D4: nat] :
            ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B3),D4) )
           => aa(nat,$o,P,D4) ) ) ) ).

% nat_diff_split
tff(fact_894_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),J)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,J),K)),I)
      <=> aa(nat,$o,ord_less(nat,J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ) ).

% less_diff_conv2
tff(fact_895_bounded__Max__nat,axiom,
    ! [P: fun(nat,$o),Xb: nat,M8: nat] :
      ( aa(nat,$o,P,Xb)
     => ( ! [X2: nat] :
            ( aa(nat,$o,P,X2)
           => aa(nat,$o,ord_less_eq(nat,X2),M8) )
       => ~ ! [M3: nat] :
              ( aa(nat,$o,P,M3)
             => ~ ! [X3: nat] :
                    ( aa(nat,$o,P,X3)
                   => aa(nat,$o,ord_less_eq(nat,X3),M3) ) ) ) ) ).

% bounded_Max_nat
tff(fact_896_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))] :
      ~ ! [F4: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc: A] : Xb != aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),F4),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A2),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B2),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_897_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux: list(vEBT_VEBT),Uy: vEBT_VEBT,Uz: nat] : ~ vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux,Uy),Uz) ).

% VEBT_internal.membermima.simps(2)
tff(fact_898_nat__1__add__1,axiom,
    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% nat_1_add_1
tff(fact_899_add__eq__if,axiom,
    ! [M: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) = $ite(M = zero_zero(nat),N,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,M),one_one(nat))),N))) ).

% add_eq_if
tff(fact_900_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,Xb: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf((A3),(B3)),Xb)
    <=> $ite(
          Xb = zero_zero(nat),
          (A3),
          $ite(Xb = one_one(nat),(B3),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_901_num_Osize_I5_J,axiom,
    ! [X22: num] : aa(num,nat,size_size(num),bit0(X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_902_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_903_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_left
tff(fact_904_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ).

% div_exp_eq
tff(fact_905_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list(vEBT_VEBT),Vb: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va,Vb),Xb)
    <=> ( ( Xb = Mi )
        | ( Xb = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_906_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A))
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_power2_le_zero_iff
tff(fact_907_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% sum_power2_ge_zero
tff(fact_908_sum__power2__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
        <=> ( ( Xb != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_power2_gt_zero_iff
tff(fact_909_not__sum__power2__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] : ~ aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(A)) ) ).

% not_sum_power2_lt_zero
tff(fact_910_ex__power__ivl2,axiom,
    ! [B3: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
     => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
       => ? [N3: nat] :
            ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),N3)),K)
            & aa(nat,$o,ord_less_eq(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),one_one(nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_911_ex__power__ivl1,axiom,
    ! [B3: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
     => ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),K)
       => ? [N3: nat] :
            ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),N3)),K)
            & aa(nat,$o,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),one_one(nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_912_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xb: nat,N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
       => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
         => aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
tff(fact_913_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xb: nat,N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
       => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
         => aa(nat,$o,ord_less(nat,vEBT_VEBT_low(Xb,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_914_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less(set(A),set_or1337092689740270186AtMost(A,A3,B3)),set_or1337092689740270186AtMost(A,C2,D))
        <=> ( ( ~ aa(A,$o,ord_less_eq(A,A3),B3)
              | ( aa(A,$o,ord_less_eq(A,C2),A3)
                & aa(A,$o,ord_less_eq(A,B3),D)
                & ( aa(A,$o,ord_less(A,C2),A3)
                  | aa(A,$o,ord_less(A,B3),D) ) ) )
            & aa(A,$o,ord_less_eq(A,C2),D) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_915_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member(vEBT_VEBT,X2,set2(vEBT_VEBT,TreeList))
         => vEBT_invar_vebt(X2,N) )
     => ( vEBT_invar_vebt(Summary2,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = N )
           => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_1)
               => ( ! [X2: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X2,set2(vEBT_VEBT,TreeList))
                     => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_1) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary2),Deg2) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_916_vebt__delete_Osimps_I3_J,axiom,
    ! [A3: $o,B3: $o,N: nat] : vEBT_vebt_delete(vEBT_Leaf((A3),(B3)),aa(nat,nat,suc,aa(nat,nat,suc,N))) = vEBT_Leaf((A3),(B3)) ).

% vebt_delete.simps(3)
tff(fact_917_vebt__delete_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] : vEBT_vebt_delete(vEBT_Leaf((A3),(B3)),zero_zero(nat)) = vEBT_Leaf($false,(B3)) ).

% vebt_delete.simps(1)
tff(fact_918_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Uu: nat] : vEBT_vebt_delete(vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary2),Uu) = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary2) ).

% vebt_delete.simps(4)
tff(fact_919_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member(vEBT_VEBT,X2,set2(vEBT_VEBT,TreeList))
         => vEBT_invar_vebt(X2,N) )
     => ( vEBT_invar_vebt(Summary2,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = aa(nat,nat,suc,N) )
           => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_1)
               => ( ! [X2: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X2,set2(vEBT_VEBT,TreeList))
                     => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_1) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary2),Deg2) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_920_vebt__delete_Osimps_I2_J,axiom,
    ! [A3: $o,B3: $o] : vEBT_vebt_delete(vEBT_Leaf((A3),(B3)),aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf((A3),$false) ).

% vebt_delete.simps(2)
tff(fact_921_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList: list(vEBT_VEBT),Vd: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeList,Vd),Xb)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.membermima.simps(5)
tff(fact_922_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option(product_prod(nat,nat)),V: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy,aa(nat,nat,suc,V),TreeList,S),Xb)
    <=> $let(
          pos: nat,
          pos:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ).

% VEBT_internal.naive_member.simps(3)
tff(fact_923_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList: list(vEBT_VEBT),Vc: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,V),TreeList,Vc),Xb)
    <=> ( ( Xb = Mi )
        | ( Xb = Ma )
        | $let(
            pos: nat,
            pos:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),pos),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ).

% VEBT_internal.membermima.simps(4)
tff(fact_924_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi: nat,Ma: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member(vEBT_VEBT,X2,set2(vEBT_VEBT,TreeList))
         => vEBT_invar_vebt(X2,N) )
     => ( vEBT_invar_vebt(Summary2,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = N )
           => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ! [I2: nat] :
                    ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                   => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),X_12)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I2) ) )
               => ( ( ( Mi = Ma )
                   => ! [X2: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X2,set2(vEBT_VEBT,TreeList))
                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_1) ) )
                 => ( aa(nat,$o,ord_less_eq(nat,Mi),Ma)
                   => ( aa(nat,$o,ord_less(nat,Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                             => ( ( ( vEBT_VEBT_high(Ma,N) = I2 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(Ma,N)) )
                                & ! [X2: nat] :
                                    ( ( ( vEBT_VEBT_high(X2,N) = I2 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(X2,N)) )
                                   => ( aa(nat,$o,ord_less(nat,Mi),X2)
                                      & aa(nat,$o,ord_less_eq(nat,X2),Ma) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Deg2) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_925_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list(vEBT_VEBT),N: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi: nat,Ma: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member(vEBT_VEBT,X2,set2(vEBT_VEBT,TreeList))
         => vEBT_invar_vebt(X2,N) )
     => ( vEBT_invar_vebt(Summary2,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = aa(nat,nat,suc,N) )
           => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M) )
             => ( ! [I2: nat] :
                    ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                   => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),X_12)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I2) ) )
               => ( ( ( Mi = Ma )
                   => ! [X2: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X2,set2(vEBT_VEBT,TreeList))
                       => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_1) ) )
                 => ( aa(nat,$o,ord_less_eq(nat,Mi),Ma)
                   => ( aa(nat,$o,ord_less(nat,Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2))
                     => ( ( ( Mi != Ma )
                         => ! [I2: nat] :
                              ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                             => ( ( ( vEBT_VEBT_high(Ma,N) = I2 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(Ma,N)) )
                                & ! [X2: nat] :
                                    ( ( ( vEBT_VEBT_high(X2,N) = I2 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I2)),vEBT_VEBT_low(X2,N)) )
                                   => ( aa(nat,$o,ord_less(nat,Mi),X2)
                                      & aa(nat,$o,ord_less_eq(nat,X2),Ma) ) ) ) ) )
                       => vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Deg2) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_926_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list(vEBT_VEBT),Smry: vEBT_VEBT,Xb: nat] : vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TrLst,Smry),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TrLst,Smry) ).

% vebt_delete.simps(5)
tff(fact_927_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xb,Xaa)
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)
           => ~ ( ( Xaa = Mi2 )
                | ( Xaa = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list(vEBT_VEBT)] :
              ( ? [Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc2)
             => ~ ( ( Xaa = Mi2 )
                  | ( Xaa = Ma2 )
                  | $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
         => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT)] :
                ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)
               => ~ $let(
                      pos: nat,
                      pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
tff(fact_928_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,N: nat,Va: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),N)
     => ( ( N = aa(nat,nat,suc,aa(nat,nat,suc,Va)) )
       => ( ~ aa(nat,$o,ord_less(nat,Ma),Mi)
         => ( ( Ma != Mi )
           => aa(nat,$o,ord_less(nat,vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Va),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)) ) ) ) ) ).

% nested_mint
tff(fact_929_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(Xb,Xaa) = Y )
     => ( ( ? [Uu2: $o,B2: $o] : Xb = vEBT_Leaf((Uu2),(B2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : Xb = vEBT_Leaf((Uv),(Uw))
           => ( ? [N3: nat] : Xaa = aa(nat,nat,suc,N3)
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V3: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc2,Vd2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)
                 => ( Y != one_one(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( Y != $ite(
                            aa(nat,$o,ord_less(nat,Xaa),Mi2),
                            one_one(nat),
                            $let(
                              l: nat,
                              l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                  $let(
                                    maxlow: option(nat),
                                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                    $ite(
                                      ( ( maxlow != none(nat) )
                                      & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary,h)),one_one(nat)) ) ),
                                  one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_930_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d2(Xb,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv: $o] : Xb = vEBT_Leaf((Uu2),(Uv))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A2: $o,Uw: $o] : Xb = vEBT_Leaf((A2),(Uw))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A2: $o,B2: $o] : Xb = vEBT_Leaf((A2),(B2))
             => ( ? [Va2: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va2))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( Y != $ite(
                              aa(nat,$o,ord_less(nat,Ma2),Xaa),
                              one_one(nat),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $ite(
                                    aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                    $let(
                                      minlow: option(nat),
                                      minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                      $ite(
                                        ( ( minlow != none(nat) )
                                        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary,h)),one_one(nat)) ) ),
                                    one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_931_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(Xb,Xaa) = Y )
     => ( ( ? [A2: $o,B2: $o] : Xb = vEBT_Leaf((A2),(B2))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)
           => ( Y != one_one(nat) ) )
         => ( ( ? [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                          $ite(
                            Xaa = Mi2,
                            zero_zero(nat),
                            $ite(
                              Xaa = Ma2,
                              zero_zero(nat),
                              $ite(
                                aa(nat,$o,ord_less(nat,Xaa),Mi2),
                                zero_zero(nat),
                                $ite(
                                  aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                  zero_zero(nat),
                                  $ite(
                                    ( aa(nat,$o,ord_less(nat,Mi2),Xaa)
                                    & aa(nat,$o,ord_less(nat,Xaa),Ma2) ),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)) ),
                                    zero_zero(nat) ) ) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.elims
tff(fact_932_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      vEBT_T_s_u_c_c2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xb) = $ite(
        aa(nat,$o,ord_less(nat,Xb),Mi),
        one_one(nat),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( maxlow != none(nat) )
                  & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary2,h)),one_one(nat)) ) ),
              one_one(nat) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_933_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      vEBT_T_p_r_e_d2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xb) = $ite(
        aa(nat,$o,ord_less(nat,Ma),Xb),
        one_one(nat),
        $let(
          l: nat,
          l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
              $let(
                minlow: option(nat),
                minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                $ite(
                  ( ( minlow != none(nat) )
                  & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary2,h)),one_one(nat)) ) ),
              one_one(nat) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_934_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      vEBT_T_m_e_m_b_e_r2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          Xb = Mi,
          zero_zero(nat),
          $ite(
            Xb = Ma,
            zero_zero(nat),
            $ite(
              aa(nat,$o,ord_less(nat,Xb),Mi),
              zero_zero(nat),
              $ite(
                aa(nat,$o,ord_less(nat,Ma),Xb),
                zero_zero(nat),
                $ite(
                  ( aa(nat,$o,ord_less(nat,Mi),Xb)
                  & aa(nat,$o,ord_less(nat,Xb),Ma) ),
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)) ),
                  zero_zero(nat) ) ) ) ) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(5)
tff(fact_935_nat__induct2,axiom,
    ! [P: fun(nat,$o),N: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N3: nat] :
              ( aa(nat,$o,P,N3)
             => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(nat,$o,P,N) ) ) ) ).

% nat_induct2
tff(fact_936_field__less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
         => aa(A,$o,ord_less(A,Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% field_less_half_sum
tff(fact_937_bit__concat__def,axiom,
    ! [H: nat,L: nat,D: nat] : vEBT_VEBT_bit_concat(H,L,D) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),D))),L) ).

% bit_concat_def
tff(fact_938_mul__shift,axiom,
    ! [Xb: nat,Y: nat,Z: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Xb),Y) = Z )
    <=> ( aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),Xb)),aa(nat,option(nat),some(nat),Y)) = aa(nat,option(nat),some(nat),Z) ) ) ).

% mul_shift
tff(fact_939_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V2048590022279873568_shift(nat,times_times(nat)) ).

% mul_def
tff(fact_940_high__inv,axiom,
    ! [Xb: nat,N: nat,Y: nat] :
      ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( vEBT_VEBT_high(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),Xb),N) = Y ) ) ).

% high_inv
tff(fact_941_low__inv,axiom,
    ! [Xb: nat,N: nat,Y: nat] :
      ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( vEBT_VEBT_low(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),Xb),N) = Xb ) ) ).

% low_inv
tff(fact_942_mult__zero__left,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% mult_zero_left
tff(fact_943_mult__zero__right,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% mult_zero_right
tff(fact_944_mult__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            | ( B3 = zero_zero(A) ) ) ) ) ).

% mult_eq_0_iff
tff(fact_945_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B3 ) ) ) ) ).

% mult_cancel_left
tff(fact_946_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A3: A,C2: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = B3 ) ) ) ) ).

% mult_cancel_right
tff(fact_947_mult__numeral__left__semiring__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [V: num,W: num,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W))),Z) ) ).

% mult_numeral_left_semiring_numeral
tff(fact_948_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ) ).

% numeral_times_numeral
tff(fact_949_mult_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),one_one(A)) = A3 ) ).

% mult.right_neutral
tff(fact_950_mult__1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A3) = A3 ) ).

% mult_1
tff(fact_951_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B3: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),C2) ) ).

% times_divide_eq_left
tff(fact_952_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).

% divide_divide_eq_left
tff(fact_953_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ).

% divide_divide_eq_right
tff(fact_954_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2) ) ).

% times_divide_eq_right
tff(fact_955_mult__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        | ( N = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_956_mult__0__right,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),zero_zero(nat)) = zero_zero(nat) ).

% mult_0_right
tff(fact_957_mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
    <=> ( ( M = N )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_958_mult__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K) )
    <=> ( ( M = N )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_959_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
    <=> ( ( M = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_960_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = one_one(nat) )
    <=> ( ( M = one_one(nat) )
        & ( N = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_961_semiring__norm_I6_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),bit0(N)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% semiring_norm(6)
tff(fact_962_sum__squares__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_eq_zero_iff
tff(fact_963_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B3: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B3 = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_964_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = one_one(A) ) ) ) ) ).

% mult_cancel_left2
tff(fact_965_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B3: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B3 = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_966_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A3: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A3 = one_one(A) ) ) ) ) ).

% mult_cancel_right2
tff(fact_967_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A3: A,B3: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(num,A,numeral_numeral(A),V))) ) ).

% distrib_right_numeral
tff(fact_968_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V: num,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% distrib_left_numeral
tff(fact_969_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_970_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_971_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_972_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_973_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_974_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ).

% div_mult_mult1
tff(fact_975_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ).

% div_mult_mult2
tff(fact_976_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B3: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ).

% div_mult_mult1_if
tff(fact_977_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),A3) = B3 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_978_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),B3) = A3 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_979_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V: num,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,minus_minus(A,B3),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% right_diff_distrib_numeral
tff(fact_980_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A3: A,B3: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A3),B3)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(num,A,numeral_numeral(A),V))) ) ).

% left_diff_distrib_numeral
tff(fact_981_one__eq__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_982_mult__eq__1__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_983_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
        & aa(nat,$o,ord_less(nat,M),N) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_984_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
        & aa(nat,$o,ord_less(nat,M),N) ) ) ).

% mult_less_cancel2
tff(fact_985_nat__0__less__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
        & aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ) ).

% nat_0_less_mult_iff
tff(fact_986_mult__Suc__right,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,suc,N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ).

% mult_Suc_right
tff(fact_987_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = $ite(K = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)) ).

% nat_mult_div_cancel_disj
tff(fact_988_semiring__norm_I2_J,axiom,
    aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),one2) = bit0(one2) ).

% semiring_norm(2)
tff(fact_989_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less(int,W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))
    <=> aa(int,$o,ord_less_eq(int,W),Z) ) ).

% zle_add1_eq_le
tff(fact_990_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,W: num,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(num,A,numeral_numeral(A),W)) = A3 )
        <=> $ite(aa(num,A,numeral_numeral(A),W) != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W)),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_991_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,W: num] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(num,A,numeral_numeral(A),W)) )
        <=> $ite(aa(num,A,numeral_numeral(A),W) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W)) = B3,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_992_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,W: num,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(num,A,numeral_numeral(A),W))),A3)
        <=> aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_993_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,W: num] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))),B3) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_994_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,W: num,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(num,A,numeral_numeral(A),W))),A3)
        <=> aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))) ) ) ).

% divide_less_eq_numeral1(1)
tff(fact_995_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,W: num] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))),B3) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_996_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ) ) ).

% div_mult_self1
tff(fact_997_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ) ) ).

% div_mult_self2
tff(fact_998_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,C2: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)),A3)),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ) ) ).

% div_mult_self3
tff(fact_999_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,C2: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)),A3)),B3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ) ) ).

% div_mult_self4
tff(fact_1000_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B3) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1001_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1002_one__le__mult__iff,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N))
    <=> ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),M)
        & aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),N) ) ) ).

% one_le_mult_iff
tff(fact_1003_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => aa(nat,$o,ord_less_eq(nat,M),N) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_1004_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => aa(nat,$o,ord_less_eq(nat,M),N) ) ) ).

% mult_le_cancel2
tff(fact_1005_div__mult__self__is__m,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),N) = M ) ) ).

% div_mult_self_is_m
tff(fact_1006_div__mult__self1__is__m,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M)),N) = M ) ) ).

% div_mult_self1_is_m
tff(fact_1007_Suc__numeral,axiom,
    ! [N: num] : aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),N)) = aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ).

% Suc_numeral
tff(fact_1008_power__add__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N))) ) ).

% power_add_numeral
tff(fact_1009_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: num,N: num,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),M))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),N))),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)))),B3) ) ).

% power_add_numeral2
tff(fact_1010_mult__not__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) != zero_zero(A) )
         => ( ( A3 != zero_zero(A) )
            & ( B3 != zero_zero(A) ) ) ) ) ).

% mult_not_zero
tff(fact_1011_divisors__zero,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = zero_zero(A) )
         => ( ( A3 = zero_zero(A) )
            | ( B3 = zero_zero(A) ) ) ) ) ).

% divisors_zero
tff(fact_1012_no__zero__divisors,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) != zero_zero(A) ) ) ) ) ).

% no_zero_divisors
tff(fact_1013_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) )
          <=> ( A3 = B3 ) ) ) ) ).

% mult_left_cancel
tff(fact_1014_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2) )
          <=> ( A3 = B3 ) ) ) ) ).

% mult_right_cancel
tff(fact_1015_add__One__commute,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N) = aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2) ).

% add_One_commute
tff(fact_1016_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).

% ring_class.ring_distribs(2)
tff(fact_1017_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).

% ring_class.ring_distribs(1)
tff(fact_1018_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( comm_semiring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).

% comm_semiring_class.distrib
tff(fact_1019_distrib__left,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).

% distrib_left
tff(fact_1020_distrib__right,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).

% distrib_right
tff(fact_1021_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A3: A,E3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E3)),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),E3)),C2) ) ).

% combine_common_factor
tff(fact_1022_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A3) = A3 ) ).

% comm_monoid_mult_class.mult_1
tff(fact_1023_mult_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),one_one(A)) = A3 ) ).

% mult.comm_neutral
tff(fact_1024_left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A3),B3)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ).

% left_diff_distrib
tff(fact_1025_right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,minus_minus(A,B3),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).

% right_diff_distrib
tff(fact_1026_left__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [B3: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B3),C2)),A3) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) ) ).

% left_diff_distrib'
tff(fact_1027_right__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,minus_minus(A,B3),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ).

% right_diff_distrib'
tff(fact_1028_times__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Y: A,Z: A,W: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),W)) ) ).

% times_divide_times_eq
tff(fact_1029_divide__divide__times__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Y: A,Z: A,W: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Z),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),W)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ).

% divide_divide_times_eq
tff(fact_1030_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ).

% divide_divide_eq_left'
tff(fact_1031_power__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_commutes
tff(fact_1032_power__mult__distrib,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N)) ) ).

% power_mult_distrib
tff(fact_1033_power__commuting__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xb: A,Y: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N)) ) ) ) ).

% power_commuting_commutes
tff(fact_1034_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N) )
    <=> ( M = N ) ) ).

% Suc_mult_cancel1
tff(fact_1035_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
    <=> ( ( K = zero_zero(nat) )
        | ( M = N ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_1036_mult__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),N) = zero_zero(nat) ).

% mult_0
tff(fact_1037_power__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: nat,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),N) ) ).

% power_mult
tff(fact_1038_plus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),zero_zero(int)),L) = L ).

% plus_int_code(2)
tff(fact_1039_plus__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),K),zero_zero(int)) = K ).

% plus_int_code(1)
tff(fact_1040_add__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)) ).

% add_mult_distrib
tff(fact_1041_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% add_mult_distrib2
tff(fact_1042_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)),U)),K) ).

% left_add_mult_distrib
tff(fact_1043_le__cube,axiom,
    ! [M: nat] : aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M))) ).

% le_cube
tff(fact_1044_le__square,axiom,
    ! [M: nat] : aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M)) ).

% le_square
tff(fact_1045_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,ord_less_eq(nat,K),L)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),L)) ) ) ).

% mult_le_mono
tff(fact_1046_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ).

% mult_le_mono1
tff(fact_1047_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ).

% mult_le_mono2
tff(fact_1048_nat__mult__1,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),N) = N ).

% nat_mult_1
tff(fact_1049_nat__mult__1__right,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),one_one(nat)) = N ).

% nat_mult_1_right
tff(fact_1050_diff__mult__distrib,axiom,
    ! [M: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,M),N)),K) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)) ).

% diff_mult_distrib
tff(fact_1051_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,minus_minus(nat,M),N)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) ).

% diff_mult_distrib2
tff(fact_1052_div__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),Q2) ).

% div_mult2_eq
tff(fact_1053_iadd__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),M),N) = zero_zero(extended_enat) )
    <=> ( ( M = zero_zero(extended_enat) )
        & ( N = zero_zero(extended_enat) ) ) ) ).

% iadd_is_0
tff(fact_1054_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_fr(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_1055_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_fs(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_1056_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),D)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D)) ) ) ) ) ) ).

% mult_mono
tff(fact_1057_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),D)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D)) ) ) ) ) ) ).

% mult_mono'
tff(fact_1058_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)) ) ).

% zero_le_square
tff(fact_1059_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B3: A] :
          ( ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) )
            | ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) ) )
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ).

% split_mult_pos_le
tff(fact_1060_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).

% mult_left_mono_neg
tff(fact_1061_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,B3),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_1062_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).

% mult_left_mono
tff(fact_1063_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ) ).

% mult_right_mono_neg
tff(fact_1064_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ) ).

% mult_right_mono
tff(fact_1065_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) )
            | ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) ) ) ) ) ).

% mult_le_0_iff
tff(fact_1066_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) )
            | ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) ) )
         => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ).

% split_mult_neg_le
tff(fact_1067_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_1068_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,B3),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_1069_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_1070_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,B3),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_1071_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) )
            | ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_1072_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_1073_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ) ).

% mult_neg_neg
tff(fact_1074_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A] : ~ aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)),zero_zero(A)) ) ).

% not_square_less_zero
tff(fact_1075_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less(A,B3),zero_zero(A)) )
            | ( aa(A,$o,ord_less(A,A3),zero_zero(A))
              & aa(A,$o,ord_less(A,zero_zero(A)),B3) ) ) ) ) ).

% mult_less_0_iff
tff(fact_1076_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ) ).

% mult_neg_pos
tff(fact_1077_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A)) ) ) ) ).

% mult_pos_neg
tff(fact_1078_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ) ) ).

% mult_pos_pos
tff(fact_1079_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),zero_zero(A)) ) ) ) ).

% mult_pos_neg2
tff(fact_1080_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
              & aa(A,$o,ord_less(A,zero_zero(A)),B3) )
            | ( aa(A,$o,ord_less(A,A3),zero_zero(A))
              & aa(A,$o,ord_less(A,B3),zero_zero(A)) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_1081_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
           => aa(A,$o,ord_less(A,zero_zero(A)),B3) ) ) ) ).

% zero_less_mult_pos
tff(fact_1082_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
           => aa(A,$o,ord_less(A,zero_zero(A)),B3) ) ) ) ).

% zero_less_mult_pos2
tff(fact_1083_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
          <=> aa(A,$o,ord_less(A,B3),A3) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_1084_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
          <=> aa(A,$o,ord_less(A,A3),B3) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_1085_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( aa(A,$o,ord_less(A,C2),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_1086_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).

% mult_strict_left_mono
tff(fact_1087_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
              & aa(A,$o,ord_less(A,A3),B3) )
            | ( aa(A,$o,ord_less(A,C2),zero_zero(A))
              & aa(A,$o,ord_less(A,B3),A3) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_1088_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( aa(A,$o,ord_less(A,C2),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_1089_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ) ).

% mult_strict_right_mono
tff(fact_1090_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
              & aa(A,$o,ord_less(A,A3),B3) )
            | ( aa(A,$o,ord_less(A,C2),zero_zero(A))
              & aa(A,$o,ord_less(A,B3),A3) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_1091_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_1092_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: A,N: A] :
          ( aa(A,$o,ord_less(A,one_one(A)),M)
         => ( aa(A,$o,ord_less(A,one_one(A)),N)
           => aa(A,$o,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M),N)) ) ) ) ).

% less_1_mult
tff(fact_1093_mult__numeral__1,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),one2)),A3) = A3 ) ).

% mult_numeral_1
tff(fact_1094_mult__numeral__1__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),one2)) = A3 ) ).

% mult_numeral_1_right
tff(fact_1095_frac__eq__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z) )
            <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),W),Y) ) ) ) ) ) ).

% frac_eq_eq
tff(fact_1096_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C2: A,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) = A3 )
        <=> $ite(C2 != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq
tff(fact_1097_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B3,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq
tff(fact_1098_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B3: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) = A3 ) ) ) ) ).

% divide_eq_imp
tff(fact_1099_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B3 )
           => ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_1100_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B3: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) = A3 )
          <=> ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1101_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A3: A,B3: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = B3 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1102_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E3: A,C2: A,B3: A,D: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E3)),D) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A3),B3)),E3)),C2) = D ) ) ) ).

% eq_add_iff1
tff(fact_1103_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,E3: A,C2: A,B3: A,D: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E3)),D) )
        <=> ( C2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B3),A3)),E3)),D) ) ) ) ).

% eq_add_iff2
tff(fact_1104_square__diff__square__factored,axiom,
    ! [A: $tType] :
      ( comm_ring(A)
     => ! [Xb: A,Y: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),aa(A,A,minus_minus(A,Xb),Y)) ) ).

% square_diff_square_factored
tff(fact_1105_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xb: A,Y: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) = one_one(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_1106_power__Suc2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),A3) ) ).

% power_Suc2
tff(fact_1107_power__Suc,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_Suc
tff(fact_1108_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N))
    <=> aa(nat,$o,ord_less(nat,M),N) ) ).

% Suc_mult_less_cancel1
tff(fact_1109_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: nat,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_add
tff(fact_1110_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N) )
      <=> ( M = N ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_1111_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))
      <=> aa(nat,$o,ord_less(nat,M),N) ) ) ).

% nat_mult_less_cancel1
tff(fact_1112_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),K)) ) ) ).

% mult_less_mono1
tff(fact_1113_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),I)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),J)) ) ) ).

% mult_less_mono2
tff(fact_1114_mult__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ).

% mult_Suc
tff(fact_1115_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),N))
    <=> aa(nat,$o,ord_less_eq(nat,M),N) ) ).

% Suc_mult_le_cancel1
tff(fact_1116_mult__eq__self__implies__10,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) )
     => ( ( N = one_one(nat) )
        | ( M = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_1117_mlex__snd__decrI,axiom,
    ! [A3: nat,A5: nat,B3: nat,B4: nat,N2: nat] :
      ( ( A3 = A5 )
     => ( aa(nat,$o,ord_less(nat,B3),B4)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N2)),B3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A5),N2)),B4)) ) ) ).

% mlex_snd_decrI
tff(fact_1118_mlex__fst__decrI,axiom,
    ! [A3: nat,A5: nat,B3: nat,N2: nat,B4: nat] :
      ( aa(nat,$o,ord_less(nat,A3),A5)
     => ( aa(nat,$o,ord_less(nat,B3),N2)
       => ( aa(nat,$o,ord_less(nat,B4),N2)
         => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N2)),B3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A5),N2)),B4)) ) ) ) ).

% mlex_fst_decrI
tff(fact_1119_mlex__bound,axiom,
    ! [A3: nat,A4: nat,B3: nat,N2: nat] :
      ( aa(nat,$o,ord_less(nat,A3),A4)
     => ( aa(nat,$o,ord_less(nat,B3),N2)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N2)),B3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A4),N2)) ) ) ).

% mlex_bound
tff(fact_1120_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),N))
     => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),I) ) ).

% less_mult_imp_div_less
tff(fact_1121_mlex__leI,axiom,
    ! [A3: nat,A5: nat,B3: nat,B4: nat,N2: nat] :
      ( aa(nat,$o,ord_less_eq(nat,A3),A5)
     => ( aa(nat,$o,ord_less_eq(nat,B3),B4)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N2)),B3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A5),N2)),B4)) ) ) ).

% mlex_leI
tff(fact_1122_div__times__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),N)),M) ).

% div_times_less_eq_dividend
tff(fact_1123_times__div__less__eq__dividend,axiom,
    ! [N: nat,M: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N))),M) ).

% times_div_less_eq_dividend
tff(fact_1124_div__mult__le,axiom,
    ! [A3: nat,B3: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),B3)),B3)),A3) ).

% div_mult_le
tff(fact_1125_odd__nonzero,axiom,
    ! [Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z) != zero_zero(int) ).

% odd_nonzero
tff(fact_1126_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,ord_less_eq(int,K),I)
     => ( aa(int,$o,P,K)
       => ( ! [I2: int] :
              ( aa(int,$o,ord_less_eq(int,K),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_ge_induct
tff(fact_1127_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,K),I)
     => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))
       => ( ! [I2: int] :
              ( aa(int,$o,ord_less(int,K),I2)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_gr_induct
tff(fact_1128_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less(int,W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),one_one(int)))
    <=> ( aa(int,$o,ord_less(int,W),Z)
        | ( W = Z ) ) ) ).

% zless_add1_eq
tff(fact_1129_power__odd__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% power_odd_eq
tff(fact_1130_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,Xb: extended_enat] :
      ( aa(extended_enat,$o,ord_less_eq(extended_enat,Z),Y)
     => ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Xb),aa(extended_enat,extended_enat,minus_minus(extended_enat,Y),Z)) = aa(extended_enat,extended_enat,minus_minus(extended_enat,aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),Xb),Y)),Z) ) ) ).

% add_diff_assoc_enat
tff(fact_1131_Suc__double__not__eq__double,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) != aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).

% Suc_double_not_eq_double
tff(fact_1132_double__not__eq__Suc__double,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% double_not_eq_Suc_double
tff(fact_1133_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less_eq(A,A3),B3) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_1134_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less_eq(A,A3),B3) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_1135_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less(A,A3),B3) ) ) ) ).

% mult_left_less_imp_less
tff(fact_1136_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,C2),D)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_1137_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A3),B3) )
            & ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B3),A3) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_1138_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less(A,A3),B3) ) ) ) ).

% mult_right_less_imp_less
tff(fact_1139_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,C2),D)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_1140_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A3),B3) )
            & ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B3),A3) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_1141_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
          <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_1142_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
          <=> aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_1143_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% mult_left_le_imp_le
tff(fact_1144_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
           => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% mult_right_le_imp_le
tff(fact_1145_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less(A,C2),D)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_1146_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),D)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
             => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_1147_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A))
        <=> ( ( Xb = zero_zero(A) )
            & ( Y = zero_zero(A) ) ) ) ) ).

% sum_squares_le_zero_iff
tff(fact_1148_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Y: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))) ) ).

% sum_squares_ge_zero
tff(fact_1149_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => ( aa(A,$o,ord_less_eq(A,Y),one_one(A))
             => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb)),Xb) ) ) ) ) ).

% mult_left_le_one_le
tff(fact_1150_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
           => ( aa(A,$o,ord_less_eq(A,Y),one_one(A))
             => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y)),Xb) ) ) ) ) ).

% mult_right_le_one_le
tff(fact_1151_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),one_one(A))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => ( aa(A,$o,ord_less_eq(A,B3),one_one(A))
             => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),one_one(A)) ) ) ) ) ).

% mult_le_one
tff(fact_1152_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),one_one(A))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),A3) ) ) ) ).

% mult_left_le
tff(fact_1153_sum__squares__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))
        <=> ( ( Xb != zero_zero(A) )
            | ( Y != zero_zero(A) ) ) ) ) ).

% sum_squares_gt_zero_iff
tff(fact_1154_not__sum__squares__lt__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Y: A] : ~ aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y))),zero_zero(A)) ) ).

% not_sum_squares_lt_zero
tff(fact_1155_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_1156_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),A3)
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3),aa(A,$o,ord_less(A,zero_zero(A)),A3)) ) ) ) ).

% divide_less_eq
tff(fact_1157_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,ord_less(A,A3),zero_zero(A))) ) ) ) ).

% less_divide_eq
tff(fact_1158_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),A3)
          <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ) ) ).

% neg_divide_less_eq
tff(fact_1159_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
          <=> aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_less_divide_eq
tff(fact_1160_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),A3)
          <=> aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_divide_less_eq
tff(fact_1161_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
          <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ) ) ).

% pos_less_divide_eq
tff(fact_1162_mult__imp__div__pos__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less(A,Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),Z) ) ) ) ).

% mult_imp_div_pos_less
tff(fact_1163_mult__imp__less__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xb)
           => aa(A,$o,ord_less(A,Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% mult_imp_less_div_pos
tff(fact_1164_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B3)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_1165_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,C2),zero_zero(A))
           => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B3)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1166_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) = aa(num,A,numeral_numeral(A),W) )
        <=> $ite(C2 != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2),aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(1)
tff(fact_1167_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B3: A,C2: A] :
          ( ( aa(num,A,numeral_numeral(A),W) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2) = B3,aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_1168_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C2: A,B3: A,D: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E3)),D))
        <=> aa(A,$o,ord_less_eq(A,C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B3),A3)),E3)),D)) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_1169_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C2: A,B3: A,D: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E3)),D))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A3),B3)),E3)),C2)),D) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_1170_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C2: A,B3: A,D: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E3)),D))
        <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A3),B3)),E3)),C2)),D) ) ) ).

% less_add_iff1
tff(fact_1171_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A3: A,E3: A,C2: A,B3: A,D: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),E3)),D))
        <=> aa(A,$o,ord_less(A,C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B3),A3)),E3)),D)) ) ) ).

% less_add_iff2
tff(fact_1172_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z: A,B3: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z)),B3) = $ite(Z = zero_zero(A),B3,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),Z))),Z)) ) ).

% add_divide_eq_if_simps(2)
tff(fact_1173_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,Z: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),Z)) = $ite(Z = zero_zero(A),A3,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z)),B3)),Z)) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1174_add__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% add_frac_eq
tff(fact_1175_add__frac__num,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_frac_num
tff(fact_1176_add__num__frac,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A] :
          ( ( Y != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))),Y) ) ) ) ).

% add_num_frac
tff(fact_1177_add__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),Y)),Z) ) ) ) ).

% add_divide_eq_iff
tff(fact_1178_divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_add_eq_iff
tff(fact_1179_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,Z: A] :
          aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),Z)) = $ite(Z = zero_zero(A),A3,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),Z)),B3)),Z)) ) ).

% add_divide_eq_if_simps(4)
tff(fact_1180_diff__frac__eq,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z)) ) ) ) ) ).

% diff_frac_eq
tff(fact_1181_diff__divide__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),Y)),Z) ) ) ) ).

% diff_divide_eq_iff
tff(fact_1182_divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z)),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% divide_diff_eq_iff
tff(fact_1183_square__diff__one__factored,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),one_one(A))),aa(A,A,minus_minus(A,Xb),one_one(A))) ) ).

% square_diff_one_factored
tff(fact_1184_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A3)
         => aa(A,$o,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).

% power_gt1_lemma
tff(fact_1185_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A3)
         => aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ) ).

% power_less_power_Suc
tff(fact_1186_one__less__mult,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),N)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ) ) ).

% one_less_mult
tff(fact_1187_n__less__m__mult__n,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) ) ) ).

% n_less_m_mult_n
tff(fact_1188_n__less__n__mult__m,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M)) ) ) ).

% n_less_n_mult_m
tff(fact_1189_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))
      <=> aa(nat,$o,ord_less_eq(nat,M),N) ) ) ).

% nat_mult_le_cancel1
tff(fact_1190_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) ) ) ).

% nat_mult_div_cancel1
tff(fact_1191_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Q2)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Q2)),N)
      <=> aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_1192_td__gal__lt,axiom,
    ! [C2: nat,A3: nat,B3: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => ( aa(nat,$o,ord_less(nat,A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),C2))
      <=> aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),C2)),B3) ) ) ).

% td_gal_lt
tff(fact_1193_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),N)) ) ) ).

% nat_diff_add_eq2
tff(fact_1194_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,J),I)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),N) ) ) ).

% nat_diff_add_eq1
tff(fact_1195_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N))
      <=> aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),N)) ) ) ).

% nat_le_add_iff2
tff(fact_1196_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,J),I)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N))
      <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),N) ) ) ).

% nat_le_add_iff1
tff(fact_1197_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N) )
      <=> ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),N) ) ) ) ).

% nat_eq_add_iff2
tff(fact_1198_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,J),I)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N) )
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M) = N ) ) ) ).

% nat_eq_add_iff1
tff(fact_1199_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,Z),zero_zero(int)) ) ).

% odd_less_0_iff
tff(fact_1200_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z)
    <=> aa(int,$o,ord_less(int,W),Z) ) ).

% add1_zle_eq
tff(fact_1201_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less(int,W),Z)
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),W),one_one(int))),Z) ) ).

% zless_imp_add1_zle
tff(fact_1202_int__induct,axiom,
    ! [P: fun(int,$o),K: int,I: int] :
      ( aa(int,$o,P,K)
     => ( ! [I2: int] :
            ( aa(int,$o,ord_less_eq(int,K),I2)
           => ( aa(int,$o,P,I2)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))) ) )
       => ( ! [I2: int] :
              ( aa(int,$o,ord_less_eq(int,I2),K)
             => ( aa(int,$o,P,I2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I2),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_induct
tff(fact_1203_power__numeral__even,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z: A,W: num] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),bit0(W))) = $let(
            w: A,
            w:= aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),W)),
            aa(A,A,aa(A,fun(A,A),times_times(A),w),w) ) ) ).

% power_numeral_even
tff(fact_1204_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,Xb: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Leaf((A3),(B3)),Xb) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_1205_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( ! [Z2: A] :
              ( aa(A,$o,ord_less(A,zero_zero(A)),Z2)
             => ( aa(A,$o,ord_less(A,Z2),one_one(A))
               => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Z2),Xb)),Y) ) )
         => aa(A,$o,ord_less_eq(A,Xb),Y) ) ) ).

% field_le_mult_one_interval
tff(fact_1206_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less_eq(A,one_one(A)),B3) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,B3),one_one(A)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1207_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),C2)
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less_eq(A,A3),one_one(A)) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,one_one(A)),A3) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1208_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less_eq(A,one_one(A)),B3) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,B3),one_one(A)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1209_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),C2)
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less_eq(A,A3),one_one(A)) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,one_one(A)),A3) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1210_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,one_one(A)),B3) )
            & ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B3),one_one(A)) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1211_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),C2)
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A3),one_one(A)) )
            & ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,one_one(A)),A3) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1212_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,one_one(A)),B3) )
            & ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B3),one_one(A)) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1213_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),C2)
        <=> ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A3),one_one(A)) )
            & ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,one_one(A)),A3) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1214_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [Xb: A,A3: A,Y: A,U: A,V: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),A3)
         => ( aa(A,$o,ord_less_eq(A,Y),A3)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),U)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),V)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A3) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1215_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),A3)
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3),aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)) ) ) ) ).

% divide_le_eq
tff(fact_1216_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,ord_less_eq(A,A3),zero_zero(A))) ) ) ) ).

% le_divide_eq
tff(fact_1217_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B3)) ) ) ) ) ).

% divide_left_mono
tff(fact_1218_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),A3)
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ) ) ).

% neg_divide_le_eq
tff(fact_1219_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
          <=> aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_le_divide_eq
tff(fact_1220_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),A3)
          <=> aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_divide_le_eq
tff(fact_1221_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ) ) ).

% pos_le_divide_eq
tff(fact_1222_mult__imp__div__pos__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A,Z: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less_eq(A,Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),Z) ) ) ) ).

% mult_imp_div_pos_le
tff(fact_1223_mult__imp__le__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xb)
           => aa(A,$o,ord_less_eq(A,Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% mult_imp_le_div_pos
tff(fact_1224_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
           => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B3)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1225_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,W: num] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),aa(num,A,numeral_numeral(A),W))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B3),aa(A,$o,ord_less(A,zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1226_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B3),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1227_mult__2,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Z) ) ).

% mult_2
tff(fact_1228_mult__2__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Z) ) ).

% mult_2_right
tff(fact_1229_left__add__twice,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),B3) ) ).

% left_add_twice
tff(fact_1230_frac__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z))
            <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A)) ) ) ) ) ).

% frac_le_eq
tff(fact_1231_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,A3),one_one(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ).

% power_Suc_less
tff(fact_1232_frac__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Z: A,Xb: A,W: A] :
          ( ( Y != zero_zero(A) )
         => ( ( Z != zero_zero(A) )
           => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),divide_divide(A),W),Z))
            <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),zero_zero(A)) ) ) ) ) ).

% frac_less_eq
tff(fact_1233_Suc__nat__number__of__add,axiom,
    ! [V: num,N: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),one2))),N) ).

% Suc_nat_number_of_add
tff(fact_1234_power4__eq__xxxx,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xb: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb)),Xb)),Xb) ) ).

% power4_eq_xxxx
tff(fact_1235_power2__eq__square,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).

% power2_eq_square
tff(fact_1236_power__even__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power_even_eq
tff(fact_1237_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)),M)
     => ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q2)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_1238_split__div,axiom,
    ! [P: fun(nat,$o),M: nat,N: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N))
    <=> ( ( ( N = zero_zero(nat) )
         => aa(nat,$o,P,zero_zero(nat)) )
        & ( ( N != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( aa(nat,$o,ord_less(nat,J3),N)
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I4)),J3) )
               => aa(nat,$o,P,I4) ) ) ) ) ) ).

% split_div
tff(fact_1239_dividend__less__div__times,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),N))) ) ).

% dividend_less_div_times
tff(fact_1240_dividend__less__times__div,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)))) ) ).

% dividend_less_times_div
tff(fact_1241_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Q2)
     => ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),Q2))
      <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)),N) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1242_td__gal,axiom,
    ! [C2: nat,B3: nat,A3: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),C2)),A3)
      <=> aa(nat,$o,ord_less_eq(nat,B3),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),C2)) ) ) ).

% td_gal
tff(fact_1243_mult__eq__if,axiom,
    ! [M: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N) = $ite(M = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,M),one_one(nat))),N))) ).

% mult_eq_if
tff(fact_1244_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,J),I)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N))
      <=> aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,I),J)),U)),M)),N) ) ) ).

% nat_less_add_iff1
tff(fact_1245_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),N))
      <=> aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,J),I)),U)),N)) ) ) ).

% nat_less_add_iff2
tff(fact_1246_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => aa(int,$o,ord_less(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ).

% le_imp_0_less
tff(fact_1247_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A3: $o,B3: $o,Va: nat] : vEBT_T_p_r_e_d2(vEBT_Leaf((A3),(B3)),aa(nat,nat,suc,aa(nat,nat,suc,Va))) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
tff(fact_1248_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o] : vEBT_T_p_r_e_d2(vEBT_Leaf((Uu),(Uv2)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
tff(fact_1249_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,N: nat] : vEBT_T_s_u_c_c2(vEBT_Leaf((Uv2),(Uw2)),aa(nat,nat,suc,N)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
tff(fact_1250_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list(vEBT_VEBT),Va: vEBT_VEBT,Vb: nat] : vEBT_T_p_r_e_d2(vEBT_Node(none(product_prod(nat,nat)),Uy,Uz,Va),Vb) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
tff(fact_1251_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu: $o,B3: $o] : vEBT_T_s_u_c_c2(vEBT_Leaf((Uu),(B3)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
tff(fact_1252_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Va: nat] : vEBT_T_s_u_c_c2(vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz),Va) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
tff(fact_1253_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [Xb: A,A3: A,Y: A,U: A,V: A] :
          ( aa(A,$o,ord_less(A,Xb),A3)
         => ( aa(A,$o,ord_less(A,Y),A3)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),U)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),V)
               => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),U),V) = one_one(A) )
                 => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),U),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),V),Y))),A3) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1254_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,W: num] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),aa(num,A,numeral_numeral(A),W))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B3),aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_1255_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B3),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_1256_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V: A,R2: A,S: A] :
          ( aa(A,$o,ord_less_eq(A,U),V)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),R2)
           => ( aa(A,$o,ord_less_eq(A,R2),S)
             => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(A,A,minus_minus(A,V),U))),S))),V) ) ) ) ) ).

% scaling_mono
tff(fact_1257_power__eq__if,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [P2: A,M: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),M) = $ite(M = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(nat,A,aa(A,fun(nat,A),power_power(A),P2),aa(nat,nat,minus_minus(nat,M),one_one(nat))))) ) ).

% power_eq_if
tff(fact_1258_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [N: nat,A3: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,minus_minus(nat,N),one_one(nat)))),A3) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) ) ) ) ).

% power_minus_mult
tff(fact_1259_split__div_H,axiom,
    ! [P: fun(nat,$o),M: nat,N: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N))
    <=> ( ( ( N = zero_zero(nat) )
          & aa(nat,$o,P,zero_zero(nat)) )
        | ? [Q4: nat] :
            ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q4)),M)
            & aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,Q4)))
            & aa(nat,$o,P,Q4) ) ) ) ).

% split_div'
tff(fact_1260_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Xb: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv2,Uw2),Xb) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_1261_power2__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xb: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)),Y)) ) ).

% power2_sum
tff(fact_1262_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A3: $o,Uw2: $o] : vEBT_T_p_r_e_d2(vEBT_Leaf((A3),(Uw2)),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
tff(fact_1263_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ) ).

% zero_le_even_power'
tff(fact_1264_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I5_J,axiom,
    ! [V: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT,Vf: nat] : vEBT_T_p_r_e_d2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vd,Ve),Vf) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(5)
tff(fact_1265_nat__bit__induct,axiom,
    ! [P: fun(nat,$o),N: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N3: nat] :
            ( aa(nat,$o,P,N3)
           => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N3)
             => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)) ) )
       => ( ! [N3: nat] :
              ( aa(nat,$o,P,N3)
             => aa(nat,$o,P,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3))) )
         => aa(nat,$o,P,N) ) ) ) ).

% nat_bit_induct
tff(fact_1266_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve: nat] : vEBT_T_s_u_c_c2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc,Vd),Ve) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
tff(fact_1267_axxdiv2,axiom,
    ! [Xb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Xb)),Xb)),aa(num,int,numeral_numeral(int),bit0(one2))) = Xb )
      & ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),zero_zero(int)),Xb)),Xb)),aa(num,int,numeral_numeral(int),bit0(one2))) = Xb ) ) ).

% axxdiv2
tff(fact_1268_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),L)
     => ( aa(int,$o,ord_less_eq(int,L),K)
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,K),L)),L)),one_one(int)) ) ) ) ).

% div_pos_geq
tff(fact_1269_field__lbound__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D1: A,D22: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),D1)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),D22)
           => ? [E2: A] :
                ( aa(A,$o,ord_less(A,zero_zero(A)),E2)
                & aa(A,$o,ord_less(A,E2),D1)
                & aa(A,$o,ord_less(A,E2),D22) ) ) ) ) ).

% field_lbound_gt_zero
tff(fact_1270_two__realpow__ge__one,axiom,
    ! [N: nat] : aa(real,$o,ord_less_eq(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N)) ).

% two_realpow_ge_one
tff(fact_1271_power2__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,minus_minus(A,Xb),Y)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)),Y)) ) ).

% power2_diff
tff(fact_1272_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))))
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_1273_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),zero_zero(A)) ) ) ).

% odd_power_less_zero
tff(fact_1274_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I3_J,axiom,
    ! [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Xb: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz),Xb) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(3)
tff(fact_1275_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I6_J,axiom,
    ! [V: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT,Vj: nat] : vEBT_T_p_r_e_d2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi),Vj) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(6)
tff(fact_1276_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I5_J,axiom,
    ! [V: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT,Vi: nat] : vEBT_T_s_u_c_c2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh),Vi) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(5)
tff(fact_1277_nat__div__eq__Suc__0__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( aa(nat,$o,ord_less_eq(nat,M),N)
        & aa(nat,$o,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% nat_div_eq_Suc_0_iff
tff(fact_1278_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Xb: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc),Xb) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_1279_field__sum__of__halves,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),aa(num,A,numeral_numeral(A),bit0(one2)))) = Xb ) ).

% field_sum_of_halves
tff(fact_1280_arith__geo__mean,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,Xb: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),U),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) )
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
           => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
             => aa(A,$o,ord_less_eq(A,U),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ) ) ).

% arith_geo_mean
tff(fact_1281_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q2: A,R2: A] :
          unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R2)) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),L)),R2),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q2)),one_one(A))),aa(A,A,minus_minus(A,R2),aa(num,A,numeral_numeral(A),L))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Q2)),R2)) ) ).

% divmod_step_eq
tff(fact_1282_power__2__mult__step__le,axiom,
    ! [N5: nat,N: nat,K4: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,N5),N)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5)),K4)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),K))
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N5)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K4),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),K)) ) ) ).

% power_2_mult_step_le
tff(fact_1283_nat__less__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
     => ( aa(nat,$o,ord_less_eq(nat,K),M)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% nat_less_power_trans
tff(fact_1284_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] : aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% sum_squares_bound
tff(fact_1285_nat__le__power__trans,axiom,
    ! [N: nat,M: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
     => ( aa(nat,$o,ord_less_eq(nat,K),M)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% nat_le_power_trans
tff(fact_1286_nat__power__less__diff,axiom,
    ! [N: nat,Q2: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
     => aa(nat,$o,ord_less(nat,Q2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),N))) ) ).

% nat_power_less_diff
tff(fact_1287_nat__add__offset__less,axiom,
    ! [Y: nat,N: nat,Xb: nat,M: nat,Sz: nat] :
      ( aa(nat,$o,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
       => ( ( Sz = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) )
         => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),Y)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Sz)) ) ) ) ).

% nat_add_offset_less
tff(fact_1288_two__powr__height__bound__deg,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).

% two_powr_height_bound_deg
tff(fact_1289_height__compose__summary,axiom,
    ! [Summary2: vEBT_VEBT,Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT)] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Summary2))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info2,Deg2,TreeList,Summary2))) ).

% height_compose_summary
tff(fact_1290_height__compose__child,axiom,
    ! [Tb: vEBT_VEBT,TreeList: list(vEBT_VEBT),Info2: option(product_prod(nat,nat)),Deg2: nat,Summary2: vEBT_VEBT] :
      ( member(vEBT_VEBT,Tb,set2(vEBT_VEBT,TreeList))
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info2,Deg2,TreeList,Summary2))) ) ).

% height_compose_child
tff(fact_1291_delete__bound__height_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => aa(nat,$o,ord_less_eq(nat,vEBT_V1232361888498592333_e_t_e(Tb,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb))) ) ).

% delete_bound_height'
tff(fact_1292_semiring__norm_I13_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M)),bit0(N)) = bit0(bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ).

% semiring_norm(13)
tff(fact_1293_semiring__norm_I11_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),times_times(num),M),one2) = M ).

% semiring_norm(11)
tff(fact_1294_semiring__norm_I12_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),one2),N) = N ).

% semiring_norm(12)
tff(fact_1295_num__double,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(one2)),N) = bit0(N) ).

% num_double
tff(fact_1296_power__mult__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A,M: num,N: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),M))),aa(num,nat,numeral_numeral(nat),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ) ).

% power_mult_numeral
tff(fact_1297_less__eq__real__def,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,Xb),Y)
    <=> ( aa(real,$o,ord_less(real,Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_real_def
tff(fact_1298_real__arch__pow__inv,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
     => ( aa(real,$o,ord_less(real,Xb),one_one(real))
       => ? [N3: nat] : aa(real,$o,ord_less(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N3)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_1299_real__arch__pow,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xb)
     => ? [N3: nat] : aa(real,$o,ord_less(real,Y),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N3)) ) ).

% real_arch_pow
tff(fact_1300_times__int__code_I1_J,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),times_times(int),K),zero_zero(int)) = zero_zero(int) ).

% times_int_code(1)
tff(fact_1301_times__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,aa(int,fun(int,int),times_times(int),zero_zero(int)),L) = zero_zero(int) ).

% times_int_code(2)
tff(fact_1302_imult__is__0,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),N) = zero_zero(extended_enat) )
    <=> ( ( M = zero_zero(extended_enat) )
        | ( N = zero_zero(extended_enat) ) ) ) ).

% imult_is_0
tff(fact_1303_four__x__squared,axiom,
    ! [Xb: real] : aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% four_x_squared
tff(fact_1304_L2__set__mult__ineq__lemma,axiom,
    ! [A3: real,C2: real,B3: real,D: real] : aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),times_times(real),A3),C2))),aa(real,real,aa(real,fun(real,real),times_times(real),B3),D))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),C2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% L2_set_mult_ineq_lemma
tff(fact_1305_div__mult2__numeral__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,K: num,L: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),L)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),K),L))) ) ).

% div_mult2_numeral_eq
tff(fact_1306_zmult__zless__mono2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,ord_less(int,I),J)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),K)
       => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),I)),aa(int,int,aa(int,fun(int,int),times_times(int),K),J)) ) ) ).

% zmult_zless_mono2
tff(fact_1307_zdiv__mult__self,axiom,
    ! [M: int,A3: int,N: int] :
      ( ( M != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),aa(int,int,aa(int,fun(int,int),times_times(int),M),N))),M) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),M)),N) ) ) ).

% zdiv_mult_self
tff(fact_1308_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N: extended_enat] :
      ( aa(extended_enat,$o,ord_less(extended_enat,zero_zero(extended_enat)),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),N))
    <=> ( aa(extended_enat,$o,ord_less(extended_enat,zero_zero(extended_enat)),M)
        & aa(extended_enat,$o,ord_less(extended_enat,zero_zero(extended_enat)),N) ) ) ).

% enat_0_less_mult_iff
tff(fact_1309_realpow__pos__nth2,axiom,
    ! [A3: real,N: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ? [R4: real] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),R4)
          & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R4),aa(nat,nat,suc,N)) = A3 ) ) ) ).

% realpow_pos_nth2
tff(fact_1310_unique__quotient__lemma__neg,axiom,
    ! [B3: int,Q5: int,R5: int,Q2: int,R2: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q5)),R5)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q2)),R2))
     => ( aa(int,$o,ord_less_eq(int,R2),zero_zero(int))
       => ( aa(int,$o,ord_less(int,B3),R2)
         => ( aa(int,$o,ord_less(int,B3),R5)
           => aa(int,$o,ord_less_eq(int,Q2),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1311_unique__quotient__lemma,axiom,
    ! [B3: int,Q5: int,R5: int,Q2: int,R2: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q5)),R5)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q2)),R2))
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R5)
       => ( aa(int,$o,ord_less(int,R5),B3)
         => ( aa(int,$o,ord_less(int,R2),B3)
           => aa(int,$o,ord_less_eq(int,Q5),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1312_zdiv__mono2__neg__lemma,axiom,
    ! [B3: int,Q2: int,R2: int,B4: int,Q5: int,R5: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q2)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q5)),R5) )
     => ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q5)),R5)),zero_zero(int))
       => ( aa(int,$o,ord_less(int,R2),B3)
         => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R5)
           => ( aa(int,$o,ord_less(int,zero_zero(int)),B4)
             => ( aa(int,$o,ord_less_eq(int,B4),B3)
               => aa(int,$o,ord_less_eq(int,Q5),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1313_zdiv__mono2__lemma,axiom,
    ! [B3: int,Q2: int,R2: int,B4: int,Q5: int,R5: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q2)),R2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q5)),R5) )
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q5)),R5))
       => ( aa(int,$o,ord_less(int,R5),B4)
         => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R2)
           => ( aa(int,$o,ord_less(int,zero_zero(int)),B4)
             => ( aa(int,$o,ord_less_eq(int,B4),B3)
               => aa(int,$o,ord_less_eq(int,Q2),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1314_q__pos__lemma,axiom,
    ! [B4: int,Q5: int,R5: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q5)),R5))
     => ( aa(int,$o,ord_less(int,R5),B4)
       => ( aa(int,$o,ord_less(int,zero_zero(int)),B4)
         => aa(int,$o,ord_less_eq(int,zero_zero(int)),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_1315_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),M)
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
      <=> ( ( M = one_one(int) )
          & ( N = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1316_zdiv__zmult2__eq,axiom,
    ! [C2: int,A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),C2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),aa(int,int,aa(int,fun(int,int),times_times(int),B3),C2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_1317_int__div__pos__eq,axiom,
    ! [A3: int,B3: int,Q2: int,R2: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q2)),R2) )
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R2)
       => ( aa(int,$o,ord_less(int,R2),B3)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1318_int__div__neg__eq,axiom,
    ! [A3: int,B3: int,Q2: int,R2: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q2)),R2) )
     => ( aa(int,$o,ord_less_eq(int,R2),zero_zero(int))
       => ( aa(int,$o,ord_less(int,B3),R2)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1319_split__zdiv,axiom,
    ! [P: fun(int,$o),N: int,K: int] :
      ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,zero_zero(int)) )
        & ( aa(int,$o,ord_less(int,zero_zero(int)),K)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),J3)
                & aa(int,$o,ord_less(int,J3),K)
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,I4) ) )
        & ( aa(int,$o,ord_less(int,K),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,ord_less(int,K),J3)
                & aa(int,$o,ord_less_eq(int,J3),zero_zero(int))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,I4) ) ) ) ) ).

% split_zdiv
tff(fact_1320_divmod__step__int__def,axiom,
    ! [L: num,Qr: product_prod(int,int)] : unique1321980374590559556d_step(int,L,Qr) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_ft(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_1321_z1pdiv2,axiom,
    ! [B3: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3)),one_one(int))),aa(num,int,numeral_numeral(int),bit0(one2))) = B3 ).

% z1pdiv2
tff(fact_1322_pred__bound__height_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_p_r_e_d2(Tb,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb))) ) ).

% pred_bound_height'
tff(fact_1323_succ_H__bound__height,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_s_u_c_c2(Tb,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb))) ) ).

% succ'_bound_height
tff(fact_1324_member__bound__height_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_m_e_m_b_e_r2(Tb,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb))) ) ).

% member_bound_height'
tff(fact_1325_realpow__pos__nth__unique,axiom,
    ! [N: nat,A3: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
       => ? [X2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),X2)
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),X2),N) = A3 )
            & ! [Y4: real] :
                ( ( aa(real,$o,ord_less(real,zero_zero(real)),Y4)
                  & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),N) = A3 ) )
               => ( Y4 = X2 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1326_realpow__pos__nth,axiom,
    ! [N: nat,A3: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
       => ? [R4: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),R4)
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R4),N) = A3 ) ) ) ) ).

% realpow_pos_nth
tff(fact_1327_height__node,axiom,
    ! [Mi: nat,Ma: nat,Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),N)
     => aa(nat,$o,ord_less_eq(nat,one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2))) ) ).

% height_node
tff(fact_1328_diff__diff__less,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,I),aa(nat,nat,minus_minus(nat,M),aa(nat,nat,minus_minus(nat,M),N)))
    <=> ( aa(nat,$o,ord_less(nat,I),M)
        & aa(nat,$o,ord_less(nat,I),N) ) ) ).

% diff_diff_less
tff(fact_1329_neg__zdiv__mult__2,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),zero_zero(int))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B3),one_one(int))),A3) ) ) ).

% neg_zdiv_mult_2
tff(fact_1330_pos__zdiv__mult__2,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B3),A3) ) ) ).

% pos_zdiv_mult_2
tff(fact_1331_pos2,axiom,
    aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% pos2
tff(fact_1332_n__less__equal__power__2,axiom,
    ! [N: nat] : aa(nat,$o,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% n_less_equal_power_2
tff(fact_1333_msrevs_I1_J,axiom,
    ! [N: nat,K: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)),M)),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),K) ) ) ).

% msrevs(1)
tff(fact_1334_nat__mult__power__less__eq,axiom,
    ! [B3: nat,A3: nat,N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),B3)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),N))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),M))
      <=> aa(nat,$o,ord_less(nat,A3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),aa(nat,nat,minus_minus(nat,M),N))) ) ) ).

% nat_mult_power_less_eq
tff(fact_1335_divmod__step__nat__def,axiom,
    ! [L: num,Qr: product_prod(nat,nat)] : unique1321980374590559556d_step(nat,L,Qr) = aa(product_prod(nat,nat),product_prod(nat,nat),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_fu(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_1336_divmod__step__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Qr: product_prod(A,A)] : unique1321980374590559556d_step(A,L,Qr) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_fv(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_1337_cnt__bound_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => aa(real,$o,ord_less_eq(real,aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Tb)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N)),one_one(real)))) ) ).

% cnt_bound'
tff(fact_1338_real__average__minus__first,axiom,
    ! [A3: real,B3: real] : aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),A3),B3)),aa(num,real,numeral_numeral(real),bit0(one2)))),A3) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,B3),A3)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% real_average_minus_first
tff(fact_1339_real__average__minus__second,axiom,
    ! [B3: real,A3: real] : aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),B3),A3)),aa(num,real,numeral_numeral(real),bit0(one2)))),A3) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,B3),A3)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% real_average_minus_second
tff(fact_1340_bset_I6_J,axiom,
    ! [D5: int,B6: set(int),Tb: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D5)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B6)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
         => ( aa(int,$o,ord_less_eq(int,X3),Tb)
           => aa(int,$o,ord_less_eq(int,aa(int,int,minus_minus(int,X3),D5)),Tb) ) ) ) ).

% bset(6)
tff(fact_1341_bset_I8_J,axiom,
    ! [D5: int,Tb: int,B6: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D5)
     => ( member(int,aa(int,int,minus_minus(int,Tb),one_one(int)),B6)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B6)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( aa(int,$o,ord_less_eq(int,Tb),X3)
             => aa(int,$o,ord_less_eq(int,Tb),aa(int,int,minus_minus(int,X3),D5)) ) ) ) ) ).

% bset(8)
tff(fact_1342_aset_I6_J,axiom,
    ! [D5: int,Tb: int,A4: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D5)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Tb),one_one(int)),A4)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A4)
                   => ( X3 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
           => ( aa(int,$o,ord_less_eq(int,X3),Tb)
             => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5)),Tb) ) ) ) ) ).

% aset(6)
tff(fact_1343_aset_I8_J,axiom,
    ! [D5: int,A4: set(int),Tb: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D5)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A4)
                 => ( X3 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
         => ( aa(int,$o,ord_less_eq(int,Tb),X3)
           => aa(int,$o,ord_less_eq(int,Tb),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5)) ) ) ) ).

% aset(8)
tff(fact_1344_pred__empty,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( vEBT_vebt_pred(Tb,Xb) = none(nat) )
      <=> ( collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_fw(vEBT_VEBT,fun(nat,fun(nat,$o)),Tb),Xb)) = bot_bot(set(nat)) ) ) ) ).

% pred_empty
tff(fact_1345_cnt__non__neg,axiom,
    ! [Tb: vEBT_VEBT] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Tb)) ).

% cnt_non_neg
tff(fact_1346_mint__corr__help__empty,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( vEBT_vebt_mint(Tb) = none(nat) )
       => ( vEBT_VEBT_set_vebt(Tb) = bot_bot(set(nat)) ) ) ) ).

% mint_corr_help_empty
tff(fact_1347_maxt__corr__help__empty,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( vEBT_vebt_maxt(Tb) = none(nat) )
       => ( vEBT_VEBT_set_vebt(Tb) = bot_bot(set(nat)) ) ) ) ).

% maxt_corr_help_empty
tff(fact_1348_succ__empty,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( vEBT_vebt_succ(Tb,Xb) = none(nat) )
      <=> ( collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_fx(vEBT_VEBT,fun(nat,fun(nat,$o)),Tb),Xb)) = bot_bot(set(nat)) ) ) ) ).

% succ_empty
tff(fact_1349_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( ( set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% atLeastatMost_empty_iff
tff(fact_1350_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A3,B3) )
        <=> ~ aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_1351_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_1352_not__real__square__gt__zero,axiom,
    ! [Xb: real] :
      ( ~ aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Xb))
    <=> ( Xb = zero_zero(real) ) ) ).

% not_real_square_gt_zero
tff(fact_1353_minf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F5: B] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z2)
         => ( F5 = F5 ) ) ) ).

% minf(11)
tff(fact_1354_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Tb: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z2)
         => ~ aa(A,$o,ord_less(A,Tb),X3) ) ) ).

% minf(7)
tff(fact_1355_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Tb: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z2)
         => aa(A,$o,ord_less(A,X3),Tb) ) ) ).

% minf(5)
tff(fact_1356_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Tb: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z2)
         => ( X3 != Tb ) ) ) ).

% minf(4)
tff(fact_1357_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Tb: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z2)
         => ( X3 != Tb ) ) ) ).

% minf(3)
tff(fact_1358_minf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q6: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X2: A] :
              ( aa(A,$o,ord_less(A,X2),Z4)
             => ( aa(A,$o,P,X2)
              <=> aa(A,$o,P3,X2) ) )
         => ( ? [Z4: A] :
              ! [X2: A] :
                ( aa(A,$o,ord_less(A,X2),Z4)
               => ( aa(A,$o,Q,X2)
                <=> aa(A,$o,Q6,X2) ) )
           => ? [Z2: A] :
              ! [X3: A] :
                ( aa(A,$o,ord_less(A,X3),Z2)
               => ( ( aa(A,$o,P,X3)
                    | aa(A,$o,Q,X3) )
                <=> ( aa(A,$o,P3,X3)
                    | aa(A,$o,Q6,X3) ) ) ) ) ) ) ).

% minf(2)
tff(fact_1359_minf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q6: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X2: A] :
              ( aa(A,$o,ord_less(A,X2),Z4)
             => ( aa(A,$o,P,X2)
              <=> aa(A,$o,P3,X2) ) )
         => ( ? [Z4: A] :
              ! [X2: A] :
                ( aa(A,$o,ord_less(A,X2),Z4)
               => ( aa(A,$o,Q,X2)
                <=> aa(A,$o,Q6,X2) ) )
           => ? [Z2: A] :
              ! [X3: A] :
                ( aa(A,$o,ord_less(A,X3),Z2)
               => ( ( aa(A,$o,P,X3)
                    & aa(A,$o,Q,X3) )
                <=> ( aa(A,$o,P3,X3)
                    & aa(A,$o,Q6,X3) ) ) ) ) ) ) ).

% minf(1)
tff(fact_1360_pinf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F5: B] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z2),X3)
         => ( F5 = F5 ) ) ) ).

% pinf(11)
tff(fact_1361_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Tb: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z2),X3)
         => aa(A,$o,ord_less(A,Tb),X3) ) ) ).

% pinf(7)
tff(fact_1362_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Tb: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z2),X3)
         => ~ aa(A,$o,ord_less(A,X3),Tb) ) ) ).

% pinf(5)
tff(fact_1363_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Tb: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z2),X3)
         => ( X3 != Tb ) ) ) ).

% pinf(4)
tff(fact_1364_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Tb: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z2),X3)
         => ( X3 != Tb ) ) ) ).

% pinf(3)
tff(fact_1365_pinf_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q6: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X2: A] :
              ( aa(A,$o,ord_less(A,Z4),X2)
             => ( aa(A,$o,P,X2)
              <=> aa(A,$o,P3,X2) ) )
         => ( ? [Z4: A] :
              ! [X2: A] :
                ( aa(A,$o,ord_less(A,Z4),X2)
               => ( aa(A,$o,Q,X2)
                <=> aa(A,$o,Q6,X2) ) )
           => ? [Z2: A] :
              ! [X3: A] :
                ( aa(A,$o,ord_less(A,Z2),X3)
               => ( ( aa(A,$o,P,X3)
                    | aa(A,$o,Q,X3) )
                <=> ( aa(A,$o,P3,X3)
                    | aa(A,$o,Q6,X3) ) ) ) ) ) ) ).

% pinf(2)
tff(fact_1366_pinf_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o),P3: fun(A,$o),Q: fun(A,$o),Q6: fun(A,$o)] :
          ( ? [Z4: A] :
            ! [X2: A] :
              ( aa(A,$o,ord_less(A,Z4),X2)
             => ( aa(A,$o,P,X2)
              <=> aa(A,$o,P3,X2) ) )
         => ( ? [Z4: A] :
              ! [X2: A] :
                ( aa(A,$o,ord_less(A,Z4),X2)
               => ( aa(A,$o,Q,X2)
                <=> aa(A,$o,Q6,X2) ) )
           => ? [Z2: A] :
              ! [X3: A] :
                ( aa(A,$o,ord_less(A,Z2),X3)
               => ( ( aa(A,$o,P,X3)
                    & aa(A,$o,Q,X3) )
                <=> ( aa(A,$o,P3,X3)
                    & aa(A,$o,Q6,X3) ) ) ) ) ) ) ).

% pinf(1)
tff(fact_1367_mult__commute__abs,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [C2: A] : aTP_Lamp_fy(A,fun(A,A),C2) = aa(A,fun(A,A),times_times(A),C2) ) ).

% mult_commute_abs
tff(fact_1368_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Tb: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z2)
         => ~ aa(A,$o,ord_less_eq(A,Tb),X3) ) ) ).

% minf(8)
tff(fact_1369_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Tb: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z2)
         => aa(A,$o,ord_less_eq(A,X3),Tb) ) ) ).

% minf(6)
tff(fact_1370_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Tb: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z2),X3)
         => aa(A,$o,ord_less_eq(A,Tb),X3) ) ) ).

% pinf(8)
tff(fact_1371_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Tb: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z2),X3)
         => ~ aa(A,$o,ord_less_eq(A,X3),Tb) ) ) ).

% pinf(6)
tff(fact_1372_imp__le__cong,axiom,
    ! [Xb: int,X4: int,P: $o,P3: $o] :
      ( ( Xb = X4 )
     => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X4)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
           => (P) )
        <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X4)
           => (P3) ) ) ) ) ).

% imp_le_cong
tff(fact_1373_conj__le__cong,axiom,
    ! [Xb: int,X4: int,P: $o,P3: $o] :
      ( ( Xb = X4 )
     => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X4)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
            & (P) )
        <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X4)
            & (P3) ) ) ) ) ).

% conj_le_cong
tff(fact_1374_plusinfinity,axiom,
    ! [D: int,P3: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D)
     => ( ! [X2: int,K2: int] :
            ( aa(int,$o,P3,X2)
          <=> aa(int,$o,P3,aa(int,int,minus_minus(int,X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D))) )
       => ( ? [Z4: int] :
            ! [X2: int] :
              ( aa(int,$o,ord_less(int,Z4),X2)
             => ( aa(int,$o,P,X2)
              <=> aa(int,$o,P3,X2) ) )
         => ( ? [X_13: int] : aa(int,$o,P3,X_13)
           => ? [X_1: int] : aa(int,$o,P,X_1) ) ) ) ) ).

% plusinfinity
tff(fact_1375_minusinfinity,axiom,
    ! [D: int,P1: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D)
     => ( ! [X2: int,K2: int] :
            ( aa(int,$o,P1,X2)
          <=> aa(int,$o,P1,aa(int,int,minus_minus(int,X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D))) )
       => ( ? [Z4: int] :
            ! [X2: int] :
              ( aa(int,$o,ord_less(int,X2),Z4)
             => ( aa(int,$o,P,X2)
              <=> aa(int,$o,P1,X2) ) )
         => ( ? [X_13: int] : aa(int,$o,P1,X_13)
           => ? [X_1: int] : aa(int,$o,P,X_1) ) ) ) ) ).

% minusinfinity
tff(fact_1376_bset_I1_J,axiom,
    ! [D5: int,B6: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B6)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X2),D5)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B6)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,minus_minus(int,X2),D5)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B6)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( ( aa(int,$o,P,X3)
                & aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,minus_minus(int,X3),D5))
                & aa(int,$o,Q,aa(int,int,minus_minus(int,X3),D5)) ) ) ) ) ) ).

% bset(1)
tff(fact_1377_bset_I2_J,axiom,
    ! [D5: int,B6: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B6)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X2),D5)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B6)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,minus_minus(int,X2),D5)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B6)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( ( aa(int,$o,P,X3)
                | aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,minus_minus(int,X3),D5))
                | aa(int,$o,Q,aa(int,int,minus_minus(int,X3),D5)) ) ) ) ) ) ).

% bset(2)
tff(fact_1378_aset_I1_J,axiom,
    ! [D5: int,A4: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A4)
                 => ( X2 != aa(int,int,minus_minus(int,Xb3),Xa) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D5)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A4)
                   => ( X2 != aa(int,int,minus_minus(int,Xb3),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D5)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A4)
                   => ( X3 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
           => ( ( aa(int,$o,P,X3)
                & aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5)) ) ) ) ) ) ).

% aset(1)
tff(fact_1379_aset_I2_J,axiom,
    ! [D5: int,A4: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A4)
                 => ( X2 != aa(int,int,minus_minus(int,Xb3),Xa) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D5)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A4)
                   => ( X2 != aa(int,int,minus_minus(int,Xb3),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D5)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A4)
                   => ( X3 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
           => ( ( aa(int,$o,P,X3)
                | aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5)) ) ) ) ) ) ).

% aset(2)
tff(fact_1380_incr__mult__lemma,axiom,
    ! [D: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D)
     => ( ! [X2: int] :
            ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D)) )
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
         => ! [X3: int] :
              ( aa(int,$o,P,X3)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1381_decr__mult__lemma,axiom,
    ! [D: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D)
     => ( ! [X2: int] :
            ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X2),D)) )
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
         => ! [X3: int] :
              ( aa(int,$o,P,X3)
             => aa(int,$o,P,aa(int,int,minus_minus(int,X3),aa(int,int,aa(int,fun(int,int),times_times(int),K),D))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1382_periodic__finite__ex,axiom,
    ! [D: int,P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D)
     => ( ! [X2: int,K2: int] :
            ( aa(int,$o,P,X2)
          <=> aa(int,$o,P,aa(int,int,minus_minus(int,X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D))) )
       => ( ? [X_12: int] : aa(int,$o,P,X_12)
        <=> ? [X: int] :
              ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D))
              & aa(int,$o,P,X) ) ) ) ) ).

% periodic_finite_ex
tff(fact_1383_aset_I7_J,axiom,
    ! [D5: int,A4: set(int),Tb: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D5)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A4)
                 => ( X3 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
         => ( aa(int,$o,ord_less(int,Tb),X3)
           => aa(int,$o,ord_less(int,Tb),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5)) ) ) ) ).

% aset(7)
tff(fact_1384_aset_I5_J,axiom,
    ! [D5: int,Tb: int,A4: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D5)
     => ( member(int,Tb,A4)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A4)
                   => ( X3 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
           => ( aa(int,$o,ord_less(int,X3),Tb)
             => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5)),Tb) ) ) ) ) ).

% aset(5)
tff(fact_1385_aset_I4_J,axiom,
    ! [D5: int,Tb: int,A4: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D5)
     => ( member(int,Tb,A4)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A4)
                   => ( X3 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
           => ( ( X3 != Tb )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5) != Tb ) ) ) ) ) ).

% aset(4)
tff(fact_1386_aset_I3_J,axiom,
    ! [D5: int,Tb: int,A4: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D5)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Tb),one_one(int)),A4)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A4)
                   => ( X3 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
           => ( ( X3 = Tb )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5) = Tb ) ) ) ) ) ).

% aset(3)
tff(fact_1387_bset_I7_J,axiom,
    ! [D5: int,Tb: int,B6: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D5)
     => ( member(int,Tb,B6)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B6)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( aa(int,$o,ord_less(int,Tb),X3)
             => aa(int,$o,ord_less(int,Tb),aa(int,int,minus_minus(int,X3),D5)) ) ) ) ) ).

% bset(7)
tff(fact_1388_bset_I5_J,axiom,
    ! [D5: int,B6: set(int),Tb: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D5)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B6)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
         => ( aa(int,$o,ord_less(int,X3),Tb)
           => aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,X3),D5)),Tb) ) ) ) ).

% bset(5)
tff(fact_1389_bset_I4_J,axiom,
    ! [D5: int,Tb: int,B6: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D5)
     => ( member(int,Tb,B6)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B6)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( ( X3 != Tb )
             => ( aa(int,int,minus_minus(int,X3),D5) != Tb ) ) ) ) ) ).

% bset(4)
tff(fact_1390_bset_I3_J,axiom,
    ! [D5: int,Tb: int,B6: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D5)
     => ( member(int,aa(int,int,minus_minus(int,Tb),one_one(int)),B6)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B6)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
           => ( ( X3 = Tb )
             => ( aa(int,int,minus_minus(int,X3),D5) = Tb ) ) ) ) ) ).

% bset(3)
tff(fact_1391_cppi,axiom,
    ! [D5: int,P: fun(int,$o),P3: fun(int,$o),A4: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D5)
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa(int,$o,ord_less(int,Z4),X2)
           => ( aa(int,$o,P,X2)
            <=> aa(int,$o,P3,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D5))
                 => ! [Xb3: int] :
                      ( member(int,Xb3,A4)
                     => ( X2 != aa(int,int,minus_minus(int,Xb3),Xa) ) ) )
             => ( aa(int,$o,P,X2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D5)) ) )
         => ( ! [X2: int,K2: int] :
                ( aa(int,$o,P3,X2)
              <=> aa(int,$o,P3,aa(int,int,minus_minus(int,X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D5))) )
           => ( ? [X_12: int] : aa(int,$o,P,X_12)
            <=> ( ? [X: int] :
                    ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & aa(int,$o,P3,X) )
                | ? [X: int] :
                    ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & ? [Xa3: int] :
                        ( member(int,Xa3,A4)
                        & aa(int,$o,P,aa(int,int,minus_minus(int,Xa3),X)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_1392_cpmi,axiom,
    ! [D5: int,P: fun(int,$o),P3: fun(int,$o),B6: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D5)
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa(int,$o,ord_less(int,X2),Z4)
           => ( aa(int,$o,P,X2)
            <=> aa(int,$o,P3,X2) ) )
       => ( ! [X2: int] :
              ( ! [Xa: int] :
                  ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D5))
                 => ! [Xb3: int] :
                      ( member(int,Xb3,B6)
                     => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa) ) ) )
             => ( aa(int,$o,P,X2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,X2),D5)) ) )
         => ( ! [X2: int,K2: int] :
                ( aa(int,$o,P3,X2)
              <=> aa(int,$o,P3,aa(int,int,minus_minus(int,X2),aa(int,int,aa(int,fun(int,int),times_times(int),K2),D5))) )
           => ( ? [X_12: int] : aa(int,$o,P,X_12)
            <=> ( ? [X: int] :
                    ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & aa(int,$o,P3,X) )
                | ? [X: int] :
                    ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D5))
                    & ? [Xa3: int] :
                        ( member(int,Xa3,B6)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa3),X)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_1393_count__buildup,axiom,
    ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N))) ).

% count_buildup
tff(fact_1394_cnt__bound,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => aa(real,$o,ord_less_eq(real,aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Tb)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2)))))))) ) ).

% cnt_bound
tff(fact_1395_vebt__succ_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = $ite((B2),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(B2))),zero_zero(nat))) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( Xb = vEBT_Leaf((Uv),(Uw)) )
               => ! [N3: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N3) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv),(Uw))),aa(nat,nat,suc,N3))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = none(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xaa)) ) )
             => ( ! [V3: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc2,Vd2) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc2,Vd2)),Xaa)) ) )
               => ( ! [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2) )
                     => ( ( Y = none(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( ( Y = $ite(
                                aa(nat,$o,ord_less(nat,Xaa),Mi2),
                                aa(nat,option(nat),some(nat),Mi2),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(
                                      aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                      $let(
                                        maxlow: option(nat),
                                        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                          aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_succ(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                          $let(
                                            sc: option(nat),
                                            sc:= vEBT_vebt_succ(Summary,h),
                                            $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                                      none(nat) ) ) ) ) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_1396_psubsetI,axiom,
    ! [A: $tType,A4: set(A),B6: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A4),B6)
     => ( ( A4 != B6 )
       => aa(set(A),$o,ord_less(set(A),A4),B6) ) ) ).

% psubsetI
tff(fact_1397_pos__mult__pos__ge,axiom,
    ! [Xb: int,N: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),Xb)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),N)
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),times_times(int),N),one_one(int))),aa(int,int,aa(int,fun(int,int),times_times(int),N),Xb)) ) ) ).

% pos_mult_pos_ge
tff(fact_1398_vebt__pred_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = none(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv))),zero_zero(nat))) ) ) )
         => ( ! [A2: $o,Uw: $o] :
                ( ( Xb = vEBT_Leaf((A2),(Uw)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(Uw))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A2: $o,B2: $o] :
                  ( ( Xb = vEBT_Leaf((A2),(B2)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                     => ( ( Y = $ite(
                              (B2),
                              aa(nat,option(nat),some(nat),one_one(nat)),
                              $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3) )
                   => ( ( Y = none(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Xaa)) ) )
               => ( ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = none(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2)),Xaa)) ) )
                 => ( ! [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2) )
                       => ( ( Y = none(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                          ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = $ite(
                                  aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                  aa(nat,option(nat),some(nat),Ma2),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          $ite(
                                            ( ( minlow != none(nat) )
                                            & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                            aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),h))),vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                            $let(
                                              pr: option(nat),
                                              pr:= vEBT_vebt_pred(Summary,h),
                                              $ite(
                                                pr = none(nat),
                                                $ite(aa(nat,$o,ord_less(nat,Mi2),Xaa),aa(nat,option(nat),some(nat),Mi2),none(nat)),
                                                aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                                        none(nat) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_1399_word__size__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(word(A),nat,size_size(word(A)),W)) ) ).

% word_size_gt_0
tff(fact_1400_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),zero_zero(nat)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% set_bit_0
tff(fact_1401_buildup__nothing__in__min__max,axiom,
    ! [N: nat,Xb: nat] : ~ vEBT_VEBT_membermima(vEBT_vebt_buildup(N),Xb) ).

% buildup_nothing_in_min_max
tff(fact_1402_buildup__nothing__in__leaf,axiom,
    ! [N: nat,Xb: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(N),Xb) ).

% buildup_nothing_in_leaf
tff(fact_1403_buildup__gives__empty,axiom,
    ! [N: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(N)) = bot_bot(set(nat)) ).

% buildup_gives_empty
tff(fact_1404_buildup__gives__valid,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => vEBT_invar_vebt(vEBT_vebt_buildup(N),N) ) ).

% buildup_gives_valid
tff(fact_1405_semiring__norm_I90_J,axiom,
    ! [M: num,N: num] :
      ( ( aa(num,num,bit1,M) = aa(num,num,bit1,N) )
    <=> ( M = N ) ) ).

% semiring_norm(90)
tff(fact_1406_semiring__norm_I88_J,axiom,
    ! [M: num,N: num] : bit0(M) != aa(num,num,bit1,N) ).

% semiring_norm(88)
tff(fact_1407_semiring__norm_I89_J,axiom,
    ! [M: num,N: num] : aa(num,num,bit1,M) != bit0(N) ).

% semiring_norm(89)
tff(fact_1408_semiring__norm_I84_J,axiom,
    ! [N: num] : one2 != aa(num,num,bit1,N) ).

% semiring_norm(84)
tff(fact_1409_semiring__norm_I86_J,axiom,
    ! [M: num] : aa(num,num,bit1,M) != one2 ).

% semiring_norm(86)
tff(fact_1410_set__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_1411_set__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% set_bit_negative_int_iff
tff(fact_1412_semiring__norm_I80_J,axiom,
    ! [M: num,N: num] :
      ( aa(num,$o,ord_less(num,aa(num,num,bit1,M)),aa(num,num,bit1,N))
    <=> aa(num,$o,ord_less(num,M),N) ) ).

% semiring_norm(80)
tff(fact_1413_semiring__norm_I73_J,axiom,
    ! [M: num,N: num] :
      ( aa(num,$o,ord_less_eq(num,aa(num,num,bit1,M)),aa(num,num,bit1,N))
    <=> aa(num,$o,ord_less_eq(num,M),N) ) ).

% semiring_norm(73)
tff(fact_1414_semiring__norm_I9_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),bit0(N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% semiring_norm(9)
tff(fact_1415_semiring__norm_I7_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ).

% semiring_norm(7)
tff(fact_1416_semiring__norm_I14_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M)),aa(num,num,bit1,N)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),aa(num,num,bit1,N))) ).

% semiring_norm(14)
tff(fact_1417_semiring__norm_I15_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),bit0(N)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),N)) ).

% semiring_norm(15)
tff(fact_1418_semiring__norm_I81_J,axiom,
    ! [M: num,N: num] :
      ( aa(num,$o,ord_less(num,aa(num,num,bit1,M)),bit0(N))
    <=> aa(num,$o,ord_less(num,M),N) ) ).

% semiring_norm(81)
tff(fact_1419_semiring__norm_I72_J,axiom,
    ! [M: num,N: num] :
      ( aa(num,$o,ord_less_eq(num,bit0(M)),aa(num,num,bit1,N))
    <=> aa(num,$o,ord_less_eq(num,M),N) ) ).

% semiring_norm(72)
tff(fact_1420_semiring__norm_I77_J,axiom,
    ! [N: num] : aa(num,$o,ord_less(num,one2),aa(num,num,bit1,N)) ).

% semiring_norm(77)
tff(fact_1421_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa(num,$o,ord_less_eq(num,aa(num,num,bit1,M)),one2) ).

% semiring_norm(70)
tff(fact_1422_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,V))),aa(num,int,numeral_numeral(int),bit0(W))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),W)) ).

% zdiv_numeral_Bit1
tff(fact_1423_semiring__norm_I3_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bit0(N)) = aa(num,num,bit1,N) ).

% semiring_norm(3)
tff(fact_1424_semiring__norm_I4_J,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),aa(num,num,bit1,N)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),N),one2)) ).

% semiring_norm(4)
tff(fact_1425_semiring__norm_I5_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),one2) = aa(num,num,bit1,M) ).

% semiring_norm(5)
tff(fact_1426_semiring__norm_I8_J,axiom,
    ! [M: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),one2) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2)) ).

% semiring_norm(8)
tff(fact_1427_semiring__norm_I10_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)),one2)) ).

% semiring_norm(10)
tff(fact_1428_semiring__norm_I16_J,axiom,
    ! [M: num,N: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),aa(num,num,bit1,N)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)),bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),N)))) ).

% semiring_norm(16)
tff(fact_1429_semiring__norm_I79_J,axiom,
    ! [M: num,N: num] :
      ( aa(num,$o,ord_less(num,bit0(M)),aa(num,num,bit1,N))
    <=> aa(num,$o,ord_less_eq(num,M),N) ) ).

% semiring_norm(79)
tff(fact_1430_semiring__norm_I74_J,axiom,
    ! [M: num,N: num] :
      ( aa(num,$o,ord_less_eq(num,aa(num,num,bit1,M)),bit0(N))
    <=> aa(num,$o,ord_less(num,M),N) ) ).

% semiring_norm(74)
tff(fact_1431_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_div_eq_add3_div_numeral
tff(fact_1432_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).

% div_Suc_eq_div_add3
tff(fact_1433_set__bit__greater__eq,axiom,
    ! [K: int,N: nat] : aa(int,$o,ord_less_eq(int,K),aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K)) ).

% set_bit_greater_eq
tff(fact_1434_xor__num_Ocases,axiom,
    ! [Xb: product_prod(num,num)] :
      ( ( Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2) )
     => ( ! [N3: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),bit0(N3))
       => ( ! [N3: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,N3))
         => ( ! [M3: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M3)),one2)
           => ( ! [M3: num,N3: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M3)),bit0(N3))
             => ( ! [M3: num,N3: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M3)),aa(num,num,bit1,N3))
               => ( ! [M3: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),one2)
                 => ( ! [M3: num,N3: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),bit0(N3))
                   => ~ ! [M3: num,N3: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M3)),aa(num,num,bit1,N3)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_1435_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one2 )
     => ( ! [X23: num] : Y != bit0(X23)
       => ~ ! [X32: num] : Y != aa(num,num,bit1,X32) ) ) ).

% num.exhaust
tff(fact_1436_vebt__buildup_Osimps_I1_J,axiom,
    vEBT_vebt_buildup(zero_zero(nat)) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(1)
tff(fact_1437_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),N))),one_one(A)) ) ).

% numeral_Bit1
tff(fact_1438_eval__nat__numeral_I3_J,axiom,
    ! [N: num] : aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,N)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bit0(N))) ).

% eval_nat_numeral(3)
tff(fact_1439_le__minus,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & order(A) )
     => ! [Y: A,Xb: A,A3: word(B),C2: word(B),B3: word(B)] :
          ( aa(A,$o,ord_less_eq(A,Y),Xb)
         => ( aa(word(B),$o,ord_less_eq(word(B),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),A3),C2)),B3)
           => ( aa(word(B),$o,ord_less_eq(word(B),A3),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),A3),C2))
             => aa(word(B),$o,ord_less_eq(word(B),C2),aa(word(B),word(B),minus_minus(word(B),B3),A3)) ) ) ) ) ).

% le_minus
tff(fact_1440_size__0__same_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( ( aa(word(A),nat,size_size(word(A)),W) = zero_zero(nat) )
         => ( W = V ) ) ) ).

% size_0_same'
tff(fact_1441_lens__not__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),nat,size_size(word(A)),W) != zero_zero(nat) ) ).

% lens_not_0
tff(fact_1442_size__0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( ( aa(word(A),nat,size_size(word(A)),W) = zero_zero(nat) )
         => ( V = W ) ) ) ).

% size_0_eq
tff(fact_1443_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [N: num] :
          aa(num,A,numeral_numeral(A),aa(num,num,bit1,N)) = $let(
            m: A,
            m:= aa(num,A,numeral_numeral(A),N),
            aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),m),m)),one_one(A)) ) ) ).

% numeral_code(3)
tff(fact_1444_power__numeral__odd,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Z: A,W: num] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,W))) = $let(
            w: A,
            w:= aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(num,nat,numeral_numeral(nat),W)),
            aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),w)),w) ) ) ).

% power_numeral_odd
tff(fact_1445_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A4: set(A),B6: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A4),B6)
     => ? [B2: A] : member(A,B2,aa(set(A),set(A),minus_minus(set(A),B6),A4)) ) ).

% psubset_imp_ex_mem
tff(fact_1446_psubset__trans,axiom,
    ! [A: $tType,A4: set(A),B6: set(A),C5: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A4),B6)
     => ( aa(set(A),$o,ord_less(set(A),B6),C5)
       => aa(set(A),$o,ord_less(set(A),A4),C5) ) ) ).

% psubset_trans
tff(fact_1447_psubsetD,axiom,
    ! [A: $tType,A4: set(A),B6: set(A),C2: A] :
      ( aa(set(A),$o,ord_less(set(A),A4),B6)
     => ( member(A,C2,A4)
       => member(A,C2,B6) ) ) ).

% psubsetD
tff(fact_1448_vebt__buildup_Osimps_I2_J,axiom,
    vEBT_vebt_buildup(aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf($false,$false) ).

% vebt_buildup.simps(2)
tff(fact_1449_numeral__Bit1__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(num,A,numeral_numeral(A),N) ) ).

% numeral_Bit1_div_2
tff(fact_1450_power3__eq__cube,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3)),A3) ) ).

% power3_eq_cube
tff(fact_1451_numeral__3__eq__3,axiom,
    aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) = aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))) ).

% numeral_3_eq_3
tff(fact_1452_Suc3__eq__add__3,axiom,
    ! [N: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N) ).

% Suc3_eq_add_3
tff(fact_1453_Set_Oempty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = collect(A,aTP_Lamp_dp(A,$o)) ).

% Set.empty_def
tff(fact_1454_num_Osize_I6_J,axiom,
    ! [X33: num] : aa(num,nat,size_size(num),aa(num,num,bit1,X33)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X33)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(6)
tff(fact_1455_less__set__def,axiom,
    ! [A: $tType,A4: set(A),B6: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A4),B6)
    <=> aa(fun(A,$o),$o,ord_less(fun(A,$o),aTP_Lamp_cc(set(A),fun(A,$o),A4)),aTP_Lamp_cc(set(A),fun(A,$o),B6)) ) ).

% less_set_def
tff(fact_1456_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),N) ).

% Suc_div_eq_add3_div
tff(fact_1457_not__psubset__empty,axiom,
    ! [A: $tType,A4: set(A)] : ~ aa(set(A),$o,ord_less(set(A),A4),bot_bot(set(A))) ).

% not_psubset_empty
tff(fact_1458_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A4: set(A),B6: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A4),B6)
    <=> ( aa(set(A),$o,ord_less(set(A),A4),B6)
        | ( A4 = B6 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1459_subset__psubset__trans,axiom,
    ! [A: $tType,A4: set(A),B6: set(A),C5: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),A4),B6)
     => ( aa(set(A),$o,ord_less(set(A),B6),C5)
       => aa(set(A),$o,ord_less(set(A),A4),C5) ) ) ).

% subset_psubset_trans
tff(fact_1460_subset__not__subset__eq,axiom,
    ! [A: $tType,A4: set(A),B6: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A4),B6)
    <=> ( aa(set(A),$o,ord_less_eq(set(A),A4),B6)
        & ~ aa(set(A),$o,ord_less_eq(set(A),B6),A4) ) ) ).

% subset_not_subset_eq
tff(fact_1461_psubset__subset__trans,axiom,
    ! [A: $tType,A4: set(A),B6: set(A),C5: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A4),B6)
     => ( aa(set(A),$o,ord_less_eq(set(A),B6),C5)
       => aa(set(A),$o,ord_less(set(A),A4),C5) ) ) ).

% psubset_subset_trans
tff(fact_1462_psubset__imp__subset,axiom,
    ! [A: $tType,A4: set(A),B6: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A4),B6)
     => aa(set(A),$o,ord_less_eq(set(A),A4),B6) ) ).

% psubset_imp_subset
tff(fact_1463_psubset__eq,axiom,
    ! [A: $tType,A4: set(A),B6: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A4),B6)
    <=> ( aa(set(A),$o,ord_less_eq(set(A),A4),B6)
        & ( A4 != B6 ) ) ) ).

% psubset_eq
tff(fact_1464_psubsetE,axiom,
    ! [A: $tType,A4: set(A),B6: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A4),B6)
     => ~ ( aa(set(A),$o,ord_less_eq(set(A),A4),B6)
         => aa(set(A),$o,ord_less_eq(set(A),B6),A4) ) ) ).

% psubsetE
tff(fact_1465_small__powers__of__2,axiom,
    ! [Xb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Xb)
     => aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Xb),one_one(nat)))) ) ).

% small_powers_of_2
tff(fact_1466_p2__eq__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) = one_one(word(A)) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% p2_eq_1
tff(fact_1467_word__less__two__pow__divD,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: nat,M: nat] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)))
         => ( aa(nat,$o,ord_less_eq(nat,M),N)
            & aa(word(A),$o,ord_less(word(A),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ).

% word_less_two_pow_divD
tff(fact_1468_less__1__helper,axiom,
    ! [N: int,M: int] :
      ( aa(int,$o,ord_less_eq(int,N),M)
     => aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,N),one_one(int))),M) ) ).

% less_1_helper
tff(fact_1469_space__bound,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) )
       => aa(nat,$o,ord_less_eq(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_space,Tb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),U)) ) ) ).

% space_bound
tff(fact_1470_space_H__bound,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) )
       => aa(nat,$o,ord_less_eq(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Tb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),U)) ) ) ).

% space'_bound
tff(fact_1471_delete__bound__height,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_d_e_l_e_t_e(Tb,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb))),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2))))))))) ) ).

% delete_bound_height
tff(fact_1472_tdeletemimi,axiom,
    ! [Deg2: nat,Mi: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Mi)),Deg2,TreeList,Summary2),Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))) ) ).

% tdeletemimi
tff(fact_1473_vebt__buildup__bound,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) )
     => aa(nat,$o,ord_less_eq(nat,vEBT_V8346862874174094_d_u_p(N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))),U)) ) ).

% vebt_buildup_bound
tff(fact_1474_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c2(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(B2))),zero_zero(nat))) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( Xb = vEBT_Leaf((Uv),(Uw)) )
               => ! [N3: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N3) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv),(Uw))),aa(nat,nat,suc,N3))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xaa)) ) )
             => ( ! [V3: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc2,Vd2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc2,Vd2)),Xaa)) ) )
               => ( ! [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( ( Y = $ite(
                                aa(nat,$o,ord_less(nat,Xaa),Mi2),
                                one_one(nat),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(
                                      aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                      $let(
                                        maxlow: option(nat),
                                        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary,h)),one_one(nat)) ) ),
                                      one_one(nat) ) ) ) ) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_1475_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d2(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv))),zero_zero(nat))) ) ) )
         => ( ! [A2: $o,Uw: $o] :
                ( ( Xb = vEBT_Leaf((A2),(Uw)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(Uw))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A2: $o,B2: $o] :
                  ( ( Xb = vEBT_Leaf((A2),(B2)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Xaa)) ) )
               => ( ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2)),Xaa)) ) )
                 => ( ! [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2) )
                       => ( ( Y = one_one(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                          ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = $ite(
                                  aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                  one_one(nat),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          $ite(
                                            ( ( minlow != none(nat) )
                                            & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary,h)),one_one(nat)) ) ),
                                        one_one(nat) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_1476_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r2(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = one_one(nat) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)),Xaa)) ) )
           => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xaa = Mi2,
                                zero_zero(nat),
                                $ite(
                                  Xaa = Ma2,
                                  zero_zero(nat),
                                  $ite(
                                    aa(nat,$o,ord_less(nat,Xaa),Mi2),
                                    zero_zero(nat),
                                    $ite(
                                      aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                      zero_zero(nat),
                                      $ite(
                                        ( aa(nat,$o,ord_less(nat,Mi2),Xaa)
                                        & aa(nat,$o,ord_less(nat,Xaa),Ma2) ),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $ite(aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),zero_zero(nat)) ),
                                        zero_zero(nat) ) ) ) ) )) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_1477_space__space_H,axiom,
    ! [Tb: vEBT_VEBT] : aa(nat,$o,ord_less(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_space,Tb)),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Tb)) ).

% space_space'
tff(fact_1478_word__coorder_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),A3),zero_zero(word(A)))
        <=> ( A3 = zero_zero(word(A)) ) ) ) ).

% word_coorder.extremum_unique
tff(fact_1479_word__le__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),zero_zero(word(A)))
        <=> ( Xb = zero_zero(word(A)) ) ) ) ).

% word_le_0_iff
tff(fact_1480_div__of__0__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),zero_zero(word(A))),N) = zero_zero(word(A)) ) ).

% div_of_0_id
tff(fact_1481_word__gt__0__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),aa(num,word(A),numeral_numeral(word(A)),Y))
        <=> ( zero_zero(word(A)) != aa(num,word(A),numeral_numeral(word(A)),Y) ) ) ) ).

% word_gt_0_no
tff(fact_1482_word__less__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),one_one(word(A)))
        <=> ( Xb = zero_zero(word(A)) ) ) ) ).

% word_less_1
tff(fact_1483_word__le__sub1__numberof,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] :
          ( ( aa(num,word(A),numeral_numeral(word(A)),W) != zero_zero(word(A)) )
         => ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),W))
          <=> aa(word(A),$o,ord_less_eq(word(A),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),aa(num,word(A),numeral_numeral(word(A)),W)),one_one(word(A)))) ) ) ) ).

% word_le_sub1_numberof
tff(fact_1484_word__less__sub1__numberof,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] :
          ( ( aa(num,word(A),numeral_numeral(word(A)),W) != zero_zero(word(A)) )
         => ( aa(word(A),$o,ord_less(word(A),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),W))
          <=> aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),aa(num,word(A),numeral_numeral(word(A)),W)),one_one(word(A)))) ) ) ) ).

% word_less_sub1_numberof
tff(fact_1485_word__diff__ls_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xaa: word(A),Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),Y)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),Xb)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Xb)) ) ) ) ).

% word_diff_ls(2)
tff(fact_1486_word__diff__ls_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A),W: word(A),Xaa: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),Xb)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Xb))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
           => aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)) ) ) ) ).

% word_diff_ls(1)
tff(fact_1487_Word_Oword__l__diffs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xaa: word(A),Z: word(A),Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Xb)),aa(word(A),word(A),minus_minus(word(A),Z),Xb)) ) ) ) ).

% Word.word_l_diffs(2)
tff(fact_1488_Word_Oword__l__diffs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xaa: word(A),Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Xb)),aa(word(A),word(A),minus_minus(word(A),Z),Xb))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),Z)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Z) ) ) ) ).

% Word.word_l_diffs(1)
tff(fact_1489_word__plus__mcs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),Xb: word(A),W: word(A),Xaa: word(A),Xc: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xc),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),Xc)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Xc)) ) ) ) ).

% word_plus_mcs(2)
tff(fact_1490_word__plus__mcs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),Xb: word(A),Xc: word(A),W: word(A),Xaa: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),Xc)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Xc))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xc),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)) ) ) ) ).

% word_plus_mcs(1)
tff(fact_1491_More__Word_Oword__l__diffs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb)),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb))
           => aa(word(A),$o,ord_less(word(A),W),aa(word(A),word(A),minus_minus(word(A),Z),Xb)) ) ) ) ).

% More_Word.word_l_diffs(2)
tff(fact_1492_More__Word_Oword__l__diffs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Z: word(A),Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),W),aa(word(A),word(A),minus_minus(word(A),Z),Xb))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),Z)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb)),Z) ) ) ) ).

% More_Word.word_l_diffs(1)
tff(fact_1493_word__diff__ls_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),Y)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),Xb)),W) ) ) ) ).

% word_diff_ls'(2)
tff(fact_1494_word__diff__ls_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A),W: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),Xb)),W)
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb))
           => aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb)) ) ) ) ).

% word_diff_ls'(1)
tff(fact_1495_word__l__diffs_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb)),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb)),Xb)),aa(word(A),word(A),minus_minus(word(A),Z),Xb)) ) ) ) ).

% word_l_diffs'(2)
tff(fact_1496_word__l__diffs_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb)),Xb)),aa(word(A),word(A),minus_minus(word(A),Z),Xb))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),Z)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb)),Z) ) ) ) ).

% word_l_diffs'(1)
tff(fact_1497_word__diff__ls_H_H_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),Y)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),Xb)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb)),Xb)) ) ) ) ).

% word_diff_ls''(2)
tff(fact_1498_word__diff__ls_H_H_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A),W: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),Xb)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb)),Xb))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb))
           => aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb)) ) ) ) ).

% word_diff_ls''(1)
tff(fact_1499_sub__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),minus_minus(word(A),Xb),Z))
        <=> ( ( Z = zero_zero(word(A)) )
            | aa(word(A),$o,ord_less(word(A),Xb),Z) ) ) ) ).

% sub_wrap
tff(fact_1500_less__1__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),N),one_one(word(A)))),M)
        <=> ( aa(word(A),$o,ord_less_eq(word(A),N),M)
            & ( N != zero_zero(word(A)) ) ) ) ) ).

% less_1_simp
tff(fact_1501_le__m1__iff__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Xb)
        <=> ( aa(word(A),$o,ord_less_eq(word(A),Y),aa(word(A),word(A),minus_minus(word(A),Xb),one_one(word(A))))
          <=> aa(word(A),$o,ord_less(word(A),Y),Xb) ) ) ) ).

% le_m1_iff_lt
tff(fact_1502_word__div__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Y),Xb)
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Y)
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),minus_minus(word(A),Xb),Y)),Y) = aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xb),Y)),one_one(word(A))) ) ) ) ) ).

% word_div_sub
tff(fact_1503_word__less__nowrapI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Z: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),minus_minus(word(A),Z),K))
         => ( aa(word(A),$o,ord_less_eq(word(A),K),Z)
           => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),K)
             => aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),K)) ) ) ) ) ).

% word_less_nowrapI
tff(fact_1504_word__diff__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),N)
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),M)
           => ( aa(word(A),$o,ord_less_eq(word(A),N),M)
             => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),M),N)),M) ) ) ) ) ).

% word_diff_less
tff(fact_1505_plus__minus__not__NULL,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,ord_less_eq(word(A),C2),Ab)
           => ( ( C2 != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),C2) != zero_zero(word(A)) ) ) ) ) ) ).

% plus_minus_not_NULL
tff(fact_1506_word__less__add__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),minus_minus(word(A),Y),Z))
         => ( aa(word(A),$o,ord_less_eq(word(A),Z),Y)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Z)),Y) ) ) ) ).

% word_less_add_right
tff(fact_1507_word__less__sub__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Z))
         => ( aa(word(A),$o,ord_less_eq(word(A),Y),Xb)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Xb),Y)),Z) ) ) ) ).

% word_less_sub_right
tff(fact_1508_word__subset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),R2: word(A),Y: word(A),S: word(A)] :
          ( aa(set(word(A)),$o,ord_less_eq(set(word(A)),set_or1337092689740270186AtMost(word(A),Xb,aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),R2)),one_one(word(A))))),set_or1337092689740270186AtMost(word(A),Y,aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),S)),one_one(word(A)))))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),R2)),one_one(word(A))))
           => ( aa(word(A),$o,ord_less_eq(word(A),Y),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),S)),one_one(word(A))))
             => ( ( S != zero_zero(word(A)) )
               => aa(word(A),$o,ord_less_eq(word(A),R2),S) ) ) ) ) ) ).

% word_subset_less
tff(fact_1509_plus__minus__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,ord_less_eq(word(A),C2),Ab)
           => aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),C2)) ) ) ) ).

% plus_minus_no_overflow
tff(fact_1510_plus__minus__not__NULL__ab,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,ord_less_eq(word(A),C2),Ab)
           => ( ( C2 != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),C2) != zero_zero(word(A)) ) ) ) ) ) ).

% plus_minus_not_NULL_ab
tff(fact_1511_word__less__minus__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),Xb)),aa(word(A),word(A),minus_minus(word(A),Z),Xb))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),Z)
           => aa(word(A),$o,ord_less(word(A),Y),Z) ) ) ) ).

% word_less_minus_cancel
tff(fact_1512_word__less__nowrapI_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Z: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),minus_minus(word(A),Z),K))
         => ( aa(word(A),$o,ord_less_eq(word(A),K),Z)
           => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),K)
             => aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),K)) ) ) ) ) ).

% word_less_nowrapI'
tff(fact_1513_word__less__minus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),Y)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Y),Xb)),aa(word(A),word(A),minus_minus(word(A),Z),Xb)) ) ) ) ).

% word_less_minus_mono_left
tff(fact_1514_word__le__minus__one__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),Y)
         => aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A)))) ) ) ).

% word_le_minus_one_leq
tff(fact_1515_word__leq__le__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),Y)
         => ( ( Xb != zero_zero(word(A)) )
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Xb),one_one(word(A)))),Y) ) ) ) ).

% word_leq_le_minus_one
tff(fact_1516_word__leq__minus__one__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A))))
           => aa(word(A),$o,ord_less(word(A),Xb),Y) ) ) ) ).

% word_leq_minus_one_le
tff(fact_1517_word__minus__one__le__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Xb),one_one(word(A)))),Y)
         => aa(word(A),$o,ord_less_eq(word(A),Xb),Y) ) ) ).

% word_minus_one_le_leq
tff(fact_1518_word__less__imp__diff__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),N: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),K),N)
         => ( aa(word(A),$o,ord_less(word(A),N),M)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),N),K)),M) ) ) ) ).

% word_less_imp_diff_less
tff(fact_1519_word__sub__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N5: word(A),N: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),N5),N)
         => ( ( N5 != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),minus_minus(word(A),N),N5)),one_one(word(A))) != zero_zero(word(A)) ) ) ) ) ).

% word_sub_plus_one_nonzero
tff(fact_1520_word__induct__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),M: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [N3: word(A)] :
                ( aa(word(A),$o,ord_less(word(A),N3),M)
               => ( aa(word(A),$o,P,N3)
                 => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N3)) ) )
           => aa(word(A),$o,P,M) ) ) ) ).

% word_induct_less
tff(fact_1521_word__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),one_one(word(A))))
          | ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% word_overflow
tff(fact_1522_word__gr0__conv__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),M)
         => ? [N3: word(A)] : M = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N3),one_one(word(A))) ) ) ).

% word_gr0_conv_Suc
tff(fact_1523_less__is__non__zero__p1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A3),K)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A3),one_one(word(A))) != zero_zero(word(A)) ) ) ) ).

% less_is_non_zero_p1
tff(fact_1524_word__induct2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),N: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [N3: word(A)] :
                ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N3) != zero_zero(word(A)) )
               => ( aa(word(A),$o,P,N3)
                 => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N3)) ) )
           => aa(word(A),$o,P,N) ) ) ) ).

% word_induct2
tff(fact_1525_word__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),M: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [N3: word(A)] :
                ( aa(word(A),$o,P,N3)
               => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N3)) )
           => aa(word(A),$o,P,M) ) ) ) ).

% word_induct
tff(fact_1526_plus__le__left__cancel__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y5: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y5)),Xb)
         => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y)),Xb)
           => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y5)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y))
            <=> aa(word(A),$o,ord_less(word(A),Y5),Y) ) ) ) ) ).

% plus_le_left_cancel_wrap
tff(fact_1527_word__plus__strict__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),Z)
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Z))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Z)) ) ) ) ).

% word_plus_strict_mono_right
tff(fact_1528_plus__le__left__cancel__nowrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y5: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y5))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y))
           => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y5)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y))
            <=> aa(word(A),$o,ord_less(word(A),Y5),Y) ) ) ) ) ).

% plus_le_left_cancel_nowrap
tff(fact_1529_word__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y))
         => ( ( Y != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),one_one(word(A))) != zero_zero(word(A)) ) ) ) ) ).

% word_plus_one_nonzero
tff(fact_1530_word__le__not__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),A3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),B3),A3)
        <=> ~ aa(word(A),$o,ord_less(word(A),A3),B3) ) ) ).

% word_le_not_less
tff(fact_1531_plus__one__helper2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),N)
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N),one_one(word(A))) != zero_zero(word(A)) )
           => aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N),one_one(word(A)))) ) ) ) ).

% plus_one_helper2
tff(fact_1532_plus__one__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N),one_one(word(A))))
         => aa(word(A),$o,ord_less_eq(word(A),Xb),N) ) ) ).

% plus_one_helper
tff(fact_1533_word__le__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),N: word(A),A3: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),N))
         => ( aa(word(A),$o,ord_less(word(A),A3),N)
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),A3)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),A3)),one_one(word(A)))) ) ) ) ).

% word_le_plus_1
tff(fact_1534_neq__0__no__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y))
         => ( ( Xb != zero_zero(word(A)) )
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y) != zero_zero(word(A)) ) ) ) ) ).

% neq_0_no_wrap
tff(fact_1535_word__le__plus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A3),B3))
         => ( aa(word(A),$o,ord_less(word(A),C2),B3)
           => aa(word(A),$o,ord_less_eq(word(A),A3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A3),C2)) ) ) ) ).

% word_le_plus
tff(fact_1536_word__not__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,ord_less_eq(word(A),Xb),Y)
        <=> aa(word(A),$o,ord_less(word(A),Y),Xb) ) ) ).

% word_not_le
tff(fact_1537_word__le__less__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),Y)
        <=> ( ( Xb = Y )
            | aa(word(A),$o,ord_less(word(A),Xb),Y) ) ) ) ).

% word_le_less_eq
tff(fact_1538_div__le__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xb: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xb))
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Xb)
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),Xb)),K) ) ) ) ).

% div_le_mult
tff(fact_1539_inc__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),I),M)
         => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A)))),M) ) ) ).

% inc_le
tff(fact_1540_inc__i,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),I)
         => ( aa(word(A),$o,ord_less(word(A),I),M)
           => ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A))))
              & aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),one_one(word(A)))),M) ) ) ) ) ).

% inc_i
tff(fact_1541_div__lt__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xb))
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Xb)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),Xb)),K) ) ) ) ).

% div_lt_mult
tff(fact_1542_More__Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C2: word(A),A3: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),C2)
         => ( aa(word(A),$o,ord_less(word(A),A3),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),B3),C2))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A3),C2)),B3) ) ) ) ).

% More_Word.word_div_mult
tff(fact_1543_div__less__dividend__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: word(A)] :
          ( ( Xb != zero_zero(word(A)) )
         => ( ( N != one_one(word(A)) )
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xb),N)),Xb) ) ) ) ).

% div_less_dividend_word
tff(fact_1544_word__less__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),Y)
         => ( ( Xb = aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A))) )
            | aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A)))) ) ) ) ).

% word_less_cases
tff(fact_1545_gt0__iff__gem1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Xb)
        <=> aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Xb),one_one(word(A)))),Xb) ) ) ).

% gt0_iff_gem1
tff(fact_1546_word__less__sub1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( Xb != zero_zero(word(A)) )
         => ( aa(word(A),$o,ord_less(word(A),one_one(word(A))),Xb)
          <=> aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),Xb),one_one(word(A)))) ) ) ) ).

% word_less_sub1
tff(fact_1547_div__word__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),W) = one_one(word(A)) ) ) ) ).

% div_word_self
tff(fact_1548_word__coorder_Oextremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),$o,ord_less_eq(word(A),zero_zero(word(A))),A3) ) ).

% word_coorder.extremum
tff(fact_1549_word__coorder_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),A3),zero_zero(word(A)))
         => ( A3 = zero_zero(word(A)) ) ) ) ).

% word_coorder.extremum_uniqueI
tff(fact_1550_word__zero__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,ord_less_eq(word(A),zero_zero(word(A))),Y) ) ).

% word_zero_le
tff(fact_1551_lt1__neq0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),Xb)
        <=> ( Xb != zero_zero(word(A)) ) ) ) ).

% lt1_neq0
tff(fact_1552_word__must__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),minus_minus(word(A),N),one_one(word(A))))
         => ( aa(word(A),$o,ord_less_eq(word(A),N),Xb)
           => ( N = zero_zero(word(A)) ) ) ) ) ).

% word_must_wrap
tff(fact_1553_word__sub__1__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( Xb != zero_zero(word(A)) )
         => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Xb),one_one(word(A)))),Xb) ) ) ).

% word_sub_1_le
tff(fact_1554_word__le__sub1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( Xb != zero_zero(word(A)) )
         => ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),Xb)
          <=> aa(word(A),$o,ord_less_eq(word(A),zero_zero(word(A))),aa(word(A),word(A),minus_minus(word(A),Xb),one_one(word(A)))) ) ) ) ).

% word_le_sub1
tff(fact_1555_div__by__0__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xb),zero_zero(word(A))) = zero_zero(word(A)) ) ).

% div_by_0_word
tff(fact_1556_set__diff__eq,axiom,
    ! [A: $tType,A4: set(A),B6: set(A)] : aa(set(A),set(A),minus_minus(set(A),A4),B6) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_fz(set(A),fun(set(A),fun(A,$o)),A4),B6)) ).

% set_diff_eq
tff(fact_1557_minus__set__def,axiom,
    ! [A: $tType,A4: set(A),B6: set(A)] : aa(set(A),set(A),minus_minus(set(A),A4),B6) = collect(A,aa(fun(A,$o),fun(A,$o),minus_minus(fun(A,$o),aTP_Lamp_cc(set(A),fun(A,$o),A4)),aTP_Lamp_cc(set(A),fun(A,$o),B6))) ).

% minus_set_def
tff(fact_1558_sub__wrap__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),minus_minus(word(A),Xb),Z))
        <=> aa(word(A),$o,ord_less(word(A),Xb),Z) ) ) ).

% sub_wrap_lt
tff(fact_1559_word__sub__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),minus_minus(word(A),Xb),Y))
        <=> aa(word(A),$o,ord_less(word(A),Xb),Y) ) ) ).

% word_sub_less_iff
tff(fact_1560_word__less__minus__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),C2: word(A),D: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A3),C2)
         => ( aa(word(A),$o,ord_less(word(A),D),B3)
           => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),A3),B3)),A3)
             => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),C2),D)),C2)
               => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),A3),B3)),aa(word(A),word(A),minus_minus(word(A),C2),D)) ) ) ) ) ) ).

% word_less_minus_mono
tff(fact_1561_word__greater__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),A3)
        <=> ( A3 != zero_zero(word(A)) ) ) ) ).

% word_greater_zero_iff
tff(fact_1562_word__div__lt__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),Y)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xb),Y) = zero_zero(word(A)) ) ) ) ).

% word_div_lt_eq_0
tff(fact_1563_word__gt__a__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),N: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A3),N)
         => aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),N) ) ) ).

% word_gt_a_gt_0
tff(fact_1564_word__less__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xb),Y) = zero_zero(word(A)) )
         => ( ( Y = zero_zero(word(A)) )
            | aa(word(A),$o,ord_less(word(A),Xb),Y) ) ) ) ).

% word_less_div
tff(fact_1565_word__div__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),W),V)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),V) = zero_zero(word(A)) ) ) ) ).

% word_div_less
tff(fact_1566_word__neq__0__conv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
        <=> aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),W) ) ) ).

% word_neq_0_conv
tff(fact_1567_word__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Y)
        <=> ( zero_zero(word(A)) != Y ) ) ) ).

% word_gt_0
tff(fact_1568_word__coorder_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : ~ aa(word(A),$o,ord_less(word(A),A3),zero_zero(word(A))) ) ).

% word_coorder.extremum_strict
tff(fact_1569_word__not__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : ~ aa(word(A),$o,ord_less(word(A),Xb),zero_zero(word(A))) ) ).

% word_not_simps(1)
tff(fact_1570_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I3_J,axiom,
    ! [A3: $o,B3: $o,N: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Leaf((A3),(B3)),aa(nat,nat,suc,aa(nat,nat,suc,N))) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
tff(fact_1571_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] : vEBT_T_d_e_l_e_t_e(vEBT_Leaf((A3),(B3)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
tff(fact_1572_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I4_J,axiom,
    ! [Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Uu: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary2),Uu) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
tff(fact_1573_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I2_J,axiom,
    ! [A3: $o,B3: $o] : vEBT_T_d_e_l_e_t_e(vEBT_Leaf((A3),(B3)),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(2)
tff(fact_1574_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TreeList,Summary2),Xb) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
tff(fact_1575_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary2),Xb) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
tff(fact_1576_T__vebt__buildupi__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) )
     => aa(nat,$o,ord_less_eq(nat,vEBT_V441764108873111860ildupi(N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),U)) ) ).

% T_vebt_buildupi_univ
tff(fact_1577_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(Xb,Xaa) = Y )
     => ( ( ? [A2: $o,B2: $o] : Xb = vEBT_Leaf((A2),(B2))
         => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)
           => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
         => ( ( ? [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)
             => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
           => ( ( ? [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)
               => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT)] :
                    ( ? [Summary: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $ite(
                            Xaa = Mi2,
                            one_one(nat),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xaa = Ma2,
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    aa(nat,$o,ord_less(nat,Xaa),Mi2),
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                      $ite(
                                        aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                        one_one(nat),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),
                                          $let(
                                            h: nat,
                                            h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                            $ite(aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat)) )) )) )) )) )) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.elims
tff(fact_1578_vebt__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A2),
                    $ite(Xaa = one_one(nat),(B2),$false) ) ) )
         => ( ! [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)),Xaa)) )
           => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),Xaa)) )
             => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa))
                       => $ite(
                            Xaa = Mi2,
                            $true,
                            $ite(
                              Xaa = Ma2,
                              $true,
                              $ite(
                                aa(nat,$o,ord_less(nat,Xaa),Mi2),
                                $false,
                                $ite(
                                  aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                  $false,
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $ite(aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_1579_vebt__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A2),
                      $ite(Xaa = one_one(nat),(B2),$false) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw) )
               => ( ~ (Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)),Xaa)) ) )
           => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2) )
                 => ( ~ (Y)
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ~ (Y)
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi2,
                              $true,
                              $ite(
                                Xaa = Ma2,
                                $true,
                                $ite(
                                  aa(nat,$o,ord_less(nat,Xaa),Mi2),
                                  $false,
                                  $ite(
                                    aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                    $false,
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_1580_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A2),
                    $ite(Xaa = one_one(nat),(B2),$false) ) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero(nat),Uv,Uw) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv,Uw)),Xaa)) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2)),Xaa))
                   => $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
tff(fact_1581_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A2),
                      $ite(Xaa = one_one(nat),(B2),$false) ) ) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2)),Xaa))
                 => ~ $let(
                        pos: nat,
                        pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                        $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
tff(fact_1582_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xb,Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A2),
                      $ite(Xaa = one_one(nat),(B2),$false) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),Xaa)) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero(nat),Uv,Uw) )
               => ( ~ (Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv,Uw)),Xaa)) ) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2) )
                 => ( ( (Y)
                    <=> $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList2,S2)),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_1583_vebt__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A2),
                      $ite(Xaa = one_one(nat),(B2),$false) ) ) )
         => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa))
                 => ~ $ite(
                        Xaa = Mi2,
                        $true,
                        $ite(
                          Xaa = Ma2,
                          $true,
                          $ite(
                            aa(nat,$o,ord_less(nat,Xaa),Mi2),
                            $false,
                            $ite(
                              aa(nat,$o,ord_less(nat,Ma2),Xaa),
                              $false,
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_1584_T__vebt__buildupi__gq__0,axiom,
    ! [N: nat] : aa(nat,$o,ord_less(nat,zero_zero(nat)),vEBT_V441764108873111860ildupi(N)) ).

% T_vebt_buildupi_gq_0
tff(fact_1585_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT,Xb: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv2,Uw2),Xb) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(2)
tff(fact_1586_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I3_J,axiom,
    ! [V: product_prod(nat,nat),Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Xb: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Uy,Uz),Xb) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(3)
tff(fact_1587_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Xb: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vb,Vc),Xb) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(4)
tff(fact_1588_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,Xb: nat] :
      vEBT_T_m_e_m_b_e_r(vEBT_Leaf((A3),(B3)),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
        $ite(Xb = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(1)
tff(fact_1589_member__bound__height,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_m_e_m_b_e_r(Tb,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))) ) ).

% member_bound_height
tff(fact_1590_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      vEBT_T_m_e_m_b_e_r(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
        $ite(
          Xb = Mi,
          one_one(nat),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
            $ite(
              Xb = Ma,
              one_one(nat),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                $ite(
                  aa(nat,$o,ord_less(nat,Xb),Mi),
                  one_one(nat),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                    $ite(
                      aa(nat,$o,ord_less(nat,Ma),Xb),
                      one_one(nat),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),
                        $let(
                          h: nat,
                          h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat)) )) )) )) )) )) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.simps(5)
tff(fact_1591_Tb__T__vebt__buildupi_H_H,axiom,
    ! [N: nat] : aa(nat,$o,ord_less_eq(nat,vEBT_V441764108873111860ildupi(N)),aa(nat,nat,minus_minus(nat,vEBT_VEBT_Tb2(N)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% Tb_T_vebt_buildupi''
tff(fact_1592_T_092_060_094sub_062m_092_060_094sub_062e_092_060_094sub_062m_092_060_094sub_062b_092_060_094sub_062e_092_060_094sub_062r_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_m_e_m_b_e_r(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                      $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw) )
               => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)),Xaa)) ) )
           => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2) )
                 => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc2)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              $ite(
                                Xaa = Mi2,
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    Xaa = Ma2,
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                      $ite(
                                        aa(nat,$o,ord_less(nat,Xaa),Mi2),
                                        one_one(nat),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                          $ite(
                                            aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                            one_one(nat),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),
                                              $let(
                                                h: nat,
                                                h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                                $ite(aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_e_m_b_e_r(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),one_one(nat)) )) )) )) )) )) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_1593_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xb,Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv)) )
             => ( ~ (Y)
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv))),Xaa)) ) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ( ~ (Y)
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xaa)) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 ) ) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa)) ) )
             => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc2) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi2 )
                          | ( Xaa = Ma2 )
                          | $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc2)),Xaa)) ) )
               => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2) )
                     => ( ( (Y)
                        <=> $let(
                              pos: nat,
                              pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_1594_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv)) )
             => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv))),Xaa)) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xaa)) )
           => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa))
                   => ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc2) )
                   => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc2)),Xaa))
                     => ( ( Xaa = Mi2 )
                        | ( Xaa = Ma2 )
                        | $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
               => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2) )
                     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)),Xaa))
                       => $let(
                            pos: nat,
                            pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
tff(fact_1595_space__2__pow__bound,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Tb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit0(aa(num,num,bit1,one2))))),aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N)),one_one(real)))) ) ).

% space_2_pow_bound
tff(fact_1596_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Mi2: nat,Ma2: nat,Va3: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),Va3,Vb2)),Xaa))
               => ~ ( ( Xaa = Mi2 )
                    | ( Xaa = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc2) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,V3),TreeList2,Vc2)),Xaa))
                 => ~ ( ( Xaa = Mi2 )
                      | ( Xaa = Ma2 )
                      | $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) )
           => ~ ! [V3: nat,TreeList2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2) )
                 => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList2,Vd2)),Xaa))
                   => ~ $let(
                          pos: nat,
                          pos:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                          $ite(aa(nat,$o,ord_less(nat,pos),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),pos),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,V3)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
tff(fact_1597_t__buildup__cnt,axiom,
    ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_V8346862874174094_d_u_p(N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(N))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))) ).

% t_buildup_cnt
tff(fact_1598_T__vebt__buildupi__cnt_H,axiom,
    ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_V441764108873111860ildupi(N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(N)))) ).

% T_vebt_buildupi_cnt'
tff(fact_1599_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat,N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( M = N ) ) ) ).

% of_nat_eq_iff
tff(fact_1600_two__realpow__ge__two,axiom,
    ! [N: nat] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),aa(nat,real,semiring_1_of_nat(real),N))
     => aa(real,$o,ord_less_eq(real,aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N)) ) ).

% two_realpow_ge_two
tff(fact_1601_count__buildup_H,axiom,
    ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(N))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) ).

% count_buildup'
tff(fact_1602_space__cnt,axiom,
    ! [Tb: vEBT_VEBT] : aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Tb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2)))),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Tb))) ).

% space_cnt
tff(fact_1603_semiring__1__class_Oof__nat__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),zero_zero(nat)) = zero_zero(A) ) ) ).

% semiring_1_class.of_nat_0
tff(fact_1604_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( zero_zero(nat) = N ) ) ) ).

% of_nat_0_eq_iff
tff(fact_1605_of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = zero_zero(A) )
        <=> ( M = zero_zero(nat) ) ) ) ).

% of_nat_eq_0_iff
tff(fact_1606_of__nat__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: num] : aa(nat,A,semiring_1_of_nat(A),aa(num,nat,numeral_numeral(nat),N)) = aa(num,A,numeral_numeral(A),N) ) ).

% of_nat_numeral
tff(fact_1607_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))
        <=> aa(nat,$o,ord_less(nat,M),N) ) ) ).

% of_nat_less_iff
tff(fact_1608_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))
        <=> aa(nat,$o,ord_less_eq(nat,M),N) ) ) ).

% of_nat_le_iff
tff(fact_1609_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_add
tff(fact_1610_of__nat__1,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),one_one(nat)) = one_one(A) ) ) ).

% of_nat_1
tff(fact_1611_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),N) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_1612_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),N) = one_one(A) )
        <=> ( N = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_1613_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_mult
tff(fact_1614_semiring__1__class_Oof__nat__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),M)),N) ) ).

% semiring_1_class.of_nat_power
tff(fact_1615_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [B3: nat,W: nat,Xb: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B3)),W) = aa(nat,A,semiring_1_of_nat(A),Xb) )
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),W) = Xb ) ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_1616_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Xb: nat,B3: nat,W: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Xb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B3)),W) )
        <=> ( Xb = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),W) ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_1617_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A))
        <=> ( M = zero_zero(nat) ) ) ) ).

% of_nat_le_0_iff
tff(fact_1618_of__nat__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M)) ) ).

% of_nat_Suc
tff(fact_1619_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N))
        <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ) ).

% of_nat_0_less_iff
tff(fact_1620_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Xb: num,N: nat,Y: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),N) = aa(nat,A,semiring_1_of_nat(A),Y) )
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),N) = Y ) ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_1621_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Y: nat,Xb: num,N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Y) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),N) )
        <=> ( Y = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),N) ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_1622_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: nat,W: nat,Xb: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B3)),W)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),W)),Xb) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_1623_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,B3: nat,W: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B3)),W))
        <=> aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),W)) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_1624_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B3: nat,W: nat,Xb: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B3)),W)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),W)),Xb) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_1625_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,B3: nat,W: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B3)),W))
        <=> aa(nat,$o,ord_less_eq(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),W)) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_1626_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N: nat] :
      ( aa(real,$o,ord_less(real,aa(num,real,numeral_numeral(real),W)),aa(nat,real,semiring_1_of_nat(real),N))
    <=> aa(nat,$o,ord_less(nat,aa(num,nat,numeral_numeral(nat),W)),N) ) ).

% numeral_less_real_of_nat_iff
tff(fact_1627_real__of__nat__less__numeral__iff,axiom,
    ! [N: nat,W: num] :
      ( aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),N)),aa(num,real,numeral_numeral(real),W))
    <=> aa(nat,$o,ord_less(nat,N),aa(num,nat,numeral_numeral(nat),W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_1628_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M: nat] :
      ( aa(real,$o,ord_less_eq(real,aa(num,real,numeral_numeral(real),N)),aa(nat,real,semiring_1_of_nat(real),M))
    <=> aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),N)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_1629_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,N: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),Xb)),N))
        <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Xb)
            | ( N = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_1630_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,N: nat,Xb: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N)),Xb) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_1631_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,I: num,N: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N))
        <=> aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N)) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_1632_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,N: nat,Xb: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N)),Xb) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_1633_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,I: num,N: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),N))
        <=> aa(nat,$o,ord_less_eq(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),N)) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_1634_mult__of__nat__commute,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Xb: nat,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Xb)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(nat,A,semiring_1_of_nat(A),Xb)) ) ).

% mult_of_nat_commute
tff(fact_1635_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [N: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_0_le_iff
tff(fact_1636_of__nat__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] : ~ aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),M)),zero_zero(A)) ) ).

% of_nat_less_0_iff
tff(fact_1637_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N)) != zero_zero(A) ) ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_1638_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( aa(nat,$o,ord_less(nat,M),N)
         => aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ) ).

% less_imp_of_nat_less
tff(fact_1639_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,N: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))
         => aa(nat,$o,ord_less(nat,M),N) ) ) ).

% of_nat_less_imp_less
tff(fact_1640_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),M))),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% div_mult2_eq'
tff(fact_1641_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I: nat,J: nat] :
          ( aa(nat,$o,ord_less_eq(nat,I),J)
         => aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),I)),aa(nat,A,semiring_1_of_nat(A),J)) ) ) ).

% of_nat_mono
tff(fact_1642_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_1643_of__nat__gt__0,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [K: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),K) != zero_zero(A) )
         => aa(nat,$o,ord_less(nat,zero_zero(nat)),K) ) ) ).

% of_nat_gt_0
tff(fact_1644_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [N: nat,M: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N),M)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,M),N)) = aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ) ) ).

% of_nat_diff
tff(fact_1645_reals__Archimedean3,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ! [Y4: real] :
        ? [N3: nat] : aa(real,$o,ord_less(real,Y4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N3)),Xb)) ) ).

% reals_Archimedean3
tff(fact_1646_real__of__nat__div4,axiom,
    ! [N: nat,Xb: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),Xb))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,semiring_1_of_nat(real),Xb))) ).

% real_of_nat_div4
tff(fact_1647_nat__less__real__le,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,N),M)
    <=> aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),N)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),M)) ) ).

% nat_less_real_le
tff(fact_1648_nat__le__real__less,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,N),M)
    <=> aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),M)),one_one(real))) ) ).

% nat_le_real_less
tff(fact_1649_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% of_nat_less_two_power
tff(fact_1650_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [N: nat,M: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N),M)
         => ( ( N != zero_zero(nat) )
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),M))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),N))) ) ) ) ).

% inverse_of_nat_le
tff(fact_1651_real__archimedian__rdiv__eq__0,axiom,
    ! [Xb: real,C2: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),C2)
       => ( ! [M3: nat] :
              ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M3)
             => aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M3)),Xb)),C2) )
         => ( Xb = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_1652_real__of__nat__div2,axiom,
    ! [N: nat,Xb: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,semiring_1_of_nat(real),Xb))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),Xb)))) ).

% real_of_nat_div2
tff(fact_1653_real__of__nat__div3,axiom,
    ! [N: nat,Xb: nat] : aa(real,$o,ord_less_eq(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,semiring_1_of_nat(real),Xb))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),Xb)))),one_one(real)) ).

% real_of_nat_div3
tff(fact_1654_Tb_H__cnt,axiom,
    ! [N: nat] : aa(nat,$o,ord_less_eq(nat,vEBT_VEBT_Tb2(N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_vebt_buildup(N)))) ).

% Tb'_cnt
tff(fact_1655_cnt__cnt__eq,axiom,
    ! [Tb: vEBT_VEBT] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Tb) = aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Tb)) ).

% cnt_cnt_eq
tff(fact_1656_t__build__cnt,axiom,
    ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_V8646137997579335489_i_l_d(N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(N))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))) ).

% t_build_cnt
tff(fact_1657_linear__plus__1__le__power,axiom,
    ! [Xb: real,N: nat] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),Xb)),one_one(real))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),one_one(real))),N)) ) ).

% linear_plus_1_le_power
tff(fact_1658_delete__bound__size__univ,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N) )
       => aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_d_e_l_e_t_e(Tb,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2))))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2)))))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% delete_bound_size_univ
tff(fact_1659_TBOUND__vebt__predi,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : time_TBOUND(option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),vEBT_VEBT_vebt_predi,Tb),Tib),Xb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb)))) ).

% TBOUND_vebt_predi
tff(fact_1660_TBOUND__vebt__succi,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : time_TBOUND(option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),vEBT_VEBT_vebt_succi,Tb),Tib),Xb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb)))) ).

% TBOUND_vebt_succi
tff(fact_1661_buildup__build__time,axiom,
    ! [N: nat] : aa(nat,$o,ord_less(nat,vEBT_V8346862874174094_d_u_p(N)),vEBT_V8646137997579335489_i_l_d(N)) ).

% buildup_build_time
tff(fact_1662_TBOUND__vebt__minti,axiom,
    ! [Tb: vEBT_VEBTi] : time_TBOUND(option(nat),vEBT_vebt_minti(Tb),one_one(nat)) ).

% TBOUND_vebt_minti
tff(fact_1663_TBOUND__vebt__maxti,axiom,
    ! [Tb: vEBT_VEBTi] : time_TBOUND(option(nat),vEBT_vebt_maxti(Tb),one_one(nat)) ).

% TBOUND_vebt_maxti
tff(fact_1664_TBOUND__replicate,axiom,
    ! [A: $tType,Xb: heap_Time_Heap(A),C2: nat,N: nat] :
      ( time_TBOUND(A,Xb,C2)
     => time_TBOUND(list(A),vEBT_VEBT_replicatei(A,N,Xb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),C2)),N))) ) ).

% TBOUND_replicate
tff(fact_1665_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = aa(num,int,numeral_numeral(int),V) )
    <=> ( M = aa(num,nat,numeral_numeral(nat),V) ) ) ).

% int_eq_iff_numeral
tff(fact_1666_delete__bound__size__univ_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N) )
       => aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_V1232361888498592333_e_t_e(Tb,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% delete_bound_size_univ'
tff(fact_1667_height__double__log__univ__size,axiom,
    ! [U: real,Deg2: nat,Tb: vEBT_VEBT] :
      ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Deg2) )
     => ( vEBT_invar_vebt(Tb,Deg2)
       => aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% height_double_log_univ_size
tff(fact_1668_Abs__fnat__hom__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( zero_zero(word(A)) = aa(nat,word(A),semiring_1_of_nat(word(A)),zero_zero(nat)) ) ) ).

% Abs_fnat_hom_0
tff(fact_1669_zle__int,axiom,
    ! [M: nat,N: nat] :
      ( aa(int,$o,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N))
    <=> aa(nat,$o,ord_less_eq(nat,M),N) ) ).

% zle_int
tff(fact_1670_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ? [N3: nat] : K = aa(nat,int,semiring_1_of_nat(int),N3) ) ).

% zero_le_imp_eq_int
tff(fact_1671_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ~ ! [N3: nat] : K != aa(nat,int,semiring_1_of_nat(int),N3) ) ).

% nonneg_int_cases
tff(fact_1672_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less_eq(int,W),Z)
    <=> ? [N8: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),N8)) ) ).

% zle_iff_zadd
tff(fact_1673_zdiv__int,axiom,
    ! [A3: nat,B3: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),B3)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)) ).

% zdiv_int
tff(fact_1674_Abs__fnat__hom__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( one_one(word(A)) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,suc,zero_zero(nat))) ) ) ).

% Abs_fnat_hom_1
tff(fact_1675_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less(int,W),Z)
    <=> ? [N8: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N8))) ) ).

% zless_iff_Suc_zadd
tff(fact_1676_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ? [N3: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N3)
          & ( K = aa(nat,int,semiring_1_of_nat(int),N3) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_1677_pos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ~ ! [N3: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N3) )
           => ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),N3) ) ) ).

% pos_int_cases
tff(fact_1678_zmult__zless__mono2__lemma,axiom,
    ! [I: int,J: int,K: nat] :
      ( aa(int,$o,ord_less(int,I),J)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),I)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),K)),J)) ) ) ).

% zmult_zless_mono2_lemma
tff(fact_1679_word__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),Xb: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A3),one_one(word(A)))),B3)
         => ( aa(word(A),$o,ord_less(word(A),A3),aa(nat,word(A),semiring_1_of_nat(word(A)),Xb))
           => aa(word(A),$o,ord_less(word(A),A3),B3) ) ) ) ).

% word_1_0
tff(fact_1680_zdiff__int__split,axiom,
    ! [P: fun(int,$o),Xb: nat,Y: nat] :
      ( aa(int,$o,P,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,Xb),Y)))
    <=> ( ( aa(nat,$o,ord_less_eq(nat,Y),Xb)
         => aa(int,$o,P,aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),Xb)),aa(nat,int,semiring_1_of_nat(int),Y))) )
        & ( aa(nat,$o,ord_less(nat,Xb),Y)
         => aa(int,$o,P,zero_zero(int)) ) ) ) ).

% zdiff_int_split
tff(fact_1681_word__unat__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).

% word_unat_power
tff(fact_1682_pred__bound__size__univ_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N) )
       => aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d2(Tb,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% pred_bound_size_univ'
tff(fact_1683_succ__bound__size__univ_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N) )
       => aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c2(Tb,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U)))) ) ) ).

% succ_bound_size_univ'
tff(fact_1684_member__bound__size__univ,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N) )
       => aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_m_e_m_b_e_r(Tb,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% member_bound_size_univ
tff(fact_1685_Bolzano,axiom,
    ! [A3: real,B3: real,P: fun(real,fun(real,$o))] :
      ( aa(real,$o,ord_less_eq(real,A3),B3)
     => ( ! [A2: real,B2: real,C3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),P,A2),B2)
           => ( aa(real,$o,aa(real,fun(real,$o),P,B2),C3)
             => ( aa(real,$o,ord_less_eq(real,A2),B2)
               => ( aa(real,$o,ord_less_eq(real,B2),C3)
                 => aa(real,$o,aa(real,fun(real,$o),P,A2),C3) ) ) ) )
       => ( ! [X2: real] :
              ( aa(real,$o,ord_less_eq(real,A3),X2)
             => ( aa(real,$o,ord_less_eq(real,X2),B3)
               => ? [D6: real] :
                    ( aa(real,$o,ord_less(real,zero_zero(real)),D6)
                    & ! [A2: real,B2: real] :
                        ( ( aa(real,$o,ord_less_eq(real,A2),X2)
                          & aa(real,$o,ord_less_eq(real,X2),B2)
                          & aa(real,$o,ord_less(real,aa(real,real,minus_minus(real,B2),A2)),D6) )
                       => aa(real,$o,aa(real,fun(real,$o),P,A2),B2) ) ) ) )
         => aa(real,$o,aa(real,fun(real,$o),P,A3),B3) ) ) ) ).

% Bolzano
tff(fact_1686_log__pow__cancel,axiom,
    ! [A3: real,B3: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log(A3),aa(nat,real,aa(real,fun(nat,real),power_power(real),A3),B3)) = aa(nat,real,semiring_1_of_nat(real),B3) ) ) ) ).

% log_pow_cancel
tff(fact_1687_Tb__T__vebt__buildupi,axiom,
    ! [N: nat] : aa(int,$o,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),vEBT_V441764108873111860ildupi(N))),aa(int,int,minus_minus(int,vEBT_VEBT_Tb(N)),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% Tb_T_vebt_buildupi
tff(fact_1688_zero__le__log__cancel__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,log(A3),Xb))
        <=> aa(real,$o,ord_less_eq(real,one_one(real)),Xb) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_1689_log__le__zero__cancel__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,log(A3),Xb)),zero_zero(real))
        <=> aa(real,$o,ord_less_eq(real,Xb),one_one(real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_1690_one__le__log__cancel__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less_eq(real,one_one(real)),aa(real,real,log(A3),Xb))
        <=> aa(real,$o,ord_less_eq(real,A3),Xb) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_1691_log__le__one__cancel__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,log(A3),Xb)),one_one(real))
        <=> aa(real,$o,ord_less_eq(real,Xb),A3) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_1692_log__le__cancel__iff,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
         => ( aa(real,$o,ord_less_eq(real,aa(real,real,log(A3),Xb)),aa(real,real,log(A3),Y))
          <=> aa(real,$o,ord_less_eq(real,Xb),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_1693_log2__of__power__le,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
       => aa(real,$o,ord_less_eq(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ).

% log2_of_power_le
tff(fact_1694_TBOUND__highi,axiom,
    ! [Xb: nat,N: nat] : time_TBOUND(nat,vEBT_VEBT_highi(Xb,N),one_one(nat)) ).

% TBOUND_highi
tff(fact_1695_TBOUND__lowi,axiom,
    ! [Xb: nat,N: nat] : time_TBOUND(nat,vEBT_VEBT_lowi(Xb,N),one_one(nat)) ).

% TBOUND_lowi
tff(fact_1696_Tb__Tb_H,axiom,
    ! [Tb: nat] : vEBT_VEBT_Tb(Tb) = aa(nat,int,semiring_1_of_nat(int),vEBT_VEBT_Tb2(Tb)) ).

% Tb_Tb'
tff(fact_1697_log__one,axiom,
    ! [A3: real] : aa(real,real,log(A3),one_one(real)) = zero_zero(real) ).

% log_one
tff(fact_1698_zero__less__log__cancel__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,log(A3),Xb))
        <=> aa(real,$o,ord_less(real,one_one(real)),Xb) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_1699_log__less__zero__cancel__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,aa(real,real,log(A3),Xb)),zero_zero(real))
        <=> aa(real,$o,ord_less(real,Xb),one_one(real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_1700_one__less__log__cancel__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,one_one(real)),aa(real,real,log(A3),Xb))
        <=> aa(real,$o,ord_less(real,A3),Xb) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_1701_log__less__one__cancel__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,aa(real,real,log(A3),Xb)),one_one(real))
        <=> aa(real,$o,ord_less(real,Xb),A3) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_1702_log__less__cancel__iff,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
         => ( aa(real,$o,ord_less(real,aa(real,real,log(A3),Xb)),aa(real,real,log(A3),Y))
          <=> aa(real,$o,ord_less(real,Xb),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_1703_log__eq__one,axiom,
    ! [A3: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log(A3),A3) = one_one(real) ) ) ) ).

% log_eq_one
tff(fact_1704_TBOUND__vebt__memberi,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : time_TBOUND($o,aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),vEBT_V854960066525838166emberi,Tb),Tib),Xb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb)))) ).

% TBOUND_vebt_memberi
tff(fact_1705_log__base__change,axiom,
    ! [A3: real,B3: real,Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log(B3),Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A3),Xb)),aa(real,real,log(A3),B3)) ) ) ) ).

% log_base_change
tff(fact_1706_log__of__power__eq,axiom,
    ! [M: nat,B3: real,N: nat] :
      ( ( aa(nat,real,semiring_1_of_nat(real),M) = aa(nat,real,aa(real,fun(nat,real),power_power(real),B3),N) )
     => ( aa(real,$o,ord_less(real,one_one(real)),B3)
       => ( aa(nat,real,semiring_1_of_nat(real),N) = aa(real,real,log(B3),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ) ).

% log_of_power_eq
tff(fact_1707_less__log__of__power,axiom,
    ! [B3: real,N: nat,M: real] :
      ( aa(real,$o,ord_less(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),B3),N)),M)
     => ( aa(real,$o,ord_less(real,one_one(real)),B3)
       => aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B3),M)) ) ) ).

% less_log_of_power
tff(fact_1708_log__mult,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
           => ( aa(real,real,log(A3),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(A3),Xb)),aa(real,real,log(A3),Y)) ) ) ) ) ) ).

% log_mult
tff(fact_1709_le__log__of__power,axiom,
    ! [B3: real,N: nat,M: real] :
      ( aa(real,$o,ord_less_eq(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),B3),N)),M)
     => ( aa(real,$o,ord_less(real,one_one(real)),B3)
       => aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B3),M)) ) ) ).

% le_log_of_power
tff(fact_1710_log__divide,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
           => ( aa(real,real,log(A3),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(real,real,minus_minus(real,aa(real,real,log(A3),Xb)),aa(real,real,log(A3),Y)) ) ) ) ) ) ).

% log_divide
tff(fact_1711_log__base__pow,axiom,
    ! [A3: real,N: nat,Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( aa(real,real,log(aa(nat,real,aa(real,fun(nat,real),power_power(real),A3),N)),Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A3),Xb)),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ).

% log_base_pow
tff(fact_1712_log__nat__power,axiom,
    ! [Xb: real,B3: real,N: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,real,log(B3),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B3),Xb)) ) ) ).

% log_nat_power
tff(fact_1713_log2__of__power__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) )
     => ( aa(nat,real,semiring_1_of_nat(real),N) = aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ).

% log2_of_power_eq
tff(fact_1714_log__of__power__less,axiom,
    ! [M: nat,B3: real,N: nat] :
      ( aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B3),N))
     => ( aa(real,$o,ord_less(real,one_one(real)),B3)
       => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
         => aa(real,$o,ord_less(real,aa(real,real,log(B3),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ) ).

% log_of_power_less
tff(fact_1715_log__of__power__le,axiom,
    ! [M: nat,B3: real,N: nat] :
      ( aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B3),N))
     => ( aa(real,$o,ord_less(real,one_one(real)),B3)
       => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
         => aa(real,$o,ord_less_eq(real,aa(real,real,log(B3),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ) ).

% log_of_power_le
tff(fact_1716_less__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),M)
     => aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% less_log2_of_power
tff(fact_1717_le__log2__of__power,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),M)
     => aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% le_log2_of_power
tff(fact_1718_log2__of__power__less,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
       => aa(real,$o,ord_less(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ).

% log2_of_power_less
tff(fact_1719_Tb__T__vebt__buildupi_H,axiom,
    ! [N: nat] : aa(int,$o,ord_less_eq(int,vEBT_V9176841429113362141ildupi(N)),aa(int,int,minus_minus(int,vEBT_VEBT_Tb(N)),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% Tb_T_vebt_buildupi'
tff(fact_1720_TBOUND__buildupi,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => time_TBOUND(vEBT_VEBTi,vEBT_vebt_buildupi(N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ) ).

% TBOUND_buildupi
tff(fact_1721_Tbuildupi__buildupi_H,axiom,
    ! [N: nat] : aa(nat,int,semiring_1_of_nat(int),vEBT_V441764108873111860ildupi(N)) = vEBT_V9176841429113362141ildupi(N) ).

% Tbuildupi_buildupi'
tff(fact_1722_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arcosh(A),one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_1723_minNull__delete__time__bound,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( vEBT_VEBT_minNull(vEBT_vebt_delete(Tb,Xb))
       => aa(nat,$o,ord_less_eq(nat,vEBT_T_d_e_l_e_t_e(Tb,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))) ) ) ).

% minNull_delete_time_bound
tff(fact_1724_VEBT__internal_OTb_Osimps_I2_J,axiom,
    vEBT_VEBT_Tb(aa(nat,nat,suc,zero_zero(nat))) = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) ).

% VEBT_internal.Tb.simps(2)
tff(fact_1725_VEBT__internal_OTb_H_Osimps_I2_J,axiom,
    vEBT_VEBT_Tb2(aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.Tb'.simps(2)
tff(fact_1726_not__min__Null__member,axiom,
    ! [Tb: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Tb)
     => ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Tb),X_1) ) ).

% not_min_Null_member
tff(fact_1727_min__Null__member,axiom,
    ! [Tb: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_minNull(Tb)
     => ~ aa(nat,$o,vEBT_vebt_member(Tb),Xb) ) ).

% min_Null_member
tff(fact_1728_minminNull,axiom,
    ! [Tb: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(Tb) = none(nat) )
     => vEBT_VEBT_minNull(Tb) ) ).

% minminNull
tff(fact_1729_minNullmin,axiom,
    ! [Tb: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Tb)
     => ( vEBT_vebt_mint(Tb) = none(nat) ) ) ).

% minNullmin
tff(fact_1730_minNull__delete__time__bound_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( vEBT_VEBT_minNull(vEBT_vebt_delete(Tb,Xb))
       => aa(nat,$o,ord_less_eq(nat,vEBT_V1232361888498592333_e_t_e(Tb,Xb)),one_one(nat)) ) ) ).

% minNull_delete_time_bound'
tff(fact_1731_vebt__memberi__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xb: nat,Tb: vEBT_VEBT] : refine_Imp_refines($o,aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),vEBT_vebt_memberi,Tib),Xb),aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),vEBT_V854960066525838166emberi,Tb),Tib),Xb)) ).

% vebt_memberi_refines
tff(fact_1732_TBOUND__minNull,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] :
      ( vEBT_VEBT_minNull(Tb)
     => time_TBOUND(vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),vEBT_V3964819847710782039nserti,Tb),Tib),Xb),one_one(nat)) ) ).

% TBOUND_minNull
tff(fact_1733_TBOUND__vebt__inserti,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] :
      time_TBOUND(vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),vEBT_V3964819847710782039nserti,Tb),Tib),Xb),
        $ite(vEBT_VEBT_minNull(Tb),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb))))) ).

% TBOUND_vebt_inserti
tff(fact_1734_VEBT__internal_OT__vebt__buildupi_H_Osimps_I1_J,axiom,
    vEBT_V9176841429113362141ildupi(zero_zero(nat)) = one_one(int) ).

% VEBT_internal.T_vebt_buildupi'.simps(1)
tff(fact_1735_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod(nat,nat),Va: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] : ~ vEBT_VEBT_minNull(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va,Vb,Vc)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_1736_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : vEBT_VEBT_minNull(vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux,Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_1737_VEBT__internal_OT__vebt__buildupi_H_Osimps_I2_J,axiom,
    vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,zero_zero(nat))) = one_one(int) ).

% VEBT_internal.T_vebt_buildupi'.simps(2)
tff(fact_1738_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xb)
     => ( ! [Uv: $o] : Xb != vEBT_Leaf($true,(Uv))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_1739_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xb)
     => ( ( Xb != vEBT_Leaf($false,$false) )
       => ~ ! [Uw: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uw,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_1740_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xb)
      <=> (Y) )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv: $o] : Xb = vEBT_Leaf($true,(Uv))
           => (Y) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => (Y) )
           => ( ( ? [Uw: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uw,Ux2,Uy2)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_1741_VEBTi_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),F2: fun($o,fun($o,B)),VEBTi: vEBT_VEBTi] : aa(B,A,H,vEBT_case_VEBTi(B,F1,F2,VEBTi)) = vEBT_case_VEBTi(A,aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),aTP_Lamp_ga(fun(B,A),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A))))),H),F1),aa(fun($o,fun($o,B)),fun($o,fun($o,A)),aTP_Lamp_fo(fun(B,A),fun(fun($o,fun($o,B)),fun($o,fun($o,A))),H),F2),VEBTi) ).

% VEBTi.case_distrib
tff(fact_1742_VEBT__internal_Oreplicatei_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(nat,heap_Time_Heap(A))] :
      ( ! [X2: heap_Time_Heap(A)] : Xb != aa(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A)),aa(nat,fun(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A))),product_Pair(nat,heap_Time_Heap(A)),zero_zero(nat)),X2)
     => ~ ! [N3: nat,X2: heap_Time_Heap(A)] : Xb != aa(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A)),aa(nat,fun(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A))),product_Pair(nat,heap_Time_Heap(A)),aa(nat,nat,suc,N3)),X2) ) ).

% VEBT_internal.replicatei.cases
tff(fact_1743_VEBT__internal_OT__vebt__buildupi_Osimps_I1_J,axiom,
    vEBT_V441764108873111860ildupi(zero_zero(nat)) = aa(nat,nat,suc,zero_zero(nat)) ).

% VEBT_internal.T_vebt_buildupi.simps(1)
tff(fact_1744_VEBT__internal_OT__vebt__buildupi_Osimps_I2_J,axiom,
    vEBT_V441764108873111860ildupi(aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,suc,zero_zero(nat)) ).

% VEBT_internal.T_vebt_buildupi.simps(2)
tff(fact_1745_VEBT__internal_OTb_H_Osimps_I1_J,axiom,
    vEBT_VEBT_Tb2(zero_zero(nat)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.Tb'.simps(1)
tff(fact_1746_VEBT__internal_OTb_Osimps_I1_J,axiom,
    vEBT_VEBT_Tb(zero_zero(nat)) = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) ).

% VEBT_internal.Tb.simps(1)
tff(fact_1747_del__x__mia,axiom,
    ! [Xb: nat,Mi: nat,Ma: nat,Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( ( ( Xb = Mi )
        & aa(nat,$o,ord_less(nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = $let(
                xn: nat,
                xn:= aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $ite(
                    aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                    $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                      $let(
                        newlist: list(vEBT_VEBT),
                        newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                        $ite(
                          vEBT_VEBT_minNull(newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary2,h),
                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),xn),
                                  $ite(
                                    xn = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                    Ma ))),Deg2,newlist,sn) ),
                          vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),xn),
                                $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma))),Deg2,newlist,Summary2) ) ) ),
                    vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2) ) ) ) ) ) ) ) ).

% del_x_mia
tff(fact_1748_del__x__mi__lets__in__minNull,axiom,
    ! [Xb: nat,Mi: nat,Ma: nat,Deg2: nat,Xn: nat,H: nat,Summary2: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT),Sn: vEBT_VEBT] :
      ( ( ( Xb = Mi )
        & aa(nat,$o,ord_less(nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( vEBT_VEBT_minNull(Newnode)
                       => ( ( Sn = vEBT_vebt_delete(Summary2,H) )
                         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(Sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                      Ma ))),Deg2,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_1749_del__x__mi__lets__in,axiom,
    ! [Xb: nat,Mi: nat,Ma: nat,Deg2: nat,Xn: nat,H: nat,Summary2: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( Xb = Mi )
        & aa(nat,$o,ord_less(nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = $ite(
                            vEBT_VEBT_minNull(Newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary2,H),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                      Ma ))),Deg2,Newlist,sn) ),
                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg2,Newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_1750_del__x__mi,axiom,
    ! [Xb: nat,Mi: nat,Ma: nat,Deg2: nat,Xn: nat,H: nat,Summary2: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat] :
      ( ( ( Xb = Mi )
        & aa(nat,$o,ord_less(nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L),
                        $let(
                          newlist: list(vEBT_VEBT),
                          newlist:= list_update(vEBT_VEBT,TreeList,H,newnode),
                          $ite(
                            vEBT_VEBT_minNull(newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary2,H),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                    $ite(
                                      Xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),Xn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                      Ma ))),Deg2,newlist,sn) ),
                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),H)))),Ma))),Deg2,newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_1751_del__in__range,axiom,
    ! [Mi: nat,Xb: nat,Ma: nat,Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( ( aa(nat,$o,ord_less_eq(nat,Mi),Xb)
        & aa(nat,$o,ord_less_eq(nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
         => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = $let(
                xn: nat,
                xn:= 
                  $ite(Xb = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xb),
                $let(
                  minn: nat,
                  minn:= 
                    $ite(Xb = Mi,xn,Mi),
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $ite(
                      aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                      $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                        $let(
                          newlist: list(vEBT_VEBT),
                          newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                          $ite(
                            vEBT_VEBT_minNull(newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summary2,h),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                    $ite(
                                      xn = Ma,
                                      $let(
                                        maxs: option(nat),
                                        maxs:= vEBT_vebt_maxt(sn),
                                        $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                      Ma ))),Deg2,newlist,sn) ),
                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                  $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma))),Deg2,newlist,Summary2) ) ) ),
                      vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2) ) ) ) ) ) ) ) ) ).

% del_in_range
tff(fact_1752_del__x__not__mia,axiom,
    ! [Mi: nat,Xb: nat,Ma: nat,Deg2: nat,H: nat,L: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( ( aa(nat,$o,ord_less(nat,Mi),Xb)
        & aa(nat,$o,ord_less_eq(nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
         => ( ( vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
               => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L),
                      $let(
                        newlist: list(vEBT_VEBT),
                        newlist:= list_update(vEBT_VEBT,TreeList,H,newnode),
                        $ite(
                          vEBT_VEBT_minNull(newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary2,H),
                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                  $ite(
                                    Xb = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                    Ma ))),Deg2,newlist,sn) ),
                          vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                $ite(Xb = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),H)))),Ma))),Deg2,newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_1753_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,Xb: nat,Ma: nat,Deg2: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Sn: vEBT_VEBT,Summary2: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( aa(nat,$o,ord_less(nat,Mi),Xb)
        & aa(nat,$o,ord_less_eq(nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
         => ( ( vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( vEBT_VEBT_minNull(Newnode)
                 => ( ( Sn = vEBT_vebt_delete(Summary2,H) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                  $ite(
                                    Xb = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(Sn),
                                      $ite(maxs = none(nat),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                    Ma ))),Deg2,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_1754_vebt__inserti__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xb: nat,Tb: vEBT_VEBT] : refine_Imp_refines(vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),vEBT_vebt_inserti,Tib),Xb),aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),vEBT_V3964819847710782039nserti,Tb),Tib),Xb)) ).

% vebt_inserti_refines
tff(fact_1755_nth__update__invalid,axiom,
    ! [A: $tType,I: nat,L: list(A),J: nat,Xb: A] :
      ( ~ aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(nat,A,nth(A,list_update(A,L,J,Xb)),I) = aa(nat,A,nth(A,L),I) ) ) ).

% nth_update_invalid
tff(fact_1756_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,Xb: nat,Ma: nat,Deg2: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( ( aa(nat,$o,ord_less(nat,Mi),Xb)
        & aa(nat,$o,ord_less_eq(nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
         => ( ( vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( ~ vEBT_VEBT_minNull(Newnode)
                 => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                   => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                     => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                $ite(Xb = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg2,Newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_1757_del__x__mi__lets__in__not__minNull,axiom,
    ! [Xb: nat,Mi: nat,Ma: nat,Deg2: nat,Xn: nat,H: nat,Summary2: vEBT_VEBT,TreeList: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( Xb = Mi )
        & aa(nat,$o,ord_less(nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                     => ( ~ vEBT_VEBT_minNull(Newnode)
                       => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xn),
                                  $ite(Xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg2,Newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_1758_del__x__not__mi,axiom,
    ! [Mi: nat,Xb: nat,Ma: nat,Deg2: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( ( aa(nat,$o,ord_less(nat,Mi),Xb)
        & aa(nat,$o,ord_less_eq(nat,Xb),Ma) )
     => ( ( Mi != Ma )
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
         => ( ( vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
               => ( ( Newlist = list_update(vEBT_VEBT,TreeList,H,Newnode) )
                 => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
                   => ( vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = $ite(
                          vEBT_VEBT_minNull(Newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summary2,H),
                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                  $ite(
                                    Xb = Ma,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                    Ma ))),Deg2,Newlist,sn) ),
                          vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),
                                $ite(Xb = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),H),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Newlist),H)))),Ma))),Deg2,Newlist,Summary2) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_1759_in__set__upd__eq,axiom,
    ! [A: $tType,I: nat,L: list(A),Xb: A,Y: A] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
     => ( member(A,Xb,set2(A,list_update(A,L,I,Y)))
      <=> ( ( Xb = Y )
          | ( member(A,Xb,set2(A,L))
            & ! [Y3: A] : member(A,Xb,set2(A,list_update(A,L,I,Y3))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_1760_in__set__upd__cases,axiom,
    ! [A: $tType,Xb: A,L: list(A),I: nat,Y: A] :
      ( member(A,Xb,set2(A,list_update(A,L,I,Y)))
     => ( ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
         => ( Xb != Y ) )
       => member(A,Xb,set2(A,L)) ) ) ).

% in_set_upd_cases
tff(fact_1761_in__set__upd__eq__aux,axiom,
    ! [A: $tType,I: nat,L: list(A),Xb: A,Y: A] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
     => ( member(A,Xb,set2(A,list_update(A,L,I,Y)))
      <=> ( ( Xb = Y )
          | ! [Y3: A] : member(A,Xb,set2(A,list_update(A,L,I,Y3))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_1762_nth__list__update_H,axiom,
    ! [A: $tType,L: list(A),I: nat,Xb: A,J: nat] :
      aa(nat,A,nth(A,list_update(A,L,I,Xb)),J) = $ite(
        ( ( I = J )
        & aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L)) ),
        Xb,
        aa(nat,A,nth(A,L),J) ) ).

% nth_list_update'
tff(fact_1763_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      vEBT_V1232361888498592333_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xb) = $ite(
        ( aa(nat,$o,ord_less(nat,Xb),Mi)
        | aa(nat,$o,ord_less(nat,Ma),Xb) ),
        one_one(nat),
        $ite(
          ( ( Xb = Mi )
          & ( Xb = Ma ) ),
          one_one(nat),
          $let(
            xn: nat,
            xn:= 
              $ite(Xb = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xb),
            $let(
              l: nat,
              l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                    $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),vEBT_V1232361888498592333_e_t_e(Summary2,h),one_one(nat))),
                  one_one(nat) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(7)
tff(fact_1764_vebt__delete_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_delete(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xb) = $ite(
        ( aa(nat,$o,ord_less(nat,Xb),Mi)
        | aa(nat,$o,ord_less(nat,Ma),Xb) ),
        vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),
        $ite(
          ( ( Xb = Mi )
          & ( Xb = Ma ) ),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),
          $let(
            xn: nat,
            xn:= 
              $ite(Xb = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xb),
            $let(
              minn: nat,
              minn:= 
                $ite(Xb = Mi,xn,Mi),
              $let(
                h: nat,
                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                $ite(
                  aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  $let(
                    newnode: vEBT_VEBT,
                    newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                    $let(
                      newlist: list(vEBT_VEBT),
                      newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                      $ite(
                        vEBT_VEBT_minNull(newnode),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= vEBT_vebt_delete(Summary2,h),
                          vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                              aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                $ite(
                                  xn = Ma,
                                  $let(
                                    maxs: option(nat),
                                    maxs:= vEBT_vebt_maxt(sn),
                                    $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                  Ma ))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,sn) ),
                        vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                            aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                              $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),newlist,Summary2) ) ) ),
                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_1765_vebt__delete_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xb,Xaa) = Y )
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ( ( Xaa = zero_zero(nat) )
             => ( Y != vEBT_Leaf($false,(B2)) ) ) )
       => ( ! [A2: $o] :
              ( ? [B2: $o] : Xb = vEBT_Leaf((A2),(B2))
             => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != vEBT_Leaf((A2),$false) ) ) )
         => ( ! [A2: $o,B2: $o] :
                ( ( Xb = vEBT_Leaf((A2),(B2)) )
               => ( ? [N3: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N3))
                 => ( Y != vEBT_Leaf((A2),(B2)) ) ) )
           => ( ! [Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary) )
                 => ( Y != vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list(vEBT_VEBT),Smry2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2) )
                   => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list(vEBT_VEBT),Sm2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                     => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( Y != $ite(
                              ( aa(nat,$o,ord_less(nat,Xaa),Mi2)
                              | aa(nat,$o,ord_less(nat,Ma2),Xaa) ),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary),
                              $ite(
                                ( ( Xaa = Mi2 )
                                & ( Xaa = Ma2 ) ),
                                vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),Xaa),
                                  $let(
                                    minn: nat,
                                    minn:= 
                                      $ite(Xaa = Mi2,xn,Mi2),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          newnode: vEBT_VEBT,
                                          newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                                          $let(
                                            newlist: list(vEBT_VEBT),
                                            newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                            $ite(
                                              vEBT_VEBT_minNull(newnode),
                                              $let(
                                                sn: vEBT_VEBT,
                                                sn:= vEBT_vebt_delete(Summary,h),
                                                vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                                    aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                      $ite(
                                                        xn = Ma2,
                                                        $let(
                                                          maxs: option(nat),
                                                          maxs:= vEBT_vebt_maxt(sn),
                                                          $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                                        Ma2 ))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),newlist,sn) ),
                                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                                  aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                    $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),newlist,Summary) ) ) ),
                                        vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_1766_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e(Xb,Xaa) = Y )
     => ( ( ? [A2: $o,B2: $o] : Xb = vEBT_Leaf((A2),(B2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A2: $o,B2: $o] : Xb = vEBT_Leaf((A2),(B2))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A2: $o,B2: $o] : Xb = vEBT_Leaf((A2),(B2))
             => ( ? [N3: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N3))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary)
               => ( Y != one_one(nat) ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( Y != $ite(
                              ( aa(nat,$o,ord_less(nat,Xaa),Mi2)
                              | aa(nat,$o,ord_less(nat,Ma2),Xaa) ),
                              one_one(nat),
                              $ite(
                                ( ( Xaa = Mi2 )
                                & ( Xaa = Ma2 ) ),
                                one_one(nat),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),Xaa),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $ite(
                                        aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                          $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),vEBT_V1232361888498592333_e_t_e(Summary,h),one_one(nat))),
                                        one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.elims
tff(fact_1767_VEBT__internal_OT_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_V1232361888498592333_e_t_e(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),zero_zero(nat))) ) ) )
         => ( ! [A2: $o,B2: $o] :
                ( ( Xb = vEBT_Leaf((A2),(B2)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A2: $o,B2: $o] :
                  ( ( Xb = vEBT_Leaf((A2),(B2)) )
                 => ! [N3: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N3)) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),aa(nat,nat,suc,aa(nat,nat,suc,N3)))) ) ) )
             => ( ! [Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary)),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary)),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary) )
                       => ( ( Y = one_one(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                          ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = $ite(
                                  ( aa(nat,$o,ord_less(nat,Xaa),Mi2)
                                  | aa(nat,$o,ord_less(nat,Ma2),Xaa) ),
                                  one_one(nat),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    one_one(nat),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),Xaa),
                                      $let(
                                        l: nat,
                                        l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $ite(
                                            aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                              $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),vEBT_V1232361888498592333_e_t_e(Summary,h),one_one(nat))),
                                            one_one(nat) ) ) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
tff(fact_1768_vebt__delete_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = vEBT_Leaf($false,(B2)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),zero_zero(nat))) ) ) )
         => ( ! [A2: $o,B2: $o] :
                ( ( Xb = vEBT_Leaf((A2),(B2)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = vEBT_Leaf((A2),$false) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A2: $o,B2: $o] :
                  ( ( Xb = vEBT_Leaf((A2),(B2)) )
                 => ! [N3: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N3)) )
                     => ( ( Y = vEBT_Leaf((A2),(B2)) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),aa(nat,nat,suc,aa(nat,nat,suc,N3)))) ) ) )
             => ( ! [Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary) )
                   => ( ( Y = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary)),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list(vEBT_VEBT),Smry2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2) )
                     => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TrLst2,Smry2)),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list(vEBT_VEBT),Sm2: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                       => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                          ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = $ite(
                                  ( aa(nat,$o,ord_less(nat,Xaa),Mi2)
                                  | aa(nat,$o,ord_less(nat,Ma2),Xaa) ),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),Xaa),
                                      $let(
                                        minn: nat,
                                        minn:= 
                                          $ite(Xaa = Mi2,xn,Mi2),
                                        $let(
                                          h: nat,
                                          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $ite(
                                            aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                            $let(
                                              newnode: vEBT_VEBT,
                                              newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                                              $let(
                                                newlist: list(vEBT_VEBT),
                                                newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                                $ite(
                                                  vEBT_VEBT_minNull(newnode),
                                                  $let(
                                                    sn: vEBT_VEBT,
                                                    sn:= vEBT_vebt_delete(Summary,h),
                                                    vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                                        aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                          $ite(
                                                            xn = Ma2,
                                                            $let(
                                                              maxs: option(nat),
                                                              maxs:= vEBT_vebt_maxt(sn),
                                                              $ite(maxs = none(nat),minn,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(option(nat),nat,the2(nat),maxs))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) ),
                                                            Ma2 ))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),newlist,sn) ),
                                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                                      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),minn),
                                                        $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),newlist,Summary) ) ) ),
                                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) ) ) ) ) ) ) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_1769_set__swap,axiom,
    ! [A: $tType,I: nat,Xs: list(A),J: nat] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,$o,ord_less(nat,J),aa(list(A),nat,size_size(list(A)),Xs))
       => ( set2(A,list_update(A,list_update(A,Xs,I,aa(nat,A,nth(A,Xs),J)),J,aa(nat,A,nth(A,Xs),I))) = set2(A,Xs) ) ) ) ).

% set_swap
tff(fact_1770_insert__simp__excp,axiom,
    ! [Mi: nat,Deg2: nat,TreeList: list(vEBT_VEBT),Xb: nat,Ma: nat,Summary2: vEBT_VEBT] :
      ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,ord_less(nat,Xb),Mi)
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
         => ( ( Xb != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),aa(nat,nat,ord_max(nat,Mi),Ma))),Deg2,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary2,vEBT_VEBT_high(Mi,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary2)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_1771_insert__simp__norm,axiom,
    ! [Xb: nat,Deg2: nat,TreeList: list(vEBT_VEBT),Mi: nat,Ma: nat,Summary2: vEBT_VEBT] :
      ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( aa(nat,$o,ord_less(nat,Mi),Xb)
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2)
         => ( ( Xb != Ma )
           => ( vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary2),Xb) = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),aa(nat,nat,ord_max(nat,Xb),Ma))),Deg2,list_update(vEBT_VEBT,TreeList,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summary2,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summary2)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_1772_nth__list__update__eq,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I,Xb)),I) = Xb ) ) ).

% nth_list_update_eq
tff(fact_1773_TBOUND__minNulli,axiom,
    ! [Tb: vEBT_VEBTi] : time_TBOUND($o,vEBT_VEBT_minNulli(Tb),one_one(nat)) ).

% TBOUND_minNulli
tff(fact_1774_length__list__update,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xb: A] : aa(list(A),nat,size_size(list(A)),list_update(A,Xs,I,Xb)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_list_update
tff(fact_1775_list__update__id,axiom,
    ! [A: $tType,Xs: list(A),I: nat] : list_update(A,Xs,I,aa(nat,A,nth(A,Xs),I)) = Xs ).

% list_update_id
tff(fact_1776_nth__list__update__neq,axiom,
    ! [A: $tType,I: nat,J: nat,Xs: list(A),Xb: A] :
      ( ( I != J )
     => ( aa(nat,A,nth(A,list_update(A,Xs,I,Xb)),J) = aa(nat,A,nth(A,Xs),J) ) ) ).

% nth_list_update_neq
tff(fact_1777_max__Suc__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,ord_max(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,M),N)) ).

% max_Suc_Suc
tff(fact_1778_max__0R,axiom,
    ! [N: nat] : aa(nat,nat,ord_max(nat,N),zero_zero(nat)) = N ).

% max_0R
tff(fact_1779_max__0L,axiom,
    ! [N: nat] : aa(nat,nat,ord_max(nat,zero_zero(nat)),N) = N ).

% max_0L
tff(fact_1780_max__nat_Oright__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,ord_max(nat,A3),zero_zero(nat)) = A3 ).

% max_nat.right_neutral
tff(fact_1781_max__nat_Oneutr__eq__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,ord_max(nat,A3),B3) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B3 = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_1782_max__nat_Oleft__neutral,axiom,
    ! [A3: nat] : aa(nat,nat,ord_max(nat,zero_zero(nat)),A3) = A3 ).

% max_nat.left_neutral
tff(fact_1783_max__nat_Oeq__neutr__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( aa(nat,nat,ord_max(nat,A3),B3) = zero_zero(nat) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B3 = zero_zero(nat) ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_1784_max__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,ord_max(A,zero_zero(A)),aa(num,A,numeral_numeral(A),Xb)) = aa(num,A,numeral_numeral(A),Xb) ) ).

% max_0_1(3)
tff(fact_1785_max__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,ord_max(A,aa(num,A,numeral_numeral(A),Xb)),zero_zero(A)) = aa(num,A,numeral_numeral(A),Xb) ) ).

% max_0_1(4)
tff(fact_1786_max__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,ord_max(A,aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),V),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(1)
tff(fact_1787_max__0__1_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,ord_max(A,zero_zero(A)),one_one(A)) = one_one(A) ) ) ).

% max_0_1(1)
tff(fact_1788_max__0__1_I2_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( aa(A,A,ord_max(A,one_one(A)),zero_zero(A)) = one_one(A) ) ) ).

% max_0_1(2)
tff(fact_1789_max__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,ord_max(A,one_one(A)),aa(num,A,numeral_numeral(A),Xb)) = aa(num,A,numeral_numeral(A),Xb) ) ).

% max_0_1(5)
tff(fact_1790_max__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : aa(A,A,ord_max(A,aa(num,A,numeral_numeral(A),Xb)),one_one(A)) = aa(num,A,numeral_numeral(A),Xb) ) ).

% max_0_1(6)
tff(fact_1791_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xb: A] :
      ( aa(nat,$o,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),Xs)),I)
     => ( list_update(A,Xs,I,Xb) = Xs ) ) ).

% list_update_beyond
tff(fact_1792_of__nat__max,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,ord_max(nat,Xb),Y)) = aa(A,A,ord_max(A,aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_max
tff(fact_1793_nat__add__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,ord_max(nat,N),Q2)) = aa(nat,nat,ord_max(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q2)) ).

% nat_add_max_right
tff(fact_1794_nat__add__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,ord_max(nat,M),N)),Q2) = aa(nat,nat,ord_max(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Q2)) ).

% nat_add_max_left
tff(fact_1795_nat__mult__max__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,ord_max(nat,N),Q2)) = aa(nat,nat,ord_max(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)) ).

% nat_mult_max_right
tff(fact_1796_nat__mult__max__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,ord_max(nat,M),N)),Q2) = aa(nat,nat,ord_max(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ).

% nat_mult_max_left
tff(fact_1797_nat__minus__add__max,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,N),M)),M) = aa(nat,nat,ord_max(nat,N),M) ).

% nat_minus_add_max
tff(fact_1798_neq__if__length__neq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) != aa(list(A),nat,size_size(list(A)),Ys) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
tff(fact_1799_Ex__list__of__length,axiom,
    ! [A: $tType,N: nat] :
    ? [Xs2: list(A)] : aa(list(A),nat,size_size(list(A)),Xs2) = N ).

% Ex_list_of_length
tff(fact_1800_length__induct,axiom,
    ! [A: $tType,P: fun(list(A),$o),Xs: list(A)] :
      ( ! [Xs2: list(A)] :
          ( ! [Ys2: list(A)] :
              ( aa(nat,$o,ord_less(nat,aa(list(A),nat,size_size(list(A)),Ys2)),aa(list(A),nat,size_size(list(A)),Xs2))
             => aa(list(A),$o,P,Ys2) )
         => aa(list(A),$o,P,Xs2) )
     => aa(list(A),$o,P,Xs) ) ).

% length_induct
tff(fact_1801_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [I4: nat] :
            ( aa(nat,$o,ord_less(nat,I4),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I4) = aa(nat,A,nth(A,Ys),I4) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1802_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: fun(nat,fun(A,$o))] :
      ( ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),K)
         => ? [X_12: A] : aa(A,$o,aa(nat,fun(A,$o),P,I4),X_12) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
          & ! [I4: nat] :
              ( aa(nat,$o,ord_less(nat,I4),K)
             => aa(A,$o,aa(nat,fun(A,$o),P,I4),aa(nat,A,nth(A,Xs3),I4)) ) ) ) ).

% Skolem_list_nth
tff(fact_1803_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,ord_less(nat,I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Ys),I2) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1804_all__nat__less,axiom,
    ! [N: nat,P: fun(nat,$o)] :
      ( ! [M6: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M6),N)
         => aa(nat,$o,P,M6) )
    <=> ! [X: nat] :
          ( member(nat,X,set_or1337092689740270186AtMost(nat,zero_zero(nat),N))
         => aa(nat,$o,P,X) ) ) ).

% all_nat_less
tff(fact_1805_ex__nat__less,axiom,
    ! [N: nat,P: fun(nat,$o)] :
      ( ? [M6: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M6),N)
          & aa(nat,$o,P,M6) )
    <=> ? [X: nat] :
          ( member(nat,X,set_or1337092689740270186AtMost(nat,zero_zero(nat),N))
          & aa(nat,$o,P,X) ) ) ).

% ex_nat_less
tff(fact_1806_length__pos__if__in__set,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,set2(A,Xs))
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_pos_if_in_set
tff(fact_1807_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,set2(A,Xs))
         => aa(A,$o,P,X) )
    <=> ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I4)) ) ) ).

% all_set_conv_all_nth
tff(fact_1808_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Xb: A] :
      ( ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I2)) )
     => ( member(A,Xb,set2(A,Xs))
       => aa(A,$o,P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_1809_in__set__conv__nth,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,set2(A,Xs))
    <=> ? [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(list(A),nat,size_size(list(A)),Xs))
          & ( aa(nat,A,nth(A,Xs),I4) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_1810_list__ball__nth,axiom,
    ! [A: $tType,N: nat,Xs: list(A),P: fun(A,$o)] :
      ( aa(nat,$o,ord_less(nat,N),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ! [X2: A] :
            ( member(A,X2,set2(A,Xs))
           => aa(A,$o,P,X2) )
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),N)) ) ) ).

% list_ball_nth
tff(fact_1811_nth__mem,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( aa(nat,$o,ord_less(nat,N),aa(list(A),nat,size_size(list(A)),Xs))
     => member(A,aa(nat,A,nth(A,Xs),N),set2(A,Xs)) ) ).

% nth_mem
tff(fact_1812_set__update__memI,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,ord_less(nat,N),aa(list(A),nat,size_size(list(A)),Xs))
     => member(A,Xb,set2(A,list_update(A,Xs,N,Xb))) ) ).

% set_update_memI
tff(fact_1813_list__update__same__conv,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ( list_update(A,Xs,I,Xb) = Xs )
      <=> ( aa(nat,A,nth(A,Xs),I) = Xb ) ) ) ).

% list_update_same_conv
tff(fact_1814_nth__list__update,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Xb: A,J: nat] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,A,nth(A,list_update(A,Xs,I,Xb)),J) = $ite(I = J,Xb,aa(nat,A,nth(A,Xs),J)) ) ) ).

% nth_list_update
tff(fact_1815_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      vEBT_vebt_insert(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,ord_less(nat,Xb),Mi),Mi,Xb),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(
            ( aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
            & ~ ( ( Xb = Mi )
                | ( Xb = Ma ) ) ),
            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                aa(nat,product_prod(nat,nat),
                  aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                    $ite(aa(nat,$o,ord_less(nat,Xb),Mi),Xb,Mi)),
                  aa(nat,nat,ord_max(nat,xn),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va)),list_update(vEBT_VEBT,TreeList,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_vebt_insert(Summary2,h),Summary2)),
            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_1816_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ( Y != $ite(
                  Xaa = zero_zero(nat),
                  vEBT_Leaf($true,(B2)),
                  $ite(Xaa = one_one(nat),vEBT_Leaf((A2),$true),vEBT_Leaf((A2),(B2))) ) ) )
       => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info,zero_zero(nat),Ts2,S2) )
             => ( Y != vEBT_Node(Info,zero_zero(nat),Ts2,S2) ) )
         => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
               => ( Y != vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) ) )
           => ( ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary) )
                 => ( Y != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,ord_less(nat,Xaa),Mi2),Mi2,Xaa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(
                              ( aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                              & ~ ( ( Xaa = Mi2 )
                                  | ( Xaa = Ma2 ) ) ),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                  aa(nat,product_prod(nat,nat),
                                    aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                                      $ite(aa(nat,$o,ord_less(nat,Xaa),Mi2),Xaa,Mi2)),
                                    aa(nat,nat,ord_max(nat,xn),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),list_update(vEBT_VEBT,TreeList2,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_vebt_insert(Summary,h),Summary)),
                              vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_1817_vebt__insert_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero(nat),
                      vEBT_Leaf($true,(B2)),
                      $ite(Xaa = one_one(nat),vEBT_Leaf((A2),$true),vEBT_Leaf((A2),(B2))) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),Xaa)) ) )
         => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info,zero_zero(nat),Ts2,S2) )
               => ( ( Y = vEBT_Node(Info,zero_zero(nat),Ts2,S2) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info,zero_zero(nat),Ts2,S2)),Xaa)) ) )
           => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                 => ( ( Y = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),Xaa)) ) )
             => ( ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary) )
                   => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,ord_less(nat,Xaa),Mi2),Mi2,Xaa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  ( aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                  & ~ ( ( Xaa = Mi2 )
                                      | ( Xaa = Ma2 ) ) ),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),
                                      aa(nat,product_prod(nat,nat),
                                        aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),
                                          $ite(aa(nat,$o,ord_less(nat,Xaa),Mi2),Xaa,Mi2)),
                                        aa(nat,nat,ord_max(nat,xn),Ma2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),list_update(vEBT_VEBT,TreeList2,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_vebt_insert(Summary,h),Summary)),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_1818_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(Xb,Xaa) = Y )
     => ( ( ? [A2: $o,B2: $o] : Xb = vEBT_Leaf((A2),(B2))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : Xb = vEBT_Node(Info,zero_zero(nat),Ts2,S2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : Xb = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V3: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,ord_less(nat,Xaa),Mi2),Mi2,Xaa),
                          $let(
                            h: nat,
                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                            $ite(
                              ( aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                              & ~ ( ( Xaa = Mi2 )
                                  | ( Xaa = Ma2 ) ) ),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t2(Summary,h),one_one(nat))),
                              one_one(nat) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_1819_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      vEBT_T_i_n_s_e_r_t2(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,ord_less(nat,Xb),Mi),Mi,Xb),
        $let(
          h: nat,
          h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
          $ite(
            ( aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
            & ~ ( ( Xb = Mi )
                | ( Xb = Ma ) ) ),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t2(Summary2,h),one_one(nat))),
            one_one(nat) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_1820_vebt__buildupi__refines,axiom,
    ! [N: nat] : refine_Imp_refines(vEBT_VEBTi,vEBT_vebt_buildupi(N),vEBT_V739175172307565963ildupi(N)) ).

% vebt_buildupi_refines
tff(fact_1821_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I2_J,axiom,
    vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(2)
tff(fact_1822_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I2_J,axiom,
    vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(2)
tff(fact_1823_TBOUND__vebt__buildupi,axiom,
    ! [N: nat] : time_TBOUND(vEBT_VEBTi,vEBT_V739175172307565963ildupi(N),vEBT_V441764108873111860ildupi(N)) ).

% TBOUND_vebt_buildupi
tff(fact_1824_max__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,ord_max(extended_enat,zero_zero(extended_enat)),Q2) = Q2 ).

% max_enat_simps(3)
tff(fact_1825_max__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : aa(extended_enat,extended_enat,ord_max(extended_enat,Q2),zero_zero(extended_enat)) = Q2 ).

% max_enat_simps(2)
tff(fact_1826_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,Xb: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Leaf((A3),(B3)),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_1827_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info2,zero_zero(nat),Ts,S),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_1828_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I3_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_1829_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList,Summary2),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_1830_insersimp_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Y: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Tb),X_1)
       => aa(nat,$o,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t2(Tb,Y)),one_one(nat)) ) ) ).

% insersimp'
tff(fact_1831_insertsimp_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,L: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( vEBT_VEBT_minNull(Tb)
       => aa(nat,$o,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t2(Tb,L)),one_one(nat)) ) ) ).

% insertsimp'
tff(fact_1832_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero(nat) )
     => ( ( Xb != aa(nat,nat,suc,zero_zero(nat)) )
       => ~ ! [Va2: nat] : Xb != aa(nat,nat,suc,aa(nat,nat,suc,Va2)) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.cases
tff(fact_1833_insert_H__bound__height,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t2(Tb,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb))) ) ).

% insert'_bound_height
tff(fact_1834_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_Leaf((A3),(B3))) = one_one(nat) ).

% VEBT_internal.cnt'.simps(1)
tff(fact_1835_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I1_J,axiom,
    vEBT_V8646137997579335489_i_l_d(zero_zero(nat)) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(1)
tff(fact_1836_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I1_J,axiom,
    vEBT_V8346862874174094_d_u_p(zero_zero(nat)) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(1)
tff(fact_1837_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_space2,vEBT_Leaf((A3),(B3))) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% VEBT_internal.space'.simps(1)
tff(fact_1838_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_space,vEBT_Leaf((A3),(B3))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.space.simps(1)
tff(fact_1839_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t2(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = one_one(nat) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),Xaa)) ) )
         => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info,zero_zero(nat),Ts2,S2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info,zero_zero(nat),Ts2,S2)),Xaa)) ) )
           => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),Xaa)) ) )
             => ( ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,ord_less(nat,Xaa),Mi2),Mi2,Xaa),
                              $let(
                                h: nat,
                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $ite(
                                  ( aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                  & ~ ( ( Xaa = Mi2 )
                                      | ( Xaa = Ma2 ) ) ),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t2(Summary,h),one_one(nat))),
                                  one_one(nat) ) ) ) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_1840_T__vebt__buildupi,axiom,
    ! [N: nat,H: heap_ext(product_unit)] : aa(nat,$o,ord_less_eq(nat,time_time(vEBT_VEBTi,vEBT_V739175172307565963ildupi(N),H)),vEBT_V441764108873111860ildupi(N)) ).

% T_vebt_buildupi
tff(fact_1841_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),E3)
         => ~ ! [N3: nat] : ~ aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N3)))),E3) ) ) ).

% nat_approx_posE
tff(fact_1842_int__ops_I6_J,axiom,
    ! [A3: nat,B3: nat] :
      aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,A3),B3)) = $ite(aa(int,$o,ord_less(int,aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)),zero_zero(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3))) ).

% int_ops(6)
tff(fact_1843_max__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,ord_max(A,Xb),Y)),Z)
        <=> ( aa(A,$o,ord_less(A,Xb),Z)
            & aa(A,$o,ord_less(A,Y),Z) ) ) ) ).

% max_less_iff_conj
tff(fact_1844_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,A,ord_max(A,A3),B3) = B3 ) ) ) ).

% max.absorb4
tff(fact_1845_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( aa(A,A,ord_max(A,A3),B3) = A3 ) ) ) ).

% max.absorb3
tff(fact_1846_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y22: num] :
      ( ( bit0(X22) = bit0(Y22) )
    <=> ( X22 = Y22 ) ) ).

% verit_eq_simplify(8)
tff(fact_1847_time__replicate,axiom,
    ! [A: $tType,Xb: heap_Time_Heap(A),C2: nat,N: nat,H: heap_ext(product_unit)] :
      ( ! [H5: heap_ext(product_unit)] : aa(nat,$o,ord_less_eq(nat,time_time(A,Xb,H5)),C2)
     => aa(nat,$o,ord_less_eq(nat,time_time(list(A),vEBT_VEBT_replicatei(A,N,Xb),H)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),C2)),N))) ) ).

% time_replicate
tff(fact_1848_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,A,ord_max(A,A3),B3) = A3 ) ) ) ).

% max.absorb1
tff(fact_1849_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,A,ord_max(A,A3),B3) = B3 ) ) ) ).

% max.absorb2
tff(fact_1850_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,ord_max(A,B3),C2)),A3)
        <=> ( aa(A,$o,ord_less_eq(A,B3),A3)
            & aa(A,$o,ord_less_eq(A,C2),A3) ) ) ) ).

% max.bounded_iff
tff(fact_1851_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
          | ~ aa(A,$o,ord_less_eq(A,A3),B3)
          | ~ aa(A,$o,ord_less_eq(A,B3),A3) ) ) ).

% verit_la_disequality
tff(fact_1852_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,A3),A3) ) ).

% verit_comp_simplify1(2)
tff(fact_1853_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A] : ~ aa(A,$o,ord_less(A,A3),A3) ) ).

% verit_comp_simplify1(1)
tff(fact_1854_verit__la__generic,axiom,
    ! [A3: int,Xb: int] :
      ( aa(int,$o,ord_less_eq(int,A3),Xb)
      | ( A3 = Xb )
      | aa(int,$o,ord_less_eq(int,Xb),A3) ) ).

% verit_la_generic
tff(fact_1855_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B4: A,A5: A] :
          ( ~ aa(A,$o,ord_less_eq(A,B4),A5)
        <=> aa(A,$o,ord_less(A,A5),B4) ) ) ).

% verit_comp_simplify1(3)
tff(fact_1856_verit__sum__simplify,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),zero_zero(A)) = A3 ) ).

% verit_sum_simplify
tff(fact_1857_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] : one2 != bit0(X22) ).

% verit_eq_simplify(10)
tff(fact_1858_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X33: num] : bit0(X22) != aa(num,num,bit1,X33) ).

% verit_eq_simplify(14)
tff(fact_1859_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [N3: nat] : aa(A,$o,ord_less_eq(A,Xb),aa(nat,A,semiring_1_of_nat(A),N3)) ) ).

% real_arch_simple
tff(fact_1860_exists__least__lemma,axiom,
    ! [P: fun(nat,$o)] :
      ( ~ aa(nat,$o,P,zero_zero(nat))
     => ( ? [X_13: nat] : aa(nat,$o,P,X_13)
       => ? [N3: nat] :
            ( ~ aa(nat,$o,P,N3)
            & aa(nat,$o,P,aa(nat,nat,suc,N3)) ) ) ) ).

% exists_least_lemma
tff(fact_1861_verit__eq__simplify_I12_J,axiom,
    ! [X33: num] : one2 != aa(num,num,bit1,X33) ).

% verit_eq_simplify(12)
tff(fact_1862_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [N3: nat] : aa(A,$o,ord_less(A,Xb),aa(nat,A,semiring_1_of_nat(A),N3)) ) ).

% reals_Archimedean2
tff(fact_1863_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,D: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),A3)
         => ( aa(A,$o,ord_less_eq(A,D),B3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,ord_max(A,C2),D)),aa(A,A,ord_max(A,A3),B3)) ) ) ) ).

% max.mono
tff(fact_1864_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( A3 = aa(A,A,ord_max(A,A3),B3) ) ) ) ).

% max.orderE
tff(fact_1865_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,ord_max(A,A3),B3) )
         => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ).

% max.orderI
tff(fact_1866_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,ord_max(A,B3),C2)),A3)
         => ~ ( aa(A,$o,ord_less_eq(A,B3),A3)
             => ~ aa(A,$o,ord_less_eq(A,C2),A3) ) ) ) ).

% max.boundedE
tff(fact_1867_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,C2),A3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,ord_max(A,B3),C2)),A3) ) ) ) ).

% max.boundedI
tff(fact_1868_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
        <=> ( A3 = aa(A,A,ord_max(A,A3),B3) ) ) ) ).

% max.order_iff
tff(fact_1869_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,A3),aa(A,A,ord_max(A,A3),B3)) ) ).

% max.cobounded1
tff(fact_1870_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] : aa(A,$o,ord_less_eq(A,B3),aa(A,A,ord_max(A,A3),B3)) ) ).

% max.cobounded2
tff(fact_1871_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Z),aa(A,A,ord_max(A,Xb),Y))
        <=> ( aa(A,$o,ord_less_eq(A,Z),Xb)
            | aa(A,$o,ord_less_eq(A,Z),Y) ) ) ) ).

% le_max_iff_disj
tff(fact_1872_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
        <=> ( aa(A,A,ord_max(A,A3),B3) = A3 ) ) ) ).

% max.absorb_iff1
tff(fact_1873_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
        <=> ( aa(A,A,ord_max(A,A3),B3) = B3 ) ) ) ).

% max.absorb_iff2
tff(fact_1874_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),A3)
         => aa(A,$o,ord_less_eq(A,C2),aa(A,A,ord_max(A,A3),B3)) ) ) ).

% max.coboundedI1
tff(fact_1875_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,C2),B3)
         => aa(A,$o,ord_less_eq(A,C2),aa(A,A,ord_max(A,A3),B3)) ) ) ).

% max.coboundedI2
tff(fact_1876_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,C2),B3)
         => aa(A,$o,ord_less(A,C2),aa(A,A,ord_max(A,A3),B3)) ) ) ).

% max.strict_coboundedI2
tff(fact_1877_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),A3)
         => aa(A,$o,ord_less(A,C2),aa(A,A,ord_max(A,A3),B3)) ) ) ).

% max.strict_coboundedI1
tff(fact_1878_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
        <=> ( ( A3 = aa(A,A,ord_max(A,A3),B3) )
            & ( A3 != B3 ) ) ) ) ).

% max.strict_order_iff
tff(fact_1879_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,ord_max(A,B3),C2)),A3)
         => ~ ( aa(A,$o,ord_less(A,B3),A3)
             => ~ aa(A,$o,ord_less(A,C2),A3) ) ) ) ).

% max.strict_boundedE
tff(fact_1880_less__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Z),aa(A,A,ord_max(A,Xb),Y))
        <=> ( aa(A,$o,ord_less(A,Z),Xb)
            | aa(A,$o,ord_less(A,Z),Y) ) ) ) ).

% less_max_iff_disj
tff(fact_1881_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X3: A,Xa: A] :
          aa(A,A,ord_max(A,X3),Xa) = $ite(aa(A,$o,ord_less_eq(A,X3),Xa),Xa,X3) ) ).

% max_def_raw
tff(fact_1882_int__ops_I3_J,axiom,
    ! [N: num] : aa(nat,int,semiring_1_of_nat(int),aa(num,nat,numeral_numeral(nat),N)) = aa(num,int,numeral_numeral(int),N) ).

% int_ops(3)
tff(fact_1883_int__ops_I1_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),zero_zero(nat)) = zero_zero(int) ).

% int_ops(1)
tff(fact_1884_nat__int__comparison_I2_J,axiom,
    ! [A3: nat,B3: nat] :
      ( aa(nat,$o,ord_less(nat,A3),B3)
    <=> aa(int,$o,ord_less(int,aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)) ) ).

% nat_int_comparison(2)
tff(fact_1885_nat__int__comparison_I3_J,axiom,
    ! [A3: nat,B3: nat] :
      ( aa(nat,$o,ord_less_eq(nat,A3),B3)
    <=> aa(int,$o,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)) ) ).

% nat_int_comparison(3)
tff(fact_1886_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_1887_nat__less__as__int,axiom,
    ! [X3: nat,Xa: nat] :
      ( aa(nat,$o,ord_less(nat,X3),Xa)
    <=> aa(int,$o,ord_less(int,aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_less_as_int
tff(fact_1888_nat__leq__as__int,axiom,
    ! [X3: nat,Xa: nat] :
      ( aa(nat,$o,ord_less_eq(nat,X3),Xa)
    <=> aa(int,$o,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa)) ) ).

% nat_leq_as_int
tff(fact_1889_ex__less__of__nat__mult,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xb)
         => ? [N3: nat] : aa(A,$o,ord_less(A,Y),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N3)),Xb)) ) ) ).

% ex_less_of_nat_mult
tff(fact_1890_int__ops_I4_J,axiom,
    ! [A3: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,A3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A3)),one_one(int)) ).

% int_ops(4)
tff(fact_1891_int__Suc,axiom,
    ! [N: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) ).

% int_Suc
tff(fact_1892_heigt__uplog__rel,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ).

% heigt_uplog_rel
tff(fact_1893_succ__bound__size__univ,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N) )
       => aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c(Tb,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% succ_bound_size_univ
tff(fact_1894_TBOUND__option__case,axiom,
    ! [B: $tType,A: $tType,Tb: option(A),Fa: heap_Time_Heap(B),Bnd: nat,F3: fun(A,heap_Time_Heap(B)),Bnd2: fun(A,nat)] :
      ( ( ( Tb = none(A) )
       => time_TBOUND(B,Fa,Bnd) )
     => ( ! [X2: A] :
            ( ( Tb = aa(A,option(A),some(A),X2) )
           => time_TBOUND(B,aa(A,heap_Time_Heap(B),F3,X2),aa(A,nat,Bnd2,X2)) )
       => time_TBOUND(B,case_option(heap_Time_Heap(B),A,Fa,F3,Tb),case_option(nat,A,Bnd,Bnd2,Tb)) ) ) ).

% TBOUND_option_case
tff(fact_1895_TBOUND__assert_H__bind__strong,axiom,
    ! [A: $tType,P: $o,M: heap_Time_Heap(A),Tb: nat] :
      ( ( (P)
       => time_TBOUND(A,M,Tb) )
     => time_TBOUND(A,heap_Time_bind(product_unit,A,refine_Imp_assert((P)),aTP_Lamp_gb(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),M)),
          $ite((P),Tb,zero_zero(nat))) ) ).

% TBOUND_assert'_bind_strong
tff(fact_1896_pred__bound__size__univ,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N) )
       => aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d(Tb,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% pred_bound_size_univ
tff(fact_1897_insert__bound__size__univ,axiom,
    ! [Tb: vEBT_VEBT,N: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N) )
       => aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),vEBT_T_i_n_s_e_r_t(Tb,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2)))))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),U))))) ) ) ).

% insert_bound_size_univ
tff(fact_1898_delete__correct,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(Tb,Xb)) = aa(set(nat),set(nat),minus_minus(set(nat),vEBT_set_vebt(Tb)),aa(set(nat),set(nat),insert(nat,Xb),bot_bot(set(nat)))) ) ) ).

% delete_correct
tff(fact_1899_delete__correct_H,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(Tb,Xb)) = aa(set(nat),set(nat),minus_minus(set(nat),vEBT_VEBT_set_vebt(Tb)),aa(set(nat),set(nat),insert(nat,Xb),bot_bot(set(nat)))) ) ) ).

% delete_correct'
tff(fact_1900_singleton__conv2,axiom,
    ! [A: $tType,A3: A] : collect(A,fequal(A,A3)) = aa(set(A),set(A),insert(A,A3),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_1901_singleton__conv,axiom,
    ! [A: $tType,A3: A] : collect(A,aTP_Lamp_gc(A,fun(A,$o),A3)) = aa(set(A),set(A),insert(A,A3),bot_bot(set(A))) ).

% singleton_conv
tff(fact_1902_ceiling__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,zero_zero(A)) = zero_zero(int) ) ) ).

% ceiling_zero
tff(fact_1903_ceiling__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archimedean_ceiling(A,aa(num,A,numeral_numeral(A),V)) = aa(num,int,numeral_numeral(int),V) ) ).

% ceiling_numeral
tff(fact_1904_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_1905_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,Xb)),zero_zero(int))
        <=> aa(A,$o,ord_less_eq(A,Xb),zero_zero(A)) ) ) ).

% ceiling_le_zero
tff(fact_1906_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,ord_less_eq(A,Xb),aa(num,A,numeral_numeral(A),V)) ) ) ).

% ceiling_le_numeral
tff(fact_1907_zero__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,ord_less(int,zero_zero(int)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),Xb) ) ) ).

% zero_less_ceiling
tff(fact_1908_numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,ord_less(int,aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),V)),Xb) ) ) ).

% numeral_less_ceiling
tff(fact_1909_ceiling__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,Xb)),one_one(int))
        <=> aa(A,$o,ord_less_eq(A,Xb),zero_zero(A)) ) ) ).

% ceiling_less_one
tff(fact_1910_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,ord_less_eq(int,one_one(int)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),Xb) ) ) ).

% one_le_ceiling
tff(fact_1911_ceiling__add__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(num,A,numeral_numeral(A),V))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_add_numeral
tff(fact_1912_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,Xb)),one_one(int))
        <=> aa(A,$o,ord_less_eq(A,Xb),one_one(A)) ) ) ).

% ceiling_le_one
tff(fact_1913_one__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,ord_less(int,one_one(int)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,ord_less(A,one_one(A)),Xb) ) ) ).

% one_less_ceiling
tff(fact_1914_ceiling__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),one_one(A))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),one_one(int)) ) ).

% ceiling_add_one
tff(fact_1915_ceiling__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] : archimedean_ceiling(A,aa(A,A,minus_minus(A,Xb),aa(num,A,numeral_numeral(A),V))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_diff_numeral
tff(fact_1916_ceiling__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: num,N: nat] : archimedean_ceiling(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),N)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),N) ) ).

% ceiling_numeral_power
tff(fact_1917_ceiling__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : archimedean_ceiling(A,aa(A,A,minus_minus(A,Xb),one_one(A))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,Xb)),one_one(int)) ) ).

% ceiling_diff_one
tff(fact_1918_ceiling__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,ord_less_eq(A,Xb),aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),V)),one_one(A))) ) ) ).

% ceiling_less_numeral
tff(fact_1919_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,ord_less_eq(int,aa(num,int,numeral_numeral(int),V)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),V)),one_one(A))),Xb) ) ) ).

% numeral_le_ceiling
tff(fact_1920_insert__compr,axiom,
    ! [A: $tType,A3: A,B6: set(A)] : aa(set(A),set(A),insert(A,A3),B6) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_gd(A,fun(set(A),fun(A,$o)),A3),B6)) ).

% insert_compr
tff(fact_1921_insert__Collect,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] : aa(set(A),set(A),insert(A,A3),collect(A,P)) = collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ge(A,fun(fun(A,$o),fun(A,$o)),A3),P)) ).

% insert_Collect
tff(fact_1922_Collect__conv__if2,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] :
      collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_gf(A,fun(fun(A,$o),fun(A,$o)),A3),P)) = $ite(aa(A,$o,P,A3),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if2
tff(fact_1923_Collect__conv__if,axiom,
    ! [A: $tType,A3: A,P: fun(A,$o)] :
      collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_gg(A,fun(fun(A,$o),fun(A,$o)),A3),P)) = $ite(aa(A,$o,P,A3),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if
tff(fact_1924_time__assert_H,axiom,
    ! [P: $o,H: heap_ext(product_unit)] : time_time(product_unit,refine_Imp_assert((P)),H) = zero_zero(nat) ).

% time_assert'
tff(fact_1925_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,ord_less_eq(A,Y),Xb)
         => aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,Y)),archimedean_ceiling(A,Xb)) ) ) ).

% ceiling_mono
tff(fact_1926_ceiling__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,Xb)),archimedean_ceiling(A,Y))
         => aa(A,$o,ord_less(A,Xb),Y) ) ) ).

% ceiling_less_cancel
tff(fact_1927_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] : aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),archimedean_ceiling(A,Y))) ) ).

% ceiling_add_le
tff(fact_1928_atLeast0__atMost__Suc,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ).

% atLeast0_atMost_Suc
tff(fact_1929_atLeastAtMost__insertL,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => ( aa(set(nat),set(nat),insert(nat,M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = set_or1337092689740270186AtMost(nat,M,N) ) ) ).

% atLeastAtMost_insertL
tff(fact_1930_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,N))
     => ( set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,N)),set_or1337092689740270186AtMost(nat,M,N)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_1931_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => ( set_or1337092689740270186AtMost(nat,M,N) = aa(set(nat),set(nat),insert(nat,M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_1932_remove__subset,axiom,
    ! [A: $tType,Xb: A,S3: set(A)] :
      ( member(A,Xb,S3)
     => aa(set(A),$o,ord_less(set(A),aa(set(A),set(A),minus_minus(set(A),S3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),S3) ) ).

% remove_subset
tff(fact_1933_psubset__insert__iff,axiom,
    ! [A: $tType,A4: set(A),Xb: A,B6: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A4),aa(set(A),set(A),insert(A,Xb),B6))
    <=> $ite(
          member(A,Xb,B6),
          aa(set(A),$o,ord_less(set(A),A4),B6),
          $ite(member(A,Xb,A4),aa(set(A),$o,ord_less(set(A),aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),B6),aa(set(A),$o,ord_less_eq(set(A),A4),B6)) ) ) ).

% psubset_insert_iff
tff(fact_1934_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info2,zero_zero(nat),Ts,S),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_1935_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o] : vEBT_T_p_r_e_d(vEBT_Leaf((Uu),(Uv2)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
tff(fact_1936_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,N: nat] : vEBT_T_s_u_c_c(vEBT_Leaf((Uv2),(Uw2)),aa(nat,nat,suc,N)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
tff(fact_1937_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list(vEBT_VEBT),Va: vEBT_VEBT,Vb: nat] : vEBT_T_p_r_e_d(vEBT_Node(none(product_prod(nat,nat)),Uy,Uz,Va),Vb) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
tff(fact_1938_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A3)),archimedean_ceiling(A,B3))) ) ) ) ).

% mult_ceiling_le
tff(fact_1939_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Va: nat] : vEBT_T_s_u_c_c(vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz),Va) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
tff(fact_1940_insert__swap__set__eq,axiom,
    ! [A: $tType,I: nat,L: list(A),Xb: A] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(set(A),set(A),insert(A,aa(nat,A,nth(A,L),I)),set2(A,list_update(A,L,I,Xb))) = aa(set(A),set(A),insert(A,Xb),set2(A,L)) ) ) ).

% insert_swap_set_eq
tff(fact_1941_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S: vEBT_VEBT,Xb: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts,S),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_1942_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A3: $o,B3: $o,Va: nat] :
      vEBT_T_p_r_e_d(vEBT_Leaf((A3),(B3)),aa(nat,nat,suc,aa(nat,nat,suc,Va))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite((B3),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
tff(fact_1943_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o,Xb: nat] :
      vEBT_T_i_n_s_e_r_t(vEBT_Leaf((A3),(B3)),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(Xb = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(1)
tff(fact_1944_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I5_J,axiom,
    ! [V: product_prod(nat,nat),Vd: list(vEBT_VEBT),Ve: vEBT_VEBT,Vf: nat] : vEBT_T_p_r_e_d(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vd,Ve),Vf) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(5)
tff(fact_1945_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu: $o,B3: $o] : vEBT_T_s_u_c_c(vEBT_Leaf((Uu),(B3)),zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(1)
tff(fact_1946_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd: vEBT_VEBT,Ve: nat] : vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),zero_zero(nat),Vc,Vd),Ve) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
tff(fact_1947_TBOUND__mono,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),Tb: nat,T3: nat] :
      ( time_TBOUND(A,C2,Tb)
     => ( aa(nat,$o,ord_less_eq(nat,Tb),T3)
       => time_TBOUND(A,C2,T3) ) ) ).

% TBOUND_mono
tff(fact_1948_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A3: $o,Uw2: $o] : vEBT_T_p_r_e_d(vEBT_Leaf((A3),(Uw2)),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(2)
tff(fact_1949_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I6_J,axiom,
    ! [V: product_prod(nat,nat),Vh: list(vEBT_VEBT),Vi: vEBT_VEBT,Vj: nat] : vEBT_T_p_r_e_d(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vh,Vi),Vj) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(6)
tff(fact_1950_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I5_J,axiom,
    ! [V: product_prod(nat,nat),Vg: list(vEBT_VEBT),Vh: vEBT_VEBT,Vi: nat] : vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V),aa(nat,nat,suc,zero_zero(nat)),Vg,Vh),Vi) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(5)
tff(fact_1951_TBOUND__refines,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),Tb: nat,D: heap_Time_Heap(A)] :
      ( time_TBOUND(A,C2,Tb)
     => ( refine_Imp_refines(A,C2,D)
       => time_TBOUND(A,D,Tb) ) ) ).

% TBOUND_refines
tff(fact_1952_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I4_J,axiom,
    ! [V: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeList,Summary2),Xb) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(4)
tff(fact_1953_insersimp,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Y: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Tb),X_1)
       => aa(nat,$o,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t(Tb,Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ) ) ).

% insersimp
tff(fact_1954_insertsimp,axiom,
    ! [Tb: vEBT_VEBT,N: nat,L: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => ( vEBT_VEBT_minNull(Tb)
       => aa(nat,$o,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t(Tb,L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ) ) ).

% insertsimp
tff(fact_1955_TBOUND__return,axiom,
    ! [A: $tType,Xb: A] : time_TBOUND(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),Xb),one_one(nat)) ).

% TBOUND_return
tff(fact_1956_time__return,axiom,
    ! [A: $tType,Xb: A,H: heap_ext(product_unit)] : time_time(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),Xb),H) = one_one(nat) ).

% time_return
tff(fact_1957_TBOUND__nth,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: array(A),I: nat] : time_TBOUND(A,aa(nat,heap_Time_Heap(A),array_nth(A,Xs),I),one_one(nat)) ) ).

% TBOUND_nth
tff(fact_1958_TBOUND__def,axiom,
    ! [A: $tType,M: heap_Time_Heap(A),Tb: nat] :
      ( time_TBOUND(A,M,Tb)
    <=> ! [H6: heap_ext(product_unit)] : aa(nat,$o,ord_less_eq(nat,time_time(A,M,H6)),Tb) ) ).

% TBOUND_def
tff(fact_1959_TBOUNDI,axiom,
    ! [A: $tType,M: heap_Time_Heap(A),Tb: nat] :
      ( ! [H5: heap_ext(product_unit)] : aa(nat,$o,ord_less_eq(nat,time_time(A,M,H5)),Tb)
     => time_TBOUND(A,M,Tb) ) ).

% TBOUNDI
tff(fact_1960_TBOUNDD,axiom,
    ! [A: $tType,M: heap_Time_Heap(A),Tb: nat,H: heap_ext(product_unit)] :
      ( time_TBOUND(A,M,Tb)
     => aa(nat,$o,ord_less_eq(nat,time_time(A,M,H)),Tb) ) ).

% TBOUNDD
tff(fact_1961_ceiling__log__nat__eq__if,axiom,
    ! [B3: nat,N: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),N)),K)
     => ( aa(nat,$o,ord_less_eq(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
         => ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B3)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_1962_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))))),one_one(int)) ) ) ).

% ceiling_log2_div2
tff(fact_1963_TBOUND__prod__case,axiom,
    ! [C: $tType,B: $tType,A: $tType,Tb: product_prod(A,B),Fa: fun(A,fun(B,heap_Time_Heap(C))),Bnd: fun(A,fun(B,nat))] :
      ( ! [A2: A,B2: B] :
          ( ( Tb = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) )
         => time_TBOUND(C,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),Fa,A2),B2),aa(B,nat,aa(A,fun(B,nat),Bnd,A2),B2)) )
     => time_TBOUND(C,aa(product_prod(A,B),heap_Time_Heap(C),product_case_prod(A,B,heap_Time_Heap(C),Fa),Tb),aa(product_prod(A,B),nat,product_case_prod(A,B,nat,Bnd),Tb)) ) ).

% TBOUND_prod_case
tff(fact_1964_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B3: nat,K: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => ( ( archimedean_ceiling(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B3)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),one_one(int)) )
        <=> ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),N)),K)
            & aa(nat,$o,ord_less_eq(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_1965_TBOUND__assert_H__weak,axiom,
    ! [A: $tType,P: $o,M: heap_Time_Heap(A),Tb: nat] :
      ( ( (P)
       => time_TBOUND(A,M,Tb) )
     => time_TBOUND(A,heap_Time_bind(product_unit,A,refine_Imp_assert((P)),aTP_Lamp_gb(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),M)),Tb) ) ).

% TBOUND_assert'_weak
tff(fact_1966_pred__bound__height,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_p_r_e_d(Tb,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))) ) ).

% pred_bound_height
tff(fact_1967_insert__bound__height,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_i_n_s_e_r_t(Tb,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))))) ) ).

% insert_bound_height
tff(fact_1968_succ__bound__height,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => aa(nat,$o,ord_less_eq(nat,vEBT_T_s_u_c_c(Tb,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))) ) ).

% succ_bound_height
tff(fact_1969_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(Xb,Xaa) = Y )
     => ( ( ? [A2: $o,B2: $o] : Xb = vEBT_Leaf((A2),(B2))
         => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : Xb = vEBT_Node(Info,zero_zero(nat),Ts2,S2)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] : Xb = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V3: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary)
               => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2)))))),
                          $let(
                            xn: nat,
                            xn:= 
                              $ite(aa(nat,$o,ord_less(nat,Xaa),Mi2),Mi2,Xaa),
                            $let(
                              h: nat,
                              h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                              $ite(
                                ( aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                & ~ ( ( Xaa = Mi2 )
                                    | ( Xaa = Ma2 ) ) ),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t(Summary,h),one_one(nat))),
                                one_one(nat) ) ) )) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.elims
tff(fact_1970_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      vEBT_T_i_n_s_e_r_t(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2)))))),
        $let(
          xn: nat,
          xn:= 
            $ite(aa(nat,$o,ord_less(nat,Xb),Mi),Mi,Xb),
          $let(
            h: nat,
            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $ite(
              ( aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
              & ~ ( ( Xb = Mi )
                  | ( Xb = Ma ) ) ),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),
                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t(Summary2,h),one_one(nat))),
              one_one(nat) ) ) )) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_1971_T_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062s_092_060_094sub_062e_092_060_094sub_062r_092_060_094sub_062t_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_i_n_s_e_r_t(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),Xaa)) ) )
         => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info,zero_zero(nat),Ts2,S2) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info,zero_zero(nat),Ts2,S2)),Xaa)) ) )
           => ( ! [Info: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts2,S2)),Xaa)) ) )
             => ( ! [V3: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList2,Summary)),Xaa)) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2)))))),
                              $let(
                                xn: nat,
                                xn:= 
                                  $ite(aa(nat,$o,ord_less(nat,Xaa),Mi2),Mi2,Xaa),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $ite(
                                    ( aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))
                                    & ~ ( ( Xaa = Mi2 )
                                        | ( Xaa = Ma2 ) ) ),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_i_n_s_e_r_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                      $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),vEBT_T_i_n_s_e_r_t(Summary,h),one_one(nat))),
                                    one_one(nat) ) ) )) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_1972_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(Xb,Xaa) = Y )
     => ( ( ? [Uu2: $o,B2: $o] : Xb = vEBT_Leaf((Uu2),(B2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] : Xb = vEBT_Leaf((Uv),(Uw))
           => ( ? [N3: nat] : Xaa = aa(nat,nat,suc,N3)
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V3: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc2,Vd2)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)
                 => ( Y != one_one(nat) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                     => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $ite(
                              aa(nat,$o,ord_less(nat,Xaa),Mi2),
                              one_one(nat),
                              $let(
                                l: nat,
                                l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                $let(
                                  h: nat,
                                  h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),
                                    $ite(
                                      aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                        $let(
                                          maxlow: option(nat),
                                          maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                            $ite(
                                              ( ( maxlow != none(nat) )
                                              & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                              $let(
                                                sc: option(nat),
                                                sc:= vEBT_vebt_succ(Summary,h),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary,h))),one_one(nat))),
                                                  $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                                      one_one(nat) )) ) ) )) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_1973_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d(Xb,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv: $o] : Xb = vEBT_Leaf((Uu2),(Uv))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A2: $o,Uw: $o] : Xb = vEBT_Leaf((A2),(Uw))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ) ) )
         => ( ! [A2: $o,B2: $o] :
                ( ( Xb = vEBT_Leaf((A2),(B2)) )
               => ( ? [Va2: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va2))
                 => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                        $ite((B2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)
               => ( Y != one_one(nat) ) )
             => ( ( ? [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                one_one(nat),
                                $let(
                                  l: nat,
                                  l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                  $let(
                                    h: nat,
                                    h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),one_one(nat))),
                                      $ite(
                                        aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),
                                            $ite(
                                              ( ( minlow != none(nat) )
                                              & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                              $let(
                                                pr: option(nat),
                                                pr:= vEBT_vebt_pred(Summary,h),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary,h))),one_one(nat))),
                                                  $ite(pr = none(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),pr)))))) ) )) ),
                                        one_one(nat) )) ) ) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_1974_log__ceil__idem,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),Xb)
     => ( archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),Xb)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(int,real,ring_1_of_int(real),archimedean_ceiling(real,Xb)))) ) ) ).

% log_ceil_idem
tff(fact_1975_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      vEBT_T_s_u_c_c(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,ord_less(nat,Xb),Mi),
          one_one(nat),
          $let(
            l: nat,
            l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $let(
              h: nat,
              h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),
                $ite(
                  aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),
                    $let(
                      maxlow: option(nat),
                      maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                        $ite(
                          ( ( maxlow != none(nat) )
                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                          $let(
                            sc: option(nat),
                            sc:= vEBT_vebt_succ(Summary2,h),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary2,h))),one_one(nat))),
                              $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                  one_one(nat) )) ) ) )) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_1976_highsimp,axiom,
    ! [Xb: nat,N: nat] : aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),vEBT_VEBT_high(Xb,N)) = vEBT_VEBT_highi(Xb,N) ).

% highsimp
tff(fact_1977_lowsimp,axiom,
    ! [Xb: nat,N: nat] : aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),vEBT_VEBT_low(Xb,N)) = vEBT_VEBT_lowi(Xb,N) ).

% lowsimp
tff(fact_1978_highi__def,axiom,
    ! [Xb: nat,N: nat] : vEBT_VEBT_highi(Xb,N) = aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% highi_def
tff(fact_1979_of__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = zero_zero(A) )
        <=> ( Z = zero_zero(int) ) ) ) ).

% of_int_eq_0_iff
tff(fact_1980_of__int__0__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( zero_zero(A) = aa(int,A,ring_1_of_int(A),Z) )
        <=> ( Z = zero_zero(int) ) ) ) ).

% of_int_0_eq_iff
tff(fact_1981_of__int__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( aa(int,A,ring_1_of_int(A),zero_zero(int)) = zero_zero(A) ) ) ).

% of_int_0
tff(fact_1982_of__int__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int,N: num] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = aa(num,A,numeral_numeral(A),N) )
        <=> ( Z = aa(num,int,numeral_numeral(int),N) ) ) ) ).

% of_int_eq_numeral_iff
tff(fact_1983_of__int__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(num,int,numeral_numeral(int),K)) = aa(num,A,numeral_numeral(A),K) ) ).

% of_int_numeral
tff(fact_1984_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less_eq(int,W),Z) ) ) ).

% of_int_le_iff
tff(fact_1985_of__int__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less(int,W),Z) ) ) ).

% of_int_less_iff
tff(fact_1986_of__int__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = one_one(A) )
        <=> ( Z = one_one(int) ) ) ) ).

% of_int_eq_1_iff
tff(fact_1987_of__int__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ( aa(int,A,ring_1_of_int(A),one_one(int)) = one_one(A) ) ) ).

% of_int_1
tff(fact_1988_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: int,B3: int,W: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Xb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B3)),W) )
        <=> ( Xb = aa(nat,int,aa(int,fun(nat,int),power_power(int),B3),W) ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_1989_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [B3: int,W: nat,Xb: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B3)),W) = aa(int,A,ring_1_of_int(A),Xb) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),B3),W) = Xb ) ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_1990_of__int__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int,N: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),Z)),N) ) ).

% of_int_power
tff(fact_1991_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,ord_less_eq(int,Z),zero_zero(int)) ) ) ).

% of_int_le_0_iff
tff(fact_1992_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),Z) ) ) ).

% of_int_0_le_iff
tff(fact_1993_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,Z: int] :
          ( aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),N)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less_eq(int,aa(num,int,numeral_numeral(int),N)),Z) ) ) ).

% of_int_numeral_le_iff
tff(fact_1994_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,N: num] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),N))
        <=> aa(int,$o,ord_less_eq(int,Z),aa(num,int,numeral_numeral(int),N)) ) ) ).

% of_int_le_numeral_iff
tff(fact_1995_of__int__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,ord_less(int,Z),zero_zero(int)) ) ) ).

% of_int_less_0_iff
tff(fact_1996_of__int__0__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less(int,zero_zero(int)),Z) ) ) ).

% of_int_0_less_iff
tff(fact_1997_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num,Z: int] :
          ( aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),N)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less(int,aa(num,int,numeral_numeral(int),N)),Z) ) ) ).

% of_int_numeral_less_iff
tff(fact_1998_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,N: num] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),N))
        <=> aa(int,$o,ord_less(int,Z),aa(num,int,numeral_numeral(int),N)) ) ) ).

% of_int_less_numeral_iff
tff(fact_1999_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,ord_less_eq(int,Z),one_one(int)) ) ) ).

% of_int_le_1_iff
tff(fact_2000_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less_eq(A,one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less_eq(int,one_one(int)),Z) ) ) ).

% of_int_1_le_iff
tff(fact_2001_of__int__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,ord_less(int,Z),one_one(int)) ) ) ).

% of_int_less_1_iff
tff(fact_2002_of__int__1__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,ord_less(A,one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less(int,one_one(int)),Z) ) ) ).

% of_int_1_less_iff
tff(fact_2003_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,Xb: num,N: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),N) )
        <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),N) ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_2004_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: num,N: nat,Y: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),N) = aa(int,A,ring_1_of_int(A),Y) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),N) = Y ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_2005_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int,B3: int,W: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B3)),W))
        <=> aa(int,$o,ord_less_eq(int,Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),B3),W)) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_2006_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: int,W: nat,Xb: int] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B3)),W)),aa(int,A,ring_1_of_int(A),Xb))
        <=> aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),B3),W)),Xb) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_2007_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int,B3: int,W: nat] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B3)),W))
        <=> aa(int,$o,ord_less(int,Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),B3),W)) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_2008_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: int,W: nat,Xb: int] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B3)),W)),aa(int,A,ring_1_of_int(A),Xb))
        <=> aa(int,$o,ord_less(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),B3),W)),Xb) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_2009_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,Xb: num,N: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),N))
        <=> aa(int,$o,ord_less_eq(int,A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),N)) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_2010_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,N: nat,A3: int] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),N)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),N)),A3) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_2011_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,Xb: num,N: nat] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),N))
        <=> aa(int,$o,ord_less(int,A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),N)) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_2012_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,N: nat,A3: int] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),N)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$o,ord_less(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),N)),A3) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_2013_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z2: int] : aa(A,$o,ord_less_eq(A,Xb),aa(int,A,ring_1_of_int(A),Z2)) ) ).

% ex_le_of_int
tff(fact_2014_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z2: int] : aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Z2)),Xb) ) ).

% ex_of_int_less
tff(fact_2015_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z2: int] : aa(A,$o,ord_less(A,Xb),aa(int,A,ring_1_of_int(A),Z2)) ) ).

% ex_less_of_int
tff(fact_2016_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,ord_less_eq(A,Xb),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xb))) ) ).

% le_of_int_ceiling
tff(fact_2017_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,Xb)),Z)
        <=> aa(A,$o,ord_less_eq(A,Xb),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% ceiling_le_iff
tff(fact_2018_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A3: int] :
          ( aa(A,$o,ord_less_eq(A,Xb),aa(int,A,ring_1_of_int(A),A3))
         => aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,Xb)),A3) ) ) ).

% ceiling_le
tff(fact_2019_less__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,ord_less(int,Z),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Z)),Xb) ) ) ).

% less_ceiling_iff
tff(fact_2020_real__of__int__div4,axiom,
    ! [N: int,Xb: int] : aa(real,$o,ord_less_eq(real,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),N),Xb))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),N)),aa(int,real,ring_1_of_int(real),Xb))) ).

% real_of_int_div4
tff(fact_2021_minNull__bound,axiom,
    ! [Tb: vEBT_VEBT] : aa(nat,$o,ord_less_eq(nat,vEBT_T_m_i_n_N_u_l_l(Tb)),one_one(nat)) ).

% minNull_bound
tff(fact_2022_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I1_J,axiom,
    vEBT_T_m_i_n_N_u_l_l(vEBT_Leaf($false,$false)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(1)
tff(fact_2023_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I2_J,axiom,
    ! [Uv2: $o] : vEBT_T_m_i_n_N_u_l_l(vEBT_Leaf($true,(Uv2))) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
tff(fact_2024_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I3_J,axiom,
    ! [Uu: $o] : vEBT_T_m_i_n_N_u_l_l(vEBT_Leaf((Uu),$true)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(3)
tff(fact_2025_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_nonneg
tff(fact_2026_of__int__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,ord_less(int,zero_zero(int)),Z)
         => aa(A,$o,ord_less(A,zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_pos
tff(fact_2027_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [X2: int] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),X2)),Xb)
          & aa(A,$o,ord_less(A,Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),one_one(int))))
          & ! [Y4: int] :
              ( ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Y4)),Xb)
                & aa(A,$o,ord_less(A,Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Y4),one_one(int)))) )
             => ( Y4 = X2 ) ) ) ) ).

% floor_exists1
tff(fact_2028_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z2: int] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z2)),Xb)
          & aa(A,$o,ord_less(A,Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z2),one_one(int)))) ) ) ).

% floor_exists
tff(fact_2029_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R2),one_one(A))) ) ).

% of_int_ceiling_le_add_one
tff(fact_2030_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R2))),one_one(A))),R2) ) ).

% of_int_ceiling_diff_one_le
tff(fact_2031_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,Xb: int] :
          ( aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),N)),aa(int,A,ring_1_of_int(A),Xb))
        <=> aa(int,$o,ord_less(int,aa(nat,int,semiring_1_of_nat(int),N)),Xb) ) ) ).

% of_nat_less_of_int_iff
tff(fact_2032_int__le__real__less,axiom,
    ! [N: int,M: int] :
      ( aa(int,$o,ord_less_eq(int,N),M)
    <=> aa(real,$o,ord_less(real,aa(int,real,ring_1_of_int(real),N)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),M)),one_one(real))) ) ).

% int_le_real_less
tff(fact_2033_int__less__real__le,axiom,
    ! [N: int,M: int] :
      ( aa(int,$o,ord_less(int,N),M)
    <=> aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),N)),one_one(real))),aa(int,real,ring_1_of_int(real),M)) ) ).

% int_less_real_le
tff(fact_2034_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N: int] :
      ( aa(int,$o,ord_less_eq(int,M),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N))
     => ( set_or1337092689740270186AtMost(int,M,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)) = aa(set(int),set(int),insert(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)),set_or1337092689740270186AtMost(int,M,N)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_2035_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      vEBT_T_m_a_x_t(vEBT_Leaf((A3),(B3))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite((B3),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
tff(fact_2036_simp__from__to,axiom,
    ! [I: int,J: int] :
      set_or1337092689740270186AtMost(int,I,J) = $ite(aa(int,$o,ord_less(int,J),I),bot_bot(set(int)),aa(set(int),set(int),insert(int,I),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J))) ).

% simp_from_to
tff(fact_2037_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : vEBT_T_m_a_x_t(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv2,Uw2)) = one_one(nat) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
tff(fact_2038_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : vEBT_T_m_i_n_t(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv2,Uw2)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
tff(fact_2039_ceiling__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xb))),one_one(A))),Xb)
          & aa(A,$o,ord_less_eq(A,Xb),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xb))) ) ) ).

% ceiling_correct
tff(fact_2040_ceiling__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xb)
         => ( aa(A,$o,ord_less_eq(A,Xb),aa(int,A,ring_1_of_int(A),Z))
           => ( archimedean_ceiling(A,Xb) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_2041_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A3: int] :
          ( ( archimedean_ceiling(A,Xb) = A3 )
        <=> ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),A3)),one_one(A))),Xb)
            & aa(A,$o,ord_less_eq(A,Xb),aa(int,A,ring_1_of_int(A),A3)) ) ) ) ).

% ceiling_eq_iff
tff(fact_2042_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Tb: A] :
          ( aa(int,$o,P,archimedean_ceiling(A,Tb))
        <=> ! [I4: int] :
              ( ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),I4)),one_one(A))),Tb)
                & aa(A,$o,ord_less_eq(A,Tb),aa(int,A,ring_1_of_int(A),I4)) )
             => aa(int,$o,P,I4) ) ) ) ).

% ceiling_split
tff(fact_2043_ceiling__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,Xb)),Z)
        <=> aa(A,$o,ord_less_eq(A,Xb),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))) ) ) ).

% ceiling_less_iff
tff(fact_2044_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,ord_less_eq(int,Z),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xb) ) ) ).

% le_ceiling_iff
tff(fact_2045_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I5_J,axiom,
    ! [Uz: product_prod(nat,nat),Va: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT] : vEBT_T_m_i_n_N_u_l_l(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va,Vb,Vc)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
tff(fact_2046_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : vEBT_T_m_i_n_N_u_l_l(vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux,Uy)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
tff(fact_2047_real__of__int__div2,axiom,
    ! [N: int,Xb: int] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),N)),aa(int,real,ring_1_of_int(real),Xb))),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),N),Xb)))) ).

% real_of_int_div2
tff(fact_2048_real__of__int__div3,axiom,
    ! [N: int,Xb: int] : aa(real,$o,ord_less_eq(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),N)),aa(int,real,ring_1_of_int(real),Xb))),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),N),Xb)))),one_one(real)) ).

% real_of_int_div3
tff(fact_2049_maxt__bound,axiom,
    ! [Tb: vEBT_VEBT] : aa(nat,$o,ord_less_eq(nat,vEBT_T_m_a_x_t(Tb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ).

% maxt_bound
tff(fact_2050_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Q2)
         => aa(A,$o,ord_less_eq(A,P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q2)))),Q2)) ) ) ).

% ceiling_divide_upper
tff(fact_2051_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_T_m_a_x_t(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux,Uy,Uz)) = one_one(nat) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_2052_mint__bound,axiom,
    ! [Tb: vEBT_VEBT] : aa(nat,$o,ord_less_eq(nat,vEBT_T_m_i_n_t(Tb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ).

% mint_bound
tff(fact_2053_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      vEBT_T_m_i_n_t(vEBT_Leaf((A3),(B3))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite((A3),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
tff(fact_2054_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT] : vEBT_T_m_i_n_t(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux,Uy,Uz)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_2055_ceiling__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Q2)
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q2)))),one_one(A))),Q2)),P2) ) ) ).

% ceiling_divide_lower
tff(fact_2056_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: int,Xb: A] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),N)),Xb)
         => ( aa(A,$o,ord_less_eq(A,Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),N)),one_one(A)))
           => ( archimedean_ceiling(A,Xb) = aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_2057_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_N_u_l_l(Xb) = Y )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uv: $o] : Xb = vEBT_Leaf($true,(Uv))
           => ( Y != one_one(nat) ) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => ( Y != one_one(nat) ) )
           => ( ( ? [Uw: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uw,Ux2,Uy2)
               => ( Y != one_one(nat) ) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)
                 => ( Y != one_one(nat) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_2058_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(Xb) = Y )
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                  $ite((B2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_2059_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(Xb) = Y )
     => ( ! [A2: $o] :
            ( ? [B2: $o] : Xb = vEBT_Leaf((A2),(B2))
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                  $ite((A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_2060_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      vEBT_T_d_e_l_e_t_e(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
        $ite(
          ( aa(nat,$o,ord_less(nat,Xb),Mi)
          | aa(nat,$o,ord_less(nat,Ma),Xb) ),
          one_one(nat),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
            $ite(
              ( ( Xb = Mi )
              & ( Xb = Ma ) ),
              aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)),
              aa(nat,nat,
                aa(nat,fun(nat,nat),plus_plus(nat),
                  aa(nat,nat,
                    aa(nat,fun(nat,nat),plus_plus(nat),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),
                        $ite(Xb = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(Summary2)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),one_one(nat)))),
                    one_one(nat))),
                $let(
                  xn: nat,
                  xn:= 
                    $ite(Xb = Mi,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary2)))))),Xb),
                  $let(
                    l: nat,
                    l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                    $let(
                      h: nat,
                      h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                      $ite(
                        aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_d_e_l_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l))),
                          $let(
                            newnode: vEBT_VEBT,
                            newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l),
                            $let(
                              newlist: list(vEBT_VEBT),
                              newlist:= list_update(vEBT_VEBT,TreeList,h,newnode),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_i_n_N_u_l_l(newnode))),
                                $ite(
                                  vEBT_VEBT_minNull(newnode),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_d_e_l_e_t_e(Summary2,h))),
                                    $let(
                                      sn: vEBT_VEBT,
                                      sn:= vEBT_vebt_delete(Summary2,h),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                        $ite(
                                          xn = Ma,
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(sn))),
                                            $let(
                                              maxs: option(nat),
                                              maxs:= vEBT_vebt_maxt(sn),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                                $ite(maxs = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) )),
                                          one_one(nat) )) )),
                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                    $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h))),one_one(nat))) )) ) )),
                        one_one(nat) ) ) ) )) )) )) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(7)
tff(fact_2061_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_d_e_l_e_t_e(Xb,Xaa) = Y )
     => ( ( ? [A2: $o,B2: $o] : Xb = vEBT_Leaf((A2),(B2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A2: $o,B2: $o] : Xb = vEBT_Leaf((A2),(B2))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A2: $o,B2: $o] : Xb = vEBT_Leaf((A2),(B2))
             => ( ? [N3: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N3))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary)
               => ( Y != one_one(nat) ) )
             => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                              $ite(
                                ( aa(nat,$o,ord_less(nat,Xaa),Mi2)
                                | aa(nat,$o,ord_less(nat,Ma2),Xaa) ),
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                  $ite(
                                    ( ( Xaa = Mi2 )
                                    & ( Xaa = Ma2 ) ),
                                    aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)),
                                    aa(nat,nat,
                                      aa(nat,fun(nat,nat),plus_plus(nat),
                                        aa(nat,nat,
                                          aa(nat,fun(nat,nat),plus_plus(nat),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),
                                              $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(Summary)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),one_one(nat)))),
                                          one_one(nat))),
                                      $let(
                                        xn: nat,
                                        xn:= 
                                          $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),Xaa),
                                        $let(
                                          l: nat,
                                          l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                          $let(
                                            h: nat,
                                            h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                            $ite(
                                              aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_d_e_l_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l))),
                                                $let(
                                                  newnode: vEBT_VEBT,
                                                  newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l),
                                                  $let(
                                                    newlist: list(vEBT_VEBT),
                                                    newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_i_n_N_u_l_l(newnode))),
                                                      $ite(
                                                        vEBT_VEBT_minNull(newnode),
                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_d_e_l_e_t_e(Summary,h))),
                                                          $let(
                                                            sn: vEBT_VEBT,
                                                            sn:= vEBT_vebt_delete(Summary,h),
                                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                              $ite(
                                                                xn = Ma2,
                                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(sn))),
                                                                  $let(
                                                                    maxs: option(nat),
                                                                    maxs:= vEBT_vebt_maxt(sn),
                                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                                                      $ite(maxs = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) )),
                                                                one_one(nat) )) )),
                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                          $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h))),one_one(nat))) )) ) )),
                                              one_one(nat) ) ) ) )) )) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.elims
tff(fact_2062_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: nat] :
      vEBT_T_p_r_e_d(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),TreeList,Summary2),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,ord_less(nat,Ma),Xb),
          one_one(nat),
          $let(
            l: nat,
            l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
            $let(
              h: nat,
              h:= vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),one_one(nat))),
                $ite(
                  aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList)),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),
                      $ite(
                        ( ( minlow != none(nat) )
                        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                        $let(
                          pr: option(nat),
                          pr:= vEBT_vebt_pred(Summary2,h),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary2,h))),one_one(nat))),
                            $ite(pr = none(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))))) ) )) ),
                  one_one(nat) )) ) ) )) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_2063_T_092_060_094sub_062d_092_060_094sub_062e_092_060_094sub_062l_092_060_094sub_062e_092_060_094sub_062t_092_060_094sub_062e_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_d_e_l_e_t_e(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),zero_zero(nat))) ) ) )
         => ( ! [A2: $o,B2: $o] :
                ( ( Xb = vEBT_Leaf((A2),(B2)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A2: $o,B2: $o] :
                  ( ( Xb = vEBT_Leaf((A2),(B2)) )
                 => ! [N3: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N3)) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),aa(nat,nat,suc,aa(nat,nat,suc,N3)))) ) ) )
             => ( ! [Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg,TreeList2,Summary)),Xaa)) ) )
               => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),zero_zero(nat),TreeList2,Summary)),Xaa)) ) )
                 => ( ! [Mi2: nat,Ma2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary) )
                       => ( ( Y = one_one(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,zero_zero(nat)),TreeList2,Summary)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                          ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                  $ite(
                                    ( aa(nat,$o,ord_less(nat,Xaa),Mi2)
                                    | aa(nat,$o,ord_less(nat,Ma2),Xaa) ),
                                    one_one(nat),
                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                      $ite(
                                        ( ( Xaa = Mi2 )
                                        & ( Xaa = Ma2 ) ),
                                        aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)),
                                        aa(nat,nat,
                                          aa(nat,fun(nat,nat),plus_plus(nat),
                                            aa(nat,nat,
                                              aa(nat,fun(nat,nat),plus_plus(nat),
                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),
                                                  $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_m_i_n_t(Summary)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),one_one(nat)))),
                                              one_one(nat))),
                                          $let(
                                            xn: nat,
                                            xn:= 
                                              $ite(Xaa = Mi2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),Xaa),
                                            $let(
                                              l: nat,
                                              l:= vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                              $let(
                                                h: nat,
                                                h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                                $ite(
                                                  aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_d_e_l_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l))),
                                                    $let(
                                                      newnode: vEBT_VEBT,
                                                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l),
                                                      $let(
                                                        newlist: list(vEBT_VEBT),
                                                        newlist:= list_update(vEBT_VEBT,TreeList2,h,newnode),
                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_i_n_N_u_l_l(newnode))),
                                                          $ite(
                                                            vEBT_VEBT_minNull(newnode),
                                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_d_e_l_e_t_e(Summary,h))),
                                                              $let(
                                                                sn: vEBT_VEBT,
                                                                sn:= vEBT_vebt_delete(Summary,h),
                                                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                                  $ite(
                                                                    xn = Ma2,
                                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(sn))),
                                                                      $let(
                                                                        maxs: option(nat),
                                                                        maxs:= vEBT_vebt_maxt(sn),
                                                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                                                          $ite(maxs = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),aa(option(nat),nat,the2(nat),maxs)))))) )),
                                                                    one_one(nat) )) )),
                                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                                              $ite(xn = Ma2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h))),one_one(nat))) )) ) )),
                                                  one_one(nat) ) ) ) )) )) )) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_2064_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_s_u_c_c(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(B2))),zero_zero(nat))) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( Xb = vEBT_Leaf((Uv),(Uw)) )
               => ! [N3: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N3) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uv),(Uw))),aa(nat,nat,suc,N3))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xaa)) ) )
             => ( ! [V3: product_prod(nat,nat),Vc2: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc2,Vd2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc2,Vd2)),Xaa)) ) )
               => ( ! [V3: product_prod(nat,nat),Vg2: list(vEBT_VEBT),Vh2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vg2,Vh2)),Xaa)) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                       => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                $ite(
                                  aa(nat,$o,ord_less(nat,Xaa),Mi2),
                                  one_one(nat),
                                  $let(
                                    l: nat,
                                    l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                    $let(
                                      h: nat,
                                      h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),
                                        $ite(
                                          aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),
                                            $let(
                                              maxlow: option(nat),
                                              maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),
                                                $ite(
                                                  ( ( maxlow != none(nat) )
                                                  & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_s_u_c_c(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                                  $let(
                                                    sc: option(nat),
                                                    sc:= vEBT_vebt_succ(Summary,h),
                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c(Summary,h))),one_one(nat))),
                                                      $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                                          one_one(nat) )) ) ) )) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_2065_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: nat] :
      ( ( vEBT_T_p_r_e_d(Xb,Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv))),zero_zero(nat))) ) ) )
         => ( ! [A2: $o,Uw: $o] :
                ( ( Xb = vEBT_Leaf((A2),(Uw)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) )
                   => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(Uw))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A2: $o,B2: $o] :
                  ( ( Xb = vEBT_Leaf((A2),(B2)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite((B2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A2),(B2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va3: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va3)),Xaa)) ) )
               => ( ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2)),Xaa)) ) )
                 => ( ! [V3: product_prod(nat,nat),Vh2: list(vEBT_VEBT),Vi2: vEBT_VEBT] :
                        ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2) )
                       => ( ( Y = one_one(nat) )
                         => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vh2,Vi2)),Xaa)) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                          ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary) )
                         => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    aa(nat,$o,ord_less(nat,Ma2),Xaa),
                                    one_one(nat),
                                    $let(
                                      l: nat,
                                      l:= vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                      $let(
                                        h: nat,
                                        h:= vEBT_VEBT_high(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),one_one(nat))),
                                          $ite(
                                            aa(nat,$o,ord_less(nat,h),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2)),
                                            $let(
                                              minlow: option(nat),
                                              minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)),
                                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h)))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)))),
                                                $ite(
                                                  ( ( minlow != none(nat) )
                                                  & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_p_r_e_d(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),h),l)),
                                                  $let(
                                                    pr: option(nat),
                                                    pr:= vEBT_vebt_pred(Summary,h),
                                                    aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d(Summary,h))),one_one(nat))),
                                                      $ite(pr = none(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),aa(option(nat),nat,the2(nat),pr)))))) ) )) ),
                                            one_one(nat) )) ) ) )) )
                           => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList2,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_2066_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M),bit0(N)) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_gh(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).

% divmod_algorithm_code(6)
tff(fact_2067_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K5: real,N: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,Z)),K5)
           => ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H))),K5)
             => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),N))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),K5),aa(nat,nat,minus_minus(nat,N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),real_V7770717601297561774m_norm(A,H))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_2068_foldr__zero,axiom,
    ! [Xs: list(nat),D: nat] :
      ( ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(list(nat),nat,size_size(list(nat)),Xs))
         => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,nth(nat,Xs),I2)) )
     => aa(nat,$o,ord_less_eq(nat,aa(list(nat),nat,size_size(list(nat)),Xs)),aa(nat,nat,minus_minus(nat,foldr(nat,nat,plus_plus(nat),Xs,D)),D)) ) ).

% foldr_zero
tff(fact_2069_rel__of__empty,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o)] : rel_of(A,B,aTP_Lamp_gi(A,option(B)),P) = bot_bot(set(product_prod(A,B))) ).

% rel_of_empty
tff(fact_2070_foldr__one,axiom,
    ! [D: nat,Ys: list(nat)] : aa(nat,$o,ord_less_eq(nat,D),foldr(nat,nat,plus_plus(nat),Ys,D)) ).

% foldr_one
tff(fact_2071_foldr__same__int,axiom,
    ! [Xs: list(nat),Y: nat] :
      ( ! [X2: nat,Y2: nat] :
          ( member(nat,X2,set2(nat,Xs))
         => ( member(nat,Y2,set2(nat,Xs))
           => ( X2 = Y2 ) ) )
     => ( ! [X2: nat] :
            ( member(nat,X2,set2(nat,Xs))
           => ( X2 = Y ) )
       => ( foldr(nat,nat,plus_plus(nat),Xs,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(nat),nat,size_size(list(nat)),Xs)),Y) ) ) ) ).

% foldr_same_int
tff(fact_2072_foldr__mono,axiom,
    ! [Xs: list(nat),Ys: list(nat),C2: nat,D: nat] :
      ( ( aa(list(nat),nat,size_size(list(nat)),Xs) = aa(list(nat),nat,size_size(list(nat)),Ys) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,ord_less(nat,I2),aa(list(nat),nat,size_size(list(nat)),Xs))
           => aa(nat,$o,ord_less(nat,aa(nat,nat,nth(nat,Xs),I2)),aa(nat,nat,nth(nat,Ys),I2)) )
       => ( aa(nat,$o,ord_less_eq(nat,C2),D)
         => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),foldr(nat,nat,plus_plus(nat),Xs,C2)),aa(list(nat),nat,size_size(list(nat)),Ys))),foldr(nat,nat,plus_plus(nat),Ys,D)) ) ) ) ).

% foldr_mono
tff(fact_2073_word__of__int__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(int,word(A),ring_1_of_int(word(A)),aa(num,int,numeral_numeral(int),Bin)) = aa(num,word(A),numeral_numeral(word(A)),Bin) ) ).

% word_of_int_numeral
tff(fact_2074_word__of__int__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,word(A),ring_1_of_int(word(A)),zero_zero(int)) = zero_zero(word(A)) ) ) ).

% word_of_int_0
tff(fact_2075_word__of__int__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,word(A),ring_1_of_int(word(A)),one_one(int)) = one_one(word(A)) ) ) ).

% word_of_int_1
tff(fact_2076_numeral__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K: num,L: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,K,L)) ) ).

% numeral_div_numeral
tff(fact_2077_foldr__length,axiom,
    ! [A: $tType,L: list(A)] : foldr(A,nat,aTP_Lamp_gj(A,fun(nat,nat)),L,zero_zero(nat)) = aa(list(A),nat,size_size(list(A)),L) ).

% foldr_length
tff(fact_2078_divmod__algorithm__code_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num] : unique8689654367752047608divmod(A,M,one2) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(num,A,numeral_numeral(A),M)),zero_zero(A)) ) ).

% divmod_algorithm_code(2)
tff(fact_2079_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one2,bit0(N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(3)
tff(fact_2080_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit1,N)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),one2)) ) ).

% divmod_algorithm_code(4)
tff(fact_2081_Suc__0__div__numeral,axiom,
    ! [K: num] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_fst(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ).

% Suc_0_div_numeral
tff(fact_2082_one__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,one2,N)) ) ).

% one_div_numeral
tff(fact_2083_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,bit0(M),bit0(N)) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_gk(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,N)) ) ).

% divmod_algorithm_code(5)
tff(fact_2084_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          unique8689654367752047608divmod(A,bit0(M),aa(num,num,bit1,N)) = $ite(aa(num,$o,ord_less_eq(num,M),N),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),bit0(M))),unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,bit0(M),bit0(aa(num,num,bit1,N))))) ) ).

% divmod_algorithm_code(7)
tff(fact_2085_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,N)) = $ite(aa(num,$o,ord_less(num,M),N),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),unique1321980374590559556d_step(A,aa(num,num,bit1,N),unique8689654367752047608divmod(A,aa(num,num,bit1,M),bit0(aa(num,num,bit1,N))))) ) ).

% divmod_algorithm_code(8)
tff(fact_2086_word__numeral__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(num,word(A),numeral_numeral(word(A)),B3) = aa(int,word(A),ring_1_of_int(word(A)),aa(num,int,numeral_numeral(int),B3)) ) ).

% word_numeral_alt
tff(fact_2087_word__of__int__power__hom,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: int,N: nat] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A3)),N) = aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),A3),N)) ) ).

% word_of_int_power_hom
tff(fact_2088_foldr__length__aux,axiom,
    ! [A: $tType,L: list(A),A3: nat] : foldr(A,nat,aTP_Lamp_gj(A,fun(nat,nat)),L,A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),aa(list(A),nat,size_size(list(A)),L)) ).

% foldr_length_aux
tff(fact_2089_fst__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,M,N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) ) ).

% fst_divmod
tff(fact_2090_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K6: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K6)
              & ! [N8: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,X5,N8))),K6) )
        <=> ? [N9: nat] :
            ! [N8: A] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,aa(A,B,X5,N8))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N9))) ) ) ).

% lemma_NBseq_def2
tff(fact_2091_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K6: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K6)
              & ! [N8: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,X5,N8))),K6) )
        <=> ? [N9: nat] :
            ! [N8: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,X5,N8))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N9))) ) ) ).

% lemma_NBseq_def
tff(fact_2092_word__of__int__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) ) ).

% word_of_int_2p
tff(fact_2093_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          unique8689654367752047608divmod(A,M,N) = $ite(aa(num,$o,ord_less(num,M),N),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),zero_zero(A)),aa(num,A,numeral_numeral(A),M)),unique1321980374590559556d_step(A,N,unique8689654367752047608divmod(A,M,bit0(N)))) ) ).

% divmod_divmod_step
tff(fact_2094_rel__of__def,axiom,
    ! [B: $tType,A: $tType,M: fun(A,option(B)),P: fun(product_prod(A,B),$o)] : rel_of(A,B,M,P) = collect(product_prod(A,B),product_case_prod(A,B,$o,aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),aTP_Lamp_gl(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),M),P))) ).

% rel_of_def
tff(fact_2095_norm__divide__numeral,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A3)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_divide_numeral
tff(fact_2096_norm__mult__numeral1,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [W: num,A3: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),W)),real_V7770717601297561774m_norm(A,A3)) ) ).

% norm_mult_numeral1
tff(fact_2097_norm__mult__numeral2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A3)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_mult_numeral2
tff(fact_2098_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,Xb)),zero_zero(real))
        <=> ( Xb = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_2099_zero__less__norm__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),real_V7770717601297561774m_norm(A,Xb))
        <=> ( Xb != zero_zero(A) ) ) ) ).

% zero_less_norm_iff
tff(fact_2100_norm__numeral,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [W: num] : real_V7770717601297561774m_norm(A,aa(num,A,numeral_numeral(A),W)) = aa(num,real,numeral_numeral(real),W) ) ).

% norm_numeral
tff(fact_2101_norm__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( real_V7770717601297561774m_norm(A,one_one(A)) = one_one(real) ) ) ).

% norm_one
tff(fact_2102_norm__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( real_V7770717601297561774m_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).

% norm_zero
tff(fact_2103_norm__eq__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( ( real_V7770717601297561774m_norm(A,Xb) = zero_zero(real) )
        <=> ( Xb = zero_zero(A) ) ) ) ).

% norm_eq_zero
tff(fact_2104_norm__not__less__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] : ~ aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xb)),zero_zero(real)) ) ).

% norm_not_less_zero
tff(fact_2105_norm__ge__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] : aa(real,$o,ord_less_eq(real,zero_zero(real)),real_V7770717601297561774m_norm(A,Xb)) ) ).

% norm_ge_zero
tff(fact_2106_norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: A,B3: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A3)),real_V7770717601297561774m_norm(A,B3)) ) ).

% norm_divide
tff(fact_2107_norm__power,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A,N: nat] : real_V7770717601297561774m_norm(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Xb)),N) ) ).

% norm_power
tff(fact_2108_power__eq__imp__eq__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,N: nat,Z: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),N) )
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
           => ( real_V7770717601297561774m_norm(A,W) = real_V7770717601297561774m_norm(A,Z) ) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_2109_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A3)),real_V7770717601297561774m_norm(A,B3)) ) ) ) ).

% nonzero_norm_divide
tff(fact_2110_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xb: A,R2: real,Y: A,S: real] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xb)),R2)
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Y)),S)
           => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y))),aa(real,real,aa(real,fun(real,real),times_times(real),R2),S)) ) ) ) ).

% norm_mult_less
tff(fact_2111_norm__triangle__lt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E3: real] :
          ( aa(real,$o,ord_less(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))),E3)
         => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),E3) ) ) ).

% norm_triangle_lt
tff(fact_2112_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,R2: real,Y: A,S: real] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xb)),R2)
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Y)),S)
           => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R2),S)) ) ) ) ).

% norm_add_less
tff(fact_2113_norm__diff__triangle__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Xb),Y))),E1)
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y),Z))),E22)
           => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Xb),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),E1),E22)) ) ) ) ).

% norm_diff_triangle_less
tff(fact_2114_norm__power__ineq,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xb: A,N: nat] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Xb)),N)) ) ).

% norm_power_ineq
tff(fact_2115_power__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W),N) = one_one(A) )
         => ( ( real_V7770717601297561774m_norm(A,W) = one_one(real) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% power_eq_1_iff
tff(fact_2116_square__norm__one,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
         => ( real_V7770717601297561774m_norm(A,Xb) = one_one(real) ) ) ) ).

% square_norm_one
tff(fact_2117_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A,W: A,M: nat] :
          ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,W)),one_one(real))
           => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),W),M)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Z),W)))) ) ) ) ).

% norm_power_diff
tff(fact_2118_lowi__def,axiom,
    ! [Xb: nat,N: nat] : vEBT_VEBT_lowi(Xb,N) = aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% lowi_def
tff(fact_2119_foldr__same,axiom,
    ! [Xs: list(real),Y: real] :
      ( ! [X2: real,Y2: real] :
          ( member(real,X2,set2(real,Xs))
         => ( member(real,Y2,set2(real,Xs))
           => ( X2 = Y2 ) ) )
     => ( ! [X2: real] :
            ( member(real,X2,set2(real,Xs))
           => ( X2 = Y ) )
       => ( foldr(real,real,plus_plus(real),Xs,zero_zero(real)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(list(real),nat,size_size(list(real)),Xs))),Y) ) ) ) ).

% foldr_same
tff(fact_2120_list__every__elemnt__bound__sum__bound,axiom,
    ! [A: $tType,Xs: list(A),Fa: fun(A,nat),Bound: nat,I: nat] :
      ( ! [X2: A] :
          ( member(A,X2,set2(A,Xs))
         => aa(nat,$o,ord_less_eq(nat,aa(A,nat,Fa,X2)),Bound) )
     => aa(nat,$o,ord_less_eq(nat,foldr(nat,nat,plus_plus(nat),map(A,nat,Fa,Xs),I)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound
tff(fact_2121_neg__eucl__rel__int__mult__2,axiom,
    ! [B3: int,A3: int,Q2: int,R2: int] :
      ( aa(int,$o,ord_less_eq(int,B3),zero_zero(int))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),one_one(int)),B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R2)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_2122_of__nat__code__if,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          aa(nat,A,semiring_1_of_nat(A),N) = $ite(N = zero_zero(nat),zero_zero(A),aa(product_prod(nat,nat),A,product_case_prod(nat,nat,A,aTP_Lamp_gm(nat,fun(nat,A))),divmod_nat(N,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% of_nat_code_if
tff(fact_2123_Suc__if__eq,axiom,
    ! [A: $tType,Fa: fun(nat,A),H: fun(nat,A),G: A,N: nat] :
      ( ! [N3: nat] : aa(nat,A,Fa,aa(nat,nat,suc,N3)) = aa(nat,A,H,N3)
     => ( ( aa(nat,A,Fa,zero_zero(nat)) = G )
       => ( aa(nat,A,Fa,N) = $ite(N = zero_zero(nat),G,aa(nat,A,H,aa(nat,nat,minus_minus(nat,N),one_one(nat)))) ) ) ) ).

% Suc_if_eq
tff(fact_2124_foldr0,axiom,
    ! [Xs: list(real),C2: real,D: real] : foldr(real,real,plus_plus(real),Xs,aa(real,real,aa(real,fun(real,real),plus_plus(real),C2),D)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),foldr(real,real,plus_plus(real),Xs,D)),C2) ).

% foldr0
tff(fact_2125_mod__mod__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),B3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) ) ).

% mod_mod_trivial
tff(fact_2126_low__def,axiom,
    ! [Xb: nat,N: nat] : vEBT_VEBT_low(Xb,N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% low_def
tff(fact_2127_map__ident,axiom,
    ! [A: $tType,X3: list(A)] : map(A,A,aTP_Lamp_ct(A,A),X3) = X3 ).

% map_ident
tff(fact_2128_bits__mod__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% bits_mod_0
tff(fact_2129_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% mod_0
tff(fact_2130_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),zero_zero(A)) = A3 ) ).

% mod_by_0
tff(fact_2131_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),A3) = zero_zero(A) ) ).

% mod_self
tff(fact_2132_mod__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3)),B3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) ) ).

% mod_add_self1
tff(fact_2133_mod__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),B3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) ) ).

% mod_add_self2
tff(fact_2134_minus__mod__self2,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,A3),B3)),B3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) ) ).

% minus_mod_self2
tff(fact_2135_length__map,axiom,
    ! [A: $tType,B: $tType,Fa: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),map(B,A,Fa,Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% length_map
tff(fact_2136_mod__less,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N) = M ) ) ).

% mod_less
tff(fact_2137_nat__mod__eq_H,axiom,
    ! [A3: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,A3),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A3),N) = A3 ) ) ).

% nat_mod_eq'
tff(fact_2138_bits__mod__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),one_one(A)) = zero_zero(A) ) ).

% bits_mod_by_1
tff(fact_2139_mod__by__1,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),one_one(A)) = zero_zero(A) ) ).

% mod_by_1
tff(fact_2140_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),B3) = zero_zero(A) ) ).

% mod_mult_self1_is_0
tff(fact_2141_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),B3) = zero_zero(A) ) ).

% mod_mult_self2_is_0
tff(fact_2142_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),B3) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_2143_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),B3) = zero_zero(A) ) ).

% mod_div_trivial
tff(fact_2144_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B3: A,C2: A,A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)),A3)),B3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) ) ).

% mod_mult_self4
tff(fact_2145_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B3: A,A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)),A3)),B3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) ) ).

% mod_mult_self3
tff(fact_2146_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))),B3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) ) ).

% mod_mult_self2
tff(fact_2147_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))),B3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) ) ).

% mod_mult_self1
tff(fact_2148_mod__by__Suc__0,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% mod_by_Suc_0
tff(fact_2149_fst__divmod__nat,axiom,
    ! [M: nat,N: nat] : aa(product_prod(nat,nat),nat,product_fst(nat,nat),divmod_nat(M,N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) ).

% fst_divmod_nat
tff(fact_2150_snd__divmod__nat,axiom,
    ! [M: nat,N: nat] : aa(product_prod(nat,nat),nat,product_snd(nat,nat),divmod_nat(M,N)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N) ).

% snd_divmod_nat
tff(fact_2151_nth__map,axiom,
    ! [B: $tType,A: $tType,N: nat,Xs: list(A),Fa: fun(A,B)] :
      ( aa(nat,$o,ord_less(nat,N),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,B,nth(B,map(A,B,Fa,Xs)),N) = aa(A,B,Fa,aa(nat,A,nth(A,Xs),N)) ) ) ).

% nth_map
tff(fact_2152_Suc__mod__mult__self4,axiom,
    ! [N: nat,K: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)),M))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),N) ).

% Suc_mod_mult_self4
tff(fact_2153_Suc__mod__mult__self3,axiom,
    ! [K: nat,N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)),M))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),N) ).

% Suc_mod_mult_self3
tff(fact_2154_Suc__mod__mult__self2,axiom,
    ! [M: nat,N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),N) ).

% Suc_mod_mult_self2
tff(fact_2155_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),N) ).

% Suc_mod_mult_self1
tff(fact_2156_numeral__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [K: num,L: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),K)),aa(num,A,numeral_numeral(A),L)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,K,L)) ) ).

% numeral_mod_numeral
tff(fact_2157_bits__one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% bits_one_mod_two_eq_one
tff(fact_2158_one__mod__two__eq__one,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% one_mod_two_eq_one
tff(fact_2159_mod2__Suc__Suc,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,M))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% mod2_Suc_Suc
tff(fact_2160_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,N: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) != one_one(nat) )
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),K)),N))),aa(num,nat,numeral_numeral(nat),K)) = one_one(nat) ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_2161_not__mod__2__eq__0__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_2162_not__mod__2__eq__1__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_2163_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))) != aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_2164_add__self__mod__2,axiom,
    ! [M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),M)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ).

% add_self_mod_2
tff(fact_2165_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,N)))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),N)) ).

% mod_Suc_eq_mod_add3
tff(fact_2166_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),aa(num,nat,numeral_numeral(nat),V)) ).

% Suc_mod_eq_add3_mod_numeral
tff(fact_2167_one__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(num,A,numeral_numeral(A),N)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,one2,N)) ) ).

% one_mod_numeral
tff(fact_2168_mod2__gr__0,axiom,
    ! [M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(nat) ) ) ).

% mod2_gr_0
tff(fact_2169_Suc__0__mod__numeral,axiom,
    ! [K: num] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),K)) = aa(product_prod(nat,nat),nat,product_snd(nat,nat),unique8689654367752047608divmod(nat,one2,K)) ).

% Suc_0_mod_numeral
tff(fact_2170_list_Omap__ident,axiom,
    ! [A: $tType,Tb: list(A)] : map(A,A,aTP_Lamp_ct(A,A),Tb) = Tb ).

% list.map_ident
tff(fact_2171_of__nat__mod,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_mod
tff(fact_2172_map__eq__imp__length__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,Fa: fun(B,A),Xs: list(B),G: fun(C,A),Ys: list(C)] :
      ( ( map(B,A,Fa,Xs) = map(C,A,G,Ys) )
     => ( aa(list(B),nat,size_size(list(B)),Xs) = aa(list(C),nat,size_size(list(C)),Ys) ) ) ).

% map_eq_imp_length_eq
tff(fact_2173_map__eq__nth__eq,axiom,
    ! [A: $tType,B: $tType,Fa: fun(B,A),L: list(B),L3: list(B),I: nat] :
      ( ( map(B,A,Fa,L) = map(B,A,Fa,L3) )
     => ( aa(B,A,Fa,aa(nat,B,nth(B,L),I)) = aa(B,A,Fa,aa(nat,B,nth(B,L3),I)) ) ) ).

% map_eq_nth_eq
tff(fact_2174_mod__add__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) ) ).

% mod_add_eq
tff(fact_2175_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,A5: A,B3: A,B4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A5),C2) )
         => ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B4),C2) )
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A5),B4)),C2) ) ) ) ) ).

% mod_add_cong
tff(fact_2176_mod__add__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2)),B3)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) ) ).

% mod_add_left_eq
tff(fact_2177_mod__add__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) ) ).

% mod_add_right_eq
tff(fact_2178_mod__mult__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2) ) ).

% mod_mult_eq
tff(fact_2179_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,A5: A,B3: A,B4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A5),C2) )
         => ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B4),C2) )
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A5),B4)),C2) ) ) ) ) ).

% mod_mult_cong
tff(fact_2180_mod__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),C2) ) ).

% mod_mult_mult2
tff(fact_2181_mult__mod__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ).

% mult_mod_right
tff(fact_2182_mod__mult__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,C2: A,B3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2)),B3)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2) ) ).

% mod_mult_left_eq
tff(fact_2183_mod__mult__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2) ) ).

% mod_mult_right_eq
tff(fact_2184_mod__diff__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,B3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,A3),B3)),C2) ) ).

% mod_diff_eq
tff(fact_2185_mod__diff__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,A5: A,B3: A,B4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A5),C2) )
         => ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B4),C2) )
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,A5),B4)),C2) ) ) ) ) ).

% mod_diff_cong
tff(fact_2186_mod__diff__left__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,B3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2)),B3)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,A3),B3)),C2) ) ).

% mod_diff_left_eq
tff(fact_2187_mod__diff__right__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,A3),B3)),C2) ) ).

% mod_diff_right_eq
tff(fact_2188_power__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A3: A,B3: A,N: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),N)),B3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),B3) ) ).

% power_mod
tff(fact_2189_mod__Suc__eq,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),N) ).

% mod_Suc_eq
tff(fact_2190_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,M))),N) ).

% mod_Suc_Suc_eq
tff(fact_2191_nat__mod__eq,axiom,
    ! [B3: nat,N: nat,A3: nat] :
      ( aa(nat,$o,ord_less(nat,B3),N)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A3),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B3),N) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A3),N) = B3 ) ) ) ).

% nat_mod_eq
tff(fact_2192_mod__plus__right,axiom,
    ! [A3: nat,Xb: nat,M: nat,B3: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),Xb)),M) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B3),Xb)),M) )
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A3),M) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B3),M) ) ) ).

% mod_plus_right
tff(fact_2193_mod__less__eq__dividend,axiom,
    ! [M: nat,N: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)),M) ).

% mod_less_eq_dividend
tff(fact_2194_divmod__nat__def,axiom,
    ! [M: nat,N: nat] : divmod_nat(M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)) ).

% divmod_nat_def
tff(fact_2195_unique__quotient,axiom,
    ! [A3: int,B3: int,Q2: int,R2: int,Q5: int,R5: int] :
      ( eucl_rel_int(A3,B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
     => ( eucl_rel_int(A3,B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R5))
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
tff(fact_2196_unique__remainder,axiom,
    ! [A3: int,B3: int,Q2: int,R2: int,Q5: int,R5: int] :
      ( eucl_rel_int(A3,B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
     => ( eucl_rel_int(A3,B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R5))
       => ( R2 = R5 ) ) ) ).

% unique_remainder
tff(fact_2197_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),A3) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2198_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),B3) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2199_cong__exp__iff__simps_I9_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,N: num] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit0(M))),aa(num,A,numeral_numeral(A),bit0(Q2))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit0(N))),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Q2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_2200_cong__exp__iff__simps_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),one2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),one2)) ) ).

% cong_exp_iff_simps(4)
tff(fact_2201_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) = A3 )
        <=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2202_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),C2) )
         => ~ ! [D3: A] : B3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3)) ) ) ).

% mod_eqE
tff(fact_2203_div__add1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),C2))),C2)) ) ).

% div_add1_eq
tff(fact_2204_mod__Suc,axiom,
    ! [M: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),N) = $ite(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)) = N,zero_zero(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N))) ).

% mod_Suc
tff(fact_2205_mod__induct,axiom,
    ! [P: fun(nat,$o),N: nat,P2: nat,M: nat] :
      ( aa(nat,$o,P,N)
     => ( aa(nat,$o,ord_less(nat,N),P2)
       => ( aa(nat,$o,ord_less(nat,M),P2)
         => ( ! [N3: nat] :
                ( aa(nat,$o,ord_less(nat,N3),P2)
               => ( aa(nat,$o,P,N3)
                 => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,N3)),P2)) ) )
           => aa(nat,$o,P,M) ) ) ) ) ).

% mod_induct
tff(fact_2206_mod__less__divisor,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)),N) ) ).

% mod_less_divisor
tff(fact_2207_nat__mod__lem,axiom,
    ! [N: nat,B3: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,$o,ord_less(nat,B3),N)
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B3),N) = B3 ) ) ) ).

% nat_mod_lem
tff(fact_2208_mod__Suc__le__divisor,axiom,
    ! [M: nat,N: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,suc,N))),N) ).

% mod_Suc_le_divisor
tff(fact_2209_word__rot__lem,axiom,
    ! [L: nat,K: nat,D: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),L),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),D),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),K),L)) )
     => ( aa(nat,$o,ord_less(nat,N),L)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),D),N)),L) = N ) ) ) ).

% word_rot_lem
tff(fact_2210_nat__minus__mod,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),M))),M) = zero_zero(nat) ).

% nat_minus_mod
tff(fact_2211_mod__if,axiom,
    ! [M: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N) = $ite(aa(nat,$o,ord_less(nat,M),N),M,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,minus_minus(nat,M),N)),N)) ).

% mod_if
tff(fact_2212_mod__nat__sub,axiom,
    ! [Xb: nat,Z: nat,Y: nat] :
      ( aa(nat,$o,ord_less(nat,Xb),Z)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,minus_minus(nat,Xb),Y)),Z) = aa(nat,nat,minus_minus(nat,Xb),Y) ) ) ).

% mod_nat_sub
tff(fact_2213_mod__geq,axiom,
    ! [M: nat,N: nat] :
      ( ~ aa(nat,$o,ord_less(nat,M),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,minus_minus(nat,M),N)),N) ) ) ).

% mod_geq
tff(fact_2214_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),D) = zero_zero(nat) )
     => ? [Q3: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D),Q3) ) ).

% mod_eq_0D
tff(fact_2215_nat__minus__mod__plus__right,axiom,
    ! [N: nat,Xb: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),M))),M) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),M) ).

% nat_minus_mod_plus_right
tff(fact_2216_le__mod__geq,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,N),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,minus_minus(nat,M),N)),N) ) ) ).

% le_mod_geq
tff(fact_2217_nat__mod__eq__iff,axiom,
    ! [Xb: nat,N: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Y),N) )
    <=> ? [Q1: nat,Q22: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_2218_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) = A3 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2219_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2220_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num,Q2: num] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit0(N))),aa(num,A,numeral_numeral(A),bit0(Q2))) = zero_zero(A) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2221_cong__exp__iff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),one2)) = zero_zero(A) ) ).

% cong_exp_iff_simps(1)
tff(fact_2222_cong__exp__iff__simps_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit0(M))),aa(num,A,numeral_numeral(A),bit0(Q2))) != aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),one2)),aa(num,A,numeral_numeral(A),bit0(Q2))) ) ).

% cong_exp_iff_simps(8)
tff(fact_2223_cong__exp__iff__simps_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,N: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),one2)),aa(num,A,numeral_numeral(A),bit0(Q2))) != aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit0(N))),aa(num,A,numeral_numeral(A),bit0(Q2))) ) ).

% cong_exp_iff_simps(6)
tff(fact_2224_cong__exp__iff__simps_I13_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,N: num] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),aa(num,A,numeral_numeral(A),bit0(Q2))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Q2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_2225_cong__exp__iff__simps_I12_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,N: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),aa(num,A,numeral_numeral(A),bit0(Q2))) != aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit0(N))),aa(num,A,numeral_numeral(A),bit0(Q2))) ) ).

% cong_exp_iff_simps(12)
tff(fact_2226_cong__exp__iff__simps_I10_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,N: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit0(M))),aa(num,A,numeral_numeral(A),bit0(Q2))) != aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))),aa(num,A,numeral_numeral(A),bit0(Q2))) ) ).

% cong_exp_iff_simps(10)
tff(fact_2227_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B3: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ).

% cancel_div_mod_rules(2)
tff(fact_2228_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),B3)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),C2) ) ).

% cancel_div_mod_rules(1)
tff(fact_2229_mod__div__decomp,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),B3)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) ) ).

% mod_div_decomp
tff(fact_2230_div__mult__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),B3)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) = A3 ) ).

% div_mult_mod_eq
tff(fact_2231_mod__div__mult__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),B3)) = A3 ) ).

% mod_div_mult_eq
tff(fact_2232_mod__mult__div__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3))) = A3 ) ).

% mod_mult_div_eq
tff(fact_2233_mult__div__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [B3: A,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) = A3 ) ).

% mult_div_mod_eq
tff(fact_2234_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),C2))),C2)) ) ).

% div_mult1_eq
tff(fact_2235_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),B3)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) ) ).

% minus_div_mult_eq_mod
tff(fact_2236_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),B3) ) ).

% minus_mod_eq_div_mult
tff(fact_2237_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ).

% minus_mod_eq_mult_div
tff(fact_2238_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] : aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) ) ).

% minus_mult_div_eq_mod
tff(fact_2239_zmde,axiom,
    ! [A: $tType] :
      ( ( group_add(A)
        & semiring_modulo(A) )
     => ! [B3: A,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) = aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) ) ).

% zmde
tff(fact_2240_mod__le__divisor,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)),N) ) ).

% mod_le_divisor
tff(fact_2241_div__less__mono,axiom,
    ! [A4: nat,B6: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,A4),B6)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A4),N) = zero_zero(nat) )
         => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B6),N) = zero_zero(nat) )
           => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A4),N)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B6),N)) ) ) ) ) ).

% div_less_mono
tff(fact_2242_mod__nat__add,axiom,
    ! [Xb: nat,Z: nat,Y: nat] :
      ( aa(nat,$o,ord_less(nat,Xb),Z)
     => ( aa(nat,$o,ord_less(nat,Y),Z)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xb),Y)),Z) = $ite(aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xb),Y)),Z),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xb),Y),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xb),Y)),Z)) ) ) ) ).

% mod_nat_add
tff(fact_2243_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,N: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Y),N) )
     => ( aa(nat,$o,ord_less_eq(nat,Y),Xb)
       => ? [Q3: nat] : Xb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_2244_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),Q2) )
     => ( aa(nat,$o,ord_less_eq(nat,M),N)
       => ~ ! [S2: nat] : N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S2)) ) ) ).

% mod_eq_nat2E
tff(fact_2245_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),Q2) )
     => ( aa(nat,$o,ord_less_eq(nat,N),M)
       => ~ ! [S2: nat] : M != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S2)) ) ) ).

% mod_eq_nat1E
tff(fact_2246_mod__mult2__eq,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),Q2))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)) ).

% mod_mult2_eq
tff(fact_2247_div__mod__decomp,axiom,
    ! [A4: nat,N: nat] : A4 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A4),N)),N)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A4),N)) ).

% div_mod_decomp
tff(fact_2248_modulo__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),N)) ).

% modulo_nat_def
tff(fact_2249_eucl__rel__int__by0,axiom,
    ! [K: int] : eucl_rel_int(K,zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K)) ).

% eucl_rel_int_by0
tff(fact_2250_snd__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,M,N)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N)) ) ).

% snd_divmod
tff(fact_2251_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = Q2 ) ) ).

% div_int_unique
tff(fact_2252_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [N: num,Q2: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))),aa(num,A,numeral_numeral(A),bit0(Q2))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_2253_map__upd__eq,axiom,
    ! [B: $tType,A: $tType,I: nat,L: list(A),Fa: fun(A,B),Xb: A] :
      ( ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
       => ( aa(A,B,Fa,aa(nat,A,nth(A,L),I)) = aa(A,B,Fa,Xb) ) )
     => ( map(A,B,Fa,list_update(A,L,I,Xb)) = map(A,B,Fa,L) ) ) ).

% map_upd_eq
tff(fact_2254_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),M))),aa(nat,A,semiring_1_of_nat(A),N)))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(nat,A,semiring_1_of_nat(A),M))) ) ).

% mod_mult2_eq'
tff(fact_2255_field__char__0__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N)))),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% field_char_0_class.of_nat_div
tff(fact_2256_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] : unique8689654367752047608divmod(A,M,N) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))) ) ).

% divmod_def
tff(fact_2257_split__mod,axiom,
    ! [P: fun(nat,$o),M: nat,N: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N))
    <=> ( ( ( N = zero_zero(nat) )
         => aa(nat,$o,P,M) )
        & ( ( N != zero_zero(nat) )
         => ! [I4: nat,J3: nat] :
              ( aa(nat,$o,ord_less(nat,J3),N)
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),I4)),J3) )
               => aa(nat,$o,P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_2258_mod__lemma,axiom,
    ! [C2: nat,R2: nat,B3: nat,Q2: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => ( aa(nat,$o,ord_less(nat,R2),B3)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Q2),C2))),R2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),C2)) ) ) ).

% mod_lemma
tff(fact_2259_real__of__nat__div__aux,axiom,
    ! [Xb: nat,D: nat] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Xb)),aa(nat,real,semiring_1_of_nat(real),D)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),D))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),D))),aa(nat,real,semiring_1_of_nat(real),D))) ).

% real_of_nat_div_aux
tff(fact_2260_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q2: int] :
      ( ( L != zero_zero(int) )
     => ( ( K = aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L) )
       => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),zero_zero(int))) ) ) ).

% eucl_rel_int_dividesI
tff(fact_2261_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2262_cong__exp__iff__simps_I11_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),aa(num,A,numeral_numeral(A),bit0(Q2))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),one2)),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(11)
tff(fact_2263_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,N: num] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),one2)),aa(num,A,numeral_numeral(A),bit0(Q2))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))),aa(num,A,numeral_numeral(A),bit0(Q2))) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),N)),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_2264_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,M)))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),M)),N) ).

% Suc_mod_eq_add3_mod
tff(fact_2265_divmod_H__nat__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(nat,M,N) = aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))) ).

% divmod'_nat_def
tff(fact_2266_Suc__times__mod__eq,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N))),M) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_2267_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))),B3)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2268_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = A3 )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_2269_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat,M: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% div_exp_mod_exp_eq
tff(fact_2270_power__mod__div,axiom,
    ! [Xb: nat,N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M))) ).

% power_mod_div
tff(fact_2271_verit__le__mono__div,axiom,
    ! [A4: nat,B6: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,A4),B6)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
       => aa(nat,$o,
            ord_less_eq(nat,
              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A4),N)),
                $ite(aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B6),N) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B6),N)) ) ) ).

% verit_le_mono_div
tff(fact_2272_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_Node(Info2,Deg2,TreeList,Summary2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Summary2))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2,TreeList),zero_zero(nat))) ).

% VEBT_internal.cnt'.simps(2)
tff(fact_2273_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))),B3)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2274_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat,A3: A] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ) ) ).

% mult_exp_mod_exp_eq
tff(fact_2275_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Xb) = Y )
     => ( ( ? [A2: $o,B2: $o] : Xb = vEBT_Leaf((A2),(B2))
         => ( Y != one_one(nat) ) )
       => ~ ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info,Deg,TreeList2,Summary) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Summary))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2,TreeList2),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.cnt'.elims
tff(fact_2276_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,Xb: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),M)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
           => ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Xb),M) )
              | ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Xb),M)),M) ) ) ) ) ) ).

% mod_double_modulus
tff(fact_2277_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
           => ( aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)))
             => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))),B3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_2278_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),aa(nat,nat,suc,N)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% set_bit_Suc
tff(fact_2279_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
    <=> ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),L),Q2)),R2) )
        & $ite(
            aa(int,$o,ord_less(int,zero_zero(int)),L),
            ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R2)
            & aa(int,$o,ord_less(int,R2),L) ),
            $ite(
              aa(int,$o,ord_less(int,L),zero_zero(int)),
              ( aa(int,$o,ord_less(int,L),R2)
              & aa(int,$o,ord_less_eq(int,R2),zero_zero(int)) ),
              Q2 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_2280_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_space2,vEBT_Node(Info2,Deg2,TreeList,Summary2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Summary2))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2,TreeList),zero_zero(nat))) ).

% VEBT_internal.space'.simps(2)
tff(fact_2281_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
           => ( aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)))
             => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)))),one_one(A)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_2282_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Xb) = Y )
     => ( ( ? [A2: $o,B2: $o] : Xb = vEBT_Leaf((A2),(B2))
         => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
       => ~ ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info,Deg,TreeList2,Summary) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Summary))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2,TreeList2),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.space'.elims
tff(fact_2283_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_space,vEBT_Node(Info2,Deg2,TreeList,Summary2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary2))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space,TreeList),zero_zero(nat))) ).

% VEBT_internal.space.simps(2)
tff(fact_2284_VEBT__internal_Ospace_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,Xb) = Y )
     => ( ( ? [A2: $o,B2: $o] : Xb = vEBT_Leaf((A2),(B2))
         => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ~ ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info,Deg,TreeList2,Summary) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space,TreeList2),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.space.elims
tff(fact_2285_pos__eucl__rel__int__mult__2,axiom,
    ! [B3: int,A3: int,Q2: int,R2: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B3)
     => ( eucl_rel_int(A3,B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
       => eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),R2)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_2286_divmod__nat__if,axiom,
    ! [M: nat,N: nat] :
      divmod_nat(M,N) = $ite(
        ( ( N = zero_zero(nat) )
        | aa(nat,$o,ord_less(nat,M),N) ),
        aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),zero_zero(nat)),M),
        aa(product_prod(nat,nat),product_prod(nat,nat),product_case_prod(nat,nat,product_prod(nat,nat),aTP_Lamp_gn(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,minus_minus(nat,M),N),N)) ) ).

% divmod_nat_if
tff(fact_2287_div__half__nat,axiom,
    ! [Y: nat,Xb: nat] :
      ( ( Y != zero_zero(nat) )
     => ( aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),Y)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Y)) = $let(
            q: nat,
            q:= aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)),
            $let(
              r: nat,
              r:= aa(nat,nat,minus_minus(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),q),Y)),
              $ite(aa(nat,$o,ord_less_eq(nat,Y),r),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),q),one_one(nat))),aa(nat,nat,minus_minus(nat,r),Y)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),q),r)) ) ) ) ) ).

% div_half_nat
tff(fact_2288_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = zero_zero(nat) )
      | ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = one_one(nat) )
      | ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
      | ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) ) ).

% mod_exhaust_less_4
tff(fact_2289_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [A: $tType,Xs: list(A),Fa: fun(A,real),Bound: real,I: real] :
      ( ! [X2: A] :
          ( member(A,X2,set2(A,Xs))
         => aa(real,$o,ord_less_eq(real,aa(A,real,Fa,X2)),Bound) )
     => aa(real,$o,ord_less_eq(real,foldr(real,real,plus_plus(real),map(A,real,Fa,Xs),I)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(list(A),nat,size_size(list(A)),Xs))),Bound)),I)) ) ).

% list_every_elemnt_bound_sum_bound_real
tff(fact_2290_real__nat__list,axiom,
    ! [A: $tType,Fa: fun(A,nat),Xs: list(A),C2: nat] : aa(nat,real,semiring_1_of_nat(real),foldr(nat,nat,plus_plus(nat),map(A,nat,Fa,Xs),C2)) = foldr(real,real,plus_plus(real),map(A,real,aTP_Lamp_go(fun(A,nat),fun(A,real),Fa),Xs),aa(nat,real,semiring_1_of_nat(real),C2)) ).

% real_nat_list
tff(fact_2291_f__g__map__foldr__bound,axiom,
    ! [A: $tType,Xs: list(A),Fa: fun(A,real),C2: real,G: fun(A,real),D: real] :
      ( ! [X2: A] :
          ( member(A,X2,set2(A,Xs))
         => aa(real,$o,ord_less_eq(real,aa(A,real,Fa,X2)),aa(real,real,aa(real,fun(real,real),times_times(real),C2),aa(A,real,G,X2))) )
     => aa(real,$o,ord_less_eq(real,foldr(real,real,plus_plus(real),map(A,real,Fa,Xs),D)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),C2),foldr(real,real,plus_plus(real),map(A,real,G,Xs),zero_zero(real)))),D)) ) ).

% f_g_map_foldr_bound
tff(fact_2292_round__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: int] :
          ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),Y))
         => ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))))
           => ( archimedean_round(A,Xb) = Y ) ) ) ) ).

% round_unique
tff(fact_2293_mult__le__cancel__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Z)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,ord_less_eq(A,Xb),Y) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_2294_mod__word__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),W),W) = zero_zero(word(A)) ) ).

% mod_word_self
tff(fact_2295_listsum__bound,axiom,
    ! [A: $tType,Xs: list(A),Fa: fun(A,real),Y: real] :
      ( ! [X2: A] :
          ( member(A,X2,set2(A,Xs))
         => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(A,real,Fa,X2)) )
     => aa(real,$o,ord_less_eq(real,Y),foldr(real,real,plus_plus(real),map(A,real,Fa,Xs),Y)) ) ).

% listsum_bound
tff(fact_2296_map__fst__mk__snd,axiom,
    ! [B: $tType,A: $tType,K: B,L: list(A)] : map(product_prod(A,B),A,product_fst(A,B),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_gp(B,fun(A,product_prod(A,B))),K),L)) = L ).

% map_fst_mk_snd
tff(fact_2297_map__snd__mk__fst,axiom,
    ! [B: $tType,A: $tType,K: B,L: list(A)] : map(product_prod(B,A),A,product_snd(B,A),map(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),K),L)) = L ).

% map_snd_mk_fst
tff(fact_2298_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less(int,K),L)
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = K ) ) ) ).

% mod_pos_pos_trivial
tff(fact_2299_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,K),zero_zero(int))
     => ( aa(int,$o,ord_less(int,L),K)
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = K ) ) ) ).

% mod_neg_neg_trivial
tff(fact_2300_round__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,zero_zero(A)) = zero_zero(int) ) ) ).

% round_0
tff(fact_2301_round__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: num] : archimedean_round(A,aa(num,A,numeral_numeral(A),N)) = aa(num,int,numeral_numeral(int),N) ) ).

% round_numeral
tff(fact_2302_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_2303_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),bit0(V))),aa(num,int,numeral_numeral(int),bit0(W))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),W))) ).

% zmod_numeral_Bit0
tff(fact_2304_one__mod__exp__eq__one,axiom,
    ! [N: nat] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) = one_one(int) ).

% one_mod_exp_eq_one
tff(fact_2305_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,V))),aa(num,int,numeral_numeral(int),bit0(W))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),W)))),one_one(int)) ).

% zmod_numeral_Bit1
tff(fact_2306_word__mod__by__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),K),zero_zero(word(A))) = K ) ).

% word_mod_by_0
tff(fact_2307_mod__word__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),W),V)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),W),V) = W ) ) ) ).

% mod_word_less
tff(fact_2308_zmod__helper,axiom,
    ! [N: int,M: int,K: int,A3: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),M) = K )
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),A3)),M) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),A3)),M) ) ) ).

% zmod_helper
tff(fact_2309_pair__list__eqI,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
      ( ( map(product_prod(A,B),A,product_fst(A,B),Xs) = map(product_prod(A,B),A,product_fst(A,B),Ys) )
     => ( ( map(product_prod(A,B),B,product_snd(A,B),Xs) = map(product_prod(A,B),B,product_snd(A,B),Ys) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
tff(fact_2310_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),M)
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),K)),M) ) ).

% zmod_le_nonneg_dividend
tff(fact_2311_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),L)
     => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)),L) ) ).

% Euclidean_Division.pos_mod_bound
tff(fact_2312_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,L),zero_zero(int))
     => aa(int,$o,ord_less(int,L),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)) ) ).

% neg_mod_bound
tff(fact_2313_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),D) = zero_zero(int) )
     => ? [Q3: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D),Q3) ) ).

% zmod_eq_0D
tff(fact_2314_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),D) = zero_zero(int) )
    <=> ? [Q4: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D),Q4) ) ).

% zmod_eq_0_iff
tff(fact_2315_word__mod__less__divisor,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),N)
         => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),M),N)),N) ) ) ).

% word_mod_less_divisor
tff(fact_2316_zmod__int,axiom,
    ! [A3: nat,B3: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A3),B3)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)) ).

% zmod_int
tff(fact_2317_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R2: int] :
      ( eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = R2 ) ) ).

% mod_int_unique
tff(fact_2318_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),Y)
         => aa(int,$o,ord_less_eq(int,archimedean_round(A,Xb)),archimedean_round(A,Y)) ) ) ).

% round_mono
tff(fact_2319_int__mod__ge,axiom,
    ! [A3: int,N: int] :
      ( aa(int,$o,ord_less(int,A3),N)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),N)
       => aa(int,$o,ord_less_eq(int,A3),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),N)) ) ) ).

% int_mod_ge
tff(fact_2320_int__mod__lem,axiom,
    ! [N: int,B3: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),N)
     => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B3)
          & aa(int,$o,ord_less(int,B3),N) )
      <=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B3),N) = B3 ) ) ) ).

% int_mod_lem
tff(fact_2321_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),I),K) = I )
    <=> ( ( K = zero_zero(int) )
        | ( aa(int,$o,ord_less_eq(int,zero_zero(int)),I)
          & aa(int,$o,ord_less(int,I),K) )
        | ( aa(int,$o,ord_less_eq(int,I),zero_zero(int))
          & aa(int,$o,ord_less(int,K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_2322_int__mod__eq,axiom,
    ! [B3: int,N: int,A3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B3)
     => ( aa(int,$o,ord_less(int,B3),N)
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),N) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B3),N) )
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),N) = B3 ) ) ) ) ).

% int_mod_eq
tff(fact_2323_pos__mod__conj,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3))
        & aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3)),B3) ) ) ).

% pos_mod_conj
tff(fact_2324_neg__mod__conj,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,ord_less(int,B3),zero_zero(int))
     => ( aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3)),zero_zero(int))
        & aa(int,$o,ord_less(int,B3),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3)) ) ) ).

% neg_mod_conj
tff(fact_2325_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),L)
     => aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)) ) ).

% Euclidean_Division.pos_mod_sign
tff(fact_2326_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,L),zero_zero(int))
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)),zero_zero(int)) ) ).

% neg_mod_sign
tff(fact_2327_int__mod__le_H,axiom,
    ! [B3: int,N: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,minus_minus(int,B3),N))
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B3),N)),aa(int,int,minus_minus(int,B3),N)) ) ).

% int_mod_le'
tff(fact_2328_nonneg__mod__div,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B3)
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3))
          & aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)) ) ) ) ).

% nonneg_mod_div
tff(fact_2329_zdiv__mono__strict,axiom,
    ! [A4: int,B6: int,N: int] :
      ( aa(int,$o,ord_less(int,A4),B6)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),N)
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A4),N) = zero_zero(int) )
         => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B6),N) = zero_zero(int) )
           => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B6),N)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_2330_div__mod__decomp__int,axiom,
    ! [A4: int,N: int] : A4 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),N)),N)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A4),N)) ).

% div_mod_decomp_int
tff(fact_2331_divmod__int__def,axiom,
    ! [M: num,N: num] : unique8689654367752047608divmod(int,M,N) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N))) ).

% divmod_int_def
tff(fact_2332_mod__div__equality__div__eq,axiom,
    ! [A3: int,B3: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),B3) = aa(int,int,minus_minus(int,A3),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3)) ).

% mod_div_equality_div_eq
tff(fact_2333_ceiling__ge__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(int,$o,ord_less_eq(int,archimedean_round(A,Xb)),archimedean_ceiling(A,Xb)) ) ).

% ceiling_ge_round
tff(fact_2334_eucl__rel__int,axiom,
    ! [K: int,L: int] : eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L))) ).

% eucl_rel_int
tff(fact_2335_pos__mod__bound2,axiom,
    ! [A3: int] : aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% pos_mod_bound2
tff(fact_2336_int__mod__ge_H,axiom,
    ! [B3: int,N: int] :
      ( aa(int,$o,ord_less(int,B3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),N)
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),B3),N)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B3),N)) ) ) ).

% int_mod_ge'
tff(fact_2337_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L) ) ) ) ).

% mod_pos_neg_trivial
tff(fact_2338_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),L)
     => ( aa(int,$o,ord_less_eq(int,L),K)
       => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,K),L)),L) ) ) ) ).

% mod_pos_geq
tff(fact_2339_real__of__int__div__aux,axiom,
    ! [Xb: int,D: int] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Xb)),aa(int,real,ring_1_of_int(real),D)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),D))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xb),D))),aa(int,real,ring_1_of_int(real),D))) ).

% real_of_int_div_aux
tff(fact_2340_pos__mod__sign2,axiom,
    ! [A3: int] : aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% pos_mod_sign2
tff(fact_2341_nmod2,axiom,
    ! [N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int) )
      | ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) ) ) ).

% nmod2
tff(fact_2342_mod__2__neq__1__eq__eq__0,axiom,
    ! [K: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2))) != one_one(int) )
    <=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int) ) ) ).

% mod_2_neq_1_eq_eq_0
tff(fact_2343_mod__exp__less__eq__exp,axiom,
    ! [A3: int,N: nat] : aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).

% mod_exp_less_eq_exp
tff(fact_2344_mod__power__lem,axiom,
    ! [A3: int,N: nat,M: nat] :
      ( aa(int,$o,ord_less(int,one_one(int)),A3)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),A3),N)),aa(nat,int,aa(int,fun(nat,int),power_power(int),A3),M)) = $ite(aa(nat,$o,ord_less_eq(nat,M),N),zero_zero(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),A3),N)) ) ) ).

% mod_power_lem
tff(fact_2345_int__mod__pos__eq,axiom,
    ! [A3: int,B3: int,Q2: int,R2: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q2)),R2) )
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R2)
       => ( aa(int,$o,ord_less(int,R2),B3)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3) = R2 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_2346_int__mod__neg__eq,axiom,
    ! [A3: int,B3: int,Q2: int,R2: int] :
      ( ( A3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),Q2)),R2) )
     => ( aa(int,$o,ord_less_eq(int,R2),zero_zero(int))
       => ( aa(int,$o,ord_less(int,B3),R2)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3) = R2 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_2347_split__zmod,axiom,
    ! [P: fun(int,$o),N: int,K: int] :
      ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,N) )
        & ( aa(int,$o,ord_less(int,zero_zero(int)),K)
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),J3)
                & aa(int,$o,ord_less(int,J3),K)
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,J3) ) )
        & ( aa(int,$o,ord_less(int,K),zero_zero(int))
         => ! [I4: int,J3: int] :
              ( ( aa(int,$o,ord_less(int,K),J3)
                & aa(int,$o,ord_less_eq(int,J3),zero_zero(int))
                & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
             => aa(int,$o,P,J3) ) ) ) ) ).

% split_zmod
tff(fact_2348_mod__sub__if__z,axiom,
    ! [Xb: int,Z: int,Y: int] :
      ( aa(int,$o,ord_less(int,Xb),Z)
     => ( aa(int,$o,ord_less(int,Y),Z)
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
         => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
           => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
             => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,Xb),Y)),Z) = $ite(aa(int,$o,ord_less_eq(int,Y),Xb),aa(int,int,minus_minus(int,Xb),Y),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,Xb),Y)),Z)) ) ) ) ) ) ) ).

% mod_sub_if_z
tff(fact_2349_mod__add__if__z,axiom,
    ! [Xb: int,Z: int,Y: int] :
      ( aa(int,$o,ord_less(int,Xb),Z)
     => ( aa(int,$o,ord_less(int,Y),Z)
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
         => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
           => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
             => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Y)),Z) = $ite(aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Y)),Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Y),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Y)),Z)) ) ) ) ) ) ) ).

% mod_add_if_z
tff(fact_2350_zmod__zmult2__eq,axiom,
    ! [C2: int,A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),C2)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),aa(int,int,aa(int,fun(int,int),times_times(int),B3),C2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B3),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),C2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3)) ) ) ).

% zmod_zmult2_eq
tff(fact_2351_axxmod2,axiom,
    ! [Xb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Xb)),Xb)),aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) )
      & ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),zero_zero(int)),Xb)),Xb)),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int) ) ) ).

% axxmod2
tff(fact_2352_z1pmod2,axiom,
    ! [B3: int] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3)),one_one(int))),aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) ).

% z1pmod2
tff(fact_2353_verit__le__mono__div__int,axiom,
    ! [A4: int,B6: int,N: int] :
      ( aa(int,$o,ord_less(int,A4),B6)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),N)
       => aa(int,$o,
            ord_less_eq(int,
              aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),N)),
                $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B6),N) = zero_zero(int),one_one(int),zero_zero(int)))),
            aa(int,int,aa(int,fun(int,int),divide_divide(int),B6),N)) ) ) ).

% verit_le_mono_div_int
tff(fact_2354_split__pos__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),N: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),K))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),J3)
              & aa(int,$o,ord_less(int,J3),K)
              & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).

% split_pos_lemma
tff(fact_2355_split__neg__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),N: int] :
      ( aa(int,$o,ord_less(int,K),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),K)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),K))
      <=> ! [I4: int,J3: int] :
            ( ( aa(int,$o,ord_less(int,K),J3)
              & aa(int,$o,ord_less_eq(int,J3),zero_zero(int))
              & ( N = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I4)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I4),J3) ) ) ) ).

% split_neg_lemma
tff(fact_2356_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_Node(Info2,Deg2,TreeList,Summary2)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Summary2))),foldr(real,real,plus_plus(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt,TreeList),zero_zero(real))) ).

% VEBT_internal.cnt.simps(2)
tff(fact_2357_p1mod22k,axiom,
    ! [B3: int,N: nat] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3)),one_one(int))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))),one_one(int)) ).

% p1mod22k
tff(fact_2358_p1mod22k_H,axiom,
    ! [B3: int,N: nat] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))) ).

% p1mod22k'
tff(fact_2359_VEBT__internal_Ocnt_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: real] :
      ( ( aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Xb) = Y )
     => ( ( ? [A2: $o,B2: $o] : Xb = vEBT_Leaf((A2),(B2))
         => ( Y != one_one(real) ) )
       => ~ ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info,Deg,TreeList2,Summary) )
             => ( Y != aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Summary))),foldr(real,real,plus_plus(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt,TreeList2),zero_zero(real))) ) ) ) ) ).

% VEBT_internal.cnt.elims
tff(fact_2360_pos__zmod__mult__2,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B3),A3))) ) ) ).

% pos_zmod_mult_2
tff(fact_2361_sb__inc__lem,axiom,
    ! [A3: int,K: nat] :
      ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),zero_zero(int))
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))) ) ).

% sb_inc_lem
tff(fact_2362_neg__zmod__mult__2,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),zero_zero(int))
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B3))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A3)) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B3),one_one(int))),A3))),one_one(int)) ) ) ).

% neg_zmod_mult_2
tff(fact_2363_mult__less__iff1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Z)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))
          <=> aa(A,$o,ord_less(A,Xb),Y) ) ) ) ).

% mult_less_iff1
tff(fact_2364_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),archimedean_round(A,Xb))),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% of_int_round_le
tff(fact_2365_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xb))) ) ).

% of_int_round_ge
tff(fact_2366_of__int__round__gt,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(int,A,ring_1_of_int(A),archimedean_round(A,Xb))) ) ).

% of_int_round_gt
tff(fact_2367_mult__le__cancel__iff2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Z)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Z),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
          <=> aa(A,$o,ord_less_eq(A,Xb),Y) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_2368_quickcheck__narrowing__samples_Opartial__term__of__sample__def,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),Zero: A,I: code_integer] :
          code_T4081349890594273596sample(A,A_of_integer,Zero,I) = $ite(
            aa(code_integer,$o,ord_less(code_integer,I),zero_zero(code_integer)),
            undefined(A),
            $ite(
              I = zero_zero(code_integer),
              Zero,
              $ite(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))) = zero_zero(code_integer),aa(product_prod(A,A),A,product_snd(A,A),aa(code_integer,product_prod(A,A),A_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))),aa(product_prod(A,A),A,product_fst(A,A),aa(code_integer,product_prod(A,A),A_of_integer,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),one_one(code_integer))))) ) ) ) ).

% quickcheck_narrowing_samples.partial_term_of_sample_def
tff(fact_2369_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: A,R2: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R2))
        <=> ( R2 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_2370_VEBT__internal_OT__vebt__buildupi_H_Oelims,axiom,
    ! [Xb: nat,Y: int] :
      ( ( vEBT_V9176841429113362141ildupi(Xb) = Y )
     => ( ( ( Xb = zero_zero(nat) )
         => ( Y != one_one(int) ) )
       => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != one_one(int) ) )
         => ~ ! [N3: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,N3)) )
               => ( Y != $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(bit0(one2))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
tff(fact_2371_refines__assert_H__bind,axiom,
    ! [A: $tType,P2: heap_Time_Heap(A),Q2: heap_Time_Heap(A),Phi: $o] :
      ( refine_Imp_refines(A,P2,Q2)
     => refine_Imp_refines(A,P2,heap_Time_bind(product_unit,A,refine_Imp_assert((Phi)),aTP_Lamp_gb(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),Q2))) ) ).

% refines_assert'_bind
tff(fact_2372_artanh__0,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ( aa(A,A,artanh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% artanh_0
tff(fact_2373_arsinh__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,arsinh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% arsinh_0
tff(fact_2374_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : aa(A,$o,dvd_dvd(A,A3),zero_zero(A)) ) ).

% dvd_0_right
tff(fact_2375_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,zero_zero(A)),A3)
        <=> ( A3 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_2376_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))
        <=> aa(A,$o,dvd_dvd(A,A3),B3) ) ) ).

% dvd_add_triv_left_iff
tff(fact_2377_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3))
        <=> aa(A,$o,dvd_dvd(A,A3),B3) ) ) ).

% dvd_add_triv_right_iff
tff(fact_2378_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,$o,dvd_dvd(A,A3),C2)
           => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3))
            <=> aa(A,$o,dvd_dvd(A,B3),C2) ) ) ) ) ).

% div_dvd_div
tff(fact_2379_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa(nat,$o,dvd_dvd(nat,M),one_one(nat))
    <=> ( M = one_one(nat) ) ) ).

% nat_dvd_1_iff_1
tff(fact_2380_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3))
          <=> aa(A,$o,dvd_dvd(A,B3),C2) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_2381_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2))
          <=> aa(A,$o,dvd_dvd(A,B3),C2) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_2382_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
        <=> ( ( C2 = zero_zero(A) )
            | aa(A,$o,dvd_dvd(A,A3),B3) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_2383_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
        <=> ( ( C2 = zero_zero(A) )
            | aa(A,$o,dvd_dvd(A,A3),B3) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_2384_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)),B3))
        <=> aa(A,$o,dvd_dvd(A,A3),B3) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2385_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)))
        <=> aa(A,$o,dvd_dvd(A,A3),B3) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2386_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
           => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),one_one(A)) ) ) ) ).

% unit_prod
tff(fact_2387_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A3)
         => ( aa(A,$o,dvd_dvd(A,C2),B3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) ) ) ) ) ).

% div_add
tff(fact_2388_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
           => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),one_one(A)) ) ) ) ).

% unit_div
tff(fact_2389_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)),one_one(A)) ) ) ).

% unit_div_1_unit
tff(fact_2390_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)) = A3 ) ) ) ).

% unit_div_1_div_1
tff(fact_2391_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)) = B3 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_2392_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),A3) = B3 ) ) ) ).

% dvd_div_mult_self
tff(fact_2393_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A3)
         => ( aa(A,$o,dvd_dvd(A,C2),B3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,A3),B3)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) ) ) ) ) ).

% div_diff
tff(fact_2394_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),A3) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_2395_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( M = aa(nat,nat,suc,zero_zero(nat)) ) ) ).

% dvd_1_iff_1
tff(fact_2396_dvd__1__left,axiom,
    ! [K: nat] : aa(nat,$o,dvd_dvd(nat,aa(nat,nat,suc,zero_zero(nat))),K) ).

% dvd_1_left
tff(fact_2397_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))
    <=> ( ( K = zero_zero(nat) )
        | aa(nat,$o,dvd_dvd(nat,M),N) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_2398_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3) ) ) ) ).

% unit_mult_div_div
tff(fact_2399_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3)),A3) = B3 ) ) ) ).

% unit_div_mult_self
tff(fact_2400_odd__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))
        <=> ~ ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B3) ) ) ) ).

% odd_add
tff(fact_2401_even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
          <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B3) ) ) ) ).

% even_add
tff(fact_2402_even__mult__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B3) ) ) ) ).

% even_mult_iff
tff(fact_2403_even__mod__2__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ).

% even_mod_2_iff
tff(fact_2404_even__Suc,axiom,
    ! [N: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N))
    <=> ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ) ).

% even_Suc
tff(fact_2405_even__Suc__Suc__iff,axiom,
    ! [N: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,N)))
    <=> aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ) ).

% even_Suc_Suc_iff
tff(fact_2406_dvd__numeral__simp,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,N: num] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))
        <=> unique5940410009612947441es_aux(A,unique8689654367752047608divmod(A,N,M)) ) ) ).

% dvd_numeral_simp
tff(fact_2407_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)))
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ).

% even_plus_one_iff
tff(fact_2408_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,minus_minus(A,A3),B3))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ).

% even_diff
tff(fact_2409_odd__Suc__div__two,axiom,
    ! [N: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% odd_Suc_div_two
tff(fact_2410_even__Suc__div__two,axiom,
    ! [N: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ).

% even_Suc_div_two
tff(fact_2411_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_2412_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_two
tff(fact_2413_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_2
tff(fact_2414_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            & aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ) ) ).

% even_power
tff(fact_2415_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: num] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
            | ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2416_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),zero_zero(A))
        <=> ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
            & aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ) ).

% power_less_zero_eq
tff(fact_2417_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: num] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A))
        <=> ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
            & aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_2418_even__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),N))
        <=> aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ) ) ).

% even_of_nat
tff(fact_2419_odd__Suc__minus__one,axiom,
    ! [N: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))) = N ) ) ).

% odd_Suc_minus_one
tff(fact_2420_even__diff__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),N))
    <=> ( aa(nat,$o,ord_less(nat,M),N)
        | aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ) ) ).

% even_diff_nat
tff(fact_2421_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))),one_one(A)) = A3 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_2422_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A)))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_2423_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: num] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W) = zero_zero(nat) )
            | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
              & ( A3 != zero_zero(A) ) )
            | ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
              & aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_2424_odd__two__times__div__two__nat,axiom,
    ! [N: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,nat,minus_minus(nat,N),one_one(nat)) ) ) ).

% odd_two_times_div_two_nat
tff(fact_2425_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,W: num] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A))
        <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),W))
            & ( ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
                & aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) )
              | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_2426_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_2427_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_2428_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : aa(A,$o,dvd_dvd(A,A3),A3) ) ).

% dvd_refl
tff(fact_2429_dvd__trans,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,$o,dvd_dvd(A,B3),C2)
           => aa(A,$o,dvd_dvd(A,A3),C2) ) ) ) ).

% dvd_trans
tff(fact_2430_dvd__antisym,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,dvd_dvd(nat,M),N)
     => ( aa(nat,$o,dvd_dvd(nat,N),M)
       => ( M = N ) ) ) ).

% dvd_antisym
tff(fact_2431_of__nat__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))
        <=> aa(nat,$o,dvd_dvd(nat,M),N) ) ) ).

% of_nat_dvd_iff
tff(fact_2432_dvd__field__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
        <=> ( ( A3 = zero_zero(A) )
           => ( B3 = zero_zero(A) ) ) ) ) ).

% dvd_field_iff
tff(fact_2433_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,zero_zero(A)),A3)
         => ( A3 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_2434_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,$o,dvd_dvd(A,A3),C2)
           => aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2)) ) ) ) ).

% dvd_add
tff(fact_2435_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),C2)
         => ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2))
          <=> aa(A,$o,dvd_dvd(A,A3),B3) ) ) ) ).

% dvd_add_left_iff
tff(fact_2436_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),C2))
          <=> aa(A,$o,dvd_dvd(A,A3),C2) ) ) ) ).

% dvd_add_right_iff
tff(fact_2437_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A] : aa(A,$o,dvd_dvd(A,one_one(A)),A3) ) ).

% one_dvd
tff(fact_2438_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => aa(A,$o,dvd_dvd(A,B3),A3) ) ) ).

% unit_imp_dvd
tff(fact_2439_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
           => aa(A,$o,dvd_dvd(A,A3),one_one(A)) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_2440_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),A3)
         => ~ ! [K2: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),B3),K2) ) ) ).

% dvdE
tff(fact_2441_dvdI,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [A3: A,B3: A,K: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),K) )
         => aa(A,$o,dvd_dvd(A,B3),A3) ) ) ).

% dvdI
tff(fact_2442_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),A3)
        <=> ? [K3: A] : A3 = aa(A,A,aa(A,fun(A,A),times_times(A),B3),K3) ) ) ).

% dvd_def
tff(fact_2443_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),C2)
         => aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ).

% dvd_mult
tff(fact_2444_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) ) ) ).

% dvd_mult2
tff(fact_2445_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2)
         => aa(A,$o,dvd_dvd(A,A3),C2) ) ) ).

% dvd_mult_left
tff(fact_2446_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A] : aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ).

% dvd_triv_left
tff(fact_2447_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
         => ( aa(A,$o,dvd_dvd(A,C2),D)
           => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D)) ) ) ) ).

% mult_dvd_mono
tff(fact_2448_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2)
         => aa(A,$o,dvd_dvd(A,B3),C2) ) ) ).

% dvd_mult_right
tff(fact_2449_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A] : aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)) ) ).

% dvd_triv_right
tff(fact_2450_dvd__diff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,dvd_dvd(A,Xb),Y)
         => ( aa(A,$o,dvd_dvd(A,Xb),Z)
           => aa(A,$o,dvd_dvd(A,Xb),aa(A,A,minus_minus(A,Y),Z)) ) ) ) ).

% dvd_diff
tff(fact_2451_dvd__diff__commute,axiom,
    ! [A: $tType] :
      ( euclid5891614535332579305n_ring(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,minus_minus(A,C2),B3))
        <=> aa(A,$o,dvd_dvd(A,A3),aa(A,A,minus_minus(A,B3),C2)) ) ) ).

% dvd_diff_commute
tff(fact_2452_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A3)
         => ( aa(A,$o,dvd_dvd(A,C2),B3)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) )
            <=> ( A3 = B3 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_2453_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,C2: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) )
         => ( aa(A,$o,dvd_dvd(A,C2),A3)
           => ( aa(A,$o,dvd_dvd(A,C2),B3)
             => ( A3 = B3 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2454_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D: A,B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,D),B3)
         => ( aa(A,$o,dvd_dvd(A,B3),A3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),D)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),D)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ) ).

% div_div_div_same
tff(fact_2455_dvd__power__same,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xb: A,Y: A,N: nat] :
          ( aa(A,$o,dvd_dvd(A,Xb),Y)
         => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N)) ) ) ).

% dvd_power_same
tff(fact_2456_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B3)
         => ( aa(A,$o,dvd_dvd(A,C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3))
          <=> aa(A,$o,dvd_dvd(A,C2),A3) ) ) ) ).

% dvd_mod_iff
tff(fact_2457_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3))
         => ( aa(A,$o,dvd_dvd(A,C2),B3)
           => aa(A,$o,dvd_dvd(A,C2),A3) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_2458_dvd__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [K: A,M: A,N: A] :
          ( aa(A,$o,dvd_dvd(A,K),M)
         => ( aa(A,$o,dvd_dvd(A,K),N)
           => aa(A,$o,dvd_dvd(A,K),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),M),N)) ) ) ) ).

% dvd_mod
tff(fact_2459_mod__mod__cancel,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B3)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2) ) ) ) ).

% mod_mod_cancel
tff(fact_2460_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),M)
     => ( aa(nat,$o,dvd_dvd(nat,K),N)
       => aa(nat,$o,dvd_dvd(nat,K),aa(nat,nat,minus_minus(nat,M),N)) ) ) ).

% dvd_diff_nat
tff(fact_2461_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),collect(A,aTP_Lamp_gq(A,fun(A,$o),A3))),collect(A,aTP_Lamp_gq(A,fun(A,$o),B3)))
        <=> aa(A,$o,dvd_dvd(A,A3),B3) ) ) ).

% subset_divisors_dvd
tff(fact_2462_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: A,B3: A] :
          ( aa(set(A),$o,ord_less(set(A),collect(A,aTP_Lamp_gq(A,fun(A,$o),A3))),collect(A,aTP_Lamp_gq(A,fun(A,$o),B3)))
        <=> ( aa(A,$o,dvd_dvd(A,A3),B3)
            & ~ aa(A,$o,dvd_dvd(A,B3),A3) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_2463_not__is__unit__0,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ~ aa(A,$o,dvd_dvd(A,zero_zero(A)),one_one(A)) ) ).

% not_is_unit_0
tff(fact_2464_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D: A,S: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z2)
         => ( ~ aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S))
          <=> ~ aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S)) ) ) ) ).

% minf(10)
tff(fact_2465_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D: A,S: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z2)
         => ( aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S))
          <=> aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S)) ) ) ) ).

% minf(9)
tff(fact_2466_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D: A,S: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z2),X3)
         => ( ~ aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S))
          <=> ~ aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S)) ) ) ) ).

% pinf(10)
tff(fact_2467_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D: A,S: A] :
        ? [Z2: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z2),X3)
         => ( aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S))
          <=> aa(A,$o,dvd_dvd(A,D),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S)) ) ) ) ).

% pinf(9)
tff(fact_2468_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),A3)
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_2469_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) )
          <=> ( B3 = C2 ) ) ) ) ).

% unit_mult_right_cancel
tff(fact_2470_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) )
          <=> ( B3 = C2 ) ) ) ) ).

% unit_mult_left_cancel
tff(fact_2471_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2)
          <=> aa(A,$o,dvd_dvd(A,B3),C2) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_2472_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2))
          <=> aa(A,$o,dvd_dvd(A,A3),C2) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_2473_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2)
          <=> aa(A,$o,dvd_dvd(A,A3),C2) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_2474_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3))
          <=> aa(A,$o,dvd_dvd(A,A3),C2) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_2475_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),one_one(A))
        <=> ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
            & aa(A,$o,dvd_dvd(A,B3),one_one(A)) ) ) ) ).

% is_unit_mult_iff
tff(fact_2476_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_2477_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_2478_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3) )
          <=> ( B3 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_2479_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2)
          <=> aa(A,$o,dvd_dvd(A,A3),C2) ) ) ) ).

% div_unit_dvd_iff
tff(fact_2480_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B3))
          <=> aa(A,$o,dvd_dvd(A,A3),C2) ) ) ) ).

% dvd_div_unit_iff
tff(fact_2481_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)),C2) ) ) ) ).

% dvd_div_mult
tff(fact_2482_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2) ) ) ) ).

% div_mult_swap
tff(fact_2483_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B3)
         => ( aa(A,$o,dvd_dvd(A,B3),A3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2) ) ) ) ) ).

% div_div_eq_right
tff(fact_2484_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_2485_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3)
         => aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) ) ) ).

% dvd_mult_imp_div
tff(fact_2486_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,D: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),A3)
         => ( aa(A,$o,dvd_dvd(A,D),C2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),D)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),D)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_2487_div__power,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,N: nat] :
          ( aa(A,$o,dvd_dvd(A,B3),A3)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N)) ) ) ) ).

% div_power
tff(fact_2488_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) = zero_zero(A) )
         => aa(A,$o,dvd_dvd(A,B3),A3) ) ) ).

% mod_0_imp_dvd
tff(fact_2489_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),B3)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),A3) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_2490_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) = zero_zero(A) )
        <=> aa(A,$o,dvd_dvd(A,B3),A3) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_2491_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xb: A,Y: A,N: nat,M: nat] :
          ( aa(A,$o,dvd_dvd(A,Xb),Y)
         => ( aa(nat,$o,ord_less_eq(nat,N),M)
           => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),M)) ) ) ) ).

% dvd_power_le
tff(fact_2492_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat,B3: A,M: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),B3)
         => ( aa(nat,$o,ord_less_eq(nat,M),N)
           => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),B3) ) ) ) ).

% power_le_dvd
tff(fact_2493_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,N: nat,A3: A] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ).

% le_imp_power_dvd
tff(fact_2494_mod__eq__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,C2: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B3),C2) )
        <=> aa(A,$o,dvd_dvd(A,C2),aa(A,A,minus_minus(A,A3),B3)) ) ) ).

% mod_eq_dvd_iff
tff(fact_2495_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B3: A,A3: A] : aa(A,$o,dvd_dvd(A,B3),aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3))) ) ).

% dvd_minus_mod
tff(fact_2496_nat__dvd__not__less,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(nat,$o,ord_less(nat,M),N)
       => ~ aa(nat,$o,dvd_dvd(nat,N),M) ) ) ).

% nat_dvd_not_less
tff(fact_2497_dvd__minus__self,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,minus_minus(nat,N),M))
    <=> ( aa(nat,$o,ord_less(nat,N),M)
        | aa(nat,$o,dvd_dvd(nat,M),N) ) ) ).

% dvd_minus_self
tff(fact_2498_less__eq__dvd__minus,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => ( aa(nat,$o,dvd_dvd(nat,M),N)
      <=> aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,minus_minus(nat,N),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_2499_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),aa(nat,nat,minus_minus(nat,M),N))
     => ( aa(nat,$o,dvd_dvd(nat,K),M)
       => ( aa(nat,$o,ord_less_eq(nat,N),M)
         => aa(nat,$o,dvd_dvd(nat,K),N) ) ) ) ).

% dvd_diffD1
tff(fact_2500_dvd__diffD,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),aa(nat,nat,minus_minus(nat,M),N))
     => ( aa(nat,$o,dvd_dvd(nat,K),N)
       => ( aa(nat,$o,ord_less_eq(nat,N),M)
         => aa(nat,$o,dvd_dvd(nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_2501_even__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(int,A,ring_1_of_int(A),K))
        <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K) ) ) ).

% even_of_int_iff
tff(fact_2502_even__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: num] : aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,A,numeral_numeral(A),bit0(N))) ) ).

% even_numeral
tff(fact_2503_unity__coeff__ex,axiom,
    ! [A: $tType] :
      ( ( dvd(A)
        & semiring_0(A) )
     => ! [P: fun(A,$o),L: A] :
          ( ? [X: A] : aa(A,$o,P,aa(A,A,aa(A,fun(A,A),times_times(A),L),X))
        <=> ? [X: A] :
              ( aa(A,$o,dvd_dvd(A,L),aa(A,A,aa(A,fun(A,A),plus_plus(A),X),zero_zero(A)))
              & aa(A,$o,P,X) ) ) ) ).

% unity_coeff_ex
tff(fact_2504_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [C3: A] : B3 != aa(A,A,aa(A,fun(A,A),times_times(A),A3),C3) ) ) ) ).

% unit_dvdE
tff(fact_2505_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_2506_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,A3),B3)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3) = C2 )
            <=> ( B3 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_2507_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,B3),A3)
           => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2)
            <=> aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_2508_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B3: A,A3: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,C2),B3)
           => ( aa(A,$o,dvd_dvd(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
            <=> aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_2509_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A,B3: A,D: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( aa(A,$o,dvd_dvd(A,A3),B3)
             => ( aa(A,$o,dvd_dvd(A,C2),D)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),D),C2) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),D) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_2510_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = C2 )
          <=> ( A3 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ).

% unit_eq_div1
tff(fact_2511_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B3) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_2512_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,B3),A3)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_2513_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),B3) ) ) ) ).

% unit_div_commute
tff(fact_2514_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_2515_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2516_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_2517_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),one_one(A))
        <=> ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
            | ( N = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_2518_dvd__imp__le,axiom,
    ! [K: nat,N: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),N)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
       => aa(nat,$o,ord_less_eq(nat,K),N) ) ) ).

% dvd_imp_le
tff(fact_2519_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))
      <=> aa(nat,$o,dvd_dvd(nat,M),N) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_2520_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),N))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => aa(nat,$o,dvd_dvd(nat,M),N) ) ) ).

% dvd_mult_cancel
tff(fact_2521_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N))
    <=> ~ aa(nat,$o,dvd_dvd(nat,N),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_2522_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M: nat,Q2: nat] :
      ( aa(nat,$o,ord_less_eq(nat,N),M)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Q2) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),Q2) )
      <=> aa(nat,$o,dvd_dvd(nat,Q2),aa(nat,nat,minus_minus(nat,M),N)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_2523_real__of__nat__div,axiom,
    ! [D: nat,N: nat] :
      ( aa(nat,$o,dvd_dvd(nat,D),N)
     => ( aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),D)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,semiring_1_of_nat(real),D)) ) ) ).

% real_of_nat_div
tff(fact_2524_even__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),zero_zero(A)) ) ).

% even_zero
tff(fact_2525_odd__even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B3)
           => aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ) ).

% odd_even_add
tff(fact_2526_odd__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),one_one(A)) ) ).

% odd_one
tff(fact_2527_evenE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ~ ! [B2: A] : A3 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ).

% evenE
tff(fact_2528_bit__eq__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
        <=> ( ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B3) )
            & ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ) ).

% bit_eq_rec
tff(fact_2529_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ~ ( ( A3 != zero_zero(A) )
             => ! [B2: A] :
                  ( ( B2 != zero_zero(A) )
                 => ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
                   => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) = B2 )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) = A3 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B2) = one_one(A) )
                         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_2530_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B3) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_2531_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B3) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_2532_odd__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: num] : ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))) ) ).

% odd_numeral
tff(fact_2533_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Xb: A,M: nat,N: nat] :
          ( ( Xb != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N))
          <=> ( aa(A,$o,dvd_dvd(A,Xb),one_one(A))
              | aa(nat,$o,ord_less_eq(nat,M),N) ) ) ) ) ).

% dvd_power_iff
tff(fact_2534_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat,Xb: A] :
          ( ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
            | ( Xb = one_one(A) ) )
         => aa(A,$o,dvd_dvd(A,Xb),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N)) ) ) ).

% dvd_power
tff(fact_2535_refines__If,axiom,
    ! [A: $tType,B3: $o,Tb: heap_Time_Heap(A),T3: heap_Time_Heap(A),E3: heap_Time_Heap(A),E4: heap_Time_Heap(A)] :
      ( ( (B3)
       => refine_Imp_refines(A,Tb,T3) )
     => ( ( ~ (B3)
         => refine_Imp_refines(A,E3,E4) )
       => refine_Imp_refines(A,
            $ite((B3),Tb,E3),
            $ite((B3),T3,E4)) ) ) ).

% refines_If
tff(fact_2536_refines__refl,axiom,
    ! [A: $tType,P2: heap_Time_Heap(A)] : refine_Imp_refines(A,P2,P2) ).

% refines_refl
tff(fact_2537_even__even__mod__4__iff,axiom,
    ! [N: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
    <=> aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))))) ) ).

% even_even_mod_4_iff
tff(fact_2538_dvd__mult__cancel1,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N)),M)
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_2539_dvd__mult__cancel2,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),M)),M)
      <=> ( N = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_2540_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),N))
     => ( aa(nat,$o,ord_less(nat,one_one(nat)),I)
       => aa(nat,$o,ord_less_eq(nat,M),N) ) ) ).

% power_dvd_imp_le
tff(fact_2541_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R2: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,Q2),N)
     => ( aa(nat,$o,ord_less_eq(nat,Q2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),M))
       => ( aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,minus_minus(nat,N),Q2))
        <=> aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R2),M)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_2542_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,R2),N)
     => ( aa(nat,$o,ord_less_eq(nat,R2),M)
       => ( aa(nat,$o,dvd_dvd(nat,N),aa(nat,nat,minus_minus(nat,M),R2))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N) = R2 ) ) ) ) ).

% mod_nat_eqI
tff(fact_2543_diff__mod__le,axiom,
    ! [A3: nat,D: nat,B3: nat] :
      ( aa(nat,$o,ord_less(nat,A3),D)
     => ( aa(nat,$o,dvd_dvd(nat,B3),D)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,A3),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A3),B3))),aa(nat,nat,minus_minus(nat,D),B3)) ) ) ).

% diff_mod_le
tff(fact_2544_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) = A3 ) ) ) ).

% even_two_times_div_two
tff(fact_2545_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_2546_odd__iff__mod__2__eq__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_2547_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A,B3: A] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
         => ( aa(A,$o,ord_less_eq(A,A3),B3)
           => aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N)) ) ) ) ).

% power_mono_odd
tff(fact_2548_odd__pos,axiom,
    ! [N: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ).

% odd_pos
tff(fact_2549_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A3))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            & ( M != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_2550_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),N))
      <=> aa(nat,$o,ord_less_eq(nat,M),N) ) ) ).

% dvd_power_iff_le
tff(fact_2551_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ~ ! [B2: A] : A3 != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)),one_one(A)) ) ) ).

% oddE
tff(fact_2552_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = $ite(aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3),zero_zero(A),one_one(A)) ) ).

% mod2_eq_if
tff(fact_2553_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] :
          ( ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) ) )
         => ~ ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
             => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_2554_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))
        <=> ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
            | ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ) ) ).

% zero_le_power_eq
tff(fact_2555_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))
          <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ) ).

% zero_le_odd_power
tff(fact_2556_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ).

% zero_le_even_power
tff(fact_2557_divides__aux__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Qr: product_prod(A,A)] :
          ( unique5940410009612947441es_aux(A,Qr)
        <=> ( aa(product_prod(A,A),A,product_snd(A,A),Qr) = zero_zero(A) ) ) ) ).

% divides_aux_def
tff(fact_2558_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))
        <=> ( ( N = zero_zero(nat) )
            | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
              & ( A3 != zero_zero(A) ) )
            | ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
              & aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ) ) ).

% zero_less_power_eq
tff(fact_2559_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)))
        <=> aa(nat,$o,ord_less_eq(nat,M),N) ) ) ).

% even_mask_div_iff'
tff(fact_2560_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),zero_zero(A))
        <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
            & ( ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
                & aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) )
              | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
                & ( A3 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_2561_even__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(nat,nat,suc,zero_zero(nat)) )
     => aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% even_mod_4_div_2
tff(fact_2562_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
            | aa(nat,$o,ord_less_eq(nat,M),N) ) ) ) ).

% even_mask_div_iff
tff(fact_2563_odd__mod__4__div__2,axiom,
    ! [N: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
     => ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% odd_mod_4_div_2
tff(fact_2564_refines__bind,axiom,
    ! [B: $tType,A: $tType,M: heap_Time_Heap(A),M5: heap_Time_Heap(A),Fa: fun(A,heap_Time_Heap(B)),F3: fun(A,heap_Time_Heap(B))] :
      ( refine_Imp_refines(A,M,M5)
     => ( ! [X2: A] : refine_Imp_refines(B,aa(A,heap_Time_Heap(B),Fa,X2),aa(A,heap_Time_Heap(B),F3,X2))
       => refine_Imp_refines(B,heap_Time_bind(A,B,M,Fa),heap_Time_bind(A,B,M5,F3)) ) ) ).

% refines_bind
tff(fact_2565_Bernoulli__inequality__even,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
     => aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),Xb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb)),N)) ) ).

% Bernoulli_inequality_even
tff(fact_2566_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [N: nat] :
      vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
tff(fact_2567_prod__case__refines,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: product_prod(A,B),P4: product_prod(A,B),Fa: fun(A,fun(B,heap_Time_Heap(C))),F3: fun(A,fun(B,heap_Time_Heap(C)))] :
      ( ( P2 = P4 )
     => ( ! [A2: A,B2: B] : refine_Imp_refines(C,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),Fa,A2),B2),aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F3,A2),B2))
       => refine_Imp_refines(C,aa(product_prod(A,B),heap_Time_Heap(C),product_case_prod(A,B,heap_Time_Heap(C),Fa),P2),aa(product_prod(A,B),heap_Time_Heap(C),product_case_prod(A,B,heap_Time_Heap(C),F3),P4)) ) ) ).

% prod_case_refines
tff(fact_2568_refines__case__prod__right,axiom,
    ! [C: $tType,B: $tType,A: $tType,M: heap_Time_Heap(C),M5: fun(A,fun(B,heap_Time_Heap(C))),Tb: product_prod(A,B)] :
      ( ! [A2: A,B2: B] : refine_Imp_refines(C,M,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),M5,A2),B2))
     => refine_Imp_refines(C,M,aa(product_prod(A,B),heap_Time_Heap(C),product_case_prod(A,B,heap_Time_Heap(C),M5),Tb)) ) ).

% refines_case_prod_right
tff(fact_2569_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M: nat,N: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)))
        <=> ( aa(nat,$o,ord_less(nat,N),M)
            | ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
            | ( aa(nat,$o,ord_less_eq(nat,M),N)
              & aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M)))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_2570_refines__option,axiom,
    ! [B: $tType,A: $tType,A3: option(A),A5: option(A),M1: heap_Time_Heap(B),M12: heap_Time_Heap(B),M22: fun(A,heap_Time_Heap(B)),M23: fun(A,heap_Time_Heap(B))] :
      ( ( A3 = A5 )
     => ( refine_Imp_refines(B,M1,M12)
       => ( ! [X2: A] : refine_Imp_refines(B,aa(A,heap_Time_Heap(B),M22,X2),aa(A,heap_Time_Heap(B),M23,X2))
         => refine_Imp_refines(B,case_option(heap_Time_Heap(B),A,M1,M22,A3),case_option(heap_Time_Heap(B),A,M12,M23,A5)) ) ) ) ).

% refines_option
tff(fact_2571_VEBT__internal_OT__vebt__buildupi_Oelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( vEBT_V441764108873111860ildupi(Xb) = Y )
     => ( ( ( Xb = zero_zero(nat) )
         => ( Y != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(nat,nat,suc,zero_zero(nat)) ) )
         => ~ ! [N3: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,N3)) )
               => ( Y != $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N3),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_2572_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% VEBT_internal.Tb'.simps(3)
tff(fact_2573_VEBT__internal_OTb_H_Oelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( vEBT_VEBT_Tb2(Xb) = Y )
     => ( ( ( Xb = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
         => ~ ! [N3: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,N3)) )
               => ( Y != $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
tff(fact_2574_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [N: nat] :
      vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% VEBT_internal.Tb.simps(3)
tff(fact_2575_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [N: nat] :
      vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(bit0(one2))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
tff(fact_2576_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Osimps_I3_J,axiom,
    ! [Va: nat] :
      vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = $ite(
        aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
          $let(
            half: nat,
            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.simps(3)
tff(fact_2577_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [Va: nat] :
      vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = $ite(
        aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
          $let(
            half: nat,
            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.simps(3)
tff(fact_2578_VEBT__internal_OTb_Oelims,axiom,
    ! [Xb: nat,Y: int] :
      ( ( vEBT_VEBT_Tb(Xb) = Y )
     => ( ( ( Xb = zero_zero(nat) )
         => ( Y != aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) ) )
       => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) ) )
         => ~ ! [N3: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,N3)) )
               => ( Y != $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
tff(fact_2579_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Oelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(Xb) = Y )
     => ( ( ( Xb = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
               => ( Y != $ite(
                      aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                        $let(
                          half: nat,
                          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.elims
tff(fact_2580_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Oelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(Xb) = Y )
     => ( ( ( Xb = zero_zero(nat) )
         => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
       => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
               => ( Y != $ite(
                      aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                        $let(
                          half: nat,
                          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.elims
tff(fact_2581_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [N: nat,A3: A,B3: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
         => ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N))
          <=> aa(A,$o,dvd_dvd(A,A3),B3) ) ) ) ).

% pow_divides_pow_iff
tff(fact_2582_divmod__step__integer__def,axiom,
    ! [L: num,Qr: product_prod(code_integer,code_integer)] : unique1321980374590559556d_step(code_integer,L,Qr) = aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_gr(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_2583_artanh__def,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A] : aa(A,A,artanh(A),Xb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),Xb)),aa(A,A,minus_minus(A,one_one(A)),Xb)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% artanh_def
tff(fact_2584_div2__even__ext__nat,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
     => ( ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Xb)
        <=> aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Y) )
       => ( Xb = Y ) ) ) ).

% div2_even_ext_nat
tff(fact_2585_bezout__add__strong__nat,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 != zero_zero(nat) )
     => ? [D3: nat,X2: nat,Y2: nat] :
          ( aa(nat,$o,dvd_dvd(nat,D3),A3)
          & aa(nat,$o,dvd_dvd(nat,D3),B3)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),Y2)),D3) ) ) ) ).

% bezout_add_strong_nat
tff(fact_2586_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,N),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,N,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% flip_bit_Suc
tff(fact_2587_ln__inj__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => ( ( aa(real,real,ln_ln(real),Xb) = aa(real,real,ln_ln(real),Y) )
        <=> ( Xb = Y ) ) ) ) ).

% ln_inj_iff
tff(fact_2588_ln__less__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less(real,aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,ord_less(real,Xb),Y) ) ) ) ).

% ln_less_cancel_iff
tff(fact_2589_flip__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),bit_se8732182000553998342ip_bit(int,N,K))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_2590_flip__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less(int,bit_se8732182000553998342ip_bit(int,N,K)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% flip_bit_negative_int_iff
tff(fact_2591_ln__le__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,ord_less_eq(real,Xb),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_2592_ln__eq__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( ( aa(real,real,ln_ln(real),Xb) = zero_zero(real) )
      <=> ( Xb = one_one(real) ) ) ) ).

% ln_eq_zero_iff
tff(fact_2593_ln__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,ln_ln(real),Xb))
      <=> aa(real,$o,ord_less(real,one_one(real)),Xb) ) ) ).

% ln_gt_zero_iff
tff(fact_2594_ln__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,aa(real,real,ln_ln(real),Xb)),zero_zero(real))
      <=> aa(real,$o,ord_less(real,Xb),one_one(real)) ) ) ).

% ln_less_zero_iff
tff(fact_2595_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_2596_ln__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),Xb)),zero_zero(real))
      <=> aa(real,$o,ord_less_eq(real,Xb),one_one(real)) ) ) ).

% ln_le_zero_iff
tff(fact_2597_ln__ge__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,ln_ln(real),Xb))
      <=> aa(real,$o,ord_less_eq(real,one_one(real)),Xb) ) ) ).

% ln_ge_zero_iff
tff(fact_2598_ln__less__self,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => aa(real,$o,ord_less(real,aa(real,real,ln_ln(real),Xb)),Xb) ) ).

% ln_less_self
tff(fact_2599_zdvd__antisym__nonneg,axiom,
    ! [M: int,N: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),M)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),N)
       => ( aa(int,$o,dvd_dvd(int,M),N)
         => ( aa(int,$o,dvd_dvd(int,N),M)
           => ( M = N ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_2600_zdvd__not__zless,axiom,
    ! [M: int,N: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),M)
     => ( aa(int,$o,ord_less(int,M),N)
       => ~ aa(int,$o,dvd_dvd(int,N),M) ) ) ).

% zdvd_not_zless
tff(fact_2601_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N: int] :
      ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K),N))
     => ( ( K != zero_zero(int) )
       => aa(int,$o,dvd_dvd(int,M),N) ) ) ).

% zdvd_mult_cancel
tff(fact_2602_zdvd__mono,axiom,
    ! [K: int,M: int,Tb: int] :
      ( ( K != zero_zero(int) )
     => ( aa(int,$o,dvd_dvd(int,M),Tb)
      <=> aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K),Tb)) ) ) ).

% zdvd_mono
tff(fact_2603_ln__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),Xb)),Xb) ) ).

% ln_bound
tff(fact_2604_ln__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xb)
     => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,ln_ln(real),Xb)) ) ).

% ln_gt_zero
tff(fact_2605_ln__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),one_one(real))
       => aa(real,$o,ord_less(real,aa(real,real,ln_ln(real),Xb)),zero_zero(real)) ) ) ).

% ln_less_zero
tff(fact_2606_ln__gt__zero__imp__gt__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,ln_ln(real),Xb))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => aa(real,$o,ord_less(real,one_one(real)),Xb) ) ) ).

% ln_gt_zero_imp_gt_one
tff(fact_2607_ln__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),Xb)
     => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,ln_ln(real),Xb)) ) ).

% ln_ge_zero
tff(fact_2608_zdvd__imp__le,axiom,
    ! [Z: int,N: int] :
      ( aa(int,$o,dvd_dvd(int,Z),N)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),N)
       => aa(int,$o,ord_less_eq(int,Z),N) ) ) ).

% zdvd_imp_le
tff(fact_2609_zero__natural_Orsp,axiom,
    zero_zero(nat) = zero_zero(nat) ).

% zero_natural.rsp
tff(fact_2610_one__natural_Orsp,axiom,
    one_one(nat) = one_one(nat) ).

% one_natural.rsp
tff(fact_2611_real__of__int__div,axiom,
    ! [D: int,N: int] :
      ( aa(int,$o,dvd_dvd(int,D),N)
     => ( aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),N),D)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),N)),aa(int,real,ring_1_of_int(real),D)) ) ) ).

% real_of_int_div
tff(fact_2612_zero__integer_Orsp,axiom,
    zero_zero(int) = zero_zero(int) ).

% zero_integer.rsp
tff(fact_2613_one__integer_Orsp,axiom,
    one_one(int) = one_one(int) ).

% one_integer.rsp
tff(fact_2614_prod__decode__aux_Ocases,axiom,
    ! [Xb: product_prod(nat,nat)] :
      ~ ! [K2: nat,M3: nat] : Xb != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K2),M3) ).

% prod_decode_aux.cases
tff(fact_2615_times__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),K),zero_zero(code_integer)) = zero_zero(code_integer) ).

% times_integer_code(1)
tff(fact_2616_times__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),zero_zero(code_integer)),L) = zero_zero(code_integer) ).

% times_integer_code(2)
tff(fact_2617_less__eq__integer__code_I1_J,axiom,
    aa(code_integer,$o,ord_less_eq(code_integer,zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_eq_integer_code(1)
tff(fact_2618_ln__ge__zero__imp__ge__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,ln_ln(real),Xb))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => aa(real,$o,ord_less_eq(real,one_one(real)),Xb) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_2619_ln__add__one__self__le__self,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb))),Xb) ) ).

% ln_add_one_self_le_self
tff(fact_2620_ln__mult,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_mult
tff(fact_2621_ln__eq__minus__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( ( aa(real,real,ln_ln(real),Xb) = aa(real,real,minus_minus(real,Xb),one_one(real)) )
       => ( Xb = one_one(real) ) ) ) ).

% ln_eq_minus_one
tff(fact_2622_ln__div,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => ( aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y)) ) ) ) ).

% ln_div
tff(fact_2623_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L))
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        | ( ( L = zero_zero(int) )
          & aa(int,$o,ord_less_eq(int,zero_zero(int)),K) )
        | aa(int,$o,ord_less(int,zero_zero(int)),L) ) ) ).

% mod_int_pos_iff
tff(fact_2624_int__div__sub__1,axiom,
    ! [M: int,N: int] :
      ( aa(int,$o,ord_less_eq(int,one_one(int)),M)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,N),one_one(int))),M) = $ite(aa(int,$o,dvd_dvd(int,M),N),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),N),M)),one_one(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),N),M)) ) ) ).

% int_div_sub_1
tff(fact_2625_bset_I9_J,axiom,
    ! [D: int,D5: int,B6: set(int),Tb: int] :
      ( aa(int,$o,dvd_dvd(int,D),D5)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B6)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
         => ( aa(int,$o,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),Tb))
           => aa(int,$o,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,X3),D5)),Tb)) ) ) ) ).

% bset(9)
tff(fact_2626_bset_I10_J,axiom,
    ! [D: int,D5: int,B6: set(int),Tb: int] :
      ( aa(int,$o,dvd_dvd(int,D),D5)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B6)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa2) ) ) )
         => ( ~ aa(int,$o,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),Tb))
           => ~ aa(int,$o,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,X3),D5)),Tb)) ) ) ) ).

% bset(10)
tff(fact_2627_aset_I9_J,axiom,
    ! [D: int,D5: int,A4: set(int),Tb: int] :
      ( aa(int,$o,dvd_dvd(int,D),D5)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A4)
                 => ( X3 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
         => ( aa(int,$o,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),Tb))
           => aa(int,$o,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5)),Tb)) ) ) ) ).

% aset(9)
tff(fact_2628_aset_I10_J,axiom,
    ! [D: int,D5: int,A4: set(int),Tb: int] :
      ( aa(int,$o,dvd_dvd(int,D),D5)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D5))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A4)
                 => ( X3 != aa(int,int,minus_minus(int,Xb2),Xa2) ) ) )
         => ( ~ aa(int,$o,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),Tb))
           => ~ aa(int,$o,dvd_dvd(int,D),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D5)),Tb)) ) ) ) ).

% aset(10)
tff(fact_2629_ln__2__less__1,axiom,
    aa(real,$o,ord_less(real,aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),bit0(one2)))),one_one(real)) ).

% ln_2_less_1
tff(fact_2630_ln__le__minus__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),Xb)),aa(real,real,minus_minus(real,Xb),one_one(real))) ) ).

% ln_le_minus_one
tff(fact_2631_ln__diff__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => aa(real,$o,ord_less_eq(real,aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,Xb),Y)),Y)) ) ) ).

% ln_diff_le
tff(fact_2632_ln__realpow,axiom,
    ! [Xb: real,N: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,real,ln_ln(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,ln_ln(real),Xb)) ) ) ).

% ln_realpow
tff(fact_2633_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,minus_minus(int,K),L))
    <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_diff_iff
tff(fact_2634_log__eq__div__ln__mult__log,axiom,
    ! [A3: real,B3: real,Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
         => ( ( B3 != one_one(real) )
           => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
             => ( aa(real,real,log(A3),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B3)),aa(real,real,ln_ln(real),A3))),aa(real,real,log(B3),Xb)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_2635_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero(nat) )
     => ~ ! [N3: nat] : Xb != aa(nat,nat,suc,N3) ) ).

% list_decode.cases
tff(fact_2636_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),A3: nat,B3: nat] :
      ( ! [A2: nat,B2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),P,A2),B2)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),P,B2),A2) )
     => ( ! [A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,A2),zero_zero(nat))
       => ( ! [A2: nat,B2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,A2),B2)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,A3),B3) ) ) ) ).

% Euclid_induct
tff(fact_2637_gcd__nat_Oextremum,axiom,
    ! [A3: nat] : aa(nat,$o,dvd_dvd(nat,A3),zero_zero(nat)) ).

% gcd_nat.extremum
tff(fact_2638_gcd__nat_Oextremum__strict,axiom,
    ! [A3: nat] :
      ~ ( aa(nat,$o,dvd_dvd(nat,zero_zero(nat)),A3)
        & ( zero_zero(nat) != A3 ) ) ).

% gcd_nat.extremum_strict
tff(fact_2639_gcd__nat_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( aa(nat,$o,dvd_dvd(nat,zero_zero(nat)),A3)
    <=> ( A3 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_unique
tff(fact_2640_gcd__nat_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero(nat) )
    <=> ( aa(nat,$o,dvd_dvd(nat,A3),zero_zero(nat))
        & ( A3 != zero_zero(nat) ) ) ) ).

% gcd_nat.not_eq_extremum
tff(fact_2641_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( aa(nat,$o,dvd_dvd(nat,zero_zero(nat)),A3)
     => ( A3 = zero_zero(nat) ) ) ).

% gcd_nat.extremum_uniqueI
tff(fact_2642_less__integer__code_I1_J,axiom,
    ~ aa(code_integer,$o,ord_less(code_integer,zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_integer_code(1)
tff(fact_2643_plus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),K),zero_zero(code_integer)) = K ).

% plus_integer_code(1)
tff(fact_2644_plus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),zero_zero(code_integer)),L) = L ).

% plus_integer_code(2)
tff(fact_2645_eme1p,axiom,
    ! [N: int,D: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),N)
     => ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),D)
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),D)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),N)),D) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),D)) ) ) ) ) ).

% eme1p
tff(fact_2646_emep1,axiom,
    ! [N: int,D: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),N)
     => ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),D)
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),D)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int))),D) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),D)),one_one(int)) ) ) ) ) ).

% emep1
tff(fact_2647_minus__integer__code_I1_J,axiom,
    ! [K: code_integer] : aa(code_integer,code_integer,minus_minus(code_integer,K),zero_zero(code_integer)) = K ).

% minus_integer_code(1)
tff(fact_2648_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,M,A3))
        <=> ~ ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            <=> ( M = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_2649_ln__one__plus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,minus_minus(real,Xb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb))) ) ) ).

% ln_one_plus_pos_lower_bound
tff(fact_2650_dvd__pos__nat,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,$o,dvd_dvd(nat,M),N)
       => aa(nat,$o,ord_less(nat,zero_zero(nat)),M) ) ) ).

% dvd_pos_nat
tff(fact_2651_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),M: nat,N: nat] :
      ( ! [M3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,M3),zero_zero(nat))
     => ( ! [M3: nat,N3: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N3)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),P,N3),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M3),N3))
             => aa(nat,$o,aa(nat,fun(nat,$o),P,M3),N3) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),P,M),N) ) ) ).

% gcd_nat_induct
tff(fact_2652_VEBT__internal_Ovebt__memberi_H_Osimps,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),vEBT_V854960066525838166emberi,Tb),Tib),Xb) = vEBT_case_VEBTi(heap_Time_Heap($o),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))),aTP_Lamp_hf(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))))),Tb),Xb),aTP_Lamp_hg(nat,fun($o,fun($o,heap_Time_Heap($o))),Xb),Tib) ).

% VEBT_internal.vebt_memberi'.simps
tff(fact_2653_vebt__memberi_Osimps,axiom,
    ! [Tb: vEBT_VEBTi,Xb: nat] : aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),vEBT_vebt_memberi,Tb),Xb) = vEBT_case_VEBTi(heap_Time_Heap($o),aTP_Lamp_hm(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))),Xb),aTP_Lamp_hg(nat,fun($o,fun($o,heap_Time_Heap($o))),Xb),Tb) ).

% vebt_memberi.simps
tff(fact_2654_VEBT__internal_Ovebt__memberi_H_Oraw__induct,axiom,
    ! [P: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o))))))),Xaa: product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: $o,N: nat] :
      ( ! [Vebt_memberi: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))))] :
          ( ! [A6: vEBT_VEBT,B5: vEBT_VEBTi,Ba: nat,H3: heap_ext(product_unit),H4: heap_ext(product_unit),R3: $o,N4: nat] :
              ( heap_Time_effect($o,aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),Vebt_memberi,A6),B5),Ba),H3,H4,(R3),N4)
             => aa(nat,$o,aa($o,fun(nat,$o),aa(heap_ext(product_unit),fun($o,fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o))),aa(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o)))),aa(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o))))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o)))))),P,A6),B5),Ba),H3),H4),(R3)),N4) )
         => ! [T2: vEBT_VEBT,Ti2: vEBT_VEBTi,X2: nat,Ta: heap_ext(product_unit),Tia: heap_ext(product_unit),Xa2: $o,N3: nat] :
              ( heap_Time_effect($o,vEBT_case_VEBTi(heap_Time_Heap($o),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))))),aTP_Lamp_ia(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))))),Vebt_memberi),T2),X2),aTP_Lamp_hg(nat,fun($o,fun($o,heap_Time_Heap($o))),X2),Ti2),Ta,Tia,(Xa2),N3)
             => aa(nat,$o,aa($o,fun(nat,$o),aa(heap_ext(product_unit),fun($o,fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o))),aa(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o)))),aa(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o))))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o)))))),P,T2),Ti2),X2),Ta),Tia),(Xa2)),N3) ) )
     => ( heap_Time_effect($o,aa(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o),product_case_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap($o),product_case_prod(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),vEBT_V854960066525838166emberi)),Xaa),H,H2,(R2),N)
       => aa(nat,$o,aa($o,fun(nat,$o),aa(heap_ext(product_unit),fun($o,fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o))),aa(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o)))),product_case_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o)))),product_case_prod(vEBT_VEBT,vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o))))),P)),Xaa),H),H2),(R2)),N) ) ) ).

% VEBT_internal.vebt_memberi'.raw_induct
tff(fact_2655_bezw_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod(int,int)] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Xaa)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Xaa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),Xaa)))))) ) ) ).

% bezw.elims
tff(fact_2656_bezw_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      bezw(Xb,Y) = $ite(Y = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Y)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),Y)))))) ).

% bezw.simps
tff(fact_2657_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),zero_zero(nat)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% unset_bit_0
tff(fact_2658_unset__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_2659_unset__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% unset_bit_negative_int_iff
tff(fact_2660_bezw__0,axiom,
    ! [Xb: nat] : bezw(Xb,zero_zero(nat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)) ).

% bezw_0
tff(fact_2661_unset__bit__less__eq,axiom,
    ! [N: nat,K: int] : aa(int,$o,ord_less_eq(int,aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K)),K) ).

% unset_bit_less_eq
tff(fact_2662_time__array__len,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P2: array(A),H: heap_ext(product_unit)] : time_time(nat,array_len(A,P2),H) = one_one(nat) ) ).

% time_array_len
tff(fact_2663_TBOUND__len,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: array(A)] : time_TBOUND(nat,array_len(A,Xs),one_one(nat)) ) ).

% TBOUND_len
tff(fact_2664_vebt__memberi_Oraw__induct,axiom,
    ! [P: fun(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o)))))),Xaa: product_prod(vEBT_VEBTi,nat),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: $o,N: nat] :
      ( ! [Vebt_memberi2: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))] :
          ( ! [A6: vEBT_VEBTi,B5: nat,H3: heap_ext(product_unit),H4: heap_ext(product_unit),R3: $o,N4: nat] :
              ( heap_Time_effect($o,aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),Vebt_memberi2,A6),B5),H3,H4,(R3),N4)
             => aa(nat,$o,aa($o,fun(nat,$o),aa(heap_ext(product_unit),fun($o,fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o))),aa(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o)))),aa(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o))))),P,A6),B5),H3),H4),(R3)),N4) )
         => ! [T2: vEBT_VEBTi,X2: nat,Ta: heap_ext(product_unit),Xa2: heap_ext(product_unit),R4: $o,N3: nat] :
              ( heap_Time_effect($o,vEBT_case_VEBTi(heap_Time_Heap($o),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))),aTP_Lamp_ig(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))))),Vebt_memberi2),X2),aTP_Lamp_hg(nat,fun($o,fun($o,heap_Time_Heap($o))),X2),T2),Ta,Xa2,(R4),N3)
             => aa(nat,$o,aa($o,fun(nat,$o),aa(heap_ext(product_unit),fun($o,fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o))),aa(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o)))),aa(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o))))),P,T2),X2),Ta),Xa2),(R4)),N3) ) )
     => ( heap_Time_effect($o,aa(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o),product_case_prod(vEBT_VEBTi,nat,heap_Time_Heap($o),vEBT_vebt_memberi),Xaa),H,H2,(R2),N)
       => aa(nat,$o,aa($o,fun(nat,$o),aa(heap_ext(product_unit),fun($o,fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o))),aa(product_prod(vEBT_VEBTi,nat),fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o)))),product_case_prod(vEBT_VEBTi,nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun($o,fun(nat,$o)))),P),Xaa),H),H2),(R2)),N) ) ) ).

% vebt_memberi.raw_induct
tff(fact_2665_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A3))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            | ( M = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_2666_bezw__non__0,axiom,
    ! [Y: nat,Xb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Y)
     => ( bezw(Xb,Y) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Y)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Y)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Y)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),Y))))) ) ) ).

% bezw_non_0
tff(fact_2667_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),aa(nat,nat,suc,N)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% unset_bit_Suc
tff(fact_2668_tanh__ln__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,real,tanh(real),aa(real,real,ln_ln(real),Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))) ) ) ).

% tanh_ln_real
tff(fact_2669_VEBT__internal_Ovebt__succi_H_Omono,axiom,
    ! [Xb: product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat)] : comple7038119648293358887notone(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),heap_Time_Heap(option(nat)),partial_fun_ord(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_ord(option(nat))),heap_Time_Heap_ord(option(nat)),aTP_Lamp_ix(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),fun(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),heap_Time_Heap(option(nat))),Xb)) ).

% VEBT_internal.vebt_succi'.mono
tff(fact_2670_VEBT__internal_Ovebt__predi_H_Omono,axiom,
    ! [Xb: product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat)] : comple7038119648293358887notone(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),heap_Time_Heap(option(nat)),partial_fun_ord(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_ord(option(nat))),heap_Time_Heap_ord(option(nat)),aTP_Lamp_jo(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),fun(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),heap_Time_Heap(option(nat))),Xb)) ).

% VEBT_internal.vebt_predi'.mono
tff(fact_2671_product__nth,axiom,
    ! [A: $tType,B: $tType,N: nat,Xs: list(A),Ys: list(B)] :
      ( aa(nat,$o,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)))
     => ( aa(nat,product_prod(A,B),nth(product_prod(A,B),product(A,B,Xs,Ys)),N) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(list(B),nat,size_size(list(B)),Ys)))),aa(nat,B,nth(B,Ys),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).

% product_nth
tff(fact_2672_Heap__Time__Monad_Obind__bind,axiom,
    ! [C: $tType,A: $tType,B: $tType,Fa: heap_Time_Heap(C),G: fun(C,heap_Time_Heap(B)),K: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,heap_Time_bind(C,B,Fa,G),K) = heap_Time_bind(C,A,Fa,aa(fun(B,heap_Time_Heap(A)),fun(C,heap_Time_Heap(A)),aTP_Lamp_jp(fun(C,heap_Time_Heap(B)),fun(fun(B,heap_Time_Heap(A)),fun(C,heap_Time_Heap(A))),G),K)) ).

% Heap_Time_Monad.bind_bind
tff(fact_2673_curry__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,Fa: fun(product_prod(B,C),A),A3: B,B3: C] : aa(C,A,aa(B,fun(C,A),product_curry(B,C,A,Fa),A3),B3) = aa(product_prod(B,C),A,Fa,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B3)) ).

% curry_conv
tff(fact_2674_tanh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,tanh(A),zero_zero(A)) = zero_zero(A) ) ) ).

% tanh_0
tff(fact_2675_curry__case__prod,axiom,
    ! [C: $tType,B: $tType,A: $tType,Fa: fun(A,fun(B,C))] : product_curry(A,B,C,product_case_prod(A,B,C,Fa)) = Fa ).

% curry_case_prod
tff(fact_2676_case__prod__curry,axiom,
    ! [C: $tType,B: $tType,A: $tType,Fa: fun(product_prod(A,B),C)] : product_case_prod(A,B,C,product_curry(A,B,C,Fa)) = Fa ).

% case_prod_curry
tff(fact_2677_tanh__real__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa(real,real,tanh(real),Xb) = zero_zero(real) )
    <=> ( Xb = zero_zero(real) ) ) ).

% tanh_real_zero_iff
tff(fact_2678_tanh__real__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,tanh(real),Xb)),aa(real,real,tanh(real),Y))
    <=> aa(real,$o,ord_less(real,Xb),Y) ) ).

% tanh_real_less_iff
tff(fact_2679_tanh__real__neg__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,tanh(real),Xb)),zero_zero(real))
    <=> aa(real,$o,ord_less(real,Xb),zero_zero(real)) ) ).

% tanh_real_neg_iff
tff(fact_2680_tanh__real__pos__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,tanh(real),Xb))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),Xb) ) ).

% tanh_real_pos_iff
tff(fact_2681_tanh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,tanh(real),Xb))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb) ) ).

% tanh_real_nonneg_iff
tff(fact_2682_tanh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,tanh(real),Xb)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,Xb),zero_zero(real)) ) ).

% tanh_real_nonpos_iff
tff(fact_2683_length__product,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B)] : aa(list(product_prod(A,B)),nat,size_size(list(product_prod(A,B))),product(A,B,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(B),nat,size_size(list(B)),Ys)) ).

% length_product
tff(fact_2684_Heap__Time__Monad_Obind__mono,axiom,
    ! [D2: $tType,B: $tType,A: $tType,C: $tType,B6: fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(C)),C5: fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2)))] :
      ( comple7038119648293358887notone(fun(A,heap_Time_Heap(B)),heap_Time_Heap(C),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),heap_Time_Heap_ord(C),B6)
     => ( ! [Y2: C] : comple7038119648293358887notone(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),heap_Time_Heap_ord(D2),aa(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2)),C5,Y2))
       => comple7038119648293358887notone(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),heap_Time_Heap_ord(D2),aa(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2))),fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2)),aTP_Lamp_jr(fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(C)),fun(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2))),fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2))),B6),C5)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_2685_heap_Oconst__mono,axiom,
    ! [A: $tType,B: $tType,Ord: fun(A,fun(A,$o)),C2: heap_Time_Heap(B)] : comple7038119648293358887notone(A,heap_Time_Heap(B),Ord,heap_Time_Heap_ord(B),aTP_Lamp_js(heap_Time_Heap(B),fun(A,heap_Time_Heap(B)),C2)) ).

% heap.const_mono
tff(fact_2686_curry__K,axiom,
    ! [B: $tType,A: $tType,C: $tType,C2: C,X3: A,Xa: B] : aa(B,C,aa(A,fun(B,C),product_curry(A,B,C,aTP_Lamp_jt(C,fun(product_prod(A,B),C),C2)),X3),Xa) = C2 ).

% curry_K
tff(fact_2687_tanh__real__lt__1,axiom,
    ! [Xb: real] : aa(real,$o,ord_less(real,aa(real,real,tanh(real),Xb)),one_one(real)) ).

% tanh_real_lt_1
tff(fact_2688_curry__def,axiom,
    ! [C: $tType,A: $tType,B: $tType,X3: fun(product_prod(A,B),C),Xa: A,Xb3: B] : aa(B,C,aa(A,fun(B,C),product_curry(A,B,C,X3),Xa),Xb3) = aa(product_prod(A,B),C,X3,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xa),Xb3)) ).

% curry_def
tff(fact_2689_Heap_Osize__neq,axiom,
    ! [A: $tType,Xb: heap_Time_Heap(A)] : aa(heap_Time_Heap(A),nat,size_size(heap_Time_Heap(A)),Xb) != zero_zero(nat) ).

% Heap.size_neq
tff(fact_2690_fold__if__return,axiom,
    ! [A: $tType,B3: $o,C2: A,D: A] :
      $ite((B3),aa(A,heap_Time_Heap(A),heap_Time_return(A),C2),aa(A,heap_Time_Heap(A),heap_Time_return(A),D)) = aa(A,heap_Time_Heap(A),heap_Time_return(A),
        $ite((B3),C2,D)) ).

% fold_if_return
tff(fact_2691_effect__deterministic_I3_J,axiom,
    ! [A: $tType,Fa: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),A3: A,N: nat,H7: heap_ext(product_unit),B3: A,N5: nat] :
      ( heap_Time_effect(A,Fa,H,H2,A3,N)
     => ( heap_Time_effect(A,Fa,H,H7,B3,N5)
       => ( N = N5 ) ) ) ).

% effect_deterministic(3)
tff(fact_2692_effect__deterministic_I2_J,axiom,
    ! [A: $tType,Fa: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),A3: A,N: nat,H7: heap_ext(product_unit),B3: A,N5: nat] :
      ( heap_Time_effect(A,Fa,H,H2,A3,N)
     => ( heap_Time_effect(A,Fa,H,H7,B3,N5)
       => ( H2 = H7 ) ) ) ).

% effect_deterministic(2)
tff(fact_2693_effect__deterministic_I1_J,axiom,
    ! [A: $tType,Fa: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),A3: A,N: nat,H7: heap_ext(product_unit),B3: A,N5: nat] :
      ( heap_Time_effect(A,Fa,H,H2,A3,N)
     => ( heap_Time_effect(A,Fa,H,H7,B3,N5)
       => ( A3 = B3 ) ) ) ).

% effect_deterministic(1)
tff(fact_2694_effect__ifE,axiom,
    ! [A: $tType,C2: $o,Tb: heap_Time_Heap(A),E3: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: A,N: nat] :
      ( heap_Time_effect(A,
          $ite((C2),Tb,E3),
          H,H2,R2,N)
     => ( ( (C2)
         => ~ heap_Time_effect(A,Tb,H,H2,R2,N) )
       => ~ ( ~ (C2)
           => ~ heap_Time_effect(A,E3,H,H2,R2,N) ) ) ) ).

% effect_ifE
tff(fact_2695_effect__ifI,axiom,
    ! [A: $tType,C2: $o,Tb: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: A,N: nat,E3: heap_Time_Heap(A)] :
      ( ( (C2)
       => heap_Time_effect(A,Tb,H,H2,R2,N) )
     => ( ( ~ (C2)
         => heap_Time_effect(A,E3,H,H2,R2,N) )
       => heap_Time_effect(A,
            $ite((C2),Tb,E3),
            H,H2,R2,N) ) ) ).

% effect_ifI
tff(fact_2696_distrib__if__bind,axiom,
    ! [A: $tType,B: $tType,B3: $o,C2: heap_Time_Heap(B),D: heap_Time_Heap(B),Fa: fun(B,heap_Time_Heap(A))] :
      heap_Time_bind(B,A,
        $ite((B3),C2,D),
        Fa) = $ite((B3),heap_Time_bind(B,A,C2,Fa),heap_Time_bind(B,A,D,Fa)) ).

% distrib_if_bind
tff(fact_2697_effect__LetI,axiom,
    ! [B: $tType,A: $tType,Xb: A,Tb: A,Fa: fun(A,heap_Time_Heap(B)),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: B,N: nat] :
      ( ( Xb = Tb )
     => ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),Fa,Xb),H,H2,R2,N)
       => heap_Time_effect(B,aa(A,heap_Time_Heap(B),Fa,Tb),H,H2,R2,N) ) ) ).

% effect_LetI
tff(fact_2698_effect__returnI,axiom,
    ! [A: $tType,H: heap_ext(product_unit),H2: heap_ext(product_unit),Xb: A] :
      ( ( H = H2 )
     => heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),Xb),H,H2,Xb,one_one(nat)) ) ).

% effect_returnI
tff(fact_2699_effect__returnE,axiom,
    ! [A: $tType,Xb: A,H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: A,N: nat] :
      ( heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),Xb),H,H2,R2,N)
     => ~ ( ( R2 = Xb )
         => ( ( H2 = H )
           => ( N != one_one(nat) ) ) ) ) ).

% effect_returnE
tff(fact_2700_effect__bindE,axiom,
    ! [B: $tType,A: $tType,Fa: heap_Time_Heap(B),G: fun(B,heap_Time_Heap(A)),H: heap_ext(product_unit),H7: heap_ext(product_unit),R5: A,N: nat] :
      ( heap_Time_effect(A,heap_Time_bind(B,A,Fa,G),H,H7,R5,N)
     => ~ ! [H8: heap_ext(product_unit),R4: B,N1: nat] :
            ( heap_Time_effect(B,Fa,H,H8,R4,N1)
           => ! [N22: nat] :
                ( heap_Time_effect(A,aa(B,heap_Time_Heap(A),G,R4),H8,H7,R5,N22)
               => ( N != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_2701_effect__bindI,axiom,
    ! [B: $tType,A: $tType,Fa: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: A,N: nat,G: fun(A,heap_Time_Heap(B)),H7: heap_ext(product_unit),R5: B,N5: nat] :
      ( heap_Time_effect(A,Fa,H,H2,R2,N)
     => ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),G,R2),H2,H7,R5,N5)
       => heap_Time_effect(B,heap_Time_bind(A,B,Fa,G),H,H7,R5,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N5)) ) ) ).

% effect_bindI
tff(fact_2702_VEBT__internal_Ovebt__predi_H_Ofixp__induct,axiom,
    ! [P: fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o)] :
      ( comple1908693960933563346ssible(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),partial_fun_lub(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_lub(option(nat))),partial_fun_ord(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_ord(option(nat))),aTP_Lamp_ju(fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o),fun(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),$o),P))
     => ( aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o,P,aTP_Lamp_jw(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))))
       => ( ! [F4: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))] :
              ( aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o,P,F4)
             => aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o,P,aTP_Lamp_jx(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),F4)) )
         => aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o,P,vEBT_VEBT_vebt_predi) ) ) ) ).

% VEBT_internal.vebt_predi'.fixp_induct
tff(fact_2703_VEBT__internal_Ovebt__succi_H_Ofixp__induct,axiom,
    ! [P: fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o)] :
      ( comple1908693960933563346ssible(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),partial_fun_lub(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_lub(option(nat))),partial_fun_ord(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_ord(option(nat))),aTP_Lamp_ju(fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o),fun(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),$o),P))
     => ( aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o,P,aTP_Lamp_jw(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))))
       => ( ! [F4: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))] :
              ( aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o,P,F4)
             => aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o,P,aTP_Lamp_jy(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),F4)) )
         => aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o,P,vEBT_VEBT_vebt_succi) ) ) ) ).

% VEBT_internal.vebt_succi'.fixp_induct
tff(fact_2704_vebt__succi_Omono,axiom,
    ! [Xb: product_prod(vEBT_VEBTi,nat)] : comple7038119648293358887notone(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),heap_Time_Heap(option(nat)),partial_fun_ord(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_ord(option(nat))),heap_Time_Heap_ord(option(nat)),aTP_Lamp_kg(product_prod(vEBT_VEBTi,nat),fun(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),heap_Time_Heap(option(nat))),Xb)) ).

% vebt_succi.mono
tff(fact_2705_vebt__predi_Omono,axiom,
    ! [Xb: product_prod(vEBT_VEBTi,nat)] : comple7038119648293358887notone(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),heap_Time_Heap(option(nat)),partial_fun_ord(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_ord(option(nat))),heap_Time_Heap_ord(option(nat)),aTP_Lamp_ko(product_prod(vEBT_VEBTi,nat),fun(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),heap_Time_Heap(option(nat))),Xb)) ).

% vebt_predi.mono
tff(fact_2706_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( aa(fun(A,fun(B,$o)),$o,ord_less_eq(fun(A,fun(B,$o)),aTP_Lamp_kp(set(product_prod(A,B)),fun(A,fun(B,$o)),R)),aTP_Lamp_kp(set(product_prod(A,B)),fun(A,fun(B,$o)),S3))
    <=> aa(set(product_prod(A,B)),$o,ord_less_eq(set(product_prod(A,B)),R),S3) ) ).

% pred_subset_eq2
tff(fact_2707_ln__one__minus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less_eq(real,aa(real,real,minus_minus(real,aa(real,real,uminus_uminus(real),Xb)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,ln_ln(real),aa(real,real,minus_minus(real,one_one(real)),Xb))) ) ) ).

% ln_one_minus_pos_lower_bound
tff(fact_2708_curryI,axiom,
    ! [A: $tType,B: $tType,Fa: fun(product_prod(A,B),$o),A3: A,B3: B] :
      ( aa(product_prod(A,B),$o,Fa,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3))
     => aa(B,$o,aa(A,fun(B,$o),product_curry(A,B,$o,Fa),A3),B3) ) ).

% curryI
tff(fact_2709__092_060open_062_092_060And_062xa_Atia_O_Arefines_A_Ivebt__predi_Atia_Axa_J_A_IHeap_OHeap_AMap_Oempty_J_092_060close_062,axiom,
    ! [Tiaa: vEBT_VEBTi,Xaa: nat] : refine_Imp_refines(option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_vebt_predi,Tiaa),Xaa),heap_Time_Heap2(option(nat),aTP_Lamp_jv(heap_ext(product_unit),option(product_prod(option(nat),product_prod(heap_ext(product_unit),nat)))))) ).

% \<open>\<And>xa tia. refines (vebt_predi tia xa) (Heap.Heap Map.empty)\<close>
tff(fact_2710_add_Oinverse__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ( aa(A,A,uminus_uminus(A),zero_zero(A)) = zero_zero(A) ) ) ).

% add.inverse_neutral
tff(fact_2711_neg__0__equal__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = aa(A,A,uminus_uminus(A),A3) )
        <=> ( zero_zero(A) = A3 ) ) ) ).

% neg_0_equal_iff_equal
tff(fact_2712_neg__equal__0__iff__equal,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% neg_equal_0_iff_equal
tff(fact_2713_equal__neg__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% equal_neg_zero
tff(fact_2714_neg__equal__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = A3 )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% neg_equal_zero
tff(fact_2715_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% neg_le_iff_le
tff(fact_2716_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% neg_less_iff_less
tff(fact_2717_neg__numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,N: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) )
        <=> ( M = N ) ) ) ).

% neg_numeral_eq_iff
tff(fact_2718_mult__minus__left,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),B3) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ).

% mult_minus_left
tff(fact_2719_minus__mult__minus,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) ) ).

% minus_mult_minus
tff(fact_2720_mult__minus__right,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) ) ).

% mult_minus_right
tff(fact_2721_div__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ).

% div_minus_minus
tff(fact_2722_dvd__minus__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,dvd_dvd(A,Xb),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,dvd_dvd(A,Xb),Y) ) ) ).

% dvd_minus_iff
tff(fact_2723_minus__dvd__iff,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,uminus_uminus(A),Xb)),Y)
        <=> aa(A,$o,dvd_dvd(A,Xb),Y) ) ) ).

% minus_dvd_iff
tff(fact_2724_mod__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3)) ) ).

% mod_minus_minus
tff(fact_2725_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) ) ) ).

% neg_0_le_iff_le
tff(fact_2726_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ).

% neg_le_0_iff_le
tff(fact_2727_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) ) ) ).

% less_eq_neg_nonpos
tff(fact_2728_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A3)),A3)
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ).

% neg_less_eq_nonneg
tff(fact_2729_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% less_neg_neg
tff(fact_2730_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),A3)),A3)
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ).

% neg_less_pos
tff(fact_2731_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% neg_0_less_iff_less
tff(fact_2732_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ).

% neg_less_0_iff_less
tff(fact_2733_add_Oright__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,uminus_uminus(A),A3)) = zero_zero(A) ) ).

% add.right_inverse
tff(fact_2734_ab__left__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).

% ab_left_minus
tff(fact_2735_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),N))) ) ).

% add_neg_numeral_simps(3)
tff(fact_2736_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A] : aa(A,A,minus_minus(A,zero_zero(A)),A3) = aa(A,A,uminus_uminus(A),A3) ) ).

% diff_0
tff(fact_2737_verit__minus__simplify_I3_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B3: A] : aa(A,A,minus_minus(A,zero_zero(A)),B3) = aa(A,A,uminus_uminus(A),B3) ) ).

% verit_minus_simplify(3)
tff(fact_2738_mult__minus1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),one_one(A))),Z) = aa(A,A,uminus_uminus(A),Z) ) ).

% mult_minus1
tff(fact_2739_mult__minus1__right,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Z),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),Z) ) ).

% mult_minus1_right
tff(fact_2740_divide__minus1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),Xb) ) ).

% divide_minus1
tff(fact_2741_div__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),A3) ) ).

% div_minus1_right
tff(fact_2742_minus__mod__self1,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [B3: A,A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,B3),A3)),B3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A3)),B3) ) ).

% minus_mod_self1
tff(fact_2743_real__add__minus__iff,axiom,
    ! [Xb: real,A3: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,uminus_uminus(real),A3)) = zero_zero(real) )
    <=> ( Xb = A3 ) ) ).

% real_add_minus_iff
tff(fact_2744_add__neg__numeral__special_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = zero_zero(A) ) ) ).

% add_neg_numeral_special(8)
tff(fact_2745_add__neg__numeral__special_I7_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% add_neg_numeral_special(7)
tff(fact_2746_neg__one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] :
          ( ( aa(A,A,uminus_uminus(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) )
        <=> ( N = one2 ) ) ) ).

% neg_one_eq_numeral_iff
tff(fact_2747_numeral__eq__neg__one__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( N = one2 ) ) ) ).

% numeral_eq_neg_one_iff
tff(fact_2748_diff__numeral__special_I12_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% diff_numeral_special(12)
tff(fact_2749_minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)) = one_one(A) ) ).

% minus_one_mult_self
tff(fact_2750_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),A3)) = A3 ) ).

% left_minus_one_mult_self
tff(fact_2751_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).

% mod_minus1_right
tff(fact_2752_max__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,ord_max(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(4)
tff(fact_2753_max__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,ord_max(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),V),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))) ) ).

% max_number_of(3)
tff(fact_2754_max__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          aa(A,A,ord_max(A,aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),U)) ) ).

% max_number_of(2)
tff(fact_2755_norm__neg__numeral,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [W: num] : real_V7770717601297561774m_norm(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = aa(num,real,numeral_numeral(real),W) ) ).

% norm_neg_numeral
tff(fact_2756_semiring__norm_I168_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),V),W)))),Y) ) ).

% semiring_norm(168)
tff(fact_2757_ceiling__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archimedean_ceiling(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)) ) ).

% ceiling_neg_numeral
tff(fact_2758_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N)) ) ).

% diff_numeral_simps(2)
tff(fact_2759_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,N: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),N))) ) ).

% diff_numeral_simps(3)
tff(fact_2760_semiring__norm_I172_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W))),Y) ) ).

% semiring_norm(172)
tff(fact_2761_semiring__norm_I171_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W)))),Y) ) ).

% semiring_norm(171)
tff(fact_2762_semiring__norm_I170_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [V: num,W: num,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),V),W)))),Y) ) ).

% semiring_norm(170)
tff(fact_2763_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_2764_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_2765_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,N: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),N))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_2766_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))
        <=> aa(num,$o,ord_less_eq(num,N),M) ) ) ).

% neg_numeral_le_iff
tff(fact_2767_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)))
        <=> aa(num,$o,ord_less(num,N),M) ) ) ).

% neg_numeral_less_iff
tff(fact_2768_round__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [N: num] : archimedean_round(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)) ) ).

% round_neg_numeral
tff(fact_2769_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( ~ aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)))
        <=> ( M != one2 ) ) ) ).

% not_neg_one_le_neg_numeral_iff
tff(fact_2770_neg__numeral__less__neg__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A)))
        <=> ( M != one2 ) ) ) ).

% neg_numeral_less_neg_one_iff
tff(fact_2771_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,W: num,A3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = A3 )
        <=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),A3 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_2772_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,W: num] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) )
        <=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = B3,A3 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_2773_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,W: num,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A3)
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B3) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_2774_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,W: num] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_2775_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,W: num,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A3)
        <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B3) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_2776_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,W: num] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_2777_power2__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_minus
tff(fact_2778_add__neg__numeral__special_I9_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% add_neg_numeral_special(9)
tff(fact_2779_diff__numeral__special_I11_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% diff_numeral_special(11)
tff(fact_2780_diff__numeral__special_I10_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% diff_numeral_special(10)
tff(fact_2781_minus__1__div__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% minus_1_div_2_eq
tff(fact_2782_minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( euclid8789492081693882211th_nat(A)
     => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% minus_1_mod_2_eq
tff(fact_2783_bits__minus__1__mod__2__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ).

% bits_minus_1_mod_2_eq
tff(fact_2784_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).

% Power.ring_1_class.power_minus_even
tff(fact_2785_power__minus__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,A3: A] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ) ) ).

% power_minus_odd
tff(fact_2786_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,A3: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) ) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_2787_diff__numeral__special_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),one2))) ) ).

% diff_numeral_special(4)
tff(fact_2788_diff__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,minus_minus(A,one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),N)) ) ).

% diff_numeral_special(3)
tff(fact_2789_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: num,N: nat,Y: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),N) = aa(int,A,ring_1_of_int(A),Y) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),N) = Y ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_2790_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,Xb: num,N: nat] :
          ( ( aa(int,A,ring_1_of_int(A),Y) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),N) )
        <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),N) ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_2791_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,ord_less_eq(int,archimedean_ceiling(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,ord_less_eq(A,Xb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ) ).

% ceiling_le_neg_numeral
tff(fact_2792_neg__numeral__less__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,ord_less(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),Xb) ) ) ).

% neg_numeral_less_ceiling
tff(fact_2793_ceiling__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,Xb)),zero_zero(int))
        <=> aa(A,$o,ord_less_eq(A,Xb),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% ceiling_less_zero
tff(fact_2794_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),one_one(A))),Xb) ) ) ).

% zero_le_ceiling
tff(fact_2795_power__minus1__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = one_one(A) ) ).

% power_minus1_even
tff(fact_2796_neg__one__odd__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% neg_one_odd_power
tff(fact_2797_neg__one__even__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = one_one(A) ) ) ) ).

% neg_one_even_power
tff(fact_2798_ceiling__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,ord_less(int,archimedean_ceiling(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,ord_less_eq(A,Xb),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))) ) ) ).

% ceiling_less_neg_numeral
tff(fact_2799_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archimedean_ceiling(A,Xb))
        <=> aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),Xb) ) ) ).

% neg_numeral_le_ceiling
tff(fact_2800_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,N: nat,A3: int] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),N)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),N)),A3) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_2801_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,Xb: num,N: nat] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),N))
        <=> aa(int,$o,ord_less_eq(int,A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),N)) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_2802_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,N: nat,A3: int] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),N)),aa(int,A,ring_1_of_int(A),A3))
        <=> aa(int,$o,ord_less(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),N)),A3) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_2803_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: int,Xb: num,N: nat] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),A3)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),N))
        <=> aa(int,$o,ord_less(int,A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),N)) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_2804_Heap__lub__empty,axiom,
    ! [A: $tType] : aa(set(heap_Time_Heap(A)),heap_Time_Heap(A),heap_Time_Heap_lub(A),bot_bot(set(heap_Time_Heap(A)))) = heap_Time_Heap2(A,aTP_Lamp_kq(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ).

% Heap_lub_empty
tff(fact_2805_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,uminus_uminus(A),B3))
        <=> aa(A,$o,ord_less_eq(A,B3),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% le_minus_iff
tff(fact_2806_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A3)),B3)
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),A3) ) ) ).

% minus_le_iff
tff(fact_2807_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% le_imp_neg_le
tff(fact_2808_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,uminus_uminus(A),B3))
        <=> aa(A,$o,ord_less(A,B3),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% less_minus_iff
tff(fact_2809_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),A3)),B3)
        <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),A3) ) ) ).

% minus_less_iff
tff(fact_2810_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% verit_negate_coefficient(2)
tff(fact_2811_neg__numeral__neq__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) != aa(num,A,numeral_numeral(A),N) ) ).

% neg_numeral_neq_numeral
tff(fact_2812_numeral__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,N: num] : aa(num,A,numeral_numeral(A),M) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% numeral_neq_neg_numeral
tff(fact_2813_is__num__normalize_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A3: A,B3: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B3)),aa(A,A,uminus_uminus(A),A3)) ) ).

% is_num_normalize(8)
tff(fact_2814_one__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ( one_one(A) != aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% one_neq_neg_one
tff(fact_2815_square__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),B3),B3) )
        <=> ( ( A3 = B3 )
            | ( A3 = aa(A,A,uminus_uminus(A),B3) ) ) ) ) ).

% square_eq_iff
tff(fact_2816_minus__mult__commute,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A3)),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,uminus_uminus(A),B3)) ) ).

% minus_mult_commute
tff(fact_2817_minus__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B3)) ) ).

% minus_divide_right
tff(fact_2818_minus__divide__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ).

% minus_divide_divide
tff(fact_2819_minus__divide__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),B3) ) ).

% minus_divide_left
tff(fact_2820_div__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),B3) ) ).

% div_minus_right
tff(fact_2821_mod__minus__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3))),B3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A3)),B3) ) ).

% mod_minus_eq
tff(fact_2822_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B3: A,A5: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),B3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A5),B3) )
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A3)),B3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A5)),B3) ) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
tff(fact_2823_mod__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A3)),B3)) ) ).

% mod_minus_right
tff(fact_2824_of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K)) ) ).

% of_int_neg_numeral
tff(fact_2825_curryD,axiom,
    ! [A: $tType,B: $tType,Fa: fun(product_prod(A,B),$o),A3: A,B3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),product_curry(A,B,$o,Fa),A3),B3)
     => aa(product_prod(A,B),$o,Fa,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3)) ) ).

% curryD
tff(fact_2826_curryE,axiom,
    ! [A: $tType,B: $tType,Fa: fun(product_prod(A,B),$o),A3: A,B3: B] :
      ( aa(B,$o,aa(A,fun(B,$o),product_curry(A,B,$o,Fa),A3),B3)
     => aa(product_prod(A,B),$o,Fa,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3)) ) ).

% curryE
tff(fact_2827_TBOUND__empty,axiom,
    ! [A: $tType,Tb: nat] : time_TBOUND(A,heap_Time_Heap2(A,aTP_Lamp_kq(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),Tb) ).

% TBOUND_empty
tff(fact_2828_refines__empty,axiom,
    ! [A: $tType,M: heap_Time_Heap(A)] : refine_Imp_refines(A,M,heap_Time_Heap2(A,aTP_Lamp_kq(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))))) ).

% refines_empty
tff(fact_2829_zero__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : zero_zero(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% zero_neq_neg_numeral
tff(fact_2830_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) ) ).

% neg_numeral_le_numeral
tff(fact_2831_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : ~ aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) ) ).

% not_numeral_le_neg_numeral
tff(fact_2832_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),N)) ) ).

% neg_numeral_less_numeral
tff(fact_2833_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,N: num] : ~ aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) ) ).

% not_numeral_less_neg_numeral
tff(fact_2834_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,uminus_uminus(A),A3) = B3 )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = zero_zero(A) ) ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_2835_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),B3) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = zero_zero(A) ) ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_2836_add_Oinverse__unique,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),A3) = B3 ) ) ) ).

% add.inverse_unique
tff(fact_2837_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),A3) = zero_zero(A) ) ).

% ab_group_add_class.ab_left_minus
tff(fact_2838_add__eq__0__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = zero_zero(A) )
        <=> ( B3 = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% add_eq_0_iff
tff(fact_2839_zero__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ( zero_zero(A) != aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% zero_neq_neg_one
tff(fact_2840_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(4)
tff(fact_2841_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% le_minus_one_simps(2)
tff(fact_2842_less__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,ord_less(A,one_one(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% less_minus_one_simps(4)
tff(fact_2843_less__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),one_one(A))),one_one(A)) ) ).

% less_minus_one_simps(2)
tff(fact_2844_numeral__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),N) != aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% numeral_neq_neg_one
tff(fact_2845_one__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [N: num] : one_one(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)) ) ).

% one_neq_neg_numeral
tff(fact_2846_numeral__times__minus__swap,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [W: num,Xb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,uminus_uminus(A),Xb)) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ).

% numeral_times_minus_swap
tff(fact_2847_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_2848_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_2849_square__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [Xb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Xb) = one_one(A) )
        <=> ( ( Xb = one_one(A) )
            | ( Xb = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% square_eq_1_iff
tff(fact_2850_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,uminus_uminus(A),B3)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ) ) ).

% dvd_div_neg
tff(fact_2851_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,dvd_dvd(A,B3),A3)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A3)),B3) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) ) ) ) ).

% dvd_neg_div
tff(fact_2852_Heap_Osize_I2_J,axiom,
    ! [A: $tType,Xb: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : aa(heap_Time_Heap(A),nat,size_size(heap_Time_Heap(A)),heap_Time_Heap2(A,Xb)) = aa(nat,nat,suc,zero_zero(nat)) ).

% Heap.size(2)
tff(fact_2853_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A)) ) ).

% neg_numeral_le_zero
tff(fact_2854_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) ) ).

% not_zero_le_neg_numeral
tff(fact_2855_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))),zero_zero(A)) ) ).

% neg_numeral_less_zero
tff(fact_2856_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : ~ aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) ) ).

% not_zero_less_neg_numeral
tff(fact_2857_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% le_minus_one_simps(3)
tff(fact_2858_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% le_minus_one_simps(1)
tff(fact_2859_less__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% less_minus_one_simps(3)
tff(fact_2860_less__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),one_one(A))),zero_zero(A)) ) ).

% less_minus_one_simps(1)
tff(fact_2861_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) ) ).

% neg_numeral_le_one
tff(fact_2862_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).

% neg_one_le_numeral
tff(fact_2863_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% neg_numeral_le_neg_one
tff(fact_2864_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_le_neg_one
tff(fact_2865_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_one_le_neg_numeral
tff(fact_2866_neg__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),one_one(A)) ) ).

% neg_numeral_less_one
tff(fact_2867_neg__one__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),M)) ) ).

% neg_one_less_numeral
tff(fact_2868_not__numeral__less__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% not_numeral_less_neg_one
tff(fact_2869_not__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,ord_less(A,one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_one_less_neg_numeral
tff(fact_2870_not__neg__one__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% not_neg_one_less_neg_numeral
tff(fact_2871_uminus__numeral__One,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% uminus_numeral_One
tff(fact_2872_mult__1s__ring__1_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B3),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) = aa(A,A,uminus_uminus(A),B3) ) ).

% mult_1s_ring_1(2)
tff(fact_2873_mult__1s__ring__1_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))),B3) = aa(A,A,uminus_uminus(A),B3) ) ).

% mult_1s_ring_1(1)
tff(fact_2874_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( B3 != zero_zero(A) )
            & ( A3 = aa(A,A,uminus_uminus(A),B3) ) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_2875_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2) = aa(A,A,uminus_uminus(A),B3),A3 = zero_zero(A)) ) ) ).

% eq_minus_divide_eq
tff(fact_2876_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C2: A,A3: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)) = A3 )
        <=> $ite(C2 != zero_zero(A),aa(A,A,uminus_uminus(A),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2),A3 = zero_zero(A)) ) ) ).

% minus_divide_eq_eq
tff(fact_2877_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,A3: A,C2: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) = C2 )
          <=> ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_2878_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C2: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( ( C2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_2879_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_minus
tff(fact_2880_power__minus__Bit0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xb)),aa(num,nat,numeral_numeral(nat),bit0(K))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(K))) ) ).

% power_minus_Bit0
tff(fact_2881_power__minus__Bit1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: A,K: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K))) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,K)))) ) ).

% power_minus_Bit1
tff(fact_2882_real__add__less__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)),zero_zero(real))
    <=> aa(real,$o,ord_less(real,Y),aa(real,real,uminus_uminus(real),Xb)) ) ).

% real_add_less_0_iff
tff(fact_2883_real__0__less__add__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y))
    <=> aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),Xb)),Y) ) ).

% real_0_less_add_iff
tff(fact_2884_real__0__le__add__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y))
    <=> aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),Xb)),Y) ) ).

% real_0_le_add_iff
tff(fact_2885_real__add__le__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,Y),aa(real,real,uminus_uminus(real),Xb)) ) ).

% real_add_le_0_iff
tff(fact_2886_tanh__real__gt__neg1,axiom,
    ! [Xb: real] : aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(real,real,tanh(real),Xb)) ).

% tanh_real_gt_neg1
tff(fact_2887_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,ord_less(A,A3),zero_zero(A))) ) ) ) ).

% less_minus_divide_eq
tff(fact_2888_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))),A3)
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)),aa(A,$o,ord_less(A,zero_zero(A)),A3)) ) ) ) ).

% minus_divide_less_eq
tff(fact_2889_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)))
          <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_2890_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))),A3)
          <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_2891_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)))
          <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_2892_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))),A3)
          <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_2893_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B3: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) )
        <=> $ite(C2 != zero_zero(A),B3 = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral(2)
tff(fact_2894_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B3: A,C2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2) = B3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_2895_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z: A,B3: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z))),B3) = $ite(Z = zero_zero(A),B3,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),Z))),Z)) ) ).

% add_divide_eq_if_simps(3)
tff(fact_2896_minus__divide__add__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_add_eq_iff
tff(fact_2897_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z: A,B3: A] :
          aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z))),B3) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),B3),Z))),Z)) ) ).

% add_divide_eq_if_simps(6)
tff(fact_2898_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,Z: A,B3: A] :
          aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),Z)),B3) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B3),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,A3),aa(A,A,aa(A,fun(A,A),times_times(A),B3),Z))),Z)) ) ).

% add_divide_eq_if_simps(5)
tff(fact_2899_minus__divide__diff__eq__iff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( ( Z != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Z))),Y) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Z))),Z) ) ) ) ).

% minus_divide_diff_eq_iff
tff(fact_2900_even__minus,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,uminus_uminus(A),A3))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ).

% even_minus
tff(fact_2901_power2__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) )
        <=> ( ( Xb = Y )
            | ( Xb = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% power2_eq_iff
tff(fact_2902_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( ! [X: A,Xa3: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa3),R)
        <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X),Xa3),S3) )
    <=> ( R = S3 ) ) ).

% pred_equals_eq2
tff(fact_2903_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,$o,ord_less_eq(A,A3),zero_zero(A))) ) ) ) ).

% le_minus_divide_eq
tff(fact_2904_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))),A3)
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)),aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)) ) ) ) ).

% minus_divide_le_eq
tff(fact_2905_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)))
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_2906_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))),A3)
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_2907_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)))
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_2908_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))),A3)
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_2909_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,W: num] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B3),aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ).

% divide_less_eq_numeral(2)
tff(fact_2910_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B3),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(2)
tff(fact_2911_power2__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A3: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
        <=> ( ( A3 = one_one(A) )
            | ( A3 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% power2_eq_1_iff
tff(fact_2912_uminus__power__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,N: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N))) ) ).

% uminus_power_if
tff(fact_2913_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,K),N)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,minus_minus(nat,N),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_2914_realpow__square__minus__le,axiom,
    ! [U: real,Xb: real] : aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),U),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% realpow_square_minus_le
tff(fact_2915_ln__add__one__self__le__self2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb))),Xb) ) ).

% ln_add_one_self_le_self2
tff(fact_2916_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,C2: A,W: num] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B3),aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ) ).

% divide_le_eq_numeral(2)
tff(fact_2917_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B3),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less_eq(A,B3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(2)
tff(fact_2918_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),one_one(A))),Xb)
         => ( aa(A,$o,ord_less_eq(A,Xb),one_one(A))
           => aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ) ) ).

% square_le_1
tff(fact_2919_minus__power__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A3: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).

% minus_power_mult_self
tff(fact_2920_minus__one__power__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% minus_one_power_iff
tff(fact_2921_ln__one__minus__pos__upper__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),aa(real,real,minus_minus(real,one_one(real)),Xb))),aa(real,real,uminus_uminus(real),Xb)) ) ) ).

% ln_one_minus_pos_upper_bound
tff(fact_2922_vebt__memberi_Omono,axiom,
    ! [Xb: product_prod(vEBT_VEBTi,nat)] : comple7038119648293358887notone(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),heap_Time_Heap($o),partial_fun_ord(heap_Time_Heap($o),heap_Time_Heap($o),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_ord($o)),heap_Time_Heap_ord($o),aTP_Lamp_ky(product_prod(vEBT_VEBTi,nat),fun(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),heap_Time_Heap($o)),Xb)) ).

% vebt_memberi.mono
tff(fact_2923_power__minus1__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% power_minus1_odd
tff(fact_2924_subrelI,axiom,
    ! [B: $tType,A: $tType,R2: set(product_prod(A,B)),S: set(product_prod(A,B))] :
      ( ! [X2: A,Y2: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2),R2)
         => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y2),S) )
     => aa(set(product_prod(A,B)),$o,ord_less_eq(set(product_prod(A,B)),R2),S) ) ).

% subrelI
tff(fact_2925_VEBT__internal_Ovebt__memberi_H_Omono,axiom,
    ! [Xb: product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat)] : comple7038119648293358887notone(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),heap_Time_Heap($o),partial_fun_ord(heap_Time_Heap($o),heap_Time_Heap($o),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_ord($o)),heap_Time_Heap_ord($o),aTP_Lamp_lo(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),fun(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),heap_Time_Heap($o)),Xb)) ).

% VEBT_internal.vebt_memberi'.mono
tff(fact_2926_of__int__code__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          aa(int,A,ring_1_of_int(A),K) = $ite(
            K = zero_zero(int),
            zero_zero(A),
            $ite(
              aa(int,$o,ord_less(int,K),zero_zero(int)),
              aa(A,A,uminus_uminus(A),aa(int,A,ring_1_of_int(A),aa(int,int,uminus_uminus(int),K))),
              $let(
                l: A,
                l:= aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))))),
                $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int),l,aa(A,A,aa(A,fun(A,A),plus_plus(A),l),one_one(A))) ) ) ) ) ).

% of_int_code_if
tff(fact_2927_vebt__predi_Ofixp__induct,axiom,
    ! [P: fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o)] :
      ( comple1908693960933563346ssible(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),partial_fun_lub(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_lub(option(nat))),partial_fun_ord(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_ord(option(nat))),aTP_Lamp_lp(fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o),fun(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),$o),P))
     => ( aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o,P,aTP_Lamp_lq(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))
       => ( ! [F4: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))] :
              ( aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o,P,F4)
             => aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o,P,aTP_Lamp_lr(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),F4)) )
         => aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o,P,vEBT_vebt_predi) ) ) ) ).

% vebt_predi.fixp_induct
tff(fact_2928_vebt__succi_Ofixp__induct,axiom,
    ! [P: fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o)] :
      ( comple1908693960933563346ssible(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),partial_fun_lub(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_lub(option(nat))),partial_fun_ord(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_ord(option(nat))),aTP_Lamp_lp(fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o),fun(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),$o),P))
     => ( aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o,P,aTP_Lamp_lq(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))
       => ( ! [F4: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))] :
              ( aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o,P,F4)
             => aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o,P,aTP_Lamp_ls(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),F4)) )
         => aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o,P,vEBT_vebt_succi) ) ) ) ).

% vebt_succi.fixp_induct
tff(fact_2929_compl__less__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,ord_less(A,Y),Xb) ) ) ).

% compl_less_compl_iff
tff(fact_2930_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,ord_less_eq(A,Y),Xb) ) ) ).

% compl_le_compl_iff
tff(fact_2931_ln__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(num,real,numeral_numeral(real),bit0(one2)))
       => ( aa(real,real,ln_ln(real),Xb) = suminf(real,aTP_Lamp_lt(real,fun(nat,real),Xb)) ) ) ) ).

% ln_series
tff(fact_2932_negative__eq__positive,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N)) = aa(nat,int,semiring_1_of_nat(int),M) )
    <=> ( ( N = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% negative_eq_positive
tff(fact_2933_negative__zle,axiom,
    ! [N: nat,M: nat] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))),aa(nat,int,semiring_1_of_nat(int),M)) ).

% negative_zle
tff(fact_2934_negative__zless,axiom,
    ! [N: nat,M: nat] : aa(int,$o,ord_less(int,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),aa(nat,int,semiring_1_of_nat(int),M)) ).

% negative_zless
tff(fact_2935_word__of__int__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin))) = aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin)) ) ).

% word_of_int_neg_numeral
tff(fact_2936_word__of__int__neg__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),one_one(int))) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ).

% word_of_int_neg_1
tff(fact_2937_int__div__minus__is__minus1,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less(int,A3),zero_zero(int))
     => ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3) = aa(int,int,uminus_uminus(int),A3) )
      <=> ( B3 = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_2938_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Fa: fun(nat,A)] : suminf(A,aTP_Lamp_lu(fun(nat,A),fun(nat,A),Fa)) = aa(nat,A,Fa,zero_zero(nat)) ) ).

% powser_zero
tff(fact_2939_ceiling__divide__eq__div__numeral,axiom,
    ! [A3: num,B3: num] : archimedean_ceiling(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A3)),aa(num,real,numeral_numeral(real),B3))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(num,int,numeral_numeral(int),B3))) ).

% ceiling_divide_eq_div_numeral
tff(fact_2940_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A3: num,B3: num] : archimedean_ceiling(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A3)),aa(num,real,numeral_numeral(real),B3)))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),A3)),aa(num,int,numeral_numeral(int),B3))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_2941_uminus__int__code_I1_J,axiom,
    aa(int,int,uminus_uminus(int),zero_zero(int)) = zero_zero(int) ).

% uminus_int_code(1)
tff(fact_2942_uminus__integer__code_I1_J,axiom,
    aa(code_integer,code_integer,uminus_uminus(code_integer),zero_zero(code_integer)) = zero_zero(code_integer) ).

% uminus_integer_code(1)
tff(fact_2943_word__neg__numeral__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_neg_numeral_alt
tff(fact_2944_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
     => ( ( M = one_one(int) )
        | ( M = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
tff(fact_2945_zmult__eq__1__iff,axiom,
    ! [M: int,N: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),N) = one_one(int) )
    <=> ( ( ( M = one_one(int) )
          & ( N = one_one(int) ) )
        | ( ( M = aa(int,int,uminus_uminus(int),one_one(int)) )
          & ( N = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_2946_minus__int__code_I2_J,axiom,
    ! [L: int] : aa(int,int,minus_minus(int,zero_zero(int)),L) = aa(int,int,uminus_uminus(int),L) ).

% minus_int_code(2)
tff(fact_2947_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),K)),L) != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) != zero_zero(int) ) ) ).

% zmod_zminus1_not_zero
tff(fact_2948_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(int,int,uminus_uminus(int),L)) != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) != zero_zero(int) ) ) ).

% zmod_zminus2_not_zero
tff(fact_2949_not__int__zless__negative,axiom,
    ! [N: nat,M: nat] : ~ aa(int,$o,ord_less(int,aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M))) ).

% not_int_zless_negative
tff(fact_2950_max__word__not__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) != zero_zero(word(A)) ) ) ).

% max_word_not_0
tff(fact_2951_word__not__simps_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : ~ aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Y) ) ).

% word_not_simps(3)
tff(fact_2952_word__order_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : ~ aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),A3) ) ).

% word_order.extremum_strict
tff(fact_2953_word__order_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( ( A3 != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
        <=> aa(word(A),$o,ord_less(word(A),A3),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ) ).

% word_order.not_eq_extremum
tff(fact_2954_max__word__not__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : ~ aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Xb) ) ).

% max_word_not_less
tff(fact_2955_minus__integer__code_I2_J,axiom,
    ! [L: code_integer] : aa(code_integer,code_integer,minus_minus(code_integer,zero_zero(code_integer)),L) = aa(code_integer,code_integer,uminus_uminus(code_integer),L) ).

% minus_integer_code(2)
tff(fact_2956_int__cases4,axiom,
    ! [M: int] :
      ( ! [N3: nat] : M != aa(nat,int,semiring_1_of_nat(int),N3)
     => ~ ! [N3: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N3)
           => ( M != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N3)) ) ) ) ).

% int_cases4
tff(fact_2957_int__zle__neg,axiom,
    ! [N: nat,M: nat] :
      ( aa(int,$o,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M)))
    <=> ( ( N = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_2958_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less_eq(int,K),zero_zero(int))
     => ~ ! [N3: nat] : K != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N3)) ) ).

% nonpos_int_cases
tff(fact_2959_negative__zle__0,axiom,
    ! [N: nat] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))),zero_zero(int)) ).

% negative_zle_0
tff(fact_2960_max__word__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),one_one(word(A))) = zero_zero(word(A)) )
         => ( Xb = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% max_word_wrap
tff(fact_2961_word__add__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))
         => aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),one_one(word(A)))) ) ) ).

% word_add_no_overflow
tff(fact_2962_less__x__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( ( Xb != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),one_one(word(A))))
          <=> ( aa(word(A),$o,ord_less(word(A),Y),Xb)
              | ( Y = Xb ) ) ) ) ) ).

% less_x_plus_1
tff(fact_2963_zmod__zminus1__eq__if,axiom,
    ! [A3: int,B3: int] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),A3)),B3) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3) = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,B3),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3))) ).

% zmod_zminus1_eq_if
tff(fact_2964_zmod__zminus2__eq__if,axiom,
    ! [A3: int,B3: int] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),aa(int,int,uminus_uminus(int),B3)) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3) = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3)),B3)) ).

% zmod_zminus2_eq_if
tff(fact_2965_no__plus__overflow__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),uminus_uminus(word(A)),Y))
         => aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y)) ) ) ).

% no_plus_overflow_neg
tff(fact_2966_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N3: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N3) )
           => ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),N3) )
       => ~ ! [N3: nat] :
              ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N3)) )
             => ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),N3) ) ) ) ).

% int_cases3
tff(fact_2967_not__zle__0__negative,axiom,
    ! [N: nat] : ~ aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))) ).

% not_zle_0_negative
tff(fact_2968_negative__zless__0,axiom,
    ! [N: nat] : aa(int,$o,ord_less(int,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N)))),zero_zero(int)) ).

% negative_zless_0
tff(fact_2969_negD,axiom,
    ! [Xb: int] :
      ( aa(int,$o,ord_less(int,Xb),zero_zero(int))
     => ? [N3: nat] : Xb = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N3))) ) ).

% negD
tff(fact_2970_verit__less__mono__div__int2,axiom,
    ! [A4: int,B6: int,N: int] :
      ( aa(int,$o,ord_less_eq(int,A4),B6)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),aa(int,int,uminus_uminus(int),N))
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),B6),N)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A4),N)) ) ) ).

% verit_less_mono_div_int2
tff(fact_2971_div__eq__minus1,axiom,
    ! [B3: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),B3) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_2972_word__Suc__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),K: word(A)] :
          ( ( Xb != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),one_one(word(A)))),K)
          <=> aa(word(A),$o,ord_less(word(A),Xb),K) ) ) ) ).

% word_Suc_le
tff(fact_2973_word__Suc__leq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),Xb: word(A)] :
          ( ( K != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),one_one(word(A))))
          <=> aa(word(A),$o,ord_less_eq(word(A),Xb),K) ) ) ) ).

% word_Suc_leq
tff(fact_2974_word__le__make__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A)] :
          ( ( Y != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),Y)
          <=> aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),one_one(word(A)))) ) ) ) ).

% word_le_make_less
tff(fact_2975_ceiling__divide__eq__div,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: int,B3: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A3)),aa(int,A,ring_1_of_int(A),B3))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A3)),B3)) ) ).

% ceiling_divide_eq_div
tff(fact_2976_neg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less(int,K),zero_zero(int))
     => ~ ! [N3: nat] :
            ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N3)) )
           => ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),N3) ) ) ).

% neg_int_cases
tff(fact_2977_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),L)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),K)),L) = aa(int,int,minus_minus(int,aa(int,int,minus_minus(int,L),one_one(int))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,K),one_one(int))),L)) ) ) ).

% minus_mod_int_eq
tff(fact_2978_zmod__minus1,axiom,
    ! [B3: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),one_one(int))),B3) = aa(int,int,minus_minus(int,B3),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_2979_zdiv__zminus2__eq__if,axiom,
    ! [B3: int,A3: int] :
      ( ( B3 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),aa(int,int,uminus_uminus(int),B3)) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3))),one_one(int))) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_2980_zdiv__zminus1__eq__if,axiom,
    ! [B3: int,A3: int] :
      ( ( B3 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A3)),B3) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3))),one_one(int))) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_2981_zminus1__lemma,axiom,
    ! [A3: int,B3: int,Q2: int,R2: int] :
      ( eucl_rel_int(A3,B3,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2))
     => ( ( B3 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A3),B3,
            aa(int,product_prod(int,int),
              aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),
                $ite(R2 = zero_zero(int),aa(int,int,uminus_uminus(int),Q2),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),Q2)),one_one(int)))),
              $ite(R2 = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,B3),R2)))) ) ) ).

% zminus1_lemma
tff(fact_2982_minus__1__div__exp__eq__int,axiom,
    ! [N: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% minus_1_div_exp_eq_int
tff(fact_2983_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)),zero_zero(int))
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% div_pos_neg_trivial
tff(fact_2984_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),Y)
         => aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),Xb)) ) ) ).

% compl_mono
tff(fact_2985_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,ord_less_eq(A,Y),aa(A,A,uminus_uminus(A),Xb))
         => aa(A,$o,ord_less_eq(A,Xb),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_le_swap1
tff(fact_2986_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),Y)),Xb)
         => aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),Xb)),Y) ) ) ).

% compl_le_swap2
tff(fact_2987_compl__less__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),Y)),Xb)
         => aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),Xb)),Y) ) ) ).

% compl_less_swap2
tff(fact_2988_compl__less__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,ord_less(A,Y),aa(A,A,uminus_uminus(A),Xb))
         => aa(A,$o,ord_less(A,Xb),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_less_swap1
tff(fact_2989_int__bit__induct,axiom,
    ! [P: fun(int,$o),K: int] :
      ( aa(int,$o,P,zero_zero(int))
     => ( aa(int,$o,P,aa(int,int,uminus_uminus(int),one_one(int)))
       => ( ! [K2: int] :
              ( aa(int,$o,P,K2)
             => ( ( K2 != zero_zero(int) )
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),bit0(one2)))) ) )
         => ( ! [K2: int] :
                ( aa(int,$o,P,K2)
               => ( ( K2 != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K2),aa(num,int,numeral_numeral(int),bit0(one2))))) ) )
           => aa(int,$o,P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_2990_m1mod2k,axiom,
    ! [N: nat] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) = aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),one_one(int)) ).

% m1mod2k
tff(fact_2991_sb__dec__lem_H,axiom,
    ! [K: nat,A3: int] :
      ( aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)),A3)
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),A3)) ) ).

% sb_dec_lem'
tff(fact_2992_m1mod22k,axiom,
    ! [N: nat] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),one_one(int)) ).

% m1mod22k
tff(fact_2993_sb__inc__lem_H,axiom,
    ! [A3: int,K: nat] :
      ( aa(int,$o,ord_less(int,A3),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)))
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,K)))) ) ).

% sb_inc_lem'
tff(fact_2994_sb__dec__lem,axiom,
    ! [K: nat,A3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),A3))
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),A3)) ) ).

% sb_dec_lem
tff(fact_2995_diff__shunt__var,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,minus_minus(A,Xb),Y) = bot_bot(A) )
        <=> aa(A,$o,ord_less_eq(A,Xb),Y) ) ) ).

% diff_shunt_var
tff(fact_2996_suminf__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => ( suminf(A,aa(A,fun(nat,A),power_power(A),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,minus_minus(A,one_one(A)),C2)) ) ) ) ).

% suminf_geometric
tff(fact_2997_minus__one__mod__numeral,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),N)) = adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,N))) ).

% minus_one_mod_numeral
tff(fact_2998_one__mod__minus__numeral,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,N)))) ).

% one_mod_minus_numeral
tff(fact_2999_VEBT__internal_Ovebt__memberi_H_Ofixp__induct,axiom,
    ! [P: fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),$o)] :
      ( comple1908693960933563346ssible(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),partial_fun_lub(heap_Time_Heap($o),heap_Time_Heap($o),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_lub($o)),partial_fun_ord(heap_Time_Heap($o),heap_Time_Heap($o),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_ord($o)),aTP_Lamp_lv(fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),$o),fun(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),$o),P))
     => ( aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),$o,P,aTP_Lamp_lx(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))))
       => ( ! [F4: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))))] :
              ( aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),$o,P,F4)
             => aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),$o,P,aTP_Lamp_ly(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),F4)) )
         => aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),$o,P,vEBT_V854960066525838166emberi) ) ) ) ).

% VEBT_internal.vebt_memberi'.fixp_induct
tff(fact_3000_one__div__minus__numeral,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).

% one_div_minus_numeral
tff(fact_3001_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_lz(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_3002_minus__numeral__div__numeral,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,N))) ).

% minus_numeral_div_numeral
tff(fact_3003_numeral__div__minus__numeral,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,N))) ).

% numeral_div_minus_numeral
tff(fact_3004_numeral__mod__minus__numeral,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M,N)))) ).

% numeral_mod_minus_numeral
tff(fact_3005_minus__numeral__mod__numeral,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = adjust_mod(aa(num,int,numeral_numeral(int),N),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M,N))) ).

% minus_numeral_mod_numeral
tff(fact_3006_minus__one__div__numeral,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,N))) ).

% minus_one_div_numeral
tff(fact_3007_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_3008_bot__option__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( bot_bot(option(A)) = none(A) ) ) ).

% bot_option_def
tff(fact_3009_bot__enat__def,axiom,
    bot_bot(extended_enat) = zero_zero(extended_enat) ).

% bot_enat_def
tff(fact_3010_bot__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X3: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),bot_bot(fun(A,fun(B,$o))),X3),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa),bot_bot(set(product_prod(A,B)))) ) ).

% bot_empty_eq2
tff(fact_3011_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R2: int] :
      adjust_mod(L,R2) = $ite(R2 = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,L),R2)) ).

% Divides.adjust_mod_def
tff(fact_3012_vebt__memberi_Ofixp__induct,axiom,
    ! [P: fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),$o)] :
      ( comple1908693960933563346ssible(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),partial_fun_lub(heap_Time_Heap($o),heap_Time_Heap($o),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_lub($o)),partial_fun_ord(heap_Time_Heap($o),heap_Time_Heap($o),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_ord($o)),aTP_Lamp_ma(fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),$o),fun(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),$o),P))
     => ( aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),$o,P,aTP_Lamp_mb(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))))
       => ( ! [F4: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))] :
              ( aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),$o,P,F4)
             => aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),$o,P,aTP_Lamp_mc(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),F4)) )
         => aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),$o,P,vEBT_vebt_memberi) ) ) ) ).

% vebt_memberi.fixp_induct
tff(fact_3013_pi__series,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = suminf(real,aTP_Lamp_md(nat,real)) ).

% pi_series
tff(fact_3014_heap_Ofixp__induct__uc,axiom,
    ! [A: $tType,B: $tType,C: $tType,U2: fun(C,fun(A,heap_Time_Heap(B))),F5: fun(C,C),C5: fun(fun(A,heap_Time_Heap(B)),C),Fa: C,P: fun(fun(A,heap_Time_Heap(B)),$o)] :
      ( ! [X2: A] : comple7038119648293358887notone(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),heap_Time_Heap_ord(B),aa(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B)),aa(fun(fun(A,heap_Time_Heap(B)),C),fun(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B))),aa(fun(C,C),fun(fun(fun(A,heap_Time_Heap(B)),C),fun(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B)))),aTP_Lamp_me(fun(C,fun(A,heap_Time_Heap(B))),fun(fun(C,C),fun(fun(fun(A,heap_Time_Heap(B)),C),fun(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B))))),U2),F5),C5),X2))
     => ( ( Fa = aa(fun(A,heap_Time_Heap(B)),C,C5,comple187402453842119260l_fixp(fun(A,heap_Time_Heap(B)),partial_fun_lub(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_lub(B)),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),aa(fun(fun(A,heap_Time_Heap(B)),C),fun(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))),aa(fun(C,C),fun(fun(fun(A,heap_Time_Heap(B)),C),fun(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B)))),aTP_Lamp_mf(fun(C,fun(A,heap_Time_Heap(B))),fun(fun(C,C),fun(fun(fun(A,heap_Time_Heap(B)),C),fun(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))))),U2),F5),C5))) )
       => ( ! [F4: fun(A,heap_Time_Heap(B))] : aa(C,fun(A,heap_Time_Heap(B)),U2,aa(fun(A,heap_Time_Heap(B)),C,C5,F4)) = F4
         => ( comple1908693960933563346ssible(fun(A,heap_Time_Heap(B)),partial_fun_lub(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_lub(B)),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),P)
           => ( aa(fun(A,heap_Time_Heap(B)),$o,P,aTP_Lamp_mh(A,heap_Time_Heap(B)))
             => ( ! [F4: C] :
                    ( aa(fun(A,heap_Time_Heap(B)),$o,P,aa(C,fun(A,heap_Time_Heap(B)),U2,F4))
                   => aa(fun(A,heap_Time_Heap(B)),$o,P,aa(C,fun(A,heap_Time_Heap(B)),U2,aa(C,C,F5,F4))) )
               => aa(fun(A,heap_Time_Heap(B)),$o,P,aa(C,fun(A,heap_Time_Heap(B)),U2,Fa)) ) ) ) ) ) ) ).

% heap.fixp_induct_uc
tff(fact_3015_integer__of__int__code,axiom,
    ! [K: int] :
      code_integer_of_int(K) = $ite(
        aa(int,$o,ord_less(int,K),zero_zero(int)),
        aa(code_integer,code_integer,uminus_uminus(code_integer),code_integer_of_int(aa(int,int,uminus_uminus(int),K))),
        $ite(
          K = zero_zero(int),
          zero_zero(code_integer),
          $let(
            l: code_integer,
            l:= aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),code_integer_of_int(aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2))))),
            $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int),l,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),l),one_one(code_integer))) ) ) ) ).

% integer_of_int_code
tff(fact_3016_dbl__dec__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))) ) ) ).

% dbl_dec_simps(4)
tff(fact_3017_dbl__dec__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_dec(A,one_one(A)) = one_one(A) ) ) ).

% dbl_dec_simps(3)
tff(fact_3018_dbl__dec__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_dec(A,zero_zero(A)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% dbl_dec_simps(2)
tff(fact_3019_pi__neq__zero,axiom,
    pi != zero_zero(real) ).

% pi_neq_zero
tff(fact_3020_zero__integer__def,axiom,
    zero_zero(code_integer) = code_integer_of_int(zero_zero(int)) ).

% zero_integer_def
tff(fact_3021_less__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      ( aa(code_integer,$o,ord_less(code_integer,code_integer_of_int(Xaa)),code_integer_of_int(Xb))
    <=> aa(int,$o,ord_less(int,Xaa),Xb) ) ).

% less_integer.abs_eq
tff(fact_3022_one__integer__def,axiom,
    one_one(code_integer) = code_integer_of_int(one_one(int)) ).

% one_integer_def
tff(fact_3023_pi__gt__zero,axiom,
    aa(real,$o,ord_less(real,zero_zero(real)),pi) ).

% pi_gt_zero
tff(fact_3024_pi__not__less__zero,axiom,
    ~ aa(real,$o,ord_less(real,pi),zero_zero(real)) ).

% pi_not_less_zero
tff(fact_3025_pi__ge__zero,axiom,
    aa(real,$o,ord_less_eq(real,zero_zero(real)),pi) ).

% pi_ge_zero
tff(fact_3026_less__eq__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      ( aa(code_integer,$o,ord_less_eq(code_integer,code_integer_of_int(Xaa)),code_integer_of_int(Xb))
    <=> aa(int,$o,ord_less_eq(int,Xaa),Xb) ) ).

% less_eq_integer.abs_eq
tff(fact_3027_divide__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),code_integer_of_int(Xaa)),code_integer_of_int(Xb)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),divide_divide(int),Xaa),Xb)) ).

% divide_integer.abs_eq
tff(fact_3028_pi__less__4,axiom,
    aa(real,$o,ord_less(real,pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% pi_less_4
tff(fact_3029_pi__ge__two,axiom,
    aa(real,$o,ord_less_eq(real,aa(num,real,numeral_numeral(real),bit0(one2))),pi) ).

% pi_ge_two
tff(fact_3030_pi__half__neq__two,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) != aa(num,real,numeral_numeral(real),bit0(one2)) ).

% pi_half_neq_two
tff(fact_3031_pi__half__neq__zero,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) != zero_zero(real) ).

% pi_half_neq_zero
tff(fact_3032_pi__half__less__two,axiom,
    aa(real,$o,ord_less(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% pi_half_less_two
tff(fact_3033_pi__half__le__two,axiom,
    aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% pi_half_le_two
tff(fact_3034_dbl__dec__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xb: A] : neg_numeral_dbl_dec(A,Xb) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb)),one_one(A)) ) ).

% dbl_dec_def
tff(fact_3035_pi__half__gt__zero,axiom,
    aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_gt_zero
tff(fact_3036_pi__half__ge__zero,axiom,
    aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% pi_half_ge_zero
tff(fact_3037_m2pi__less__pi,axiom,
    aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))),pi) ).

% m2pi_less_pi
tff(fact_3038_minus__pi__half__less__zero,axiom,
    aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),zero_zero(real)) ).

% minus_pi_half_less_zero
tff(fact_3039_heap_Ofixp__rule__uc,axiom,
    ! [A: $tType,B: $tType,C: $tType,U2: fun(C,fun(A,heap_Time_Heap(B))),F5: fun(C,C),C5: fun(fun(A,heap_Time_Heap(B)),C),Fa: C] :
      ( ! [X2: A] : comple7038119648293358887notone(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),heap_Time_Heap_ord(B),aa(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B)),aa(fun(fun(A,heap_Time_Heap(B)),C),fun(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B))),aa(fun(C,C),fun(fun(fun(A,heap_Time_Heap(B)),C),fun(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B)))),aTP_Lamp_me(fun(C,fun(A,heap_Time_Heap(B))),fun(fun(C,C),fun(fun(fun(A,heap_Time_Heap(B)),C),fun(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B))))),U2),F5),C5),X2))
     => ( ( Fa = aa(fun(A,heap_Time_Heap(B)),C,C5,comple187402453842119260l_fixp(fun(A,heap_Time_Heap(B)),partial_fun_lub(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_lub(B)),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),aa(fun(fun(A,heap_Time_Heap(B)),C),fun(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))),aa(fun(C,C),fun(fun(fun(A,heap_Time_Heap(B)),C),fun(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B)))),aTP_Lamp_mf(fun(C,fun(A,heap_Time_Heap(B))),fun(fun(C,C),fun(fun(fun(A,heap_Time_Heap(B)),C),fun(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))))),U2),F5),C5))) )
       => ( ! [F4: C] : aa(fun(A,heap_Time_Heap(B)),C,C5,aa(C,fun(A,heap_Time_Heap(B)),U2,F4)) = F4
         => ( Fa = aa(C,C,F5,Fa) ) ) ) ) ).

% heap.fixp_rule_uc
tff(fact_3040_heap_Omono__body__fixp,axiom,
    ! [B: $tType,A: $tType,F5: fun(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B)))] :
      ( ! [X2: A] : comple7038119648293358887notone(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),heap_Time_Heap_ord(B),aa(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B)),aTP_Lamp_mi(fun(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))),fun(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B))),F5),X2))
     => ( comple187402453842119260l_fixp(fun(A,heap_Time_Heap(B)),partial_fun_lub(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_lub(B)),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),F5) = aa(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B)),F5,comple187402453842119260l_fixp(fun(A,heap_Time_Heap(B)),partial_fun_lub(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_lub(B)),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),F5)) ) ) ).

% heap.mono_body_fixp
tff(fact_3041_fixp__induct__heap,axiom,
    ! [C: $tType,B: $tType,A: $tType,U2: fun(C,fun(A,heap_Time_Heap(B))),F5: fun(C,C),C5: fun(fun(A,heap_Time_Heap(B)),C),Fa: C,P: fun(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,$o))))),Xb: A,H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: B,N: nat] :
      ( ! [X2: A] : comple7038119648293358887notone(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),heap_Time_Heap_ord(B),aa(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B)),aa(fun(fun(A,heap_Time_Heap(B)),C),fun(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B))),aa(fun(C,C),fun(fun(fun(A,heap_Time_Heap(B)),C),fun(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B)))),aTP_Lamp_me(fun(C,fun(A,heap_Time_Heap(B))),fun(fun(C,C),fun(fun(fun(A,heap_Time_Heap(B)),C),fun(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B))))),U2),F5),C5),X2))
     => ( ( Fa = aa(fun(A,heap_Time_Heap(B)),C,C5,comple187402453842119260l_fixp(fun(A,heap_Time_Heap(B)),partial_fun_lub(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_lub(B)),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),aa(fun(fun(A,heap_Time_Heap(B)),C),fun(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))),aa(fun(C,C),fun(fun(fun(A,heap_Time_Heap(B)),C),fun(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B)))),aTP_Lamp_mf(fun(C,fun(A,heap_Time_Heap(B))),fun(fun(C,C),fun(fun(fun(A,heap_Time_Heap(B)),C),fun(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))))),U2),F5),C5))) )
       => ( ! [F4: fun(A,heap_Time_Heap(B))] : aa(C,fun(A,heap_Time_Heap(B)),U2,aa(fun(A,heap_Time_Heap(B)),C,C5,F4)) = F4
         => ( ! [F4: C,X2: A,H5: heap_ext(product_unit),H8: heap_ext(product_unit),R4: B,N3: nat] :
                ( ! [Xa: A,Ha: heap_ext(product_unit),H_a: heap_ext(product_unit),Ra: B,Na: nat] :
                    ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),aa(C,fun(A,heap_Time_Heap(B)),U2,F4),Xa),Ha,H_a,Ra,Na)
                   => aa(nat,$o,aa(B,fun(nat,$o),aa(heap_ext(product_unit),fun(B,fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,$o))),aa(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,$o)))),P,Xa),Ha),H_a),Ra),Na) )
               => ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),aa(C,fun(A,heap_Time_Heap(B)),U2,aa(C,C,F5,F4)),X2),H5,H8,R4,N3)
                 => aa(nat,$o,aa(B,fun(nat,$o),aa(heap_ext(product_unit),fun(B,fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,$o))),aa(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,$o)))),P,X2),H5),H8),R4),N3) ) )
           => ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),aa(C,fun(A,heap_Time_Heap(B)),U2,Fa),Xb),H,H2,R2,N)
             => aa(nat,$o,aa(B,fun(nat,$o),aa(heap_ext(product_unit),fun(B,fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,$o))),aa(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,$o)))),P,Xb),H),H2),R2),N) ) ) ) ) ) ).

% fixp_induct_heap
tff(fact_3042_sin__cos__npi,axiom,
    ! [N: nat] : sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),N) ).

% sin_cos_npi
tff(fact_3043_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),zero_zero(real))
       => aa(real,$o,ord_less_eq(real,abs_abs(real,aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb))),Xb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
tff(fact_3044_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3) = $ite(N = zero_zero(nat),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))))) ) ).

% signed_take_bit_rec
tff(fact_3045_dbl__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% dbl_simps(4)
tff(fact_3046_cos__pi__eq__zero,axiom,
    ! [M: nat] : cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))))),aa(num,real,numeral_numeral(real),bit0(one2)))) = zero_zero(real) ).

% cos_pi_eq_zero
tff(fact_3047_abs__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : abs_abs(A,abs_abs(A,A3)) = abs_abs(A,A3) ) ).

% abs_abs
tff(fact_3048_abs__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( abs_abs(A,zero_zero(A)) = zero_zero(A) ) ) ).

% abs_0
tff(fact_3049_abs__0__eq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( ( zero_zero(A) = abs_abs(A,A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_0_eq
tff(fact_3050_abs__eq__0,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( ( abs_abs(A,A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_eq_0
tff(fact_3051_abs__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ( abs_abs(A,zero_zero(A)) = zero_zero(A) ) ) ).

% abs_zero
tff(fact_3052_abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : abs_abs(A,aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),N) ) ).

% abs_numeral
tff(fact_3053_abs__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( abs_abs(A,one_one(A)) = one_one(A) ) ) ).

% abs_1
tff(fact_3054_abs__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A3)),abs_abs(A,A3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),A3) ) ).

% abs_mult_self_eq
tff(fact_3055_abs__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A3: A,B3: A] : abs_abs(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),abs_abs(A,A3)),abs_abs(A,B3)) ) ).

% abs_divide
tff(fact_3056_abs__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : abs_abs(A,aa(A,A,uminus_uminus(A),A3)) = abs_abs(A,A3) ) ).

% abs_minus
tff(fact_3057_signed__take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% signed_take_bit_of_0
tff(fact_3058_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( aa(A,$o,dvd_dvd(A,abs_abs(A,M)),K)
        <=> aa(A,$o,dvd_dvd(A,M),K) ) ) ).

% abs_dvd_iff
tff(fact_3059_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( aa(A,$o,dvd_dvd(A,M),abs_abs(A,K))
        <=> aa(A,$o,dvd_dvd(A,M),K) ) ) ).

% dvd_abs_iff
tff(fact_3060_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : abs_abs(A,aa(nat,A,semiring_1_of_nat(A),N)) = aa(nat,A,semiring_1_of_nat(A),N) ) ).

% abs_of_nat
tff(fact_3061_sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sin_zero
tff(fact_3062_dbl__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,zero_zero(A)) = zero_zero(A) ) ) ).

% dbl_simps(2)
tff(fact_3063_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( abs_abs(A,A3) = A3 ) ) ) ).

% abs_of_nonneg
tff(fact_3064_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,abs_abs(A,A3)),A3)
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ).

% abs_le_self_iff
tff(fact_3065_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,abs_abs(A,A3)),zero_zero(A))
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_3066_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),abs_abs(A,A3))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_3067_abs__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: num] : abs_abs(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),N) ) ).

% abs_neg_numeral
tff(fact_3068_abs__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( abs_abs(A,aa(A,A,uminus_uminus(A),one_one(A))) = one_one(A) ) ) ).

% abs_neg_one
tff(fact_3069_cos__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,zero_zero(A)) = one_one(A) ) ) ).

% cos_zero
tff(fact_3070_abs__power__minus,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] : abs_abs(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A3)),N)) = abs_abs(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% abs_power_minus
tff(fact_3071_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_Suc_1
tff(fact_3072_signed__take__bit__numeral__of__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(num,nat,numeral_numeral(nat),K)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_numeral_of_1
tff(fact_3073_signed__take__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% signed_take_bit_of_minus_1
tff(fact_3074_sin__pi,axiom,
    sin(real,pi) = zero_zero(real) ).

% sin_pi
tff(fact_3075_dbl__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),bit0(K)) ) ).

% dbl_simps(5)
tff(fact_3076_dbl__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_simps(1)
tff(fact_3077_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),abs_abs(A,B3))),zero_zero(A))
        <=> ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
            | ( B3 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_3078_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),abs_abs(A,B3)))
        <=> ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
            | ( B3 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_3079_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),zero_zero(A))
         => ( abs_abs(A,A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% abs_of_nonpos
tff(fact_3080_artanh__minus__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xb)),one_one(real))
     => ( aa(real,real,artanh(real),aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,artanh(real),Xb)) ) ) ).

% artanh_minus_real
tff(fact_3081_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),abs_abs(A,A3)),N))
        <=> ( ( A3 != zero_zero(A) )
            | ( N = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_3082_abs__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : abs_abs(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% abs_power2
tff(fact_3083_power2__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),abs_abs(A,A3)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_abs
tff(fact_3084_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(num,int,numeral_numeral(int),bit0(K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_Suc_bit0
tff(fact_3085_sin__cos__squared__add3,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Xb))) = one_one(A) ) ).

% sin_cos_squared_add3
tff(fact_3086_sin__npi,axiom,
    ! [N: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi)) = zero_zero(real) ).

% sin_npi
tff(fact_3087_sin__npi2,axiom,
    ! [N: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),N))) = zero_zero(real) ).

% sin_npi2
tff(fact_3088_dbl__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl(A,one_one(A)) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% dbl_simps(3)
tff(fact_3089_sin__npi__int,axiom,
    ! [N: int] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),N))) = zero_zero(real) ).

% sin_npi_int
tff(fact_3090_power__even__abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: num,A3: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),abs_abs(A,A3)),aa(num,nat,numeral_numeral(nat),W)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(num,nat,numeral_numeral(nat),W)) ) ) ) ).

% power_even_abs_numeral
tff(fact_3091_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_Suc_minus_bit0
tff(fact_3092_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,zero_zero(nat)),A3) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% signed_take_bit_0
tff(fact_3093_cos__pi__half,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) = zero_zero(real) ).

% cos_pi_half
tff(fact_3094_sin__two__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)) = zero_zero(real) ).

% sin_two_pi
tff(fact_3095_cos__two__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)) = one_one(real) ).

% cos_two_pi
tff(fact_3096_sin__pi__half,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) = one_one(real) ).

% sin_pi_half
tff(fact_3097_cos__periodic,axiom,
    ! [Xb: real] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = cos(real,Xb) ).

% cos_periodic
tff(fact_3098_sin__periodic,axiom,
    ! [Xb: real] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = sin(real,Xb) ).

% sin_periodic
tff(fact_3099_cos__2pi__minus,axiom,
    ! [Xb: real] : cos(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),Xb)) = cos(real,Xb) ).

% cos_2pi_minus
tff(fact_3100_sin__cos__squared__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% sin_cos_squared_add
tff(fact_3101_sin__cos__squared__add2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% sin_cos_squared_add2
tff(fact_3102_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_Suc_bit1
tff(fact_3103_sin__2npi,axiom,
    ! [N: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))),pi)) = zero_zero(real) ).

% sin_2npi
tff(fact_3104_cos__2npi,axiom,
    ! [N: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))),pi)) = one_one(real) ).

% cos_2npi
tff(fact_3105_sin__2pi__minus,axiom,
    ! [Xb: real] : sin(real,aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),Xb)) = aa(real,real,uminus_uminus(real),sin(real,Xb)) ).

% sin_2pi_minus
tff(fact_3106_sin__int__2pin,axiom,
    ! [N: int] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),aa(int,real,ring_1_of_int(real),N))) = zero_zero(real) ).

% sin_int_2pin
tff(fact_3107_cos__int__2pin,axiom,
    ! [N: int] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),aa(int,real,ring_1_of_int(real),N))) = one_one(real) ).

% cos_int_2pin
tff(fact_3108_cos__3over2__pi,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) = zero_zero(real) ).

% cos_3over2_pi
tff(fact_3109_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_Suc_minus_bit1
tff(fact_3110_sin__3over2__pi,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% sin_3over2_pi
tff(fact_3111_cos__npi__int,axiom,
    ! [N: int] :
      cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),N))) = $ite(aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),N),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ).

% cos_npi_int
tff(fact_3112_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( abs_abs(A,one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_3113_abs__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A,B3: A] : abs_abs(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A3)),abs_abs(A,B3)) ) ).

% abs_mult
tff(fact_3114_power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] : abs_abs(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),abs_abs(A,A3)),N) ) ).

% power_abs
tff(fact_3115_dvd__if__abs__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [L: A,K: A] :
          ( ( abs_abs(A,L) = abs_abs(A,K) )
         => aa(A,$o,dvd_dvd(A,L),K) ) ) ).

% dvd_if_abs_eq
tff(fact_3116_sin__zero__abs__cos__one,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
     => ( abs_abs(real,cos(real,Xb)) = one_one(real) ) ) ).

% sin_zero_abs_cos_one
tff(fact_3117_cos__one__sin__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cos(A,Xb) = one_one(A) )
         => ( sin(A,Xb) = zero_zero(A) ) ) ) ).

% cos_one_sin_zero
tff(fact_3118_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,A3),abs_abs(A,A3)) ) ).

% abs_ge_self
tff(fact_3119_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,abs_abs(A,A3)),B3)
         => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% abs_le_D1
tff(fact_3120_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] :
          ( ( abs_abs(A,A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_3121_signed__take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% signed_take_bit_mult
tff(fact_3122_signed__take__bit__add,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ).

% signed_take_bit_add
tff(fact_3123_signed__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,uminus_uminus(int),K)) ).

% signed_take_bit_minus
tff(fact_3124_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Y: A] :
          ( ( abs_abs(A,Xb) = abs_abs(A,Y) )
        <=> ( ( Xb = Y )
            | ( Xb = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% abs_eq_iff
tff(fact_3125_signed__take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,minus_minus(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,N),aa(int,int,minus_minus(int,K),L)) ).

% signed_take_bit_diff
tff(fact_3126_sin__zero__norm__cos__one,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sin(A,Xb) = zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,cos(A,Xb)) = one_one(real) ) ) ) ).

% sin_zero_norm_cos_one
tff(fact_3127_sin__zero__pi__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xb)),pi)
     => ( ( sin(real,Xb) = zero_zero(real) )
      <=> ( Xb = zero_zero(real) ) ) ) ).

% sin_zero_pi_iff
tff(fact_3128_sin__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,Xb))),cos(A,Xb)) ) ).

% sin_double
tff(fact_3129_sincos__principal__value,axiom,
    ! [Xb: real] :
    ? [Y2: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),Y2)
      & aa(real,$o,ord_less_eq(real,Y2),pi)
      & ( sin(real,Y2) = sin(real,Xb) )
      & ( cos(real,Y2) = cos(real,Xb) ) ) ).

% sincos_principal_value
tff(fact_3130_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),abs_abs(A,A3)) ) ).

% abs_ge_zero
tff(fact_3131_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( abs_abs(A,A3) = A3 ) ) ) ).

% abs_of_pos
tff(fact_3132_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : ~ aa(A,$o,ord_less(A,abs_abs(A,A3)),zero_zero(A)) ) ).

% abs_not_less_zero
tff(fact_3133_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,abs_abs(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))),aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A3)),abs_abs(A,B3))) ) ).

% abs_triangle_ineq
tff(fact_3134_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,C2: A,B3: A,D: A] :
          ( aa(A,$o,ord_less(A,abs_abs(A,A3)),C2)
         => ( aa(A,$o,ord_less(A,abs_abs(A,B3)),D)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A3)),abs_abs(A,B3))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D)) ) ) ) ).

% abs_mult_less
tff(fact_3135_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,abs_abs(A,A3)),abs_abs(A,B3))),abs_abs(A,aa(A,A,minus_minus(A,B3),A3))) ) ).

% abs_triangle_ineq2_sym
tff(fact_3136_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,abs_abs(A,aa(A,A,minus_minus(A,abs_abs(A,A3)),abs_abs(A,B3)))),abs_abs(A,aa(A,A,minus_minus(A,A3),B3))) ) ).

% abs_triangle_ineq3
tff(fact_3137_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,abs_abs(A,A3)),abs_abs(A,B3))),abs_abs(A,aa(A,A,minus_minus(A,A3),B3))) ) ).

% abs_triangle_ineq2
tff(fact_3138_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B3: A,A3: A] :
          ( ( B3 != zero_zero(A) )
         => ( abs_abs(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),abs_abs(A,A3)),abs_abs(A,B3)) ) ) ) ).

% nonzero_abs_divide
tff(fact_3139_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A3)),B3)
           => aa(A,$o,ord_less_eq(A,abs_abs(A,A3)),B3) ) ) ) ).

% abs_leI
tff(fact_3140_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,abs_abs(A,A3)),B3)
         => aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A3)),B3) ) ) ).

% abs_le_D2
tff(fact_3141_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,abs_abs(A,A3)),B3)
        <=> ( aa(A,$o,ord_less_eq(A,A3),B3)
            & aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A3)),B3) ) ) ) ).

% abs_le_iff
tff(fact_3142_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),A3)),abs_abs(A,A3)) ) ).

% abs_ge_minus_self
tff(fact_3143_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,abs_abs(A,A3)),B3)
        <=> ( aa(A,$o,ord_less(A,A3),B3)
            & aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),A3)),B3) ) ) ) ).

% abs_less_iff
tff(fact_3144_sin__x__le__x,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => aa(real,$o,ord_less_eq(real,sin(real,Xb)),Xb) ) ).

% sin_x_le_x
tff(fact_3145_dbl__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xb: A] : neg_numeral_dbl(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb) ) ).

% dbl_def
tff(fact_3146_cos__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cos_squared_eq
tff(fact_3147_sin__squared__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% sin_squared_eq
tff(fact_3148_dense__eq0__I,axiom,
    ! [A: $tType] :
      ( ( ordere166539214618696060dd_abs(A)
        & dense_linorder(A) )
     => ! [Xb: A] :
          ( ! [E2: A] :
              ( aa(A,$o,ord_less(A,zero_zero(A)),E2)
             => aa(A,$o,ord_less_eq(A,abs_abs(A,Xb)),E2) )
         => ( Xb = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_3149_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,Y)),Xb) = abs_abs(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb)) ) ) ) ).

% abs_mult_pos
tff(fact_3150_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A3: A,B3: A] :
          ( ( ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
              | aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) )
            & ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
              | aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) ) )
         => ( abs_abs(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A3)),abs_abs(A,B3)) ) ) ) ).

% abs_eq_mult
tff(fact_3151_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),abs_abs(A,A3))),zero_zero(A)) ) ).

% abs_minus_le_zero
tff(fact_3152_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = abs_abs(A,B3) )
        <=> ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
            & ( ( B3 = A3 )
              | ( B3 = aa(A,A,uminus_uminus(A),A3) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_3153_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A3: A,B3: A] :
          ( ( abs_abs(A,A3) = B3 )
        <=> ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
            & ( ( A3 = B3 )
              | ( A3 = aa(A,A,uminus_uminus(A),B3) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_3154_abs__div__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),abs_abs(A,Xb)),Y) = abs_abs(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% abs_div_pos
tff(fact_3155_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),abs_abs(A,A3)),N)) ) ).

% zero_le_power_abs
tff(fact_3156_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A3: A] :
          abs_abs(A,A3) = $ite(aa(A,$o,ord_less(A,A3),zero_zero(A)),aa(A,A,uminus_uminus(A),A3),A3) ) ).

% abs_if
tff(fact_3157_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( abs_abs(A,A3) = aa(A,A,uminus_uminus(A),A3) ) ) ) ).

% abs_of_neg
tff(fact_3158_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X3: A] :
          abs_abs(A,X3) = $ite(aa(A,$o,ord_less(A,X3),zero_zero(A)),aa(A,A,uminus_uminus(A),X3),X3) ) ).

% abs_if_raw
tff(fact_3159_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,A3: A,R2: A] :
          ( aa(A,$o,ord_less_eq(A,abs_abs(A,aa(A,A,minus_minus(A,Xb),A3))),R2)
        <=> ( aa(A,$o,ord_less_eq(A,aa(A,A,minus_minus(A,A3),R2)),Xb)
            & aa(A,$o,ord_less_eq(A,Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R2)) ) ) ) ).

% abs_diff_le_iff
tff(fact_3160_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A,C2: A,D: A] : aa(A,$o,ord_less_eq(A,abs_abs(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,aa(A,A,minus_minus(A,A3),C2))),abs_abs(A,aa(A,A,minus_minus(A,B3),D)))) ) ).

% abs_diff_triangle_ineq
tff(fact_3161_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,abs_abs(A,aa(A,A,minus_minus(A,A3),B3))),aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A3)),abs_abs(A,B3))) ) ).

% abs_triangle_ineq4
tff(fact_3162_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,A3: A,R2: A] :
          ( aa(A,$o,ord_less(A,abs_abs(A,aa(A,A,minus_minus(A,Xb),A3))),R2)
        <=> ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A3),R2)),Xb)
            & aa(A,$o,ord_less(A,Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),R2)) ) ) ) ).

% abs_diff_less_iff
tff(fact_3163_sin__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),pi)
       => aa(real,$o,ord_less(real,zero_zero(real)),sin(real,Xb)) ) ) ).

% sin_gt_zero
tff(fact_3164_sin__x__ge__neg__x,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),Xb)),sin(real,Xb)) ) ).

% sin_x_ge_neg_x
tff(fact_3165_sin__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),pi)
       => aa(real,$o,ord_less_eq(real,zero_zero(real)),sin(real,Xb)) ) ) ).

% sin_ge_zero
tff(fact_3166_cos__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),pi)
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),pi)
           => ( ( cos(real,Xb) = cos(real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_3167_cos__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),pi)
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),pi)
           => ( aa(real,$o,ord_less_eq(real,cos(real,Xb)),cos(real,Y))
            <=> aa(real,$o,ord_less_eq(real,Y),Xb) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_3168_cos__monotone__0__pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),Xb)
       => ( aa(real,$o,ord_less_eq(real,Xb),pi)
         => aa(real,$o,ord_less_eq(real,cos(real,Xb)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_3169_cos__diff__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,minus_minus(A,cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,Z),W)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_diff_cos
tff(fact_3170_sin__diff__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,minus_minus(A,sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_diff_sin
tff(fact_3171_sin__plus__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,W),Z)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% sin_plus_sin
tff(fact_3172_cos__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),sin(A,aa(A,A,minus_minus(A,W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cos_times_sin
tff(fact_3173_sin__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z))),sin(A,aa(A,A,minus_minus(A,W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sin_times_cos
tff(fact_3174_sin__times__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,W)),sin(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,cos(A,aa(A,A,minus_minus(A,W),Z))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sin_times_sin
tff(fact_3175_abs__real__def,axiom,
    ! [A3: real] :
      abs_abs(real,A3) = $ite(aa(real,$o,ord_less(real,A3),zero_zero(real)),aa(real,real,uminus_uminus(real),A3),A3) ).

% abs_real_def
tff(fact_3176_cos__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cos_double
tff(fact_3177_lemma__interval__lt,axiom,
    ! [A3: real,Xb: real,B3: real] :
      ( aa(real,$o,ord_less(real,A3),Xb)
     => ( aa(real,$o,ord_less(real,Xb),B3)
       => ? [D3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D3)
            & ! [Y4: real] :
                ( aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,Xb),Y4))),D3)
               => ( aa(real,$o,ord_less(real,A3),Y4)
                  & aa(real,$o,ord_less(real,Y4),B3) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_3178_cos__double__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),W)) = aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,W)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% cos_double_sin
tff(fact_3179_abs__add__one__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] : aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),abs_abs(A,Xb))) ) ).

% abs_add_one_gt_zero
tff(fact_3180_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,Xb: A] :
          ( aa(A,$o,ord_less_eq(A,abs_abs(A,aa(int,A,ring_1_of_int(A),N))),Xb)
         => ( ( N = zero_zero(int) )
            | aa(A,$o,ord_less_eq(A,one_one(A)),Xb) ) ) ) ).

% of_int_leD
tff(fact_3181_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: int,Xb: A] :
          ( aa(A,$o,ord_less(A,abs_abs(A,aa(int,A,ring_1_of_int(A),N))),Xb)
         => ( ( N = zero_zero(int) )
            | aa(A,$o,ord_less(A,one_one(A)),Xb) ) ) ) ).

% of_int_lessD
tff(fact_3182_cos__two__neq__zero,axiom,
    cos(real,aa(num,real,numeral_numeral(real),bit0(one2))) != zero_zero(real) ).

% cos_two_neq_zero
tff(fact_3183_cos__monotone__0__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ( aa(real,$o,ord_less(real,Y),Xb)
       => ( aa(real,$o,ord_less_eq(real,Xb),pi)
         => aa(real,$o,ord_less(real,cos(real,Xb)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi
tff(fact_3184_cos__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),pi)
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),pi)
           => ( aa(real,$o,ord_less(real,cos(real,Xb)),cos(real,Y))
            <=> aa(real,$o,ord_less(real,Y),Xb) ) ) ) ) ) ).

% cos_mono_less_eq
tff(fact_3185_sin__eq__0__pi,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),pi)
       => ( ( sin(real,Xb) = zero_zero(real) )
         => ( Xb = zero_zero(real) ) ) ) ) ).

% sin_eq_0_pi
tff(fact_3186_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),Xb)
       => ( aa(real,$o,ord_less_eq(real,Xb),zero_zero(real))
         => aa(real,$o,ord_less_eq(real,cos(real,Y)),cos(real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_3187_lemma__interval,axiom,
    ! [A3: real,Xb: real,B3: real] :
      ( aa(real,$o,ord_less(real,A3),Xb)
     => ( aa(real,$o,ord_less(real,Xb),B3)
       => ? [D3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D3)
            & ! [Y4: real] :
                ( aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,Xb),Y4))),D3)
               => ( aa(real,$o,ord_less_eq(real,A3),Y4)
                  & aa(real,$o,ord_less_eq(real,Y4),B3) ) ) ) ) ) ).

% lemma_interval
tff(fact_3188_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: A,M: int] : aa(A,$o,ord_less_eq(A,abs_abs(A,aa(A,A,minus_minus(A,Z),aa(int,A,ring_1_of_int(A),archimedean_round(A,Z))))),abs_abs(A,aa(A,A,minus_minus(A,Z),aa(int,A,ring_1_of_int(A),M)))) ) ).

% round_diff_minimal
tff(fact_3189_sin__zero__iff__int2,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
    <=> ? [I4: int] : Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I4)),pi) ) ).

% sin_zero_iff_int2
tff(fact_3190_sincos__total__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
       => ? [T2: real] :
            ( aa(real,$o,ord_less_eq(real,zero_zero(real)),T2)
            & aa(real,$o,ord_less_eq(real,T2),pi)
            & ( Xb = cos(real,T2) )
            & ( Y = sin(real,T2) ) ) ) ) ).

% sincos_total_pi
tff(fact_3191_sin__expansion__lemma,axiom,
    ! [Xb: real,M: nat] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M))),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) = cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% sin_expansion_lemma
tff(fact_3192_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,abs_abs(A,Xb)),abs_abs(A,Y))
        <=> aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_le_square_iff
tff(fact_3193_abs__square__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
        <=> ( abs_abs(A,Xb) = one_one(A) ) ) ) ).

% abs_square_eq_1
tff(fact_3194_power__even__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),abs_abs(A,A3)),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N) ) ) ) ).

% power_even_abs
tff(fact_3195_cos__expansion__lemma,axiom,
    ! [Xb: real,M: nat] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,M))),pi)),aa(num,real,numeral_numeral(real),bit0(one2))))) = aa(real,real,uminus_uminus(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),M)),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% cos_expansion_lemma
tff(fact_3196_sin__gt__zero__02,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,ord_less(real,zero_zero(real)),sin(real,Xb)) ) ) ).

% sin_gt_zero_02
tff(fact_3197_cos__two__less__zero,axiom,
    aa(real,$o,ord_less(real,cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_less_zero
tff(fact_3198_cos__two__le__zero,axiom,
    aa(real,$o,ord_less_eq(real,cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_le_zero
tff(fact_3199_cos__is__zero,axiom,
    ? [X2: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X2)
      & aa(real,$o,ord_less_eq(real,X2),aa(num,real,numeral_numeral(real),bit0(one2)))
      & ( cos(real,X2) = zero_zero(real) )
      & ! [Y4: real] :
          ( ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y4)
            & aa(real,$o,ord_less_eq(real,Y4),aa(num,real,numeral_numeral(real),bit0(one2)))
            & ( cos(real,Y4) = zero_zero(real) ) )
         => ( Y4 = X2 ) ) ) ).

% cos_is_zero
tff(fact_3200_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : aa(int,$o,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).

% signed_take_bit_int_less_exp
tff(fact_3201_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,ord_less(real,Y),Xb)
       => ( aa(real,$o,ord_less_eq(real,Xb),zero_zero(real))
         => aa(real,$o,ord_less(real,cos(real,Y)),cos(real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0
tff(fact_3202_cos__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ? [X2: real] :
            ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X2)
            & aa(real,$o,ord_less_eq(real,X2),pi)
            & ( cos(real,X2) = Y )
            & ! [Y4: real] :
                ( ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y4)
                  & aa(real,$o,ord_less_eq(real,Y4),pi)
                  & ( cos(real,Y4) = Y ) )
               => ( Y4 = X2 ) ) ) ) ) ).

% cos_total
tff(fact_3203_even__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,M),A3))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ).

% even_signed_take_bit_iff
tff(fact_3204_sincos__total__pi__half,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
         => ? [T2: real] :
              ( aa(real,$o,ord_less_eq(real,zero_zero(real)),T2)
              & aa(real,$o,ord_less_eq(real,T2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
              & ( Xb = cos(real,T2) )
              & ( Y = sin(real,T2) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_3205_sincos__total__2pi__le,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
     => ? [T2: real] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),T2)
          & aa(real,$o,ord_less_eq(real,T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
          & ( Xb = cos(real,T2) )
          & ( Y = sin(real,T2) ) ) ) ).

% sincos_total_2pi_le
tff(fact_3206_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Y)
         => ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
          <=> aa(A,$o,ord_less_eq(A,abs_abs(A,Xb)),Y) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_3207_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,ord_less_eq(A,abs_abs(A,Xb)),one_one(A)) ) ) ).

% abs_square_le_1
tff(fact_3208_abs__square__less__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A))
        <=> aa(A,$o,ord_less(A,abs_abs(A,Xb)),one_one(A)) ) ) ).

% abs_square_less_1
tff(fact_3209_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: A,B3: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
         => ( aa(A,$o,ord_less_eq(A,abs_abs(A,A3)),abs_abs(A,B3))
           => aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N)) ) ) ) ).

% power_mono_even
tff(fact_3210_sincos__total__2pi,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(real) )
     => ~ ! [T2: real] :
            ( aa(real,$o,ord_less_eq(real,zero_zero(real)),T2)
           => ( aa(real,$o,ord_less(real,T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( ( Xb = cos(real,T2) )
               => ( Y != sin(real,T2) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_3211_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,ord_less_eq(int,K),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))
    <=> aa(int,$o,ord_less(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_3212_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),K)
    <=> aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_3213_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K: int] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_3214_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),K)
    <=> aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_3215_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,ord_less(int,K),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))
    <=> aa(int,$o,ord_less(int,K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_3216_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => aa(real,$o,ord_less_eq(real,zero_zero(real)),sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(nat,real,semiring_1_of_nat(real),N)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_3217_cos__plus__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)),aa(num,A,numeral_numeral(A),bit0(one2)))))),cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,W),Z)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% cos_plus_cos
tff(fact_3218_cos__times__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A,Z: A] : aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,W)),cos(A,Z)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,minus_minus(A,W),Z))),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),W),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cos_times_cos
tff(fact_3219_sin__gt__zero2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less(real,zero_zero(real)),sin(real,Xb)) ) ) ).

% sin_gt_zero2
tff(fact_3220_sin__lt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,pi),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
       => aa(real,$o,ord_less(real,sin(real,Xb)),zero_zero(real)) ) ) ).

% sin_lt_zero
tff(fact_3221_sin__30,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sin_30
tff(fact_3222_cos__double__less__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(num,real,numeral_numeral(real),bit0(one2)))
       => aa(real,$o,ord_less(real,cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),Xb))),one_one(real)) ) ) ).

% cos_double_less_one
tff(fact_3223_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),K)
     => aa(int,$o,ord_less_eq(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),aa(int,int,minus_minus(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,N)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_3224_cos__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less(real,zero_zero(real)),cos(real,Xb)) ) ) ).

% cos_gt_zero
tff(fact_3225_sin__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( ( sin(real,Xb) = sin(real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% sin_inj_pi
tff(fact_3226_sin__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,ord_less_eq(real,sin(real,Xb)),sin(real,Y))
            <=> aa(real,$o,ord_less_eq(real,Xb),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_3227_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),Xb)
       => ( aa(real,$o,ord_less_eq(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,ord_less_eq(real,sin(real,Y)),sin(real,Xb)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_3228_cos__60,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% cos_60
tff(fact_3229_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = K )
    <=> ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),K)
        & aa(int,$o,ord_less(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_3230_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),K)
     => ( aa(int,$o,ord_less(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_3231_cos__one__2pi__int,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = one_one(real) )
    <=> ? [X: int] : Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),X)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi) ) ).

% cos_one_2pi_int
tff(fact_3232_cos__double__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [W: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),W)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,W)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),one_one(A)) ) ).

% cos_double_cos
tff(fact_3233_cos__treble__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))),Xb)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2))),cos(A,Xb))) ) ).

% cos_treble_cos
tff(fact_3234_sin__le__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,pi),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
       => aa(real,$o,ord_less_eq(real,sin(real,Xb)),zero_zero(real)) ) ) ).

% sin_le_zero
tff(fact_3235_sin__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),zero_zero(real))
       => aa(real,$o,ord_less(real,sin(real,Xb)),zero_zero(real)) ) ) ).

% sin_less_zero
tff(fact_3236_sin__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,ord_less(real,sin(real,Xb)),sin(real,Y))
            <=> aa(real,$o,ord_less(real,Xb),Y) ) ) ) ) ) ).

% sin_mono_less_eq
tff(fact_3237_sin__monotone__2pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,ord_less(real,Y),Xb)
       => ( aa(real,$o,ord_less_eq(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,ord_less(real,sin(real,Y)),sin(real,Xb)) ) ) ) ).

% sin_monotone_2pi
tff(fact_3238_sin__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ? [X2: real] :
            ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X2)
            & aa(real,$o,ord_less_eq(real,X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( sin(real,X2) = Y )
            & ! [Y4: real] :
                ( ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y4)
                  & aa(real,$o,ord_less_eq(real,Y4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
                  & ( sin(real,Y4) = Y ) )
               => ( Y4 = X2 ) ) ) ) ) ).

% sin_total
tff(fact_3239_cos__gt__zero__pi,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less(real,zero_zero(real)),cos(real,Xb)) ) ) ).

% cos_gt_zero_pi
tff(fact_3240_cos__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less_eq(real,zero_zero(real)),cos(real,Xb)) ) ) ).

% cos_ge_zero
tff(fact_3241_cos__one__2pi,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = one_one(real) )
    <=> ( ? [X: nat] : Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),X)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)
        | ? [X: nat] : Xb = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),X)),aa(num,real,numeral_numeral(real),bit0(one2)))),pi)) ) ) ).

% cos_one_2pi
tff(fact_3242_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,ord_less_eq(A,abs_abs(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_round(A,Xb))),Xb))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% of_int_round_abs_le
tff(fact_3243_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,N: int] :
          ( aa(A,$o,ord_less(A,abs_abs(A,aa(A,A,minus_minus(A,Xb),aa(int,A,ring_1_of_int(A),N)))),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))
         => ( archimedean_round(A,Xb) = N ) ) ) ).

% round_unique'
tff(fact_3244_sin__pi__divide__n__gt__0,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
     => aa(real,$o,ord_less(real,zero_zero(real)),sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(nat,real,semiring_1_of_nat(real),N)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_3245_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,ord_less(int,K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,N)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_3246_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,N)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_3247_sin__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
    <=> ? [I4: int] :
          ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),I4)
          & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% sin_zero_iff_int
tff(fact_3248_cos__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = zero_zero(real) )
    <=> ? [I4: int] :
          ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),I4)
          & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I4)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% cos_zero_iff_int
tff(fact_3249_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),one_one(real))
       => aa(real,$o,ord_less_eq(real,abs_abs(real,aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb))),Xb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
tff(fact_3250_sin__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( ( sin(real,Xb) = zero_zero(real) )
       => ? [N3: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)
            & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N3)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% sin_zero_lemma
tff(fact_3251_sin__zero__iff,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
    <=> ( ? [N8: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N8)
            & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N8)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N8: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N8)
            & ( Xb = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N8)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ) ).

% sin_zero_iff
tff(fact_3252_cos__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( ( cos(real,Xb) = zero_zero(real) )
       => ? [N3: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)
            & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N3)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ) ).

% cos_zero_lemma
tff(fact_3253_cos__zero__iff,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = zero_zero(real) )
    <=> ( ? [N8: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N8)
            & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N8)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N8: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N8)
            & ( Xb = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N8)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) ) ) ) ) ).

% cos_zero_iff
tff(fact_3254_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
     => aa(real,$o,ord_less_eq(real,abs_abs(real,aa(real,real,minus_minus(real,aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb))),Xb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% abs_ln_one_plus_x_minus_x_bound
tff(fact_3255_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,$o)),Xb: A] :
          ( ! [X2: A] :
              ( aa(A,$o,ord_less_eq(A,zero_zero(A)),X2)
             => aa(A,$o,aa(A,fun(A,$o),P,X2),aa(nat,A,aa(A,fun(nat,A),power_power(A),X2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(A,$o,aa(A,fun(A,$o),P,abs_abs(A,Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% abs_sqrt_wlog
tff(fact_3256_monoseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xb)),one_one(real))
     => topological_monoseq(real,aTP_Lamp_mj(real,fun(nat,real),Xb)) ) ).

% monoseq_arctan_series
tff(fact_3257_arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xb)),one_one(real))
     => ( aa(real,real,arctan,Xb) = suminf(real,aTP_Lamp_mk(real,fun(nat,real),Xb)) ) ) ).

% arctan_series
tff(fact_3258_summable__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xb)),one_one(real))
     => summable(real,aTP_Lamp_mk(real,fun(nat,real),Xb)) ) ).

% summable_arctan_series
tff(fact_3259_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_numeral_minus_bit1
tff(fact_3260_zdvd1__eq,axiom,
    ! [Xb: int] :
      ( aa(int,$o,dvd_dvd(int,Xb),one_one(int))
    <=> ( abs_abs(int,Xb) = one_one(int) ) ) ).

% zdvd1_eq
tff(fact_3261_arctan__zero__zero,axiom,
    aa(real,real,arctan,zero_zero(real)) = zero_zero(real) ).

% arctan_zero_zero
tff(fact_3262_arctan__eq__zero__iff,axiom,
    ! [Xb: real] :
      ( ( aa(real,real,arctan,Xb) = zero_zero(real) )
    <=> ( Xb = zero_zero(real) ) ) ).

% arctan_eq_zero_iff
tff(fact_3263_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_ml(nat,A)) ) ).

% summable_zero
tff(fact_3264_summable__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,Fa: fun(nat,A)] : summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mm(nat,fun(fun(nat,A),fun(nat,A)),I),Fa)) ) ).

% summable_single
tff(fact_3265_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),K: nat] :
          ( summable(A,aa(nat,fun(nat,A),aTP_Lamp_mn(fun(nat,A),fun(nat,fun(nat,A)),Fa),K))
        <=> summable(A,Fa) ) ) ).

% summable_iff_shift
tff(fact_3266_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less(int,abs_abs(int,Z)),one_one(int))
    <=> ( Z = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_3267_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_3268_Suc__eq__numeral,axiom,
    ! [N: nat,K: num] :
      ( ( aa(nat,nat,suc,N) = aa(num,nat,numeral_numeral(nat),K) )
    <=> ( N = pred_numeral(K) ) ) ).

% Suc_eq_numeral
tff(fact_3269_eq__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,N) )
    <=> ( pred_numeral(K) = N ) ) ).

% eq_numeral_Suc
tff(fact_3270_arctan__less__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,arctan,Xb)),zero_zero(real))
    <=> aa(real,$o,ord_less(real,Xb),zero_zero(real)) ) ).

% arctan_less_zero_iff
tff(fact_3271_zero__less__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,arctan,Xb))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),Xb) ) ).

% zero_less_arctan_iff
tff(fact_3272_arctan__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,arctan,Xb)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,Xb),zero_zero(real)) ) ).

% arctan_le_zero_iff
tff(fact_3273_zero__le__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,arctan,Xb))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb) ) ).

% zero_le_arctan_iff
tff(fact_3274_summable__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,Fa: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mo(A,fun(fun(nat,A),fun(nat,A)),C2),Fa))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,Fa) ) ) ) ).

% summable_cmult_iff
tff(fact_3275_summable__divide__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(nat,A),C2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_mp(fun(nat,A),fun(A,fun(nat,A)),Fa),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,Fa) ) ) ) ).

% summable_divide_iff
tff(fact_3276_pred__numeral__simps_I3_J,axiom,
    ! [K: num] : pred_numeral(aa(num,num,bit1,K)) = aa(num,nat,numeral_numeral(nat),bit0(K)) ).

% pred_numeral_simps(3)
tff(fact_3277_less__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( aa(nat,$o,ord_less(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N))
    <=> aa(nat,$o,ord_less(nat,pred_numeral(K)),N) ) ).

% less_numeral_Suc
tff(fact_3278_less__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,ord_less(nat,N),pred_numeral(K)) ) ).

% less_Suc_numeral
tff(fact_3279_le__numeral__Suc,axiom,
    ! [K: num,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N))
    <=> aa(nat,$o,ord_less_eq(nat,pred_numeral(K)),N) ) ).

% le_numeral_Suc
tff(fact_3280_le__Suc__numeral,axiom,
    ! [N: nat,K: num] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,ord_less_eq(nat,N),pred_numeral(K)) ) ).

% le_Suc_numeral
tff(fact_3281_diff__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,minus_minus(nat,N),pred_numeral(K)) ).

% diff_Suc_numeral
tff(fact_3282_diff__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)) = aa(nat,nat,minus_minus(nat,pred_numeral(K)),N) ).

% diff_numeral_Suc
tff(fact_3283_max__Suc__numeral,axiom,
    ! [N: nat,K: num] : aa(nat,nat,ord_max(nat,aa(nat,nat,suc,N)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,N),pred_numeral(K))) ).

% max_Suc_numeral
tff(fact_3284_max__numeral__Suc,axiom,
    ! [K: num,N: nat] : aa(nat,nat,ord_max(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,N)) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,pred_numeral(K)),N)) ).

% max_numeral_Suc
tff(fact_3285_summable__geometric__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( summable(A,aa(A,fun(nat,A),power_power(A),C2))
        <=> aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,C2)),one_one(real)) ) ) ).

% summable_geometric_iff
tff(fact_3286_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),bit0(K))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_numeral_bit0
tff(fact_3287_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_numeral_minus_bit0
tff(fact_3288_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,pred_numeral(L)),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% signed_take_bit_numeral_bit1
tff(fact_3289_summable__norm__cancel,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Fa: fun(nat,A)] :
          ( summable(real,aTP_Lamp_mq(fun(nat,A),fun(nat,real),Fa))
         => summable(A,Fa) ) ) ).

% summable_norm_cancel
tff(fact_3290_arctan__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,arctan,Xb)),aa(real,real,arctan,Y))
    <=> aa(real,$o,ord_less(real,Xb),Y) ) ).

% arctan_less_iff
tff(fact_3291_arctan__monotone,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,Xb),Y)
     => aa(real,$o,ord_less(real,aa(real,real,arctan,Xb)),aa(real,real,arctan,Y)) ) ).

% arctan_monotone
tff(fact_3292_summable__comparison__test_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,real),N2: nat,Fa: fun(nat,A)] :
          ( summable(real,G)
         => ( ! [N3: nat] :
                ( aa(nat,$o,ord_less_eq(nat,N2),N3)
               => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,Fa,N3))),aa(nat,real,G,N3)) )
           => summable(A,Fa) ) ) ) ).

% summable_comparison_test'
tff(fact_3293_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Fa: fun(nat,A),G: fun(nat,real)] :
          ( ? [N10: nat] :
            ! [N3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,N10),N3)
             => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,Fa,N3))),aa(nat,real,G,N3)) )
         => ( summable(real,G)
           => summable(A,Fa) ) ) ) ).

% summable_comparison_test
tff(fact_3294_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: A] :
          ( summable(A,aTP_Lamp_mr(A,fun(nat,A),C2))
        <=> ( C2 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_3295_summable__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Fa: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,Fa)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ms(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Fa),G)) ) ) ) ).

% summable_add
tff(fact_3296_summable__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Fa: fun(nat,A),C2: A] :
          ( summable(A,Fa)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_mt(fun(nat,A),fun(A,fun(nat,A)),Fa),C2)) ) ) ).

% summable_mult2
tff(fact_3297_summable__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Fa: fun(nat,A),C2: A] :
          ( summable(A,Fa)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_mu(fun(nat,A),fun(A,fun(nat,A)),Fa),C2)) ) ) ).

% summable_mult
tff(fact_3298_summable__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,Fa)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mv(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Fa),G)) ) ) ) ).

% summable_diff
tff(fact_3299_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A)] :
          ( summable(A,aTP_Lamp_mw(fun(nat,A),fun(nat,A),Fa))
        <=> summable(A,Fa) ) ) ).

% summable_Suc_iff
tff(fact_3300_summable__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(nat,A),C2: A] :
          ( summable(A,Fa)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_mp(fun(nat,A),fun(A,fun(nat,A)),Fa),C2)) ) ) ).

% summable_divide
tff(fact_3301_summable__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A)] :
          ( summable(A,aTP_Lamp_mx(fun(nat,A),fun(nat,A),Fa))
        <=> summable(A,Fa) ) ) ).

% summable_minus_iff
tff(fact_3302_summable__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A)] :
          ( summable(A,Fa)
         => summable(A,aTP_Lamp_mx(fun(nat,A),fun(nat,A),Fa)) ) ) ).

% summable_minus
tff(fact_3303_summable__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),K: nat] :
          ( summable(A,Fa)
         => summable(A,aa(nat,fun(nat,A),aTP_Lamp_mn(fun(nat,A),fun(nat,fun(nat,A)),Fa),K)) ) ) ).

% summable_ignore_initial_segment
tff(fact_3304_summable__rabs__cancel,axiom,
    ! [Fa: fun(nat,real)] :
      ( summable(real,aTP_Lamp_my(fun(nat,real),fun(nat,real),Fa))
     => summable(real,Fa) ) ).

% summable_rabs_cancel
tff(fact_3305_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Fa: fun(nat,A),Xb: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_mz(fun(nat,A),fun(A,fun(nat,A)),Fa),Xb))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xb))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_na(fun(nat,A),fun(A,fun(nat,real)),Fa),Z)) ) ) ) ).

% powser_insidea
tff(fact_3306_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Fa: fun(nat,A),G: fun(nat,A)] :
          ( ! [N3: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,Fa,N3)),aa(nat,A,G,N3))
         => ( summable(A,Fa)
           => ( summable(A,G)
             => aa(A,$o,ord_less_eq(A,suminf(A,Fa)),suminf(A,G)) ) ) ) ) ).

% suminf_le
tff(fact_3307_abs__zmult__eq__1,axiom,
    ! [M: int,N: int] :
      ( ( abs_abs(int,aa(int,int,aa(int,fun(int,int),times_times(int),M),N)) = one_one(int) )
     => ( abs_abs(int,M) = one_one(int) ) ) ).

% abs_zmult_eq_1
tff(fact_3308_summable__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,Fa: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mo(A,fun(fun(nat,A),fun(nat,A)),C2),Fa))
         => ( ( C2 != zero_zero(A) )
           => summable(A,Fa) ) ) ) ).

% summable_mult_D
tff(fact_3309_summable__zero__power,axiom,
    ! [A: $tType] :
      ( ( comm_ring_1(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aa(A,fun(nat,A),power_power(A),zero_zero(A))) ) ).

% summable_zero_power
tff(fact_3310_numeral__eq__Suc,axiom,
    ! [K: num] : aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,pred_numeral(K)) ).

% numeral_eq_Suc
tff(fact_3311_abs__div,axiom,
    ! [Y: int,Xb: int] :
      ( aa(int,$o,dvd_dvd(int,Y),Xb)
     => ( abs_abs(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),Y)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),abs_abs(int,Xb)),abs_abs(int,Y)) ) ) ).

% abs_div
tff(fact_3312_suminf__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Fa: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,Fa)
         => ( summable(A,G)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,Fa)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ms(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Fa),G)) ) ) ) ) ).

% suminf_add
tff(fact_3313_suminf__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Fa: fun(nat,A),C2: A] :
          ( summable(A,Fa)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_mu(fun(nat,A),fun(A,fun(nat,A)),Fa),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),suminf(A,Fa)) ) ) ) ).

% suminf_mult
tff(fact_3314_suminf__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Fa: fun(nat,A),C2: A] :
          ( summable(A,Fa)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,Fa)),C2) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_mt(fun(nat,A),fun(A,fun(nat,A)),Fa),C2)) ) ) ) ).

% suminf_mult2
tff(fact_3315_suminf__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,Fa)
         => ( summable(A,G)
           => ( aa(A,A,minus_minus(A,suminf(A,Fa)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mv(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Fa),G)) ) ) ) ) ).

% suminf_diff
tff(fact_3316_suminf__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(nat,A),C2: A] :
          ( summable(A,Fa)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_mp(fun(nat,A),fun(A,fun(nat,A)),Fa),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),suminf(A,Fa)),C2) ) ) ) ).

% suminf_divide
tff(fact_3317_suminf__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A)] :
          ( summable(A,Fa)
         => ( suminf(A,aTP_Lamp_mx(fun(nat,A),fun(nat,A),Fa)) = aa(A,A,uminus_uminus(A),suminf(A,Fa)) ) ) ) ).

% suminf_minus
tff(fact_3318_cos__arctan__not__zero,axiom,
    ! [Xb: real] : cos(real,aa(real,real,arctan,Xb)) != zero_zero(real) ).

% cos_arctan_not_zero
tff(fact_3319_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Fa: fun(nat,A)] :
          ( summable(A,Fa)
         => ( ! [N3: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,Fa,N3))
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),suminf(A,Fa)) ) ) ) ).

% suminf_nonneg
tff(fact_3320_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Fa: fun(nat,A)] :
          ( summable(A,Fa)
         => ( ! [N3: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,Fa,N3))
           => ( ( suminf(A,Fa) = zero_zero(A) )
            <=> ! [N8: nat] : aa(nat,A,Fa,N8) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_3321_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Fa: fun(nat,A)] :
          ( summable(A,Fa)
         => ( ! [N3: nat] : aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,Fa,N3))
           => aa(A,$o,ord_less(A,zero_zero(A)),suminf(A,Fa)) ) ) ) ).

% suminf_pos
tff(fact_3322_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Fa: fun(nat,A)] : summable(A,aTP_Lamp_nb(fun(nat,A),fun(nat,A),Fa)) ) ).

% summable_zero_power'
tff(fact_3323_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Fa: fun(nat,A)] : summable(A,aTP_Lamp_nc(fun(nat,A),fun(nat,A),Fa)) ) ).

% summable_0_powser
tff(fact_3324_powser__split__head_I3_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Fa: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_nd(fun(nat,A),fun(A,fun(nat,A)),Fa),Z))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_ne(fun(nat,A),fun(A,fun(nat,A)),Fa),Z)) ) ) ).

% powser_split_head(3)
tff(fact_3325_summable__powser__split__head,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Fa: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_nf(fun(nat,A),fun(A,fun(nat,A)),Fa),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_mz(fun(nat,A),fun(A,fun(nat,A)),Fa),Z)) ) ) ).

% summable_powser_split_head
tff(fact_3326_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Fa: fun(nat,A),M: nat,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_ng(fun(nat,A),fun(nat,fun(A,fun(nat,A))),Fa),M),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_mz(fun(nat,A),fun(A,fun(nat,A)),Fa),Z)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_3327_zabs__def,axiom,
    ! [I: int] :
      abs_abs(int,I) = $ite(aa(int,$o,ord_less(int,I),zero_zero(int)),aa(int,int,uminus_uminus(int),I),I) ).

% zabs_def
tff(fact_3328_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => aa(int,$o,ord_less(int,abs_abs(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L))),abs_abs(int,L)) ) ).

% abs_mod_less
tff(fact_3329_dvd__imp__le__int,axiom,
    ! [I: int,D: int] :
      ( ( I != zero_zero(int) )
     => ( aa(int,$o,dvd_dvd(int,D),I)
       => aa(int,$o,ord_less_eq(int,abs_abs(int,D)),abs_abs(int,I)) ) ) ).

% dvd_imp_le_int
tff(fact_3330_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),G: fun(nat,real)] :
          ( ? [N10: nat] :
            ! [N3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,N10),N3)
             => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,Fa,N3))),aa(nat,real,G,N3)) )
         => ( summable(real,G)
           => summable(real,aTP_Lamp_nh(fun(nat,A),fun(nat,real),Fa)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_3331_summable__rabs__comparison__test,axiom,
    ! [Fa: fun(nat,real),G: fun(nat,real)] :
      ( ? [N10: nat] :
        ! [N3: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N10),N3)
         => aa(real,$o,ord_less_eq(real,abs_abs(real,aa(nat,real,Fa,N3))),aa(nat,real,G,N3)) )
     => ( summable(real,G)
       => summable(real,aTP_Lamp_my(fun(nat,real),fun(nat,real),Fa)) ) ) ).

% summable_rabs_comparison_test
tff(fact_3332_pred__numeral__def,axiom,
    ! [K: num] : pred_numeral(K) = aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),K)),one_one(nat)) ).

% pred_numeral_def
tff(fact_3333_abs__integer__code,axiom,
    ! [K: code_integer] :
      abs_abs(code_integer,K) = $ite(aa(code_integer,$o,ord_less(code_integer,K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),K),K) ).

% abs_integer_code
tff(fact_3334_summable__rabs,axiom,
    ! [Fa: fun(nat,real)] :
      ( summable(real,aTP_Lamp_my(fun(nat,real),fun(nat,real),Fa))
     => aa(real,$o,ord_less_eq(real,abs_abs(real,suminf(real,Fa))),suminf(real,aTP_Lamp_my(fun(nat,real),fun(nat,real),Fa))) ) ).

% summable_rabs
tff(fact_3335_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Fa: fun(nat,A),I: nat] :
          ( summable(A,Fa)
         => ( ! [N3: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,Fa,N3))
           => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,Fa,I))
             => aa(A,$o,ord_less(A,zero_zero(A)),suminf(A,Fa)) ) ) ) ) ).

% suminf_pos2
tff(fact_3336_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Fa: fun(nat,A)] :
          ( summable(A,Fa)
         => ( ! [N3: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,Fa,N3))
           => ( aa(A,$o,ord_less(A,zero_zero(A)),suminf(A,Fa))
            <=> ? [I4: nat] : aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,Fa,I4)) ) ) ) ) ).

% suminf_pos_iff
tff(fact_3337_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Fa: fun(nat,A),Xb: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_nd(fun(nat,A),fun(A,fun(nat,A)),Fa),Xb))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xb))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_nd(fun(nat,A),fun(A,fun(nat,A)),Fa),Z)) ) ) ) ).

% powser_inside
tff(fact_3338_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A)] :
          ( summable(A,Fa)
         => ( suminf(A,aTP_Lamp_mw(fun(nat,A),fun(nat,A),Fa)) = aa(A,A,minus_minus(A,suminf(A,Fa)),aa(nat,A,Fa,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_3339_complete__algebra__summable__geometric,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xb)),one_one(real))
         => summable(A,aa(A,fun(nat,A),power_power(A),Xb)) ) ) ).

% complete_algebra_summable_geometric
tff(fact_3340_summable__geometric,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => summable(A,aa(A,fun(nat,A),power_power(A),C2)) ) ) ).

% summable_geometric
tff(fact_3341_zdvd__mult__cancel1,axiom,
    ! [M: int,N: int] :
      ( ( M != zero_zero(int) )
     => ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),M),N)),M)
      <=> ( abs_abs(int,N) = one_one(int) ) ) ) ).

% zdvd_mult_cancel1
tff(fact_3342_summable__norm,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Fa: fun(nat,A)] :
          ( summable(real,aTP_Lamp_mq(fun(nat,A),fun(nat,real),Fa))
         => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,suminf(A,Fa))),suminf(real,aTP_Lamp_mq(fun(nat,A),fun(nat,real),Fa))) ) ) ).

% summable_norm
tff(fact_3343_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),abs_abs(int,K)),L))
    <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_abs_add_iff
tff(fact_3344_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),abs_abs(int,L)))
    <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ) ).

% even_add_abs_iff
tff(fact_3345_powser__split__head_I1_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Fa: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_nd(fun(nat,A),fun(A,fun(nat,A)),Fa),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_nd(fun(nat,A),fun(A,fun(nat,A)),Fa),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Fa,zero_zero(nat))),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ne(fun(nat,A),fun(A,fun(nat,A)),Fa),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_3346_powser__split__head_I2_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Fa: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_nd(fun(nat,A),fun(A,fun(nat,A)),Fa),Z))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ne(fun(nat,A),fun(A,fun(nat,A)),Fa),Z))),Z) = aa(A,A,minus_minus(A,suminf(A,aa(A,fun(nat,A),aTP_Lamp_nd(fun(nat,A),fun(A,fun(nat,A)),Fa),Z))),aa(nat,A,Fa,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_3347_monoseq__realpow,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),one_one(real))
       => topological_monoseq(real,aa(real,fun(nat,real),power_power(real),Xb)) ) ) ).

% monoseq_realpow
tff(fact_3348_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,Fa: fun(nat,A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
         => ( summable(A,Fa)
           => ? [N11: nat] :
              ! [N4: nat] :
                ( aa(nat,$o,ord_less_eq(nat,N11),N4)
               => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_mn(fun(nat,A),fun(nat,fun(nat,A)),Fa),N4)))),R2) ) ) ) ) ).

% suminf_exist_split
tff(fact_3349_summable__power__series,axiom,
    ! [Fa: fun(nat,real),Z: real] :
      ( ! [I2: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,Fa,I2)),one_one(real))
     => ( ! [I2: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,Fa,I2))
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Z)
         => ( aa(real,$o,ord_less(real,Z),one_one(real))
           => summable(real,aa(real,fun(nat,real),aTP_Lamp_ni(fun(nat,real),fun(real,fun(nat,real)),Fa),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_3350_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real,R0: real,A3: fun(nat,A),M8: real] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),R2)
         => ( aa(real,$o,ord_less(real,R2),R0)
           => ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,A3,N3))),aa(nat,real,aa(real,fun(nat,real),power_power(real),R0),N3))),M8)
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_nj(real,fun(fun(nat,A),fun(nat,real)),R2),A3)) ) ) ) ) ).

% Abel_lemma
tff(fact_3351_nat__intermed__int__val,axiom,
    ! [M: nat,N: nat,Fa: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( ( aa(nat,$o,ord_less_eq(nat,M),I2)
            & aa(nat,$o,ord_less(nat,I2),N) )
         => aa(int,$o,ord_less_eq(int,abs_abs(int,aa(int,int,minus_minus(int,aa(nat,int,Fa,aa(nat,nat,suc,I2))),aa(nat,int,Fa,I2)))),one_one(int)) )
     => ( aa(nat,$o,ord_less_eq(nat,M),N)
       => ( aa(int,$o,ord_less_eq(int,aa(nat,int,Fa,M)),K)
         => ( aa(int,$o,ord_less_eq(int,K),aa(nat,int,Fa,N))
           => ? [I2: nat] :
                ( aa(nat,$o,ord_less_eq(nat,M),I2)
                & aa(nat,$o,ord_less_eq(nat,I2),N)
                & ( aa(nat,int,Fa,I2) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_3352_decr__lemma,axiom,
    ! [D: int,Xb: int,Z: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D)
     => aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,Xb),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),abs_abs(int,aa(int,int,minus_minus(int,Xb),Z))),one_one(int))),D))),Z) ) ).

% decr_lemma
tff(fact_3353_incr__lemma,axiom,
    ! [D: int,Z: int,Xb: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D)
     => aa(int,$o,ord_less(int,Z),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),abs_abs(int,aa(int,int,minus_minus(int,Xb),Z))),one_one(int))),D))) ) ).

% incr_lemma
tff(fact_3354_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C2: real,N2: nat,Fa: fun(nat,A)] :
          ( aa(real,$o,ord_less(real,C2),one_one(real))
         => ( ! [N3: nat] :
                ( aa(nat,$o,ord_less_eq(nat,N2),N3)
               => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,Fa,aa(nat,nat,suc,N3)))),aa(real,real,aa(real,fun(real,real),times_times(real),C2),real_V7770717601297561774m_norm(A,aa(nat,A,Fa,N3)))) )
           => summable(A,Fa) ) ) ) ).

% summable_ratio_test
tff(fact_3355_arctan__ubound,axiom,
    ! [Y: real] : aa(real,$o,ord_less(real,aa(real,real,arctan,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% arctan_ubound
tff(fact_3356_arctan__one,axiom,
    aa(real,real,arctan,one_one(real)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% arctan_one
tff(fact_3357_nat__ivt__aux,axiom,
    ! [N: nat,Fa: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),N)
         => aa(int,$o,ord_less_eq(int,abs_abs(int,aa(int,int,minus_minus(int,aa(nat,int,Fa,aa(nat,nat,suc,I2))),aa(nat,int,Fa,I2)))),one_one(int)) )
     => ( aa(int,$o,ord_less_eq(int,aa(nat,int,Fa,zero_zero(nat))),K)
       => ( aa(int,$o,ord_less_eq(int,K),aa(nat,int,Fa,N))
         => ? [I2: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I2),N)
              & ( aa(nat,int,Fa,I2) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_3358_arctan__lbound,axiom,
    ! [Y: real] : aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y)) ).

% arctan_lbound
tff(fact_3359_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y))
      & aa(real,$o,ord_less(real,aa(real,real,arctan,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% arctan_bounded
tff(fact_3360_nat0__intermed__int__val,axiom,
    ! [N: nat,Fa: fun(nat,int),K: int] :
      ( ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),N)
         => aa(int,$o,ord_less_eq(int,abs_abs(int,aa(int,int,minus_minus(int,aa(nat,int,Fa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I2),one_one(nat)))),aa(nat,int,Fa,I2)))),one_one(int)) )
     => ( aa(int,$o,ord_less_eq(int,aa(nat,int,Fa,zero_zero(nat))),K)
       => ( aa(int,$o,ord_less_eq(int,K),aa(nat,int,Fa,N))
         => ? [I2: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I2),N)
              & ( aa(nat,int,Fa,I2) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_3361_arctan__add,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xb)),one_one(real))
     => ( aa(real,$o,ord_less(real,abs_abs(real,Y)),one_one(real))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,arctan,Xb)),aa(real,real,arctan,Y)) = aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)),aa(real,real,minus_minus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)))) ) ) ) ).

% arctan_add
tff(fact_3362_machin__Euler,axiom,
    aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(one2)))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,one2))))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2))))))))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) ).

% machin_Euler
tff(fact_3363_machin,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(one2))))))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))))))) ).

% machin
tff(fact_3364_arctan__double,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xb)),one_one(real))
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,Xb)) = aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),Xb)),aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% arctan_double
tff(fact_3365_tan__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)) != zero_zero(A) )
           => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,tan(A),Xb))),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% tan_double
tff(fact_3366_even__word__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( even_word(A) = dvd_dvd(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) ) ) ).

% even_word_def
tff(fact_3367_triangle__def,axiom,
    ! [N: nat] : nat_triangle(N) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,N))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% triangle_def
tff(fact_3368_exE__realizer,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,$o)),P2: product_prod(B,A),Q: fun(C,$o),Fa: fun(B,fun(A,C))] :
      ( aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(B,A),A,product_snd(B,A),P2)),aa(product_prod(B,A),B,product_fst(B,A),P2))
     => ( ! [X2: B,Y2: A] :
            ( aa(B,$o,aa(A,fun(B,$o),P,Y2),X2)
           => aa(C,$o,Q,aa(A,C,aa(B,fun(A,C),Fa,X2),Y2)) )
       => aa(C,$o,Q,aa(product_prod(B,A),C,product_case_prod(B,A,C,Fa),P2)) ) ) ).

% exE_realizer
tff(fact_3369_VEBT__internal_Ovebt__inserti_H_Ofixp__induct,axiom,
    ! [P: fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),$o)] :
      ( comple1908693960933563346ssible(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),partial_fun_lub(heap_Time_Heap(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_lub(vEBT_VEBTi)),partial_fun_ord(heap_Time_Heap(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_ord(vEBT_VEBTi)),aTP_Lamp_nk(fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),$o),fun(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),$o),P))
     => ( aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),$o,P,aTP_Lamp_nm(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))
       => ( ! [F4: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))))] :
              ( aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),$o,P,F4)
             => aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),$o,P,aTP_Lamp_on(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),F4)) )
         => aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),$o,P,vEBT_V3964819847710782039nserti) ) ) ) ).

% VEBT_internal.vebt_inserti'.fixp_induct
tff(fact_3370_TBOUND__VEBT__case,axiom,
    ! [A: $tType,Tib: vEBT_VEBTi,Fa: fun($o,fun($o,heap_Time_Heap(A))),Bnd: fun($o,fun($o,nat)),F3: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))))),Bnd2: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,nat))))] :
      ( ! [A2: $o,B2: $o] :
          ( ( Tib = vEBT_Leafi((A2),(B2)) )
         => time_TBOUND(A,aa($o,heap_Time_Heap(A),aa($o,fun($o,heap_Time_Heap(A)),Fa,(A2)),(B2)),aa($o,nat,aa($o,fun($o,nat),Bnd,(A2)),(B2))) )
     => ( ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeArray: array(vEBT_VEBTi),Summary: vEBT_VEBTi] :
            ( ( Tib = vEBT_Nodei(Info,Deg,TreeArray,Summary) )
           => time_TBOUND(A,aa(vEBT_VEBTi,heap_Time_Heap(A),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)))),F3,Info),Deg),TreeArray),Summary),aa(vEBT_VEBTi,nat,aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,nat),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,nat)),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,nat))),Bnd2,Info),Deg),TreeArray),Summary)) )
       => time_TBOUND(A,vEBT_case_VEBTi(heap_Time_Heap(A),F3,Fa,Tib),vEBT_case_VEBTi(nat,Bnd2,Bnd,Tib)) ) ) ).

% TBOUND_VEBT_case
tff(fact_3371_tan__pi,axiom,
    aa(real,real,tan(real),pi) = zero_zero(real) ).

% tan_pi
tff(fact_3372_tan__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,tan(A),zero_zero(A)) = zero_zero(A) ) ) ).

% tan_zero
tff(fact_3373_triangle__0,axiom,
    nat_triangle(zero_zero(nat)) = zero_zero(nat) ).

% triangle_0
tff(fact_3374_tan__npi,axiom,
    ! [N: nat] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi)) = zero_zero(real) ).

% tan_npi
tff(fact_3375_tan__periodic,axiom,
    ! [Xb: real] : aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = aa(real,real,tan(real),Xb) ).

% tan_periodic
tff(fact_3376_VEBTi_Osimps_I5_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),F2: fun($o,fun($o,A)),X11: option(product_prod(nat,nat)),X12: nat,X13: array(vEBT_VEBTi),X14: vEBT_VEBTi] : vEBT_case_VEBTi(A,F1,F2,vEBT_Nodei(X11,X12,X13,X14)) = aa(vEBT_VEBTi,A,aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,A),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A))),F1,X11),X12),X13),X14) ).

% VEBTi.simps(5)
tff(fact_3377_VEBTi_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] : aa(vEBT_VEBTi,nat,size_size(vEBT_VEBTi),vEBT_Leafi((X21),(X222))) = zero_zero(nat) ).

% VEBTi.size(4)
tff(fact_3378_vebt__assn__raw_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,vEBT_VEBTi)] :
      ( ! [A2: $o,B2: $o,Ai: $o,Bi: $o] : Xb != aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),vEBT_Leaf((A2),(B2))),vEBT_Leafi((Ai),(Bi)))
     => ( ! [Mmo: option(product_prod(nat,nat)),Deg: nat,Tree_list: list(vEBT_VEBT),Summary: vEBT_VEBT,Mmoi: option(product_prod(nat,nat)),Degi: nat,Tree_array: array(vEBT_VEBTi),Summaryi: vEBT_VEBTi] : Xb != aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),vEBT_Node(Mmo,Deg,Tree_list,Summary)),vEBT_Nodei(Mmoi,Degi,Tree_array,Summaryi))
       => ( ! [V3: option(product_prod(nat,nat)),Va2: nat,Vb3: list(vEBT_VEBT),Vc3: vEBT_VEBT,Vd3: $o,Ve3: $o] : Xb != aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),vEBT_Node(V3,Va2,Vb3,Vc3)),vEBT_Leafi((Vd3),(Ve3)))
         => ~ ! [Vd3: $o,Ve3: $o,V3: option(product_prod(nat,nat)),Va2: nat,Vb3: array(vEBT_VEBTi),Vc3: vEBT_VEBTi] : Xb != aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),vEBT_Leaf((Vd3),(Ve3))),vEBT_Nodei(V3,Va2,Vb3,Vc3)) ) ) ) ).

% vebt_assn_raw.cases
tff(fact_3379_VEBT__internal_OminNulli_Ocases,axiom,
    ! [Xb: vEBT_VEBTi] :
      ( ( Xb != vEBT_Leafi($false,$false) )
     => ( ! [Uv: $o] : Xb != vEBT_Leafi($true,(Uv))
       => ( ! [Uu2: $o] : Xb != vEBT_Leafi((Uu2),$true)
         => ( ! [Uw: nat,Ux2: array(vEBT_VEBTi),Uy2: vEBT_VEBTi] : Xb != vEBT_Nodei(none(product_prod(nat,nat)),Uw,Ux2,Uy2)
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: array(vEBT_VEBTi),Vc2: vEBT_VEBTi] : Xb != vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) ) ) ) ) ).

% VEBT_internal.minNulli.cases
tff(fact_3380_VEBTi_Osimps_I6_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),F2: fun($o,fun($o,A)),X21: $o,X222: $o] : vEBT_case_VEBTi(A,F1,F2,vEBT_Leafi((X21),(X222))) = aa($o,A,aa($o,fun($o,A),F2,(X21)),(X222)) ).

% VEBTi.simps(6)
tff(fact_3381_vebt__minti_Ocases,axiom,
    ! [Xb: vEBT_VEBTi] :
      ( ! [A2: $o,B2: $o] : Xb != vEBT_Leafi((A2),(B2))
     => ( ! [Uu2: nat,Uv: array(vEBT_VEBTi),Uw: vEBT_VEBTi] : Xb != vEBT_Nodei(none(product_prod(nat,nat)),Uu2,Uv,Uw)
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array(vEBT_VEBTi),Uz2: vEBT_VEBTi] : Xb != vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) ) ) ).

% vebt_minti.cases
tff(fact_3382_TBOUND__upd,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: nat,I: A,Xb: array(A)] : time_TBOUND(array(A),array_upd(A,Xs,I,Xb),one_one(nat)) ) ).

% TBOUND_upd
tff(fact_3383_VEBT__internal_OminNulli_Oelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_Time_Heap($o)] :
      ( ( vEBT_VEBT_minNulli(Xb) = Y )
     => ( ( ( Xb = vEBT_Leafi($false,$false) )
         => ( Y != aa($o,heap_Time_Heap($o),heap_Time_return($o),$true) ) )
       => ( ( ? [Uv: $o] : Xb = vEBT_Leafi($true,(Uv))
           => ( Y != aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) ) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leafi((Uu2),$true)
             => ( Y != aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) ) )
           => ( ( ? [Uw: nat,Ux2: array(vEBT_VEBTi),Uy2: vEBT_VEBTi] : Xb = vEBT_Nodei(none(product_prod(nat,nat)),Uw,Ux2,Uy2)
               => ( Y != aa($o,heap_Time_Heap($o),heap_Time_return($o),$true) ) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va3: nat,Vb2: array(vEBT_VEBTi),Vc2: vEBT_VEBTi] : Xb = vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)
                 => ( Y != aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.elims
tff(fact_3384_VEBT__internal_Ovebt__buildupi_H_Osimps_I1_J,axiom,
    vEBT_V739175172307565963ildupi(zero_zero(nat)) = aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi($false,$false)) ).

% VEBT_internal.vebt_buildupi'.simps(1)
tff(fact_3385_vebt__buildupi_Osimps_I1_J,axiom,
    vEBT_vebt_buildupi(zero_zero(nat)) = aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi($false,$false)) ).

% vebt_buildupi.simps(1)
tff(fact_3386_VEBT__internal_Ovebt__buildupi_H_Osimps_I2_J,axiom,
    vEBT_V739175172307565963ildupi(aa(nat,nat,suc,zero_zero(nat))) = aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi($false,$false)) ).

% VEBT_internal.vebt_buildupi'.simps(2)
tff(fact_3387_vebt__buildupi_Osimps_I2_J,axiom,
    vEBT_vebt_buildupi(aa(nat,nat,suc,zero_zero(nat))) = aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi($false,$false)) ).

% vebt_buildupi.simps(2)
tff(fact_3388_VEBT__internal_OminNulli_Osimps_I5_J,axiom,
    ! [Uz: product_prod(nat,nat),Va: nat,Vb: array(vEBT_VEBTi),Vc: vEBT_VEBTi] : vEBT_VEBT_minNulli(vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Va,Vb,Vc)) = aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) ).

% VEBT_internal.minNulli.simps(5)
tff(fact_3389_tan__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X3: A] : aa(A,A,tan(A),X3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,X3)),cos(A,X3)) ) ).

% tan_def
tff(fact_3390_vebt__maxti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: array(vEBT_VEBTi),Uw2: vEBT_VEBTi] : vEBT_vebt_maxti(vEBT_Nodei(none(product_prod(nat,nat)),Uu,Uv2,Uw2)) = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)) ).

% vebt_maxti.simps(2)
tff(fact_3391_vebt__minti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv2: array(vEBT_VEBTi),Uw2: vEBT_VEBTi] : vEBT_vebt_minti(vEBT_Nodei(none(product_prod(nat,nat)),Uu,Uv2,Uw2)) = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)) ).

% vebt_minti.simps(2)
tff(fact_3392_vebt__maxti_Oelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_Time_Heap(option(nat))] :
      ( ( vEBT_vebt_maxti(Xb) = Y )
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leafi((A2),(B2)) )
           => ( Y != $ite(
                  (B2),
                  aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),one_one(nat))),
                  $ite((A2),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),zero_zero(nat))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))) ) ) )
       => ( ( ? [Uu2: nat,Uv: array(vEBT_VEBTi),Uw: vEBT_VEBTi] : Xb = vEBT_Nodei(none(product_prod(nat,nat)),Uu2,Uv,Uw)
           => ( Y != aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)) ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: array(vEBT_VEBTi),Uz2: vEBT_VEBTi] : Xb = vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),Ma2)) ) ) ) ) ) ).

% vebt_maxti.elims
tff(fact_3393_vebt__minti_Oelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_Time_Heap(option(nat))] :
      ( ( vEBT_vebt_minti(Xb) = Y )
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leafi((A2),(B2)) )
           => ( Y != $ite(
                  (A2),
                  aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),zero_zero(nat))),
                  $ite((B2),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),one_one(nat))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))) ) ) )
       => ( ( ? [Uu2: nat,Uv: array(vEBT_VEBTi),Uw: vEBT_VEBTi] : Xb = vEBT_Nodei(none(product_prod(nat,nat)),Uu2,Uv,Uw)
           => ( Y != aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)) ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: array(vEBT_VEBTi),Uz2: vEBT_VEBTi] : Xb = vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)
               => ( Y != aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),Mi2)) ) ) ) ) ) ).

% vebt_minti.elims
tff(fact_3394_vebt__maxti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: array(vEBT_VEBTi),Uz: vEBT_VEBTi] : vEBT_vebt_maxti(vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux,Uy,Uz)) = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),Ma)) ).

% vebt_maxti.simps(3)
tff(fact_3395_vebt__minti_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: array(vEBT_VEBTi),Uz: vEBT_VEBTi] : vEBT_vebt_minti(vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux,Uy,Uz)) = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),Mi)) ).

% vebt_minti.simps(3)
tff(fact_3396_tan__45,axiom,
    aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) = one_one(real) ).

% tan_45
tff(fact_3397_vebt__maxti_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      vEBT_vebt_maxti(vEBT_Leafi((A3),(B3))) = $ite(
        (B3),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),one_one(nat))),
        $ite((A3),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),zero_zero(nat))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))) ) ).

% vebt_maxti.simps(1)
tff(fact_3398_vebt__minti_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] :
      vEBT_vebt_minti(vEBT_Leafi((A3),(B3))) = $ite(
        (A3),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),zero_zero(nat))),
        $ite((B3),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),one_one(nat))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))) ) ).

% vebt_minti.simps(1)
tff(fact_3399_lemma__tan__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
     => ? [X2: real] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),X2)
          & aa(real,$o,ord_less(real,X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
          & aa(real,$o,ord_less(real,Y),aa(real,real,tan(real),X2)) ) ) ).

% lemma_tan_total
tff(fact_3400_tan__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,tan(real),Xb)) ) ) ).

% tan_gt_zero
tff(fact_3401_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X2)
      & aa(real,$o,ord_less(real,X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),X2) = Y ) ) ).

% lemma_tan_total1
tff(fact_3402_tan__mono__lt__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,ord_less(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,ord_less(real,aa(real,real,tan(real),Xb)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,ord_less(real,Xb),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_3403_tan__monotone_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,ord_less(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
         => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,ord_less(real,Y),Xb)
            <=> aa(real,$o,ord_less(real,aa(real,real,tan(real),Y)),aa(real,real,tan(real),Xb)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_3404_tan__monotone,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
     => ( aa(real,$o,ord_less(real,Y),Xb)
       => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,ord_less(real,aa(real,real,tan(real),Y)),aa(real,real,tan(real),Xb)) ) ) ) ).

% tan_monotone
tff(fact_3405_tan__total,axiom,
    ! [Y: real] :
    ? [X2: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),X2)
      & aa(real,$o,ord_less(real,X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),X2) = Y )
      & ! [Y4: real] :
          ( ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y4)
            & aa(real,$o,ord_less(real,Y4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
            & ( aa(real,real,tan(real),Y4) = Y ) )
         => ( Y4 = X2 ) ) ) ).

% tan_total
tff(fact_3406_tan__minus__45,axiom,
    aa(real,real,tan(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% tan_minus_45
tff(fact_3407_tan__inverse,axiom,
    ! [Y: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,tan(real),Y)) = aa(real,real,tan(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)) ).

% tan_inverse
tff(fact_3408_vebt__inserti_Omono,axiom,
    ! [Xb: product_prod(vEBT_VEBTi,nat)] : comple7038119648293358887notone(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),heap_Time_Heap(vEBT_VEBTi),partial_fun_ord(heap_Time_Heap(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_ord(vEBT_VEBTi)),heap_Time_Heap_ord(vEBT_VEBTi),aTP_Lamp_pc(product_prod(vEBT_VEBTi,nat),fun(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),heap_Time_Heap(vEBT_VEBTi)),Xb)) ).

% vebt_inserti.mono
tff(fact_3409_add__tan__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Y))) ) ) ) ) ).

% add_tan_eq
tff(fact_3410_exE__realizer_H,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),P2: product_prod(B,A)] :
      ( aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(B,A),A,product_snd(B,A),P2)),aa(product_prod(B,A),B,product_fst(B,A),P2))
     => ~ ! [X2: B,Y2: A] : ~ aa(B,$o,aa(A,fun(B,$o),P,Y2),X2) ) ).

% exE_realizer'
tff(fact_3411_vebt__inserti_Oraw__induct,axiom,
    ! [P: fun(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)))))),Xaa: product_prod(vEBT_VEBTi,nat),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: vEBT_VEBTi,N: nat] :
      ( ! [Vebt_inserti: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))] :
          ( ! [A6: vEBT_VEBTi,B5: nat,H3: heap_ext(product_unit),H4: heap_ext(product_unit),R3: vEBT_VEBTi,N4: nat] :
              ( heap_Time_effect(vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),Vebt_inserti,A6),B5),H3,H4,R3,N4)
             => aa(nat,$o,aa(vEBT_VEBTi,fun(nat,$o),aa(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o))),aa(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)))),aa(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o))))),P,A6),B5),H3),H4),R3),N4) )
         => ! [T2: vEBT_VEBTi,X2: nat,Ta: heap_ext(product_unit),Xa2: heap_ext(product_unit),R4: vEBT_VEBTi,N3: nat] :
              ( heap_Time_effect(vEBT_VEBTi,vEBT_case_VEBTi(heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_pp(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),Vebt_inserti),X2),aTP_Lamp_om(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),X2),T2),Ta,Xa2,R4,N3)
             => aa(nat,$o,aa(vEBT_VEBTi,fun(nat,$o),aa(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o))),aa(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)))),aa(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o))))),P,T2),X2),Ta),Xa2),R4),N3) ) )
     => ( heap_Time_effect(vEBT_VEBTi,aa(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi),product_case_prod(vEBT_VEBTi,nat,heap_Time_Heap(vEBT_VEBTi),vEBT_vebt_inserti),Xaa),H,H2,R2,N)
       => aa(nat,$o,aa(vEBT_VEBTi,fun(nat,$o),aa(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o))),aa(product_prod(vEBT_VEBTi,nat),fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)))),product_case_prod(vEBT_VEBTi,nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)))),P),Xaa),H),H2),R2),N) ) ) ).

% vebt_inserti.raw_induct
tff(fact_3412_vebt__inserti_Osimps,axiom,
    ! [Tb: vEBT_VEBTi,Xb: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),vEBT_vebt_inserti,Tb),Xb) = vEBT_case_VEBTi(heap_Time_Heap(vEBT_VEBTi),aTP_Lamp_qc(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),Xb),aTP_Lamp_om(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),Xb),Tb) ).

% vebt_inserti.simps
tff(fact_3413_vebt__inserti_Ofixp__induct,axiom,
    ! [P: fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),$o)] :
      ( comple1908693960933563346ssible(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),partial_fun_lub(heap_Time_Heap(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_lub(vEBT_VEBTi)),partial_fun_ord(heap_Time_Heap(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_ord(vEBT_VEBTi)),aTP_Lamp_qd(fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),$o),fun(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),$o),P))
     => ( aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),$o,P,aTP_Lamp_qe(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))))
       => ( ! [F4: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))] :
              ( aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),$o,P,F4)
             => aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),$o,P,aTP_Lamp_qf(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),F4)) )
         => aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),$o,P,vEBT_vebt_inserti) ) ) ) ).

% vebt_inserti.fixp_induct
tff(fact_3414_tan__pos__pi2__le,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,tan(real),Xb)) ) ) ).

% tan_pos_pi2_le
tff(fact_3415_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ? [X2: real] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X2)
          & aa(real,$o,ord_less(real,X2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
          & ( aa(real,real,tan(real),X2) = Y ) ) ) ).

% tan_total_pos
tff(fact_3416_tan__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),zero_zero(real))
       => aa(real,$o,ord_less(real,aa(real,real,tan(real),Xb)),zero_zero(real)) ) ) ).

% tan_less_zero
tff(fact_3417_tan__mono__le,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),Y)
       => ( aa(real,$o,ord_less(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,ord_less_eq(real,aa(real,real,tan(real),Xb)),aa(real,real,tan(real),Y)) ) ) ) ).

% tan_mono_le
tff(fact_3418_tan__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Y)
         => ( aa(real,$o,ord_less(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,ord_less_eq(real,aa(real,real,tan(real),Xb)),aa(real,real,tan(real),Y))
            <=> aa(real,$o,ord_less_eq(real,Xb),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_3419_tan__bound__pi2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
     => aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,tan(real),Xb))),one_one(real)) ) ).

% tan_bound_pi2
tff(fact_3420_arctan,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arctan,Y))
      & aa(real,$o,ord_less(real,aa(real,real,arctan,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( aa(real,real,tan(real),aa(real,real,arctan,Y)) = Y ) ) ).

% arctan
tff(fact_3421_arctan__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,real,arctan,aa(real,real,tan(real),Xb)) = Xb ) ) ) ).

% arctan_tan
tff(fact_3422_arctan__unique,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( ( aa(real,real,tan(real),Xb) = Y )
         => ( aa(real,real,arctan,Y) = Xb ) ) ) ) ).

% arctan_unique
tff(fact_3423_lemma__tan__add1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Y))) ) ) ) ) ).

% lemma_tan_add1
tff(fact_3424_tan__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,minus_minus(A,Xb),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,minus_minus(A,Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y))),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_3425_tan__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( ( cos(A,Y) != zero_zero(A) )
           => ( ( cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) != zero_zero(A) )
             => ( aa(A,A,tan(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y))),aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tan(A),Xb)),aa(A,A,tan(A),Y)))) ) ) ) ) ) ).

% tan_add
tff(fact_3426_tan__total__pi4,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xb)),one_one(real))
     => ? [Z2: real] :
          ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))),Z2)
          & aa(real,$o,ord_less(real,Z2),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
          & ( aa(real,real,tan(real),Z2) = Xb ) ) ) ).

% tan_total_pi4
tff(fact_3427_disjE__realizer2,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: fun(A,$o),Xb: option(A),R: fun(B,$o),Fa: B,G: fun(A,B)] :
      ( case_option($o,A,(P),Q,Xb)
     => ( ( (P)
         => aa(B,$o,R,Fa) )
       => ( ! [Q3: A] :
              ( aa(A,$o,Q,Q3)
             => aa(B,$o,R,aa(A,B,G,Q3)) )
         => aa(B,$o,R,case_option(B,A,Fa,G,Xb)) ) ) ) ).

% disjE_realizer2
tff(fact_3428_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,tan(A),Xb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb))),aa(A,A,aa(A,fun(A,A),plus_plus(A),cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb))),one_one(A))) ) ).

% tan_half
tff(fact_3429_exI__realizer,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Y: A,Xb: B] :
      ( aa(B,$o,aa(A,fun(B,$o),P,Y),Xb)
     => aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(B,A),A,product_snd(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),Y))),aa(product_prod(B,A),B,product_fst(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),Y))) ) ).

% exI_realizer
tff(fact_3430_conjI__realizer,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),P2: A,Q: fun(B,$o),Q2: B] :
      ( aa(A,$o,P,P2)
     => ( aa(B,$o,Q,Q2)
       => ( aa(A,$o,P,aa(product_prod(A,B),A,product_fst(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P2),Q2)))
          & aa(B,$o,Q,aa(product_prod(A,B),B,product_snd(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),P2),Q2))) ) ) ) ).

% conjI_realizer
tff(fact_3431_VEBT__internal_Ovebt__inserti_H_Omono,axiom,
    ! [Xb: product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat)] : comple7038119648293358887notone(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),heap_Time_Heap(vEBT_VEBTi),partial_fun_ord(heap_Time_Heap(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_ord(vEBT_VEBTi)),heap_Time_Heap_ord(vEBT_VEBTi),aTP_Lamp_re(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),fun(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),heap_Time_Heap(vEBT_VEBTi)),Xb)) ).

% VEBT_internal.vebt_inserti'.mono
tff(fact_3432_VEBT__internal_Ovebt__inserti_H_Oraw__induct,axiom,
    ! [P: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o))))))),Xaa: product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: vEBT_VEBTi,N: nat] :
      ( ! [Vebt_inserti2: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))))] :
          ( ! [A6: vEBT_VEBT,B5: vEBT_VEBTi,Ba: nat,H3: heap_ext(product_unit),H4: heap_ext(product_unit),R3: vEBT_VEBTi,N4: nat] :
              ( heap_Time_effect(vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Vebt_inserti2,A6),B5),Ba),H3,H4,R3,N4)
             => aa(nat,$o,aa(vEBT_VEBTi,fun(nat,$o),aa(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o))),aa(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)))),aa(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o))))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)))))),P,A6),B5),Ba),H3),H4),R3),N4) )
         => ! [T2: vEBT_VEBT,Ti2: vEBT_VEBTi,X2: nat,Ta: heap_ext(product_unit),Tia: heap_ext(product_unit),Xa2: vEBT_VEBTi,N3: nat] :
              ( heap_Time_effect(vEBT_VEBTi,vEBT_case_VEBTi(heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_ol(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),Vebt_inserti2),T2),X2),aTP_Lamp_om(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),X2),Ti2),Ta,Tia,Xa2,N3)
             => aa(nat,$o,aa(vEBT_VEBTi,fun(nat,$o),aa(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o))),aa(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)))),aa(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o))))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)))))),P,T2),Ti2),X2),Ta),Tia),Xa2),N3) ) )
     => ( heap_Time_effect(vEBT_VEBTi,aa(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi),product_case_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(vEBT_VEBTi),product_case_prod(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),vEBT_V3964819847710782039nserti)),Xaa),H,H2,R2,N)
       => aa(nat,$o,aa(vEBT_VEBTi,fun(nat,$o),aa(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o))),aa(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)))),product_case_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o)))),product_case_prod(vEBT_VEBT,vEBT_VEBTi,fun(nat,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(vEBT_VEBTi,fun(nat,$o))))),P)),Xaa),H),H2),R2),N) ) ) ).

% VEBT_internal.vebt_inserti'.raw_induct
tff(fact_3433_VEBT__internal_Ovebt__inserti_H_Osimps,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),vEBT_V3964819847710782039nserti,Tb),Tib),Xb) = vEBT_case_VEBTi(heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_sb(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),Tb),Xb),aTP_Lamp_om(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),Xb),Tib) ).

% VEBT_internal.vebt_inserti'.simps
tff(fact_3434_VEBT__internal_Ovebt__buildupi_H_Oelims,axiom,
    ! [Xb: nat,Y: heap_Time_Heap(vEBT_VEBTi)] :
      ( ( vEBT_V739175172307565963ildupi(Xb) = Y )
     => ( ( ( Xb = zero_zero(nat) )
         => ( Y != aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi($false,$false)) ) )
       => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi($false,$false)) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
               => ( Y != $ite(
                      aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        heap_Time_bind(list(vEBT_VEBTi),vEBT_VEBTi,vEBT_VEBT_replicatei(vEBT_VEBTi,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_V739175172307565963ildupi(half)),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sf(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va2),half)) ),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        heap_Time_bind(list(vEBT_VEBTi),vEBT_VEBTi,vEBT_VEBT_replicatei(vEBT_VEBTi,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_V739175172307565963ildupi(half)),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_si(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va2),half)) ) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.elims
tff(fact_3435_VEBT__internal_Ovebt__buildupi_H_Osimps_I3_J,axiom,
    ! [Va: nat] :
      vEBT_V739175172307565963ildupi(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = $ite(
        aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          heap_Time_bind(list(vEBT_VEBTi),vEBT_VEBTi,vEBT_VEBT_replicatei(vEBT_VEBTi,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_V739175172307565963ildupi(half)),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sf(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va),half)) ),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          heap_Time_bind(list(vEBT_VEBTi),vEBT_VEBTi,vEBT_VEBT_replicatei(vEBT_VEBTi,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_V739175172307565963ildupi(half)),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_si(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va),half)) ) ) ).

% VEBT_internal.vebt_buildupi'.simps(3)
tff(fact_3436_vebt__buildupi_Oelims,axiom,
    ! [Xb: nat,Y: heap_Time_Heap(vEBT_VEBTi)] :
      ( ( vEBT_vebt_buildupi(Xb) = Y )
     => ( ( ( Xb = zero_zero(nat) )
         => ( Y != aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi($false,$false)) ) )
       => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi($false,$false)) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
               => ( Y != $ite(
                      aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        heap_Time_bind(list(vEBT_VEBTi),vEBT_VEBTi,vEBT_VEBT_replicatei(vEBT_VEBTi,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildupi(half)),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sk(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va2),half)) ),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        heap_Time_bind(list(vEBT_VEBTi),vEBT_VEBTi,vEBT_VEBT_replicatei(vEBT_VEBTi,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildupi(half)),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sm(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va2),half)) ) ) ) ) ) ) ) ).

% vebt_buildupi.elims
tff(fact_3437_vebt__buildupi_Osimps_I3_J,axiom,
    ! [Va: nat] :
      vEBT_vebt_buildupi(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = $ite(
        aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          heap_Time_bind(list(vEBT_VEBTi),vEBT_VEBTi,vEBT_VEBT_replicatei(vEBT_VEBTi,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildupi(half)),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sk(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va),half)) ),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          heap_Time_bind(list(vEBT_VEBTi),vEBT_VEBTi,vEBT_VEBT_replicatei(vEBT_VEBTi,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildupi(half)),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sm(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va),half)) ) ) ).

% vebt_buildupi.simps(3)
tff(fact_3438_in__set__enumerate__eq,axiom,
    ! [A: $tType,P2: product_prod(nat,A),N: nat,Xs: list(A)] :
      ( member(product_prod(nat,A),P2,set2(product_prod(nat,A),enumerate(A,N,Xs)))
    <=> ( aa(nat,$o,ord_less_eq(nat,N),aa(product_prod(nat,A),nat,product_fst(nat,A),P2))
        & aa(nat,$o,ord_less(nat,aa(product_prod(nat,A),nat,product_fst(nat,A),P2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N))
        & ( aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,aa(product_prod(nat,A),nat,product_fst(nat,A),P2)),N)) = aa(product_prod(nat,A),A,product_snd(nat,A),P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_3439_length__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),nat,size_size(list(product_prod(nat,A))),enumerate(A,N,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_enumerate
tff(fact_3440_map__snd__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : map(product_prod(nat,A),A,product_snd(nat,A),enumerate(A,N,Xs)) = Xs ).

% map_snd_enumerate
tff(fact_3441_TBOUND__of__list,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A)] : time_TBOUND(array(A),array_of_list(A,Xs),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))) ) ).

% TBOUND_of_list
tff(fact_3442_time__array__of__list,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A),H: heap_ext(product_unit)] : time_time(array(A),array_of_list(A,Xs),H) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ) ).

% time_array_of_list
tff(fact_3443_nth__enumerate__eq,axiom,
    ! [A: $tType,M: nat,Xs: list(A),N: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,product_prod(nat,A),nth(product_prod(nat,A),enumerate(A,N,Xs)),M) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),aa(nat,A,nth(A,Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_3444_VEBT__internal_Ovebt__buildupi_H_Opelims,axiom,
    ! [Xb: nat,Y: heap_Time_Heap(vEBT_VEBTi)] :
      ( ( vEBT_V739175172307565963ildupi(Xb) = Y )
     => ( accp(nat,vEBT_V254170901696579886pi_rel,Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi($false,$false)) )
             => ~ accp(nat,vEBT_V254170901696579886pi_rel,zero_zero(nat)) ) )
         => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi($false,$false)) )
               => ~ accp(nat,vEBT_V254170901696579886pi_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                 => ( ( Y = $ite(
                          aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            heap_Time_bind(list(vEBT_VEBTi),vEBT_VEBTi,vEBT_VEBT_replicatei(vEBT_VEBTi,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_V739175172307565963ildupi(half)),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sf(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va2),half)) ),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            heap_Time_bind(list(vEBT_VEBTi),vEBT_VEBTi,vEBT_VEBT_replicatei(vEBT_VEBTi,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_V739175172307565963ildupi(half)),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_si(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va2),half)) ) ) )
                   => ~ accp(nat,vEBT_V254170901696579886pi_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.pelims
tff(fact_3445_VEBTi_Osize_I3_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: array(vEBT_VEBTi),X14: vEBT_VEBTi] : aa(vEBT_VEBTi,nat,size_size(vEBT_VEBTi),vEBT_Nodei(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_array(vEBT_VEBTi,size_size(vEBT_VEBTi),X13)),aa(vEBT_VEBTi,nat,size_size(vEBT_VEBTi),X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBTi.size(3)
tff(fact_3446_vebt__buildupi_Opelims,axiom,
    ! [Xb: nat,Y: heap_Time_Heap(vEBT_VEBTi)] :
      ( ( vEBT_vebt_buildupi(Xb) = Y )
     => ( accp(nat,vEBT_v1230518104690509829pi_rel,Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi($false,$false)) )
             => ~ accp(nat,vEBT_v1230518104690509829pi_rel,zero_zero(nat)) ) )
         => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi($false,$false)) )
               => ~ accp(nat,vEBT_v1230518104690509829pi_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                 => ( ( Y = $ite(
                          aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            heap_Time_bind(list(vEBT_VEBTi),vEBT_VEBTi,vEBT_VEBT_replicatei(vEBT_VEBTi,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildupi(half)),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sk(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va2),half)) ),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            heap_Time_bind(list(vEBT_VEBTi),vEBT_VEBTi,vEBT_VEBT_replicatei(vEBT_VEBTi,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildupi(half)),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sm(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va2),half)) ) ) )
                   => ~ accp(nat,vEBT_v1230518104690509829pi_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildupi.pelims
tff(fact_3447_sin__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( sin(real,Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,tan(real),Xb)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,tan(real),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% sin_tan
tff(fact_3448_cos__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( cos(real,Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,tan(real),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% cos_tan
tff(fact_3449_real__sqrt__eq__zero__cancel__iff,axiom,
    ! [Xb: real] :
      ( ( aa(real,real,sqrt,Xb) = zero_zero(real) )
    <=> ( Xb = zero_zero(real) ) ) ).

% real_sqrt_eq_zero_cancel_iff
tff(fact_3450_real__sqrt__zero,axiom,
    aa(real,real,sqrt,zero_zero(real)) = zero_zero(real) ).

% real_sqrt_zero
tff(fact_3451_real__sqrt__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,ord_less(real,Xb),Y) ) ).

% real_sqrt_less_iff
tff(fact_3452_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),Y) ) ).

% real_sqrt_gt_0_iff
tff(fact_3453_real__sqrt__lt__0__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,sqrt,Xb)),zero_zero(real))
    <=> aa(real,$o,ord_less(real,Xb),zero_zero(real)) ) ).

% real_sqrt_lt_0_iff
tff(fact_3454_real__sqrt__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,Xb)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,Xb),zero_zero(real)) ) ).

% real_sqrt_le_0_iff
tff(fact_3455_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_3456_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,ord_less(real,one_one(real)),Y) ) ).

% real_sqrt_gt_1_iff
tff(fact_3457_real__sqrt__lt__1__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,sqrt,Xb)),one_one(real))
    <=> aa(real,$o,ord_less(real,Xb),one_one(real)) ) ).

% real_sqrt_lt_1_iff
tff(fact_3458_real__sqrt__four,axiom,
    aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(bit0(one2)))) = aa(num,real,numeral_numeral(real),bit0(one2)) ).

% real_sqrt_four
tff(fact_3459_real__sqrt__abs,axiom,
    ! [Xb: real] : aa(real,real,sqrt,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = abs_abs(real,Xb) ).

% real_sqrt_abs
tff(fact_3460_real__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xb ) ) ).

% real_sqrt_pow2
tff(fact_3461_real__sqrt__pow2__iff,axiom,
    ! [Xb: real] :
      ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xb )
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb) ) ).

% real_sqrt_pow2_iff
tff(fact_3462_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [Xb: real,Y: real,Xaa: real,Ya: real] : aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xaa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xaa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% real_sqrt_sum_squares_mult_squared_eq
tff(fact_3463_real__sqrt__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,Xb),Y)
     => aa(real,$o,ord_less(real,aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_less_mono
tff(fact_3464_real__sqrt__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,sqrt,Xb)) ) ).

% real_sqrt_gt_zero
tff(fact_3465_real__sqrt__eq__zero__cancel,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( ( aa(real,real,sqrt,Xb) = zero_zero(real) )
       => ( Xb = zero_zero(real) ) ) ) ).

% real_sqrt_eq_zero_cancel
tff(fact_3466_real__sqrt__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,sqrt,Xb)) ) ).

% real_sqrt_ge_zero
tff(fact_3467_real__div__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),aa(real,real,sqrt,Xb)) = aa(real,real,sqrt,Xb) ) ) ).

% real_div_sqrt
tff(fact_3468_sqrt__add__le__add__sqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y))) ) ) ).

% sqrt_add_le_add_sqrt
tff(fact_3469_sqrt2__less__2,axiom,
    aa(real,$o,ord_less(real,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sqrt2_less_2
tff(fact_3470_real__less__rsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)
     => aa(real,$o,ord_less(real,Xb),aa(real,real,sqrt,Y)) ) ).

% real_less_rsqrt
tff(fact_3471_sqrt__le__D,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,Xb)),Y)
     => aa(real,$o,ord_less_eq(real,Xb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% sqrt_le_D
tff(fact_3472_real__le__rsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y)
     => aa(real,$o,ord_less_eq(real,Xb),aa(real,real,sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_3473_real__le__lsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less_eq(real,Xb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,Xb)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_3474_real__sqrt__unique,axiom,
    ! [Y: real,Xb: real] :
      ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Xb )
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( aa(real,real,sqrt,Xb) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_3475_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),U)
     => aa(real,$o,ord_less(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2))))),U) ) ).

% lemma_real_divide_sqrt_less
tff(fact_3476_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) = Xb )
     => ( Y = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel
tff(fact_3477_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) = Y )
     => ( Xb = zero_zero(real) ) ) ).

% real_sqrt_sum_squares_eq_cancel2
tff(fact_3478_real__sqrt__sum__squares__ge1,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,ord_less_eq(real,Xb),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_sum_squares_ge1
tff(fact_3479_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,Xb: real] : aa(real,$o,ord_less_eq(real,Y),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_sum_squares_ge2
tff(fact_3480_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A3: real,C2: real,B3: real,D: real] : aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),A3),C2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),B3),D)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),C2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_3481_sqrt__ge__absD,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xb)),aa(real,real,sqrt,Y))
     => aa(real,$o,ord_less_eq(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Y) ) ).

% sqrt_ge_absD
tff(fact_3482_cos__45,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% cos_45
tff(fact_3483_sin__45,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sin_45
tff(fact_3484_tan__60,axiom,
    aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))) ).

% tan_60
tff(fact_3485_real__less__lsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less(real,Xb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))
         => aa(real,$o,ord_less(real,aa(real,real,sqrt,Xb)),Y) ) ) ) ).

% real_less_lsqrt
tff(fact_3486_sqrt__sum__squares__le__sum,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)) ) ) ).

% sqrt_sum_squares_le_sum
tff(fact_3487_sqrt__even__pow2,axiom,
    ! [N: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
     => ( aa(real,real,sqrt,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% sqrt_even_pow2
tff(fact_3488_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),abs_abs(real,Xb)),abs_abs(real,Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_3489_real__sqrt__ge__abs2,axiom,
    ! [Y: real,Xb: real] : aa(real,$o,ord_less_eq(real,abs_abs(real,Y)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_ge_abs2
tff(fact_3490_real__sqrt__ge__abs1,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,ord_less_eq(real,abs_abs(real,Xb)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% real_sqrt_ge_abs1
tff(fact_3491_ln__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,real,ln_ln(real),aa(real,real,sqrt,Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Xb)),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% ln_sqrt
tff(fact_3492_cos__30,axiom,
    cos(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% cos_30
tff(fact_3493_sin__60,axiom,
    sin(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% sin_60
tff(fact_3494_arsinh__real__def,axiom,
    ! [Xb: real] : aa(real,real,arsinh(real),Xb) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ).

% arsinh_real_def
tff(fact_3495_real__sqrt__power__even,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,sqrt,Xb)),N) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_3496_arsinh__real__aux,axiom,
    ! [Xb: real] : aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ).

% arsinh_real_aux
tff(fact_3497_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xb: real,Y: real,Xaa: real,Ya: real] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xaa),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Ya),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_sum_squares_mult_ge_zero
tff(fact_3498_arith__geo__mean__sqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => aa(real,$o,ord_less_eq(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arith_geo_mean_sqrt
tff(fact_3499_tan__30,axiom,
    aa(real,real,tan(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,one2))))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2)))) ).

% tan_30
tff(fact_3500_cos__x__y__le__one,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,ord_less_eq(real,abs_abs(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),one_one(real)) ).

% cos_x_y_le_one
tff(fact_3501_real__sqrt__sum__squares__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))))
     => ( aa(real,$o,ord_less(real,abs_abs(real,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))))
       => aa(real,$o,ord_less(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),U) ) ) ).

% real_sqrt_sum_squares_less
tff(fact_3502_arcosh__real__def,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),Xb)
     => ( aa(real,real,arcosh(real),Xb) = aa(real,real,ln_ln(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,sqrt,aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real))))) ) ) ).

% arcosh_real_def
tff(fact_3503_cos__arctan,axiom,
    ! [Xb: real] : cos(real,aa(real,real,arctan,Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% cos_arctan
tff(fact_3504_sin__arctan,axiom,
    ! [Xb: real] : sin(real,aa(real,real,arctan,Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% sin_arctan
tff(fact_3505_sqrt__sum__squares__half__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),bit0(one2))))
     => ( aa(real,$o,ord_less(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
         => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
           => aa(real,$o,ord_less(real,aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),U) ) ) ) ) ).

% sqrt_sum_squares_half_less
tff(fact_3506_sin__cos__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),sin(real,Xb))
     => ( sin(real,Xb) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),cos(real,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_cos_sqrt
tff(fact_3507_arctan__half,axiom,
    ! [Xb: real] : aa(real,real,arctan,Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% arctan_half
tff(fact_3508_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [Xb: nat,Y: int] :
      ( ( vEBT_V9176841429113362141ildupi(Xb) = Y )
     => ( accp(nat,vEBT_V3352910403632780892pi_rel,Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = one_one(int) )
             => ~ accp(nat,vEBT_V3352910403632780892pi_rel,zero_zero(nat)) ) )
         => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = one_one(int) )
               => ~ accp(nat,vEBT_V3352910403632780892pi_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N3: nat] :
                  ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,N3)) )
                 => ( ( Y = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(bit0(one2))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))) )
                   => ~ accp(nat,vEBT_V3352910403632780892pi_rel,aa(nat,nat,suc,aa(nat,nat,suc,N3))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
tff(fact_3509_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_Opelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( vEBT_V8646137997579335489_i_l_d(Xb) = Y )
     => ( accp(nat,vEBT_V5144397997797733112_d_rel,Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
             => ~ accp(nat,vEBT_V5144397997797733112_d_rel,zero_zero(nat)) ) )
         => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
               => ~ accp(nat,vEBT_V5144397997797733112_d_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                 => ( ( Y = $ite(
                          aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $let(
                              half: nat,
                              half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),vEBT_V8646137997579335489_i_l_d(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),vEBT_V8646137997579335489_i_l_d(half))) )),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) )
                   => ~ accp(nat,vEBT_V5144397997797733112_d_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d.pelims
tff(fact_3510_VEBT__internal_OT_092_060_094sub_062b_092_060_094sub_062u_092_060_094sub_062i_092_060_094sub_062l_092_060_094sub_062d_092_060_094sub_062u_092_060_094sub_062p_Opelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( vEBT_V8346862874174094_d_u_p(Xb) = Y )
     => ( accp(nat,vEBT_V1247956027447740395_p_rel,Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
             => ~ accp(nat,vEBT_V1247956027447740395_p_rel,zero_zero(nat)) ) )
         => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ accp(nat,vEBT_V1247956027447740395_p_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                 => ( ( Y = $ite(
                          aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $let(
                              half: nat,
                              half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))),vEBT_V8346862874174094_d_u_p(half))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) )),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) )
                   => ~ accp(nat,vEBT_V1247956027447740395_p_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>b\<^sub>u\<^sub>i\<^sub>l\<^sub>d\<^sub>u\<^sub>p.pelims
tff(fact_3511_VEBT__internal_OTb_Opelims,axiom,
    ! [Xb: nat,Y: int] :
      ( ( vEBT_VEBT_Tb(Xb) = Y )
     => ( accp(nat,vEBT_VEBT_Tb_rel2,Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) )
             => ~ accp(nat,vEBT_VEBT_Tb_rel2,zero_zero(nat)) ) )
         => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) )
               => ~ accp(nat,vEBT_VEBT_Tb_rel2,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N3: nat] :
                  ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,N3)) )
                 => ( ( Y = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N3),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(int,int,aa(int,fun(int,int),times_times(int),vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) )
                   => ~ accp(nat,vEBT_VEBT_Tb_rel2,aa(nat,nat,suc,aa(nat,nat,suc,N3))) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
tff(fact_3512_VEBT__internal_OTb_H_Opelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( vEBT_VEBT_Tb2(Xb) = Y )
     => ( accp(nat,vEBT_VEBT_Tb_rel,Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
             => ~ accp(nat,vEBT_VEBT_Tb_rel,zero_zero(nat)) ) )
         => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ accp(nat,vEBT_VEBT_Tb_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N3: nat] :
                  ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,N3)) )
                 => ( ( Y = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) )
                   => ~ accp(nat,vEBT_VEBT_Tb_rel,aa(nat,nat,suc,aa(nat,nat,suc,N3))) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
tff(fact_3513_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( vEBT_V441764108873111860ildupi(Xb) = Y )
     => ( accp(nat,vEBT_V2957053500504383685pi_rel,Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = aa(nat,nat,suc,zero_zero(nat)) )
             => ~ accp(nat,vEBT_V2957053500504383685pi_rel,zero_zero(nat)) ) )
         => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(nat,nat,suc,zero_zero(nat)) )
               => ~ accp(nat,vEBT_V2957053500504383685pi_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N3: nat] :
                  ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,N3)) )
                 => ( ( Y = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N3),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))) )
                   => ~ accp(nat,vEBT_V2957053500504383685pi_rel,aa(nat,nat,suc,aa(nat,nat,suc,N3))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_3514_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: array(vEBT_VEBTi),X14: vEBT_VEBTi] : aa(vEBT_VEBTi,nat,vEBT_size_VEBTi,vEBT_Nodei(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_array(vEBT_VEBTi,vEBT_size_VEBTi,X13)),aa(vEBT_VEBTi,nat,vEBT_size_VEBTi,X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBTi.size_gen(1)
tff(fact_3515_obtain__set__succ,axiom,
    ! [Xb: nat,Z: nat,A4: set(nat),B6: set(nat)] :
      ( aa(nat,$o,ord_less(nat,Xb),Z)
     => ( vEBT_VEBT_max_in_set(A4,Z)
       => ( finite_finite2(nat,B6)
         => ( ( A4 = B6 )
           => ? [X_1: nat] : vEBT_is_succ_in_set(A4,Xb,X_1) ) ) ) ) ).

% obtain_set_succ
tff(fact_3516_obtain__set__pred,axiom,
    ! [Z: nat,Xb: nat,A4: set(nat)] :
      ( aa(nat,$o,ord_less(nat,Z),Xb)
     => ( vEBT_VEBT_min_in_set(A4,Z)
       => ( finite_finite2(nat,A4)
         => ? [X_1: nat] : vEBT_is_pred_in_set(A4,Xb,X_1) ) ) ) ).

% obtain_set_pred
tff(fact_3517_bezw_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod(int,int)] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Xaa)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Xaa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),Xaa)))))) )
           => ~ accp(product_prod(nat,nat),bezw_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa)) ) ) ) ).

% bezw.pelims
tff(fact_3518_set__vebt__finite,axiom,
    ! [Tb: vEBT_VEBT,N: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => finite_finite2(nat,vEBT_VEBT_set_vebt(Tb)) ) ).

% set_vebt_finite
tff(fact_3519_pred__none__empty,axiom,
    ! [Xs: set(nat),A3: nat] :
      ( ~ ? [X_1: nat] : vEBT_is_pred_in_set(Xs,A3,X_1)
     => ( finite_finite2(nat,Xs)
       => ~ ? [X3: nat] :
              ( member(nat,X3,Xs)
              & aa(nat,$o,ord_less(nat,X3),A3) ) ) ) ).

% pred_none_empty
tff(fact_3520_succ__none__empty,axiom,
    ! [Xs: set(nat),A3: nat] :
      ( ~ ? [X_1: nat] : vEBT_is_succ_in_set(Xs,A3,X_1)
     => ( finite_finite2(nat,Xs)
       => ~ ? [X3: nat] :
              ( member(nat,X3,Xs)
              & aa(nat,$o,ord_less(nat,A3),X3) ) ) ) ).

% succ_none_empty
tff(fact_3521_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ finite_finite2(A,set_or1337092689740270186AtMost(A,A3,B3))
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% infinite_Icc_iff
tff(fact_3522_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A4: set(nat),Fa: fun(nat,A)] :
          ( finite_finite2(nat,A4)
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_sn(set(nat),fun(fun(nat,A),fun(nat,A)),A4),Fa)) ) ) ).

% summable_If_finite_set
tff(fact_3523_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),Fa: fun(nat,A)] :
          ( finite_finite2(nat,collect(nat,P))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_so(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),Fa)) ) ) ).

% summable_If_finite
tff(fact_3524_finite__nat__set__iff__bounded,axiom,
    ! [N2: set(nat)] :
      ( finite_finite2(nat,N2)
    <=> ? [M6: nat] :
        ! [X: nat] :
          ( member(nat,X,N2)
         => aa(nat,$o,ord_less(nat,X),M6) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_3525_bounded__nat__set__is__finite,axiom,
    ! [N2: set(nat),N: nat] :
      ( ! [X2: nat] :
          ( member(nat,X2,N2)
         => aa(nat,$o,ord_less(nat,X2),N) )
     => finite_finite2(nat,N2) ) ).

% bounded_nat_set_is_finite
tff(fact_3526_finite__nat__set__iff__bounded__le,axiom,
    ! [N2: set(nat)] :
      ( finite_finite2(nat,N2)
    <=> ? [M6: nat] :
        ! [X: nat] :
          ( member(nat,X,N2)
         => aa(nat,$o,ord_less_eq(nat,X),M6) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_3527_finite__if__eq__beyond__finite,axiom,
    ! [A: $tType,S3: set(A),S4: set(A)] :
      ( finite_finite2(A,S3)
     => finite_finite2(set(A),collect(set(A),aa(set(A),fun(set(A),$o),aTP_Lamp_sp(set(A),fun(set(A),fun(set(A),$o)),S3),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_3528_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,$o),I: nat] : finite_finite2(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_sq(fun(nat,$o),fun(nat,fun(nat,$o)),P),I))) ).

% finite_M_bounded_by_nat
tff(fact_3529_finite__less__ub,axiom,
    ! [Fa: fun(nat,nat),U: nat] :
      ( ! [N3: nat] : aa(nat,$o,ord_less_eq(nat,N3),aa(nat,nat,Fa,N3))
     => finite_finite2(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_sr(fun(nat,nat),fun(nat,fun(nat,$o)),Fa),U))) ) ).

% finite_less_ub
tff(fact_3530_finite__lists__length__eq,axiom,
    ! [A: $tType,A4: set(A),N: nat] :
      ( finite_finite2(A,A4)
     => finite_finite2(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_ss(set(A),fun(nat,fun(list(A),$o)),A4),N))) ) ).

% finite_lists_length_eq
tff(fact_3531_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ~ finite_finite2(A,set_or1337092689740270186AtMost(A,A3,B3)) ) ) ).

% infinite_Icc
tff(fact_3532_finite__lists__length__le,axiom,
    ! [A: $tType,A4: set(A),N: nat] :
      ( finite_finite2(A,A4)
     => finite_finite2(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_st(set(A),fun(nat,fun(list(A),$o)),A4),N))) ) ).

% finite_lists_length_le
tff(fact_3533_summable__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N2: set(nat),Fa: fun(nat,A)] :
          ( finite_finite2(nat,N2)
         => ( ! [N3: nat] :
                ( ~ member(nat,N3,N2)
               => ( aa(nat,A,Fa,N3) = zero_zero(A) ) )
           => summable(A,Fa) ) ) ) ).

% summable_finite
tff(fact_3534_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => finite_finite2(nat,collect(nat,aTP_Lamp_su(nat,fun(nat,$o),M))) ) ).

% finite_divisors_nat
tff(fact_3535_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N2: set(nat),N: nat] :
      ( aa(set(nat),$o,ord_less_eq(set(nat),N2),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))
     => finite_finite2(nat,N2) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_3536_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),N)
         => finite_finite2(A,collect(A,aTP_Lamp_sv(nat,fun(A,$o),N))) ) ) ).

% finite_roots_unity
tff(fact_3537_VEBTi_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa(vEBT_VEBTi,nat,vEBT_size_VEBTi,vEBT_Leafi((X21),(X222))) = zero_zero(nat) ).

% VEBTi.size_gen(2)
tff(fact_3538_finite__Collect__le__nat,axiom,
    ! [K: nat] : finite_finite2(nat,collect(nat,aTP_Lamp_sw(nat,fun(nat,$o),K))) ).

% finite_Collect_le_nat
tff(fact_3539_finite__Collect__less__nat,axiom,
    ! [K: nat] : finite_finite2(nat,collect(nat,aTP_Lamp_sx(nat,fun(nat,$o),K))) ).

% finite_Collect_less_nat
tff(fact_3540_finite__Collect__subsets,axiom,
    ! [A: $tType,A4: set(A)] :
      ( finite_finite2(A,A4)
     => finite_finite2(set(A),collect(set(A),aTP_Lamp_sy(set(A),fun(set(A),$o),A4))) ) ).

% finite_Collect_subsets
tff(fact_3541_finite__induct__select,axiom,
    ! [A: $tType,S3: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,S3)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [T4: set(A)] :
              ( aa(set(A),$o,ord_less(set(A),T4),S3)
             => ( aa(set(A),$o,P,T4)
               => ? [X3: A] :
                    ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),S3),T4))
                    & aa(set(A),$o,P,aa(set(A),set(A),insert(A,X3),T4)) ) ) )
         => aa(set(A),$o,P,S3) ) ) ) ).

% finite_induct_select
tff(fact_3542_set__encode__insert,axiom,
    ! [A4: set(nat),N: nat] :
      ( finite_finite2(nat,A4)
     => ( ~ member(nat,N,A4)
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),insert(nat,N),A4)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(set(nat),nat,nat_set_encode,A4)) ) ) ) ).

% set_encode_insert
tff(fact_3543_finite__Collect__disjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( finite_finite2(A,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_sz(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( finite_finite2(A,collect(A,P))
        & finite_finite2(A,collect(A,Q)) ) ) ).

% finite_Collect_disjI
tff(fact_3544_finite__Collect__conjI,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ( finite_finite2(A,collect(A,P))
        | finite_finite2(A,collect(A,Q)) )
     => finite_finite2(A,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ta(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ).

% finite_Collect_conjI
tff(fact_3545_finite__interval__int1,axiom,
    ! [A3: int,B3: int] : finite_finite2(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_tb(int,fun(int,fun(int,$o)),A3),B3))) ).

% finite_interval_int1
tff(fact_3546_finite__interval__int4,axiom,
    ! [A3: int,B3: int] : finite_finite2(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_tc(int,fun(int,fun(int,$o)),A3),B3))) ).

% finite_interval_int4
tff(fact_3547_finite__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => finite_finite2(complex,collect(complex,aa(complex,fun(complex,$o),aTP_Lamp_td(nat,fun(complex,fun(complex,$o)),N),C2))) ) ).

% finite_nth_roots
tff(fact_3548_finite__interval__int3,axiom,
    ! [A3: int,B3: int] : finite_finite2(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_te(int,fun(int,fun(int,$o)),A3),B3))) ).

% finite_interval_int3
tff(fact_3549_finite__interval__int2,axiom,
    ! [A3: int,B3: int] : finite_finite2(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_tf(int,fun(int,fun(int,$o)),A3),B3))) ).

% finite_interval_int2
tff(fact_3550_set__encode__empty,axiom,
    aa(set(nat),nat,nat_set_encode,bot_bot(set(nat))) = zero_zero(nat) ).

% set_encode_empty
tff(fact_3551_finite__maxlen,axiom,
    ! [A: $tType,M8: set(list(A))] :
      ( finite_finite2(list(A),M8)
     => ? [N3: nat] :
        ! [X3: list(A)] :
          ( member(list(A),X3,M8)
         => aa(nat,$o,ord_less(nat,aa(list(A),nat,size_size(list(A)),X3)),N3) ) ) ).

% finite_maxlen
tff(fact_3552_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero(int) )
     => finite_finite2(int,collect(int,aTP_Lamp_tg(int,fun(int,$o),I))) ) ).

% finite_divisors_int
tff(fact_3553_set__encode__inf,axiom,
    ! [A4: set(nat)] :
      ( ~ finite_finite2(nat,A4)
     => ( aa(set(nat),nat,nat_set_encode,A4) = zero_zero(nat) ) ) ).

% set_encode_inf
tff(fact_3554_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B6: set(B),R: fun(A,fun(B,$o))] :
      ( ~ finite_finite2(A,A4)
     => ( finite_finite2(B,B6)
       => ( ! [X2: A] :
              ( member(A,X2,A4)
             => ? [Xa: B] :
                  ( member(B,Xa,B6)
                  & aa(B,$o,aa(A,fun(B,$o),R,X2),Xa) ) )
         => ? [X2: B] :
              ( member(B,X2,B6)
              & ~ finite_finite2(A,collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_th(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),A4),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3555_not__finite__existsD,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ finite_finite2(A,collect(A,P))
     => ? [X_1: A] : aa(A,$o,P,X_1) ) ).

% not_finite_existsD
tff(fact_3556_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: set(A),A3: A] :
          ( finite_finite2(A,A4)
         => ( member(A,A3,A4)
           => ? [X2: A] :
                ( member(A,X2,A4)
                & aa(A,$o,ord_less_eq(A,A3),X2)
                & ! [Xa: A] :
                    ( member(A,Xa,A4)
                   => ( aa(A,$o,ord_less_eq(A,X2),Xa)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_3557_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: set(A),A3: A] :
          ( finite_finite2(A,A4)
         => ( member(A,A3,A4)
           => ? [X2: A] :
                ( member(A,X2,A4)
                & aa(A,$o,ord_less_eq(A,X2),A3)
                & ! [Xa: A] :
                    ( member(A,Xa,A4)
                   => ( aa(A,$o,ord_less_eq(A,Xa),X2)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_3558_finite__psubset__induct,axiom,
    ! [A: $tType,A4: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,A4)
     => ( ! [A8: set(A)] :
            ( finite_finite2(A,A8)
           => ( ! [B8: set(A)] :
                  ( aa(set(A),$o,ord_less(set(A),B8),A8)
                 => aa(set(A),$o,P,B8) )
             => aa(set(A),$o,P,A8) ) )
       => aa(set(A),$o,P,A4) ) ) ).

% finite_psubset_induct
tff(fact_3559_even__set__encode__iff,axiom,
    ! [A4: set(nat)] :
      ( finite_finite2(nat,A4)
     => ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(nat),nat,nat_set_encode,A4))
      <=> ~ member(nat,zero_zero(nat),A4) ) ) ).

% even_set_encode_iff
tff(fact_3560_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: set(A)] :
          ( finite_finite2(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ? [X2: A] :
                ( member(A,X2,A4)
                & ! [Xa: A] :
                    ( member(A,Xa,A4)
                   => ( aa(A,$o,ord_less_eq(A,X2),Xa)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_3561_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A4: set(A)] :
          ( finite_finite2(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ? [X2: A] :
                ( member(A,X2,A4)
                & ! [Xa: A] :
                    ( member(A,Xa,A4)
                   => ( aa(A,$o,ord_less_eq(A,Xa),X2)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_3562_diff__preserves__multiset,axiom,
    ! [A: $tType,M8: fun(A,nat),N2: fun(A,nat)] :
      ( finite_finite2(A,collect(A,aTP_Lamp_ti(fun(A,nat),fun(A,$o),M8)))
     => finite_finite2(A,collect(A,aa(fun(A,nat),fun(A,$o),aTP_Lamp_tj(fun(A,nat),fun(fun(A,nat),fun(A,$o)),M8),N2))) ) ).

% diff_preserves_multiset
tff(fact_3563_add__mset__in__multiset,axiom,
    ! [A: $tType,M8: fun(A,nat),A3: A] :
      ( finite_finite2(A,collect(A,aTP_Lamp_ti(fun(A,nat),fun(A,$o),M8)))
     => finite_finite2(A,collect(A,aa(A,fun(A,$o),aTP_Lamp_tk(fun(A,nat),fun(A,fun(A,$o)),M8),A3))) ) ).

% add_mset_in_multiset
tff(fact_3564_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),P: fun(set(A),$o)] :
          ( finite_finite2(A,A4)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B2: A,A8: set(A)] :
                  ( finite_finite2(A,A8)
                 => ( ! [X3: A] :
                        ( member(A,X3,A8)
                       => aa(A,$o,ord_less(A,B2),X3) )
                   => ( aa(set(A),$o,P,A8)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,B2),A8)) ) ) )
             => aa(set(A),$o,P,A4) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_3565_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),P: fun(set(A),$o)] :
          ( finite_finite2(A,A4)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B2: A,A8: set(A)] :
                  ( finite_finite2(A,A8)
                 => ( ! [X3: A] :
                        ( member(A,X3,A8)
                       => aa(A,$o,ord_less(A,X3),B2) )
                   => ( aa(set(A),$o,P,A8)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,B2),A8)) ) ) )
             => aa(set(A),$o,P,A4) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_3566_finite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S3: set(A),P: fun(set(A),$o),Fa: fun(A,B)] :
          ( finite_finite2(A,S3)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [X2: A,S5: set(A)] :
                  ( finite_finite2(A,S5)
                 => ( ! [Y4: A] :
                        ( member(A,Y4,S5)
                       => aa(B,$o,ord_less_eq(B,aa(A,B,Fa,Y4)),aa(A,B,Fa,X2)) )
                   => ( aa(set(A),$o,P,S5)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X2),S5)) ) ) )
             => aa(set(A),$o,P,S3) ) ) ) ) ).

% finite_ranking_induct
tff(fact_3567_size__eq__0__iff__empty,axiom,
    ! [A: $tType,M8: multiset(A)] :
      ( ( aa(multiset(A),nat,size_size(multiset(A)),M8) = zero_zero(nat) )
    <=> ( M8 = zero_zero(multiset(A)) ) ) ).

% size_eq_0_iff_empty
tff(fact_3568_size__empty,axiom,
    ! [A: $tType] : aa(multiset(A),nat,size_size(multiset(A)),zero_zero(multiset(A))) = zero_zero(nat) ).

% size_empty
tff(fact_3569_size__union,axiom,
    ! [A: $tType,M8: multiset(A),N2: multiset(A)] : aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M8),N2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(multiset(A),nat,size_size(multiset(A)),M8)),aa(multiset(A),nat,size_size(multiset(A)),N2)) ).

% size_union
tff(fact_3570_ex__has__least__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,M: fun(A,nat)] :
      ( aa(A,$o,P,K)
     => ? [X2: A] :
          ( aa(A,$o,P,X2)
          & ! [Y4: A] :
              ( aa(A,$o,P,Y4)
             => aa(nat,$o,ord_less_eq(nat,aa(A,nat,M,X2)),aa(A,nat,M,Y4)) ) ) ) ).

% ex_has_least_nat
tff(fact_3571_nonempty__has__size,axiom,
    ! [A: $tType,S3: multiset(A)] :
      ( ( S3 != zero_zero(multiset(A)) )
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(multiset(A),nat,size_size(multiset(A)),S3)) ) ).

% nonempty_has_size
tff(fact_3572_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,Fa: fun(A,nat),B3: nat] :
      ( aa(A,$o,P,K)
     => ( ! [Y2: A] :
            ( aa(A,$o,P,Y2)
           => aa(nat,$o,ord_less(nat,aa(A,nat,Fa,Y2)),B3) )
       => ? [X2: A] :
            ( aa(A,$o,P,X2)
            & ! [Y4: A] :
                ( aa(A,$o,P,Y4)
               => aa(nat,$o,ord_less_eq(nat,aa(A,nat,Fa,Y4)),aa(A,nat,Fa,X2)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_3573_diff__size__le__size__Diff,axiom,
    ! [A: $tType,M8: multiset(A),M9: multiset(A)] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,aa(multiset(A),nat,size_size(multiset(A)),M8)),aa(multiset(A),nat,size_size(multiset(A)),M9))),aa(multiset(A),nat,size_size(multiset(A)),aa(multiset(A),multiset(A),minus_minus(multiset(A),M8),M9))) ).

% diff_size_le_size_Diff
tff(fact_3574_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S3: set(A)] :
          ( finite_finite2(A,S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ? [X2: A] :
                ( member(A,X2,S3)
                & ~ ? [Xa: A] :
                      ( member(A,Xa,S3)
                      & aa(A,$o,ord_less(A,Xa),X2) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_3575_infinite__growing,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [X5: set(A)] :
          ( ( X5 != bot_bot(set(A)) )
         => ( ! [X2: A] :
                ( member(A,X2,X5)
               => ? [Xa: A] :
                    ( member(A,Xa,X5)
                    & aa(A,$o,ord_less(A,X2),Xa) ) )
           => ~ finite_finite2(A,X5) ) ) ) ).

% infinite_growing
tff(fact_3576_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,Fa: fun(A,nat),N: nat] :
      ( aa(A,$o,P,K)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ? [Y4: A] :
                ( aa(A,$o,P,Y4)
                & ~ aa(nat,$o,ord_less_eq(nat,aa(A,nat,Fa,Y4)),aa(A,nat,Fa,X2)) ) )
       => ? [Y2: A] :
            ( aa(A,$o,P,Y2)
            & ~ aa(nat,$o,ord_less(nat,aa(A,nat,Fa,Y2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Fa,K)),N)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_3577_filter__preserves__multiset,axiom,
    ! [A: $tType,M8: fun(A,nat),P: fun(A,$o)] :
      ( finite_finite2(A,collect(A,aTP_Lamp_ti(fun(A,nat),fun(A,$o),M8)))
     => finite_finite2(A,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_tl(fun(A,nat),fun(fun(A,$o),fun(A,$o)),M8),P))) ) ).

% filter_preserves_multiset
tff(fact_3578_infinite__int__iff__unbounded,axiom,
    ! [S3: set(int)] :
      ( ~ finite_finite2(int,S3)
    <=> ! [M6: int] :
        ? [N8: int] :
          ( aa(int,$o,ord_less(int,M6),abs_abs(int,N8))
          & member(int,N8,S3) ) ) ).

% infinite_int_iff_unbounded
tff(fact_3579_infinite__int__iff__unbounded__le,axiom,
    ! [S3: set(int)] :
      ( ~ finite_finite2(int,S3)
    <=> ! [M6: int] :
        ? [N8: int] :
          ( aa(int,$o,ord_less_eq(int,M6),abs_abs(int,N8))
          & member(int,N8,S3) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_3580_old_Oprod_Orec,axiom,
    ! [B: $tType,A: $tType,C: $tType,F1: fun(B,fun(C,A)),A3: B,B3: C] : product_rec_prod(B,C,A,F1,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B3)) = aa(C,A,aa(B,fun(C,A),F1,A3),B3) ).

% old.prod.rec
tff(fact_3581_cos__arcsin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),one_one(real))
       => ( cos(real,aa(real,real,arcsin,Xb)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% cos_arcsin
tff(fact_3582_union__eq__empty,axiom,
    ! [A: $tType,M8: multiset(A),N2: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M8),N2) = zero_zero(multiset(A)) )
    <=> ( ( M8 = zero_zero(multiset(A)) )
        & ( N2 = zero_zero(multiset(A)) ) ) ) ).

% union_eq_empty
tff(fact_3583_empty__eq__union,axiom,
    ! [A: $tType,M8: multiset(A),N2: multiset(A)] :
      ( ( zero_zero(multiset(A)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M8),N2) )
    <=> ( ( M8 = zero_zero(multiset(A)) )
        & ( N2 = zero_zero(multiset(A)) ) ) ) ).

% empty_eq_union
tff(fact_3584_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
    ! [A: $tType,Xb: multiset(A),Y: multiset(A)] :
      ( ( zero_zero(multiset(A)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xb),Y) )
    <=> ( ( Xb = zero_zero(multiset(A)) )
        & ( Y = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.zero_eq_add_iff_both_eq_0
tff(fact_3585_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [A: $tType,Xb: multiset(A),Y: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xb),Y) = zero_zero(multiset(A)) )
    <=> ( ( Xb = zero_zero(multiset(A)) )
        & ( Y = zero_zero(multiset(A)) ) ) ) ).

% subset_mset.add_eq_0_iff_both_eq_0
tff(fact_3586_arcsin__0,axiom,
    aa(real,real,arcsin,zero_zero(real)) = zero_zero(real) ).

% arcsin_0
tff(fact_3587_arcsin__1,axiom,
    aa(real,real,arcsin,one_one(real)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% arcsin_1
tff(fact_3588_arcsin__minus__1,axiom,
    aa(real,real,arcsin,aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ).

% arcsin_minus_1
tff(fact_3589_diff__empty,axiom,
    ! [A: $tType,M8: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),minus_minus(multiset(A),M8),zero_zero(multiset(A))) = M8 )
      & ( aa(multiset(A),multiset(A),minus_minus(multiset(A),zero_zero(multiset(A))),M8) = zero_zero(multiset(A)) ) ) ).

% diff_empty
tff(fact_3590_Multiset_Odiff__cancel,axiom,
    ! [A: $tType,A4: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),A4),A4) = zero_zero(multiset(A)) ).

% Multiset.diff_cancel
tff(fact_3591_union__le__mono1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B6: multiset(A),D5: multiset(A),C5: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),B6),D5)
         => aa(multiset(A),$o,ord_less(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),B6),C5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),D5),C5)) ) ) ).

% union_le_mono1
tff(fact_3592_union__le__mono2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B6: multiset(A),D5: multiset(A),C5: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),B6),D5)
         => aa(multiset(A),$o,ord_less(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),B6)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),D5)) ) ) ).

% union_le_mono2
tff(fact_3593_union__less__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A4: multiset(A),C5: multiset(A),B6: multiset(A),D5: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),A4),C5)
         => ( aa(multiset(A),$o,ord_less(multiset(A),B6),D5)
           => aa(multiset(A),$o,ord_less(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B6)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),D5)) ) ) ) ).

% union_less_mono
tff(fact_3594_empty__neutral_I1_J,axiom,
    ! [A: $tType,Xb: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),zero_zero(multiset(A))),Xb) = Xb ).

% empty_neutral(1)
tff(fact_3595_empty__neutral_I2_J,axiom,
    ! [A: $tType,Xb: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Xb),zero_zero(multiset(A))) = Xb ).

% empty_neutral(2)
tff(fact_3596_union__diff__assoc,axiom,
    ! [A: $tType,C5: multiset(A),B6: multiset(A),A4: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),minus_minus(multiset(A),C5),B6) = zero_zero(multiset(A)) )
     => ( aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),B6)),C5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A4),aa(multiset(A),multiset(A),minus_minus(multiset(A),B6),C5)) ) ) ).

% union_diff_assoc
tff(fact_3597_arcsin__less__arcsin,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),Y)
       => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
         => aa(real,$o,ord_less(real,aa(real,real,arcsin,Xb)),aa(real,real,arcsin,Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_3598_arcsin__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xb)),one_one(real))
     => ( aa(real,$o,ord_less_eq(real,abs_abs(real,Y)),one_one(real))
       => ( aa(real,$o,ord_less(real,aa(real,real,arcsin,Xb)),aa(real,real,arcsin,Y))
        <=> aa(real,$o,ord_less(real,Xb),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_3599_cos__arcsin__nonzero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),one_one(real))
       => ( cos(real,aa(real,real,arcsin,Xb)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_3600_infinite__nat__iff__unbounded,axiom,
    ! [S3: set(nat)] :
      ( ~ finite_finite2(nat,S3)
    <=> ! [M6: nat] :
        ? [N8: nat] :
          ( aa(nat,$o,ord_less(nat,M6),N8)
          & member(nat,N8,S3) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_3601_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set(nat)] :
      ( ! [M3: nat] :
          ( aa(nat,$o,ord_less(nat,K),M3)
         => ? [N4: nat] :
              ( aa(nat,$o,ord_less(nat,M3),N4)
              & member(nat,N4,S3) ) )
     => ~ finite_finite2(nat,S3) ) ).

% unbounded_k_infinite
tff(fact_3602_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set(nat)] :
      ( ~ finite_finite2(nat,S3)
    <=> ! [M6: nat] :
        ? [N8: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M6),N8)
          & member(nat,N8,S3) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_3603_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less(real,Y),one_one(real))
       => ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,ord_less(real,aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_lt_bounded
tff(fact_3604_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y)) ) ) ).

% arcsin_lbound
tff(fact_3605_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arcsin_ubound
tff(fact_3606_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,ord_less_eq(real,aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ) ).

% arcsin_bounded
tff(fact_3607_arcsin__sin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => ( aa(real,real,arcsin,sin(real,Xb)) = Xb ) ) ) ).

% arcsin_sin
tff(fact_3608_le__arcsin__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,ord_less_eq(real,Y),aa(real,real,arcsin,Xb))
            <=> aa(real,$o,ord_less_eq(real,sin(real,Y)),Xb) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_3609_arcsin__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Y)
         => ( aa(real,$o,ord_less_eq(real,Y),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
           => ( aa(real,$o,ord_less_eq(real,aa(real,real,arcsin,Xb)),Y)
            <=> aa(real,$o,ord_less_eq(real,Xb),sin(real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_3610_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,ord_less_eq(real,aa(real,real,arcsin,Y)),pi)
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_3611_arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,arcsin,Y))
          & aa(real,$o,ord_less_eq(real,aa(real,real,arcsin,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
          & ( sin(real,aa(real,real,arcsin,Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_3612_vebt__buildup_Oelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( ( ( Xb = zero_zero(nat) )
         => ( Y != vEBT_Leaf($false,$false) ) )
       => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
           => ( Y != vEBT_Leaf($false,$false) ) )
         => ~ ! [Va2: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
               => ( Y != $ite(
                      aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                      $let(
                        half: nat,
                        half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
tff(fact_3613_prod_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [I5: set(A),Xb: fun(A,B),Y: fun(A,B)] :
          ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_tm(set(A),fun(fun(A,B),fun(A,$o)),I5),Xb)))
         => ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_tm(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_tn(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xb),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_3614_sum_Ofinite__Collect__op,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),Xb: fun(A,B),Y: fun(A,B)] :
          ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_to(set(A),fun(fun(A,B),fun(A,$o)),I5),Xb)))
         => ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_to(set(A),fun(fun(A,B),fun(A,$o)),I5),Y)))
           => finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_tp(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xb),Y))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_3615_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,B3: real,K: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,one_one(real)),B3)
       => ( ( archimedean_ceiling(real,aa(real,real,log(B3),Xb)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),K)),one_one(int)) )
        <=> ( aa(real,$o,ord_less(real,powr(real,B3,aa(nat,real,semiring_1_of_nat(real),K))),Xb)
            & aa(real,$o,ord_less_eq(real,Xb),powr(real,B3,aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))))) ) ) ) ) ).

% ceiling_log_eq_powr_iff
tff(fact_3616_intind,axiom,
    ! [A: $tType,I: nat,N: nat,P: fun(A,$o),Xb: A] :
      ( aa(nat,$o,ord_less(nat,I),N)
     => ( aa(A,$o,P,Xb)
       => aa(A,$o,P,aa(nat,A,nth(A,replicate(A,N,Xb)),I)) ) ) ).

% intind
tff(fact_3617_powr__0,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Z: A] : powr(A,zero_zero(A),Z) = zero_zero(A) ) ).

% powr_0
tff(fact_3618_powr__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [W: A,Z: A] :
          ( ( powr(A,W,Z) = zero_zero(A) )
        <=> ( W = zero_zero(A) ) ) ) ).

% powr_eq_0_iff
tff(fact_3619_powr__one__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [A3: A] : powr(A,one_one(A),A3) = one_one(A) ) ).

% powr_one_eq_one
tff(fact_3620_replicate__eq__replicate,axiom,
    ! [A: $tType,M: nat,Xb: A,N: nat,Y: A] :
      ( ( replicate(A,M,Xb) = replicate(A,N,Y) )
    <=> ( ( M = N )
        & ( ( M != zero_zero(nat) )
         => ( Xb = Y ) ) ) ) ).

% replicate_eq_replicate
tff(fact_3621_length__replicate,axiom,
    ! [A: $tType,N: nat,Xb: A] : aa(list(A),nat,size_size(list(A)),replicate(A,N,Xb)) = N ).

% length_replicate
tff(fact_3622_powr__zero__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A] :
          powr(A,Xb,zero_zero(A)) = $ite(Xb = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% powr_zero_eq_one
tff(fact_3623_in__set__replicate,axiom,
    ! [A: $tType,Xb: A,N: nat,Y: A] :
      ( member(A,Xb,set2(A,replicate(A,N,Y)))
    <=> ( ( Xb = Y )
        & ( N != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_3624_Bex__set__replicate,axiom,
    ! [A: $tType,N: nat,A3: A,P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,set2(A,replicate(A,N,A3)))
          & aa(A,$o,P,X) )
    <=> ( aa(A,$o,P,A3)
        & ( N != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_3625_Ball__set__replicate,axiom,
    ! [A: $tType,N: nat,A3: A,P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,set2(A,replicate(A,N,A3)))
         => aa(A,$o,P,X) )
    <=> ( aa(A,$o,P,A3)
        | ( N = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_3626_powr__gt__zero,axiom,
    ! [Xb: real,A3: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),powr(real,Xb,A3))
    <=> ( Xb != zero_zero(real) ) ) ).

% powr_gt_zero
tff(fact_3627_powr__nonneg__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,ord_less_eq(real,powr(real,A3,Xb)),zero_zero(real))
    <=> ( A3 = zero_zero(real) ) ) ).

% powr_nonneg_iff
tff(fact_3628_nth__replicate,axiom,
    ! [A: $tType,I: nat,N: nat,Xb: A] :
      ( aa(nat,$o,ord_less(nat,I),N)
     => ( aa(nat,A,nth(A,replicate(A,N,Xb)),I) = Xb ) ) ).

% nth_replicate
tff(fact_3629_powr__less__cancel__iff,axiom,
    ! [Xb: real,A3: real,B3: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xb)
     => ( aa(real,$o,ord_less(real,powr(real,Xb,A3)),powr(real,Xb,B3))
      <=> aa(real,$o,ord_less(real,A3),B3) ) ) ).

% powr_less_cancel_iff
tff(fact_3630_powr__eq__one__iff,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A3)
     => ( ( powr(real,A3,Xb) = one_one(real) )
      <=> ( Xb = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_3631_powr__one__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( ( powr(real,Xb,one_one(real)) = Xb )
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb) ) ).

% powr_one_gt_zero_iff
tff(fact_3632_powr__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( powr(real,Xb,one_one(real)) = Xb ) ) ).

% powr_one
tff(fact_3633_powr__le__cancel__iff,axiom,
    ! [Xb: real,A3: real,B3: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,powr(real,Xb,A3)),powr(real,Xb,B3))
      <=> aa(real,$o,ord_less_eq(real,A3),B3) ) ) ).

% powr_le_cancel_iff
tff(fact_3634_numeral__powr__numeral__real,axiom,
    ! [M: num,N: num] : powr(real,aa(num,real,numeral_numeral(real),M),aa(num,real,numeral_numeral(real),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),M)),aa(num,nat,numeral_numeral(nat),N)) ).

% numeral_powr_numeral_real
tff(fact_3635_set__replicate,axiom,
    ! [A: $tType,N: nat,Xb: A] :
      ( ( N != zero_zero(nat) )
     => ( set2(A,replicate(A,N,Xb)) = aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_3636_powr__log__cancel,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( powr(real,A3,aa(real,real,log(A3),Xb)) = Xb ) ) ) ) ).

% powr_log_cancel
tff(fact_3637_log__powr__cancel,axiom,
    ! [A3: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,real,log(A3),powr(real,A3,Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_3638_powr__numeral,axiom,
    ! [Xb: real,N: num] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(num,real,numeral_numeral(real),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),N)) ) ) ).

% powr_numeral
tff(fact_3639_map__fst__mk__fst,axiom,
    ! [B: $tType,A: $tType,K: A,L: list(B)] : map(product_prod(A,B),A,product_fst(A,B),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),K),L)) = replicate(A,aa(list(B),nat,size_size(list(B)),L),K) ).

% map_fst_mk_fst
tff(fact_3640_map__snd__mk__snd,axiom,
    ! [B: $tType,A: $tType,K: A,L: list(B)] : map(product_prod(B,A),A,product_snd(B,A),map(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_tq(A,fun(B,product_prod(B,A))),K),L)) = replicate(A,aa(list(B),nat,size_size(list(B)),L),K) ).

% map_snd_mk_snd
tff(fact_3641_square__powr__half,axiom,
    ! [Xb: real] : powr(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))) = abs_abs(real,Xb) ).

% square_powr_half
tff(fact_3642_mset__le__not__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M8: multiset(A)] : ~ aa(multiset(A),$o,ord_less(multiset(A),M8),M8) ) ).

% mset_le_not_refl
tff(fact_3643_mset__le__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M8: multiset(A),N2: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),M8),N2)
         => ~ aa(multiset(A),$o,ord_less(multiset(A),N2),M8) ) ) ).

% mset_le_not_sym
tff(fact_3644_mset__le__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M8: multiset(A)] : ~ aa(multiset(A),$o,ord_less(multiset(A),M8),M8) ) ).

% mset_le_irrefl
tff(fact_3645_mset__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K5: multiset(A),M8: multiset(A),N2: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),K5),M8)
         => ( aa(multiset(A),$o,ord_less(multiset(A),M8),N2)
           => aa(multiset(A),$o,ord_less(multiset(A),K5),N2) ) ) ) ).

% mset_le_trans
tff(fact_3646_mset__le__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M8: multiset(A),N2: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),M8),N2)
         => ~ aa(multiset(A),$o,ord_less(multiset(A),N2),M8) ) ) ).

% mset_le_asym
tff(fact_3647_less__eq__multiset__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M8: multiset(A),N2: multiset(A)] :
          ( aa(multiset(A),$o,ord_less_eq(multiset(A),M8),N2)
        <=> ( aa(multiset(A),$o,ord_less(multiset(A),M8),N2)
            | ( M8 = N2 ) ) ) ) ).

% less_eq_multiset_def
tff(fact_3648_powr__less__mono2__neg,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,A3),zero_zero(real))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,Xb),Y)
         => aa(real,$o,ord_less(real,powr(real,Y,A3)),powr(real,Xb,A3)) ) ) ) ).

% powr_less_mono2_neg
tff(fact_3649_powr__non__neg,axiom,
    ! [A3: real,Xb: real] : ~ aa(real,$o,ord_less(real,powr(real,A3,Xb)),zero_zero(real)) ).

% powr_non_neg
tff(fact_3650_powr__mono2,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less_eq(real,Xb),Y)
         => aa(real,$o,ord_less_eq(real,powr(real,Xb,A3)),powr(real,Y,A3)) ) ) ) ).

% powr_mono2
tff(fact_3651_powr__ge__pzero,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,ord_less_eq(real,zero_zero(real)),powr(real,Xb,Y)) ).

% powr_ge_pzero
tff(fact_3652_powr__less__cancel,axiom,
    ! [Xb: real,A3: real,B3: real] :
      ( aa(real,$o,ord_less(real,powr(real,Xb,A3)),powr(real,Xb,B3))
     => ( aa(real,$o,ord_less(real,one_one(real)),Xb)
       => aa(real,$o,ord_less(real,A3),B3) ) ) ).

% powr_less_cancel
tff(fact_3653_powr__less__mono,axiom,
    ! [A3: real,B3: real,Xb: real] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( aa(real,$o,ord_less(real,one_one(real)),Xb)
       => aa(real,$o,ord_less(real,powr(real,Xb,A3)),powr(real,Xb,B3)) ) ) ).

% powr_less_mono
tff(fact_3654_replicate__length__same,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( ! [X2: A] :
          ( member(A,X2,set2(A,Xs))
         => ( X2 = Xb ) )
     => ( replicate(A,aa(list(A),nat,size_size(list(A)),Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_3655_replicate__eqI,axiom,
    ! [A: $tType,Xs: list(A),N: nat,Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = N )
     => ( ! [Y2: A] :
            ( member(A,Y2,set2(A,Xs))
           => ( Y2 = Xb ) )
       => ( Xs = replicate(A,N,Xb) ) ) ) ).

% replicate_eqI
tff(fact_3656_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K: A,Lst: list(B)] : map(B,A,aa(A,fun(B,A),aTP_Lamp_cp(A,fun(B,A)),K),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K) ).

% map_replicate_const
tff(fact_3657_powr__less__mono2,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,Xb),Y)
         => aa(real,$o,ord_less(real,powr(real,Xb,A3)),powr(real,Y,A3)) ) ) ) ).

% powr_less_mono2
tff(fact_3658_powr__mono2_H,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,A3),zero_zero(real))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less_eq(real,Xb),Y)
         => aa(real,$o,ord_less_eq(real,powr(real,Y,A3)),powr(real,Xb,A3)) ) ) ) ).

% powr_mono2'
tff(fact_3659_gr__one__powr,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
       => aa(real,$o,ord_less(real,one_one(real)),powr(real,Xb,Y)) ) ) ).

% gr_one_powr
tff(fact_3660_powr__inj,axiom,
    ! [A3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( ( powr(real,A3,Xb) = powr(real,A3,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% powr_inj
tff(fact_3661_ge__one__powr__ge__zero,axiom,
    ! [Xb: real,A3: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
       => aa(real,$o,ord_less_eq(real,one_one(real)),powr(real,Xb,A3)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_3662_powr__mono__both,axiom,
    ! [A3: real,B3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
     => ( aa(real,$o,ord_less_eq(real,A3),B3)
       => ( aa(real,$o,ord_less_eq(real,one_one(real)),Xb)
         => ( aa(real,$o,ord_less_eq(real,Xb),Y)
           => aa(real,$o,ord_less_eq(real,powr(real,Xb,A3)),powr(real,Y,B3)) ) ) ) ) ).

% powr_mono_both
tff(fact_3663_powr__le1,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less_eq(real,Xb),one_one(real))
         => aa(real,$o,ord_less_eq(real,powr(real,Xb,A3)),one_one(real)) ) ) ) ).

% powr_le1
tff(fact_3664_powr__divide,axiom,
    ! [Xb: real,Y: real,A3: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y),A3) = aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,Xb,A3)),powr(real,Y,A3)) ) ) ) ).

% powr_divide
tff(fact_3665_powr__mult,axiom,
    ! [Xb: real,Y: real,A3: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y),A3) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,Xb,A3)),powr(real,Y,A3)) ) ) ) ).

% powr_mult
tff(fact_3666_log__base__powr,axiom,
    ! [A3: real,B3: real,Xb: real] :
      ( ( A3 != zero_zero(real) )
     => ( aa(real,real,log(powr(real,A3,B3)),Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(A3),Xb)),B3) ) ) ).

% log_base_powr
tff(fact_3667_log__powr,axiom,
    ! [Xb: real,B3: real,Y: real] :
      ( ( Xb != zero_zero(real) )
     => ( aa(real,real,log(B3),powr(real,Xb,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),aa(real,real,log(B3),Xb)) ) ) ).

% log_powr
tff(fact_3668_ln__powr,axiom,
    ! [Xb: real,Y: real] :
      ( ( Xb != zero_zero(real) )
     => ( aa(real,real,ln_ln(real),powr(real,Xb,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),aa(real,real,ln_ln(real),Xb)) ) ) ).

% ln_powr
tff(fact_3669_powr__diff,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [W: A,Z1: A,Z22: A] : powr(A,W,aa(A,A,minus_minus(A,Z1),Z22)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),powr(A,W,Z1)),powr(A,W,Z22)) ) ).

% powr_diff
tff(fact_3670_set__replicate__conv__if,axiom,
    ! [A: $tType,N: nat,Xb: A] :
      set2(A,replicate(A,N,Xb)) = $ite(N = zero_zero(nat),bot_bot(set(A)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) ).

% set_replicate_conv_if
tff(fact_3671_powr__realpow,axiom,
    ! [Xb: real,N: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(nat,real,semiring_1_of_nat(real),N)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N) ) ) ).

% powr_realpow
tff(fact_3672_powr__less__iff,axiom,
    ! [B3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,powr(real,B3,Y)),Xb)
        <=> aa(real,$o,ord_less(real,Y),aa(real,real,log(B3),Xb)) ) ) ) ).

% powr_less_iff
tff(fact_3673_less__powr__iff,axiom,
    ! [B3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,Xb),powr(real,B3,Y))
        <=> aa(real,$o,ord_less(real,aa(real,real,log(B3),Xb)),Y) ) ) ) ).

% less_powr_iff
tff(fact_3674_log__less__iff,axiom,
    ! [B3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,aa(real,real,log(B3),Xb)),Y)
        <=> aa(real,$o,ord_less(real,Xb),powr(real,B3,Y)) ) ) ) ).

% log_less_iff
tff(fact_3675_less__log__iff,axiom,
    ! [B3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,Y),aa(real,real,log(B3),Xb))
        <=> aa(real,$o,ord_less(real,powr(real,B3,Y)),Xb) ) ) ) ).

% less_log_iff
tff(fact_3676_powr__minus__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A,A3: A] : powr(A,Xb,aa(A,A,uminus_uminus(A),A3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),powr(A,Xb,A3)) ) ).

% powr_minus_divide
tff(fact_3677_powr__neg__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(real,real,uminus_uminus(real),one_one(real))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Xb) ) ) ).

% powr_neg_one
tff(fact_3678_powr__mult__base,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),Xb),powr(real,Xb,Y)) = powr(real,Xb,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Y)) ) ) ).

% powr_mult_base
tff(fact_3679_powr__le__iff,axiom,
    ! [B3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less_eq(real,powr(real,B3,Y)),Xb)
        <=> aa(real,$o,ord_less_eq(real,Y),aa(real,real,log(B3),Xb)) ) ) ) ).

% powr_le_iff
tff(fact_3680_le__powr__iff,axiom,
    ! [B3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less_eq(real,Xb),powr(real,B3,Y))
        <=> aa(real,$o,ord_less_eq(real,aa(real,real,log(B3),Xb)),Y) ) ) ) ).

% le_powr_iff
tff(fact_3681_log__le__iff,axiom,
    ! [B3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less_eq(real,aa(real,real,log(B3),Xb)),Y)
        <=> aa(real,$o,ord_less_eq(real,Xb),powr(real,B3,Y)) ) ) ) ).

% log_le_iff
tff(fact_3682_le__log__iff,axiom,
    ! [B3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B3)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less_eq(real,Y),aa(real,real,log(B3),Xb))
        <=> aa(real,$o,ord_less_eq(real,powr(real,B3,Y)),Xb) ) ) ) ).

% le_log_iff
tff(fact_3683_ln__powr__bound,axiom,
    ! [Xb: real,A3: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),Xb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
       => aa(real,$o,ord_less_eq(real,aa(real,real,ln_ln(real),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,Xb,A3)),A3)) ) ) ).

% ln_powr_bound
tff(fact_3684_ln__powr__bound2,axiom,
    ! [Xb: real,A3: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
       => aa(real,$o,ord_less_eq(real,powr(real,aa(real,real,ln_ln(real),Xb),A3)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A3,A3)),Xb)) ) ) ).

% ln_powr_bound2
tff(fact_3685_add__log__eq__powr,axiom,
    ! [B3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
     => ( ( B3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Y),aa(real,real,log(B3),Xb)) = aa(real,real,log(B3),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B3,Y)),Xb)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_3686_log__add__eq__powr,axiom,
    ! [B3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
     => ( ( B3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,log(B3),Xb)),Y) = aa(real,real,log(B3),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),powr(real,B3,Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_3687_minus__log__eq__powr,axiom,
    ! [B3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
     => ( ( B3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( aa(real,real,minus_minus(real,Y),aa(real,real,log(B3),Xb)) = aa(real,real,log(B3),aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,B3,Y)),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_3688_log__minus__eq__powr,axiom,
    ! [B3: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
     => ( ( B3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( aa(real,real,minus_minus(real,aa(real,real,log(B3),Xb)),Y) = aa(real,real,log(B3),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),powr(real,B3,aa(real,real,uminus_uminus(real),Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_3689_powr__half__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(real,real,sqrt,Xb) ) ) ).

% powr_half_sqrt
tff(fact_3690_powr__neg__numeral,axiom,
    ! [Xb: real,N: num] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),N))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),N))) ) ) ).

% powr_neg_numeral
tff(fact_3691_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Va))) = $ite(
        aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va))),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) ).

% vebt_buildup.simps(3)
tff(fact_3692_vebt__buildup_Opelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( accp(nat,vEBT_v4011308405150292612up_rel,Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ accp(nat,vEBT_v4011308405150292612up_rel,zero_zero(nat)) ) )
         => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [Va2: nat] :
                  ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                 => ( ( Y = $ite(
                          aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),half),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(half)) ),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) )
                   => ~ accp(nat,vEBT_v4011308405150292612up_rel,aa(nat,nat,suc,aa(nat,nat,suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_3693_arcosh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A] : aa(A,A,arcosh(A),Xb) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),powr(A,aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)),aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))))) ) ).

% arcosh_def
tff(fact_3694_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,M: nat,N: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_or1337092689740270186AtMost(nat,M,N)) = $ite(
            aa(nat,$o,ord_less(nat,N),M),
            zero_zero(A),
            $ite(Xb = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),M)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,N)))),aa(A,A,minus_minus(A,one_one(A)),Xb))) ) ) ).

% sum_gp
tff(fact_3695_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Y)),one_one(real))
     => ( sin(real,aa(real,real,arccos,Y)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% sin_arccos_abs
tff(fact_3696_summable__complex__of__real,axiom,
    ! [Fa: fun(nat,real)] :
      ( summable(complex,aTP_Lamp_tr(fun(nat,real),fun(nat,complex),Fa))
    <=> summable(real,Fa) ) ).

% summable_complex_of_real
tff(fact_3697_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_ts(B,A)),A4) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_3698_of__nat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Fa: fun(B,nat),A4: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,Fa),A4)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_tt(fun(B,nat),fun(B,A),Fa)),A4) ) ).

% of_nat_sum
tff(fact_3699_of__int__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Fa: fun(B,int),A4: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,groups7311177749621191930dd_sum(B,int,Fa),A4)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_tu(fun(B,int),fun(B,A),Fa)),A4) ) ).

% of_int_sum
tff(fact_3700_abs__sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Fa: fun(B,A),A4: set(B)] : abs_abs(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_tv(fun(B,A),fun(B,A),Fa)),A4)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_tv(fun(B,A),fun(B,A),Fa)),A4) ) ).

% abs_sum_abs
tff(fact_3701_of__real__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Fa: fun(B,real),S: set(B)] : aa(real,A,real_Vector_of_real(A),aa(set(B),real,groups7311177749621191930dd_sum(B,real,Fa),S)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_tw(fun(B,real),fun(B,A),Fa)),S) ) ).

% of_real_sum
tff(fact_3702_sum_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty
tff(fact_3703_sum_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( ~ finite_finite2(A,A4)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4) = zero_zero(B) ) ) ) ).

% sum.infinite
tff(fact_3704_sum__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( canoni5634975068530333245id_add(B)
     => ! [F5: set(A),Fa: fun(A,B)] :
          ( finite_finite2(A,F5)
         => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),F5) = zero_zero(B) )
          <=> ! [X: A] :
                ( member(A,X,F5)
               => ( aa(A,B,Fa,X) = zero_zero(B) ) ) ) ) ) ).

% sum_eq_0_iff
tff(fact_3705_of__real__0,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ( aa(real,A,real_Vector_of_real(A),zero_zero(real)) = zero_zero(A) ) ) ).

% of_real_0
tff(fact_3706_of__real__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real] :
          ( ( aa(real,A,real_Vector_of_real(A),Xb) = zero_zero(A) )
        <=> ( Xb = zero_zero(real) ) ) ) ).

% of_real_eq_0_iff
tff(fact_3707_of__real__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real] :
          ( ( aa(real,A,real_Vector_of_real(A),Xb) = one_one(A) )
        <=> ( Xb = one_one(real) ) ) ) ).

% of_real_eq_1_iff
tff(fact_3708_of__real__1,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ( aa(real,A,real_Vector_of_real(A),one_one(real)) = one_one(A) ) ) ).

% of_real_1
tff(fact_3709_of__real__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : aa(real,A,real_Vector_of_real(A),aa(num,real,numeral_numeral(real),W)) = aa(num,A,numeral_numeral(A),W) ) ).

% of_real_numeral
tff(fact_3710_of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [Xb: real,Y: real] : aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),Xb)),aa(real,A,real_Vector_of_real(A),Y)) ) ).

% of_real_divide
tff(fact_3711_of__real__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,N: nat] : aa(real,A,real_Vector_of_real(A),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(real,A,real_Vector_of_real(A),Xb)),N) ) ).

% of_real_power
tff(fact_3712_arccos__1,axiom,
    aa(real,real,arccos,one_one(real)) = zero_zero(real) ).

% arccos_1
tff(fact_3713_sum_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),A3: A,B3: fun(A,B)] :
          ( finite_finite2(A,S3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tx(A,fun(fun(A,B),fun(A,B)),A3),B3)),S3) = $ite(member(A,A3,S3),aa(A,B,B3,A3),zero_zero(B)) ) ) ) ).

% sum.delta'
tff(fact_3714_sum_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),A3: A,B3: fun(A,B)] :
          ( finite_finite2(A,S3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ty(A,fun(fun(A,B),fun(A,B)),A3),B3)),S3) = $ite(member(A,A3,S3),aa(A,B,B3,A3),zero_zero(B)) ) ) ) ).

% sum.delta
tff(fact_3715_sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Fa: fun(B,A),A4: set(B)] : aa(A,$o,ord_less_eq(A,abs_abs(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,Fa),A4))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_tv(fun(B,A),fun(B,A),Fa)),A4)) ) ).

% sum_abs
tff(fact_3716_sin__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,aa(real,A,real_Vector_of_real(A),pi)) = zero_zero(A) ) ) ).

% sin_of_real_pi
tff(fact_3717_sum__abs__ge__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Fa: fun(B,A),A4: set(B)] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_tv(fun(B,A),fun(B,A),Fa)),A4)) ) ).

% sum_abs_ge_zero
tff(fact_3718_of__real__neg__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : aa(real,A,real_Vector_of_real(A),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),W))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) ) ).

% of_real_neg_numeral
tff(fact_3719_cos__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,aa(real,A,real_Vector_of_real(A),pi)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% cos_of_real_pi
tff(fact_3720_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,ord_less(nat,aa(nat,nat,suc,N)),M),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N)))) ) ).

% sum.cl_ivl_Suc
tff(fact_3721_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: fun(nat,A),A4: set(nat)] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_tz(fun(nat,A),fun(nat,A),C2)),A4) = $ite(
            ( finite_finite2(nat,A4)
            & member(nat,zero_zero(nat),A4) ),
            aa(nat,A,C2,zero_zero(nat)),
            zero_zero(A) ) ) ).

% sum_zero_power
tff(fact_3722_norm__of__real__add1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: real] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,real_Vector_of_real(A),Xb)),one_one(A))) = abs_abs(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),one_one(real))) ) ).

% norm_of_real_add1
tff(fact_3723_norm__of__real__addn,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: real,B3: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,real_Vector_of_real(A),Xb)),aa(num,A,numeral_numeral(A),B3))) = abs_abs(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(num,real,numeral_numeral(real),B3))) ) ).

% norm_of_real_addn
tff(fact_3724_arccos__0,axiom,
    aa(real,real,arccos,zero_zero(real)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% arccos_0
tff(fact_3725_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: fun(nat,A),D: fun(nat,A),A4: set(nat)] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ua(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D)),A4) = $ite(
            ( finite_finite2(nat,A4)
            & member(nat,zero_zero(nat),A4) ),
            aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,C2,zero_zero(nat))),aa(nat,A,D,zero_zero(nat))),
            zero_zero(A) ) ) ).

% sum_zero_power'
tff(fact_3726_cos__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ).

% cos_of_real_pi_half
tff(fact_3727_sin__of__real__pi__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V7773925162809079976_field(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sin(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2)))) = one_one(A) ) ) ).

% sin_of_real_pi_half
tff(fact_3728_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A4: set(B)] :
          ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A4) != zero_zero(A) )
         => ~ ! [A2: B] :
                ( member(B,A2,A4)
               => ( aa(B,A,G,A2) = zero_zero(A) ) ) ) ) ).

% sum.not_neutral_contains_not_neutral
tff(fact_3729_sum_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A4)
             => ( aa(A,B,G,X2) = zero_zero(B) ) )
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4) = zero_zero(B) ) ) ) ).

% sum.neutral
tff(fact_3730_sum_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,fun(C,A)),B6: set(C),A4: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(set(C),fun(B,A),aTP_Lamp_ub(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B6)),A4) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(set(B),fun(C,A),aTP_Lamp_ud(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A4)),B6) ) ).

% sum.swap
tff(fact_3731_norm__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(B,A),A4: set(B)] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,Fa),A4))),aa(set(B),real,groups7311177749621191930dd_sum(B,real,aTP_Lamp_ue(fun(B,A),fun(B,real),Fa)),A4)) ) ).

% norm_sum
tff(fact_3732_sum__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [K5: set(A),Fa: fun(A,B),G: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,K5)
             => aa(B,$o,ord_less_eq(B,aa(A,B,Fa,I2)),aa(A,B,G,I2)) )
         => aa(B,$o,ord_less_eq(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),K5)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),K5)) ) ) ).

% sum_mono
tff(fact_3733_sum_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),H: fun(B,A),A4: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_uf(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),A4) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A4)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),A4)) ) ).

% sum.distrib
tff(fact_3734_sum__product,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [Fa: fun(B,A),A4: set(B),G: fun(C,A),B6: set(C)] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,Fa),A4)),aa(set(C),A,groups7311177749621191930dd_sum(C,A,G),B6)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_uh(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Fa),G),B6)),A4) ) ).

% sum_product
tff(fact_3735_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [Fa: fun(B,A),A4: set(B),R2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,Fa),A4)),R2) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_ui(fun(B,A),fun(A,fun(B,A)),Fa),R2)),A4) ) ).

% sum_distrib_right
tff(fact_3736_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R2: A,Fa: fun(B,A),A4: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R2),aa(set(B),A,groups7311177749621191930dd_sum(B,A,Fa),A4)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_uj(A,fun(fun(B,A),fun(B,A)),R2),Fa)),A4) ) ).

% sum_distrib_left
tff(fact_3737_sum__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Fa: fun(B,A),G: fun(B,A),A4: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_uk(fun(B,A),fun(fun(B,A),fun(B,A)),Fa),G)),A4) = aa(A,A,minus_minus(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,Fa),A4)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A4)) ) ).

% sum_subtractf
tff(fact_3738_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [Fa: fun(B,A),A4: set(B),R2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,Fa),A4)),R2) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_ul(fun(B,A),fun(A,fun(B,A)),Fa),R2)),A4) ) ).

% sum_divide_distrib
tff(fact_3739_sum__negf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Fa: fun(B,A),A4: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_um(fun(B,A),fun(B,A),Fa)),A4) = aa(A,A,uminus_uminus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,Fa),A4)) ) ).

% sum_negf
tff(fact_3740_sum_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(A),B6: set(B),G: fun(A,fun(B,C)),R: fun(A,fun(B,$o))] :
          ( finite_finite2(A,A4)
         => ( finite_finite2(B,B6)
           => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_uo(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B6),G),R)),A4) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_uq(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A4),G),R)),B6) ) ) ) ) ).

% sum.swap_restrict
tff(fact_3741_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Fa: fun(B,A),A3: A,A4: set(B)] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_ur(fun(B,A),fun(A,fun(B,A)),Fa),A3)),A4)),A3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,Fa),A4)),A3) ) ).

% mod_sum_eq
tff(fact_3742_summable__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),Fa: fun(A,fun(nat,B))] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => summable(B,aa(A,fun(nat,B),Fa,I2)) )
         => summable(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_ut(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),Fa)) ) ) ).

% summable_sum
tff(fact_3743_sum__nonpos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A4: set(A),Fa: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A4)
             => aa(B,$o,ord_less_eq(B,aa(A,B,Fa,X2)),zero_zero(B)) )
         => aa(B,$o,ord_less_eq(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),A4)),zero_zero(B)) ) ) ).

% sum_nonpos
tff(fact_3744_sum__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A4: set(A),Fa: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A4)
             => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,Fa,X2)) )
         => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),A4)) ) ) ).

% sum_nonneg
tff(fact_3745_sum__mono__inv,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [Fa: fun(B,A),I5: set(B),G: fun(B,A),I: B] :
          ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,Fa),I5) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),I5) )
         => ( ! [I2: B] :
                ( member(B,I2,I5)
               => aa(A,$o,ord_less_eq(A,aa(B,A,Fa,I2)),aa(B,A,G,I2)) )
           => ( member(B,I,I5)
             => ( finite_finite2(B,I5)
               => ( aa(B,A,Fa,I) = aa(B,A,G,I) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_3746_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A4: set(nat),Fa: fun(nat,A),G: fun(nat,A)] :
          ( ~ member(nat,zero_zero(nat),A4)
         => ( ! [X2: nat] :
                ( member(nat,aa(nat,nat,suc,X2),A4)
               => ( aa(nat,A,Fa,aa(nat,nat,suc,X2)) = aa(nat,A,G,aa(nat,nat,suc,X2)) ) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),A4) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),A4) ) ) ) ) ).

% sum_cong_Suc
tff(fact_3747_sum_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( finite_finite2(A,A4)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_uu(set(A),fun(fun(A,$o),fun(A,$o)),A4),P))) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_uv(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A4) ) ) ) ).

% sum.inter_filter
tff(fact_3748_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_uw(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_3749_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ux(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_3750_suminf__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),Fa: fun(A,fun(nat,B))] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => summable(B,aa(A,fun(nat,B),Fa,I2)) )
         => ( suminf(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_ut(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),Fa)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aTP_Lamp_uy(fun(A,fun(nat,B)),fun(A,B),Fa)),I5) ) ) ) ).

% suminf_sum
tff(fact_3751_summable__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real)] :
          ( summable(real,X5)
         => summable(A,aTP_Lamp_uz(fun(nat,real),fun(nat,A),X5)) ) ) ).

% summable_of_real
tff(fact_3752_sum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A4: set(A),Fa: fun(A,B)] :
          ( finite_finite2(A,A4)
         => ( ! [X2: A] :
                ( member(A,X2,A4)
               => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,Fa,X2)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),A4) = zero_zero(B) )
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => ( aa(A,B,Fa,X) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_3753_sum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [S: set(A),Tb: set(B),G: fun(B,C),I: fun(B,A),Fa: fun(A,C)] :
          ( finite_finite2(A,S)
         => ( finite_finite2(B,Tb)
           => ( ! [X2: B] :
                  ( member(B,X2,Tb)
                 => aa(C,$o,ord_less_eq(C,zero_zero(C)),aa(B,C,G,X2)) )
             => ( ! [X2: A] :
                    ( member(A,X2,S)
                   => ? [Xa: B] :
                        ( member(B,Xa,Tb)
                        & ( aa(B,A,I,Xa) = X2 )
                        & aa(C,$o,ord_less_eq(C,aa(A,C,Fa,X2)),aa(B,C,G,Xa)) ) )
               => aa(C,$o,ord_less_eq(C,aa(set(A),C,groups7311177749621191930dd_sum(A,C,Fa),S)),aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),Tb)) ) ) ) ) ) ).

% sum_le_included
tff(fact_3754_sum__strict__mono__ex1,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A4: set(A),Fa: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A4)
         => ( ! [X2: A] :
                ( member(A,X2,A4)
               => aa(B,$o,ord_less_eq(B,aa(A,B,Fa,X2)),aa(A,B,G,X2)) )
           => ( ? [X3: A] :
                  ( member(A,X3,A4)
                  & aa(B,$o,ord_less(B,aa(A,B,Fa,X3)),aa(A,B,G,X3)) )
             => aa(B,$o,ord_less(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),A4)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_3755_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R: fun(A,fun(A,$o)),S3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,zero_zero(A)),zero_zero(A))
         => ( ! [X15: A,Y15: A,X23: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X15),X23)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y15),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(A,A,aa(A,fun(A,A),plus_plus(A),X15),Y15)),aa(A,A,aa(A,fun(A,A),plus_plus(A),X23),Y23)) )
           => ( finite_finite2(B,S3)
             => ( ! [X2: B] :
                    ( member(B,X2,S3)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H,X2)),aa(B,A,G,X2)) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),S3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),S3)) ) ) ) ) ) ).

% sum.related
tff(fact_3756_sum__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [A4: set(A),Fa: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [X2: A] :
                  ( member(A,X2,A4)
                 => aa(B,$o,ord_less(B,aa(A,B,Fa,X2)),aa(A,B,G,X2)) )
             => aa(B,$o,ord_less(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),A4)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4)) ) ) ) ) ).

% sum_strict_mono
tff(fact_3757_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S6: set(A),T5: set(B),S3: set(A),I: fun(B,A),J: fun(A,B),T6: set(B),G: fun(A,C),H: fun(B,C)] :
          ( finite_finite2(A,S6)
         => ( finite_finite2(B,T5)
           => ( ! [A2: A] :
                  ( member(A,A2,aa(set(A),set(A),minus_minus(set(A),S3),S6))
                 => ( aa(B,A,I,aa(A,B,J,A2)) = A2 ) )
             => ( ! [A2: A] :
                    ( member(A,A2,aa(set(A),set(A),minus_minus(set(A),S3),S6))
                   => member(B,aa(A,B,J,A2),aa(set(B),set(B),minus_minus(set(B),T6),T5)) )
               => ( ! [B2: B] :
                      ( member(B,B2,aa(set(B),set(B),minus_minus(set(B),T6),T5))
                     => ( aa(A,B,J,aa(B,A,I,B2)) = B2 ) )
                 => ( ! [B2: B] :
                        ( member(B,B2,aa(set(B),set(B),minus_minus(set(B),T6),T5))
                       => member(A,aa(B,A,I,B2),aa(set(A),set(A),minus_minus(set(A),S3),S6)) )
                   => ( ! [A2: A] :
                          ( member(A,A2,S6)
                         => ( aa(A,C,G,A2) = zero_zero(C) ) )
                     => ( ! [B2: B] :
                            ( member(B,B2,T5)
                           => ( aa(B,C,H,B2) = zero_zero(C) ) )
                       => ( ! [A2: A] :
                              ( member(A,A2,S3)
                             => ( aa(B,C,H,aa(A,B,J,A2)) = aa(A,C,G,A2) ) )
                         => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,G),S3) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,H),T6) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_3758_nonzero__of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [Y: real,Xb: real] :
          ( ( Y != zero_zero(real) )
         => ( aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),Xb)),aa(real,A,real_Vector_of_real(A),Y)) ) ) ) ).

% nonzero_of_real_divide
tff(fact_3759_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S: set(A),Fa: fun(A,B),B6: B,I: A] :
          ( finite_finite2(A,S)
         => ( ! [I2: A] :
                ( member(A,I2,S)
               => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,Fa,I2)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),S) = B6 )
             => ( member(A,I,S)
               => aa(B,$o,ord_less_eq(B,aa(A,B,Fa,I)),B6) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_3760_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S: set(A),Fa: fun(A,B),I: A] :
          ( finite_finite2(A,S)
         => ( ! [I2: A] :
                ( member(A,I2,S)
               => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,Fa,I2)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),S) = zero_zero(B) )
             => ( member(A,I,S)
               => ( aa(A,B,Fa,I) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_3761_sum_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A4)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),minus_minus(set(A),A4),collect(A,aTP_Lamp_va(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_3762_sum__power__add,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,M: nat,I5: set(nat)] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_vb(A,fun(nat,fun(nat,A)),Xb),M)),I5) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),I5)) ) ).

% sum_power_add
tff(fact_3763_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,N,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_vc(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,N,M)) ) ).

% sum.atLeastAtMost_rev
tff(fact_3764_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N2: set(nat),Fa: fun(nat,A)] :
          ( finite_finite2(nat,N2)
         => ( ! [N3: nat] :
                ( ~ member(nat,N3,N2)
               => ( aa(nat,A,Fa,N3) = zero_zero(A) ) )
           => ( suminf(A,Fa) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),N2) ) ) ) ) ).

% suminf_finite
tff(fact_3765_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),I: A,Fa: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,ord_less(B,zero_zero(B)),aa(A,B,Fa,I))
             => ( ! [I2: A] :
                    ( member(A,I2,I5)
                   => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,Fa,I2)) )
               => aa(B,$o,ord_less(B,zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),I5)) ) ) ) ) ) ).

% sum_pos2
tff(fact_3766_sum__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),Fa: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,ord_less(B,zero_zero(B)),aa(A,B,Fa,I2)) )
             => aa(B,$o,ord_less(B,zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),I5)) ) ) ) ) ).

% sum_pos
tff(fact_3767_suminf__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real)] :
          ( summable(real,X5)
         => ( aa(real,A,real_Vector_of_real(A),suminf(real,X5)) = suminf(A,aTP_Lamp_uz(fun(nat,real),fun(nat,A),X5)) ) ) ) ).

% suminf_of_real
tff(fact_3768_norm__less__p1,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xb: A] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(real,A,real_Vector_of_real(A),real_V7770717601297561774m_norm(A,Xb))),one_one(A)))) ) ).

% norm_less_p1
tff(fact_3769_sum_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T6: set(A),S3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,T6)
         => ( aa(set(A),$o,ord_less_eq(set(A),S3),T6)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T6),S3))
                 => ( aa(A,B,G,X2) = zero_zero(B) ) )
             => ( ! [X2: A] :
                    ( member(A,X2,S3)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
               => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),T6) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),S3) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_3770_sum_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T6: set(A),S3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,T6)
         => ( aa(set(A),$o,ord_less_eq(set(A),S3),T6)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T6),S3))
                 => ( aa(A,B,H,X2) = zero_zero(B) ) )
             => ( ! [X2: A] :
                    ( member(A,X2,S3)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
               => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),S3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),T6) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_3771_sum_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T6: set(A),S3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,T6)
         => ( aa(set(A),$o,ord_less_eq(set(A),S3),T6)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T6),S3))
                 => ( aa(A,B,G,X2) = zero_zero(B) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),T6) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),S3) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_3772_sum_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T6: set(A),S3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,T6)
         => ( aa(set(A),$o,ord_less_eq(set(A),S3),T6)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T6),S3))
                 => ( aa(A,B,G,X2) = zero_zero(B) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),S3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),T6) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_3773_sum_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C5: set(A),A4: set(A),B6: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,C5)
         => ( aa(set(A),$o,ord_less_eq(set(A),A4),C5)
           => ( aa(set(A),$o,ord_less_eq(set(A),B6),C5)
             => ( ! [A2: A] :
                    ( member(A,A2,aa(set(A),set(A),minus_minus(set(A),C5),A4))
                   => ( aa(A,B,G,A2) = zero_zero(B) ) )
               => ( ! [B2: A] :
                      ( member(A,B2,aa(set(A),set(A),minus_minus(set(A),C5),B6))
                     => ( aa(A,B,H,B2) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),C5) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),C5) )
                   => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),B6) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_3774_sum_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C5: set(A),A4: set(A),B6: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,C5)
         => ( aa(set(A),$o,ord_less_eq(set(A),A4),C5)
           => ( aa(set(A),$o,ord_less_eq(set(A),B6),C5)
             => ( ! [A2: A] :
                    ( member(A,A2,aa(set(A),set(A),minus_minus(set(A),C5),A4))
                   => ( aa(A,B,G,A2) = zero_zero(B) ) )
               => ( ! [B2: A] :
                      ( member(A,B2,aa(set(A),set(A),minus_minus(set(A),C5),B6))
                     => ( aa(A,B,H,B2) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),B6) )
                  <=> ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),C5) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),C5) ) ) ) ) ) ) ) ) ).

% sum.same_carrier
tff(fact_3775_sum__shift__lb__Suc0__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Fa: fun(nat,A),K: nat] :
          ( ( aa(nat,A,Fa,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_3776_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_3777_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_3778_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,aa(nat,nat,suc,N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_3779_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_uw(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_3780_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,Fa: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,N))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_vd(fun(nat,A),fun(nat,A),Fa)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,minus_minus(A,aa(nat,A,Fa,aa(nat,nat,suc,N))),aa(nat,A,Fa,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_3781_sum__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [B6: set(A),A4: set(A),Fa: fun(A,B)] :
          ( finite_finite2(A,B6)
         => ( aa(set(A),$o,ord_less_eq(set(A),A4),B6)
           => ( ! [B2: A] :
                  ( member(A,B2,aa(set(A),set(A),minus_minus(set(A),B6),A4))
                 => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,Fa,B2)) )
             => aa(B,$o,ord_less_eq(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),A4)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),B6)) ) ) ) ) ).

% sum_mono2
tff(fact_3782_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,arccos,Y)) ) ) ).

% arccos_lbound
tff(fact_3783_arccos__less__arccos,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),Y)
       => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
         => aa(real,$o,ord_less(real,aa(real,real,arccos,Y)),aa(real,real,arccos,Xb)) ) ) ) ).

% arccos_less_arccos
tff(fact_3784_arccos__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xb)),one_one(real))
     => ( aa(real,$o,ord_less_eq(real,abs_abs(real,Y)),one_one(real))
       => ( aa(real,$o,ord_less(real,aa(real,real,arccos,Xb)),aa(real,real,arccos,Y))
        <=> aa(real,$o,ord_less(real,Y),Xb) ) ) ) ).

% arccos_less_mono
tff(fact_3785_arccos__cos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),pi)
       => ( aa(real,real,arccos,cos(real,Xb)) = Xb ) ) ) ).

% arccos_cos
tff(fact_3786_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A),P2: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_3787_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Fa: fun(nat,A),I5: set(nat)] :
          ( summable(A,Fa)
         => ( finite_finite2(nat,I5)
           => ( ! [N3: nat] :
                  ( member(nat,N3,aa(set(nat),set(nat),uminus_uminus(set(nat)),I5))
                 => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,Fa,N3)) )
             => aa(A,$o,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),I5)),suminf(A,Fa)) ) ) ) ) ).

% sum_le_suminf
tff(fact_3788_sum_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),A3: A,B3: fun(A,B),C2: fun(A,B)] :
          ( finite_finite2(A,S3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ve(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),B3),C2)),S3) = $ite(member(A,A3,S3),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,B3,A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,C2),aa(set(A),set(A),minus_minus(set(A),S3),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,C2),aa(set(A),set(A),minus_minus(set(A),S3),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_3789_set__encode__def,axiom,
    nat_set_encode = groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% set_encode_def
tff(fact_3790_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B6: set(A),A4: set(A),B3: A,Fa: fun(A,B)] :
          ( finite_finite2(A,B6)
         => ( aa(set(A),$o,ord_less_eq(set(A),A4),B6)
           => ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),B6),A4))
             => ( aa(B,$o,ord_less(B,zero_zero(B)),aa(A,B,Fa,B3))
               => ( ! [X2: A] :
                      ( member(A,X2,B6)
                     => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,Fa,X2)) )
                 => aa(B,$o,ord_less(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),A4)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),B6)) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_3791_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: A,A4: set(A),Fa: fun(A,B)] :
          ( member(A,I,A4)
         => ( ! [X2: A] :
                ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),insert(A,I),bot_bot(set(A)))))
               => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,Fa,X2)) )
           => ( finite_finite2(A,A4)
             => aa(B,$o,ord_less_eq(B,aa(A,B,Fa,I)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),A4)) ) ) ) ) ).

% member_le_sum
tff(fact_3792_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),Xb: fun(A,B),A3: fun(A,B),B3: B,Delta: B] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,Xb,I2)) )
         => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,Xb),I5) = one_one(B) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,ord_less_eq(B,abs_abs(B,aa(B,B,minus_minus(B,aa(A,B,A3,I2)),B3))),Delta) )
             => aa(B,$o,ord_less_eq(B,abs_abs(B,aa(B,B,minus_minus(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vf(fun(A,B),fun(fun(A,B),fun(A,B)),Xb),A3)),I5)),B3))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_3793_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less(real,Y),one_one(real))
       => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,ord_less(real,aa(real,real,arccos,Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_3794_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,ord_less_eq(real,aa(real,real,arccos,Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_3795_sin__arccos__nonzero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),one_one(real))
       => ( sin(real,aa(real,real,arccos,Xb)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_3796_arccos__cos2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,Xb),zero_zero(real))
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),pi)),Xb)
       => ( aa(real,real,arccos,cos(real,Xb)) = aa(real,real,uminus_uminus(real),Xb) ) ) ) ).

% arccos_cos2
tff(fact_3797_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Fa: fun(nat,A),M: nat,N: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_vg(fun(nat,A),fun(nat,A),Fa)),set_or1337092689740270186AtMost(nat,M,N)) = $ite(aa(nat,$o,ord_less_eq(nat,M),N),aa(A,A,minus_minus(A,aa(nat,A,Fa,M)),aa(nat,A,Fa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))),zero_zero(A)) ) ).

% sum_natinterval_diff
tff(fact_3798_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,Fa: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_vh(fun(nat,A),fun(nat,A),Fa)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N)) = aa(A,A,minus_minus(A,aa(nat,A,Fa,N)),aa(nat,A,Fa,M)) ) ) ) ).

% sum_telescope''
tff(fact_3799_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Fa: fun(nat,A),E3: real] :
          ( summable(A,Fa)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
           => ~ ! [N11: nat] :
                  ~ ! [M2: nat] :
                      ( aa(nat,$o,ord_less_eq(nat,N11),M2)
                     => ! [N4: nat] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_or1337092689740270186AtMost(nat,M2,N4)))),E3) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_3800_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [N: nat] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2)))),collect(nat,aTP_Lamp_sx(nat,fun(nat,$o),N))) ) ).

% mask_eq_sum_exp
tff(fact_3801_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N: nat,Xb: A] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),Xb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_or1337092689740270186AtMost(nat,M,N))) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,N))) ) ) ) ).

% sum_gp_multiplied
tff(fact_3802_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_vi(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.in_pairs
tff(fact_3803_arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,arccos,Y))
          & aa(real,$o,ord_less_eq(real,aa(real,real,arccos,Y)),pi)
          & ( cos(real,aa(real,real,arccos,Y)) = Y ) ) ) ) ).

% arccos
tff(fact_3804_cos__sin__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : cos(A,Xb) = sin(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2)))),Xb)) ) ).

% cos_sin_eq
tff(fact_3805_sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : sin(A,Xb) = cos(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2)))),Xb)) ) ).

% sin_cos_eq
tff(fact_3806_mask__eq__sum__exp__nat,axiom,
    ! [N: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(nat,nat,suc,zero_zero(nat))) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))),collect(nat,aTP_Lamp_sx(nat,fun(nat,$o),N))) ).

% mask_eq_sum_exp_nat
tff(fact_3807_gauss__sum__nat,axiom,
    ! [N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_vj(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,suc,N))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% gauss_sum_nat
tff(fact_3808_minus__sin__cos__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,uminus_uminus(A),sin(A,Xb)) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(real,A,real_Vector_of_real(A),pi)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% minus_sin_cos_eq
tff(fact_3809_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,D: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_vk(A,fun(A,fun(nat,A)),A3),D)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D))) ) ).

% double_arith_series
tff(fact_3810_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_3811_arith__series__nat,axiom,
    ! [A3: nat,D: nat,N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_vl(nat,fun(nat,fun(nat,nat)),A3),D)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),D)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% arith_series_nat
tff(fact_3812_Sum__Icc__nat,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_vj(nat,nat)),set_or1337092689740270186AtMost(nat,M,N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,minus_minus(nat,M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Icc_nat
tff(fact_3813_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ( aa(real,$o,ord_less_eq(real,Y),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,arccos,Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ) ).

% arccos_le_pi2
tff(fact_3814_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_3815_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A3: A,D: A,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_vm(A,fun(A,fun(nat,A)),A3),D)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),D)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% arith_series
tff(fact_3816_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum
tff(fact_3817_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,M: nat,N: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = $ite(Xb = one_one(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),M)),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,N))))),aa(A,A,minus_minus(A,one_one(A)),Xb))) ) ).

% sum_gp_offset
tff(fact_3818_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_3819_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] : aa(real,real,arccos,cos(real,Theta)) != abs_abs(real,aa(real,real,minus_minus(real,Theta),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),K2)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)))) ).

% arccos_cos_eq_abs_2pi
tff(fact_3820_arsinh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A] : aa(A,A,arsinh(A),Xb) = aa(A,A,ln_ln(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),powr(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)),aa(real,A,real_Vector_of_real(A),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))))) ) ).

% arsinh_def
tff(fact_3821_sin__arccos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),one_one(real))
       => ( sin(real,aa(real,real,arccos,Xb)) = aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% sin_arccos
tff(fact_3822_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z: A,N: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),N))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))) = aa(A,A,aa(A,fun(A,A),times_times(A),H),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_vo(A,fun(A,fun(nat,fun(nat,A))),H),Z),N)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ) ).

% lemma_termdiff2
tff(fact_3823_dbl__inc__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_inc(A,one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,one2)) ) ) ).

% dbl_inc_simps(3)
tff(fact_3824_geometric__deriv__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => aa(A,$o,sums(A,aTP_Lamp_vp(A,fun(nat,A),Z)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,minus_minus(A,one_one(A)),Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% geometric_deriv_sums
tff(fact_3825_monoseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( topological_monoseq(A,X5)
        <=> ( ! [M6: nat,N8: nat] :
                ( aa(nat,$o,ord_less_eq(nat,M6),N8)
               => aa(A,$o,ord_less_eq(A,aa(nat,A,X5,M6)),aa(nat,A,X5,N8)) )
            | ! [M6: nat,N8: nat] :
                ( aa(nat,$o,ord_less_eq(nat,M6),N8)
               => aa(A,$o,ord_less_eq(A,aa(nat,A,X5,N8)),aa(nat,A,X5,M6)) ) ) ) ) ).

% monoseq_def
tff(fact_3826_lessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_lessThan(A,K))
        <=> aa(A,$o,ord_less(A,I),K) ) ) ).

% lessThan_iff
tff(fact_3827_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => aa(A,$o,sums(A,aTP_Lamp_ml(nat,A)),zero_zero(A)) ) ).

% sums_zero
tff(fact_3828_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_ord_lessThan(A,Xb)),set_ord_lessThan(A,Y))
        <=> aa(A,$o,ord_less_eq(A,Xb),Y) ) ) ).

% lessThan_subset_iff
tff(fact_3829_lessThan__0,axiom,
    set_ord_lessThan(nat,zero_zero(nat)) = bot_bot(set(nat)) ).

% lessThan_0
tff(fact_3830_dbl__inc__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_inc(A,zero_zero(A)) = one_one(A) ) ) ).

% dbl_inc_simps(2)
tff(fact_3831_dbl__inc__simps_I4_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ( neg_numeral_dbl_inc(A,aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% dbl_inc_simps(4)
tff(fact_3832_dbl__inc__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,K)) ) ).

% dbl_inc_simps(5)
tff(fact_3833_dbl__inc__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_inc(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_inc_simps(1)
tff(fact_3834_dbl__dec__simps_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_dec(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,uminus_uminus(A),neg_numeral_dbl_inc(A,aa(num,A,numeral_numeral(A),K))) ) ).

% dbl_dec_simps(1)
tff(fact_3835_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: fun(nat,A),Xb: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_nc(fun(nat,A),fun(nat,A),A3)),Xb)
        <=> ( aa(nat,A,A3,zero_zero(nat)) = Xb ) ) ) ).

% powser_sums_zero_iff
tff(fact_3836_sums__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),Fa: fun(A,fun(nat,B)),Xb: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => aa(B,$o,sums(B,aa(A,fun(nat,B),Fa,I2)),aa(A,B,Xb,I2)) )
         => aa(B,$o,sums(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_ut(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),Fa)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,Xb),I5)) ) ) ).

% sums_sum
tff(fact_3837_sums__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real),A3: real] :
          ( aa(real,$o,sums(real,X5),A3)
         => aa(A,$o,sums(A,aTP_Lamp_uz(fun(nat,real),fun(nat,A),X5)),aa(real,A,real_Vector_of_real(A),A3)) ) ) ).

% sums_of_real
tff(fact_3838_sums__of__real__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Fa: fun(nat,real),C2: real] :
          ( aa(A,$o,sums(A,aTP_Lamp_vq(fun(nat,real),fun(nat,A),Fa)),aa(real,A,real_Vector_of_real(A),C2))
        <=> aa(real,$o,sums(real,Fa),C2) ) ) ).

% sums_of_real_iff
tff(fact_3839_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Fa: fun(nat,A),G: fun(nat,A),S: A,Tb: A] :
          ( ! [N3: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,Fa,N3)),aa(nat,A,G,N3))
         => ( aa(A,$o,sums(A,Fa),S)
           => ( aa(A,$o,sums(A,G),Tb)
             => aa(A,$o,ord_less_eq(A,S),Tb) ) ) ) ) ).

% sums_le
tff(fact_3840_int__sum,axiom,
    ! [A: $tType,Fa: fun(A,nat),A4: set(A)] : aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,Fa),A4)) = aa(set(A),int,groups7311177749621191930dd_sum(A,int,aTP_Lamp_vr(fun(A,nat),fun(A,int),Fa)),A4) ).

% int_sum
tff(fact_3841_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),N: A] :
          ( ! [X2: A] : aa(nat,$o,ord_less_eq(nat,aa(A,nat,Q,X2)),aa(A,nat,P,X2))
         => ( aa(nat,nat,minus_minus(nat,aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,P),set_ord_lessThan(A,N))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,Q),set_ord_lessThan(A,N))) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_vs(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),set_ord_lessThan(A,N)) ) ) ) ).

% sum_diff_distrib
tff(fact_3842_sums__0,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Fa: fun(nat,A)] :
          ( ! [N3: nat] : aa(nat,A,Fa,N3) = zero_zero(A)
         => aa(A,$o,sums(A,Fa),zero_zero(A)) ) ) ).

% sums_0
tff(fact_3843_sums__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,Fa: fun(nat,A)] : aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mm(nat,fun(fun(nat,A),fun(nat,A)),I),Fa)),aa(nat,A,Fa,I)) ) ).

% sums_single
tff(fact_3844_sums__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),A3: A,G: fun(nat,A),B3: A] :
          ( aa(A,$o,sums(A,Fa),A3)
         => ( aa(A,$o,sums(A,G),B3)
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mv(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Fa),G)),aa(A,A,minus_minus(A,A3),B3)) ) ) ) ).

% sums_diff
tff(fact_3845_sums__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Fa: fun(nat,A),A3: A,G: fun(nat,A),B3: A] :
          ( aa(A,$o,sums(A,Fa),A3)
         => ( aa(A,$o,sums(A,G),B3)
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ms(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Fa),G)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ) ) ).

% sums_add
tff(fact_3846_sums__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Fa: fun(nat,A),A3: A,C2: A] :
          ( aa(A,$o,sums(A,Fa),A3)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_mt(fun(nat,A),fun(A,fun(nat,A)),Fa),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),C2)) ) ) ).

% sums_mult2
tff(fact_3847_sums__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Fa: fun(nat,A),A3: A,C2: A] :
          ( aa(A,$o,sums(A,Fa),A3)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_mu(fun(nat,A),fun(A,fun(nat,A)),Fa),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) ) ) ).

% sums_mult
tff(fact_3848_sums__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(nat,A),A3: A,C2: A] :
          ( aa(A,$o,sums(A,Fa),A3)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_mp(fun(nat,A),fun(A,fun(nat,A)),Fa),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)) ) ) ).

% sums_divide
tff(fact_3849_sums__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),A3: A] :
          ( aa(A,$o,sums(A,Fa),A3)
         => aa(A,$o,sums(A,aTP_Lamp_mx(fun(nat,A),fun(nat,A),Fa)),aa(A,A,uminus_uminus(A),A3)) ) ) ).

% sums_minus
tff(fact_3850_Compl__eq,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A4) = collect(A,aTP_Lamp_vt(set(A),fun(A,$o),A4)) ).

% Compl_eq
tff(fact_3851_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] : collect(A,aTP_Lamp_vu(fun(A,$o),fun(A,$o),P)) = aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P)) ).

% Collect_neg_eq
tff(fact_3852_uminus__set__def,axiom,
    ! [A: $tType,A4: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A4) = collect(A,aa(fun(A,$o),fun(A,$o),uminus_uminus(fun(A,$o)),aTP_Lamp_cc(set(A),fun(A,$o),A4))) ).

% uminus_set_def
tff(fact_3853_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_lessThan(A,U) = collect(A,aTP_Lamp_vv(A,fun(A,$o),U)) ) ).

% lessThan_def
tff(fact_3854_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),N: nat,S: A] :
          ( aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_mn(fun(nat,A),fun(nat,fun(nat,A)),Fa),N)),S)
        <=> aa(A,$o,sums(A,Fa),aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_ord_lessThan(nat,N)))) ) ) ).

% sums_iff_shift
tff(fact_3855_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),S: A,N: nat] :
          ( aa(A,$o,sums(A,Fa),S)
         => aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_mn(fun(nat,A),fun(nat,fun(nat,A)),Fa),N)),aa(A,A,minus_minus(A,S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_ord_lessThan(nat,N)))) ) ) ).

% sums_split_initial_segment
tff(fact_3856_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),N: nat,S: A] :
          ( aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_mn(fun(nat,A),fun(nat,fun(nat,A)),Fa),N)),aa(A,A,minus_minus(A,S),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_ord_lessThan(nat,N))))
        <=> aa(A,$o,sums(A,Fa),S) ) ) ).

% sums_iff_shift'
tff(fact_3857_sum__eq__empty__iff,axiom,
    ! [A: $tType,B: $tType,A4: set(A),Fa: fun(A,multiset(B))] :
      ( finite_finite2(A,A4)
     => ( ( aa(set(A),multiset(B),groups7311177749621191930dd_sum(A,multiset(B),Fa),A4) = zero_zero(multiset(B)) )
      <=> ! [X: A] :
            ( member(A,X,A4)
           => ( aa(A,multiset(B),Fa,X) = zero_zero(multiset(B)) ) ) ) ) ).

% sum_eq_empty_iff
tff(fact_3858_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N: A] :
          ( aa(set(A),$o,ord_less(set(A),set_ord_lessThan(A,M)),set_ord_lessThan(A,N))
        <=> aa(A,$o,ord_less(A,M),N) ) ) ).

% lessThan_strict_subset_iff
tff(fact_3859_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,Fa: fun(nat,A),D: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vw(A,fun(fun(nat,A),fun(nat,A)),C2),Fa)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D))
          <=> aa(A,$o,sums(A,Fa),D) ) ) ) ).

% sums_mult_iff
tff(fact_3860_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,Fa: fun(nat,A),D: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vx(A,fun(fun(nat,A),fun(nat,A)),C2),Fa)),aa(A,A,aa(A,fun(A,A),times_times(A),D),C2))
          <=> aa(A,$o,sums(A,Fa),D) ) ) ) ).

% sums_mult2_iff
tff(fact_3861_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( set_ord_lessThan(nat,N) = bot_bot(set(nat)) )
    <=> ( N = zero_zero(nat) ) ) ).

% lessThan_empty_iff
tff(fact_3862_sum__subtractf__nat,axiom,
    ! [A: $tType,A4: set(A),G: fun(A,nat),Fa: fun(A,nat)] :
      ( ! [X2: A] :
          ( member(A,X2,A4)
         => aa(nat,$o,ord_less_eq(nat,aa(A,nat,G,X2)),aa(A,nat,Fa,X2)) )
     => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_vy(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),Fa)),A4) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,Fa),A4)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,G),A4)) ) ) ).

% sum_subtractf_nat
tff(fact_3863_sum__SucD,axiom,
    ! [A: $tType,Fa: fun(A,nat),A4: set(A),N: nat] :
      ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,Fa),A4) = aa(nat,nat,suc,N) )
     => ? [X2: A] :
          ( member(A,X2,A4)
          & aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(A,nat,Fa,X2)) ) ) ).

% sum_SucD
tff(fact_3864_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A4: set(A),Fa: fun(A,nat)] :
      ( finite_finite2(A,A4)
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,Fa),A4) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X: A] :
            ( member(A,X,A4)
            & ( aa(A,nat,Fa,X) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa3: A] :
                ( member(A,Xa3,A4)
               => ( ( X != Xa3 )
                 => ( aa(A,nat,Fa,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_3865_sum__eq__1__iff,axiom,
    ! [A: $tType,A4: set(A),Fa: fun(A,nat)] :
      ( finite_finite2(A,A4)
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,Fa),A4) = one_one(nat) )
      <=> ? [X: A] :
            ( member(A,X,A4)
            & ( aa(A,nat,Fa,X) = one_one(nat) )
            & ! [Xa3: A] :
                ( member(A,Xa3,A4)
               => ( ( X != Xa3 )
                 => ( aa(A,nat,Fa,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_3866_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,Fa: fun(nat,A),A3: A] :
          ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mo(A,fun(fun(nat,A),fun(nat,A)),C2),Fa)),A3)
         => ( ( C2 != zero_zero(A) )
           => aa(A,$o,sums(A,Fa),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),C2)) ) ) ) ).

% sums_mult_D
tff(fact_3867_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),S: A] :
          ( ( aa(nat,A,Fa,zero_zero(nat)) = zero_zero(A) )
         => ( aa(A,$o,sums(A,aTP_Lamp_mw(fun(nat,A),fun(nat,A),Fa)),S)
           => aa(A,$o,sums(A,Fa),S) ) ) ) ).

% sums_Suc_imp
tff(fact_3868_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Fa: fun(nat,A),L: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_vz(fun(nat,A),fun(nat,A),Fa)),L)
         => aa(A,$o,sums(A,Fa),aa(A,A,aa(A,fun(A,A),plus_plus(A),L),aa(nat,A,Fa,zero_zero(nat)))) ) ) ).

% sums_Suc
tff(fact_3869_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),S: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_mw(fun(nat,A),fun(nat,A),Fa)),S)
        <=> aa(A,$o,sums(A,Fa),aa(A,A,aa(A,fun(A,A),plus_plus(A),S),aa(nat,A,Fa,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_3870_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [N: nat,Fa: fun(nat,A),S: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,ord_less(nat,I2),N)
             => ( aa(nat,A,Fa,I2) = zero_zero(A) ) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wa(nat,fun(fun(nat,A),fun(nat,A)),N),Fa)),S)
          <=> aa(A,$o,sums(A,Fa),S) ) ) ) ).

% sums_zero_iff_shift
tff(fact_3871_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A4: set(nat),Fa: fun(nat,A)] :
          ( finite_finite2(nat,A4)
         => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_sn(set(nat),fun(fun(nat,A),fun(nat,A)),A4),Fa)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),A4)) ) ) ).

% sums_If_finite_set
tff(fact_3872_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),Fa: fun(nat,A)] :
          ( finite_finite2(nat,collect(nat,P))
         => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_so(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),Fa)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),collect(nat,P))) ) ) ).

% sums_If_finite
tff(fact_3873_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N2: set(nat),Fa: fun(nat,A)] :
          ( finite_finite2(nat,N2)
         => ( ! [N3: nat] :
                ( ~ member(nat,N3,N2)
               => ( aa(nat,A,Fa,N3) = zero_zero(A) ) )
           => aa(A,$o,sums(A,Fa),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),N2)) ) ) ) ).

% sums_finite
tff(fact_3874_lessThan__nat__numeral,axiom,
    ! [K: num] : set_ord_lessThan(nat,aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),insert(nat,pred_numeral(K)),set_ord_lessThan(nat,pred_numeral(K))) ).

% lessThan_nat_numeral
tff(fact_3875_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wb(fun(nat,A),fun(nat,fun(nat,A)),G),N)),set_ord_lessThan(nat,N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,N)) ) ).

% sum.nat_diff_reindex
tff(fact_3876_sum__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( aa(nat,$o,ord_less(nat,one_one(nat)),N)
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_wc(complex,complex)),collect(complex,aa(complex,fun(complex,$o),aTP_Lamp_td(nat,fun(complex,fun(complex,$o)),N),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_3877_sum__roots__unity,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,one_one(nat)),N)
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_wc(complex,complex)),collect(complex,aTP_Lamp_wd(nat,fun(complex,$o),N))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_3878_powser__sums__if,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [M: nat,Z: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_we(nat,fun(A,fun(nat,A)),M),Z)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),M)) ) ).

% powser_sums_if
tff(fact_3879_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: fun(nat,A)] : aa(A,$o,sums(A,aTP_Lamp_nc(fun(nat,A),fun(nat,A),A3)),aa(nat,A,A3,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_3880_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S3: A,A4: set(nat),S6: A,Fa: fun(nat,A)] :
          ( aa(A,$o,sums(A,G),S3)
         => ( finite_finite2(nat,A4)
           => ( ( S6 = aa(A,A,aa(A,fun(A,A),plus_plus(A),S3),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G),Fa)),A4)) )
             => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_wg(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A4),Fa)),S6) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_3881_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Fa: fun(nat,A),Xb: A] :
          ( summable(A,Fa)
         => ( ! [N3: nat] : aa(A,$o,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_ord_lessThan(nat,N3))),Xb)
           => aa(A,$o,ord_less_eq(A,suminf(A,Fa)),Xb) ) ) ) ).

% suminf_le_const
tff(fact_3882_dbl__inc__def,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Xb: A] : neg_numeral_dbl_inc(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb)),one_one(A)) ) ).

% dbl_inc_def
tff(fact_3883_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_uw(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).

% sum.lessThan_Suc_shift
tff(fact_3884_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Fa: fun(nat,A),M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_vd(fun(nat,A),fun(nat,A),Fa)),set_ord_lessThan(nat,M)) = aa(A,A,minus_minus(A,aa(nat,A,Fa,M)),aa(nat,A,Fa,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_3885_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Fa: fun(nat,A),M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_wh(fun(nat,A),fun(nat,A),Fa)),set_ord_lessThan(nat,M)) = aa(A,A,minus_minus(A,aa(nat,A,Fa,zero_zero(nat))),aa(nat,A,Fa,M)) ) ).

% sum_lessThan_telescope'
tff(fact_3886_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Fa: fun(nat,A),N: nat,R2: A] : aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_ord_lessThan(nat,N))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),R2)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_wi(fun(nat,A),fun(A,fun(nat,A)),Fa),R2)),set_ord_lessThan(nat,N)) ) ).

% sumr_diff_mult_const2
tff(fact_3887_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Fa: fun(nat,A),Xb: A] :
          ( ! [N3: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,Fa,N3))
         => ( ! [N3: nat] : aa(A,$o,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_ord_lessThan(nat,N3))),Xb)
           => summable(A,Fa) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_3888_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_uw(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_3889_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,N: nat] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N)),one_one(A)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Xb),one_one(A))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_ord_lessThan(nat,N))) ) ).

% power_diff_1_eq
tff(fact_3890_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,N: nat] : aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),Xb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_ord_lessThan(nat,N))) ) ).

% one_diff_power_eq
tff(fact_3891_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,N: nat] :
          ( ( Xb != one_one(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_ord_lessThan(nat,N)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N)),one_one(A))),aa(A,A,minus_minus(A,Xb),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_3892_suminf__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),K: nat] :
          ( summable(A,Fa)
         => ( suminf(A,Fa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_mn(fun(nat,A),fun(nat,fun(nat,A)),Fa),K))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_ord_lessThan(nat,K))) ) ) ) ).

% suminf_split_initial_segment
tff(fact_3893_suminf__minus__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),K: nat] :
          ( summable(A,Fa)
         => ( suminf(A,aa(nat,fun(nat,A),aTP_Lamp_mn(fun(nat,A),fun(nat,fun(nat,A)),Fa),K)) = aa(A,A,minus_minus(A,suminf(A,Fa)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_ord_lessThan(nat,K))) ) ) ) ).

% suminf_minus_initial_segment
tff(fact_3894_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Fa: fun(nat,A),N: nat] :
          ( summable(A,Fa)
         => ( ! [M3: nat] :
                ( aa(nat,$o,ord_less_eq(nat,N),M3)
               => aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,Fa,M3)) )
           => aa(A,$o,ord_less(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_ord_lessThan(nat,N))),suminf(A,Fa)) ) ) ) ).

% sum_less_suminf
tff(fact_3895_lemma__termdiff1,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Z: A,H: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_wj(A,fun(A,fun(nat,fun(nat,A))),Z),H),M)),set_ord_lessThan(nat,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_wk(A,fun(A,fun(nat,fun(nat,A))),Z),H),M)),set_ord_lessThan(nat,M)) ) ).

% lemma_termdiff1
tff(fact_3896_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,N: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_ord_lessThan(nat,N)) = $ite(Xb = one_one(A),aa(nat,A,semiring_1_of_nat(A),N),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N))),aa(A,A,minus_minus(A,one_one(A)),Xb))) ) ).

% sum_gp_strict
tff(fact_3897_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,N: nat,Y: A] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Xb),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_wl(A,fun(nat,fun(A,fun(nat,A))),Xb),N),Y)),set_ord_lessThan(nat,aa(nat,nat,suc,N)))) ) ).

% diff_power_eq_sum
tff(fact_3898_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,N: nat,Y: A] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Xb),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_wm(A,fun(nat,fun(A,fun(nat,A))),Xb),N),Y)),set_ord_lessThan(nat,N))) ) ).

% power_diff_sumr2
tff(fact_3899_geometric__sums,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,C2)),one_one(real))
         => aa(A,$o,sums(A,aa(A,fun(nat,A),power_power(A),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,minus_minus(A,one_one(A)),C2))) ) ) ).

% geometric_sums
tff(fact_3900_power__half__series,axiom,
    aa(real,$o,sums(real,aTP_Lamp_wn(nat,real)),one_one(real)) ).

% power_half_series
tff(fact_3901_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,Fa: fun(nat,A),K5: A,K: nat] :
          ( ! [P5: nat] :
              ( aa(nat,$o,ord_less(nat,P5),N)
             => aa(A,$o,ord_less_eq(A,aa(nat,A,Fa,P5)),K5) )
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),K5)
           => aa(A,$o,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,N),K)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),K5)) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_3902_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [Fa: fun(nat,A),N: nat,I: nat] :
          ( summable(A,Fa)
         => ( ! [M3: nat] :
                ( aa(nat,$o,ord_less_eq(nat,N),M3)
               => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,Fa,M3)) )
           => ( aa(nat,$o,ord_less_eq(nat,N),I)
             => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,Fa,I))
               => aa(A,$o,ord_less(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_ord_lessThan(nat,N))),suminf(A,Fa)) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_3903_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,N: nat] : aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),Xb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wo(A,fun(nat,fun(nat,A)),Xb),N)),set_ord_lessThan(nat,N))) ) ).

% one_diff_power_eq'
tff(fact_3904_sums__if_H,axiom,
    ! [G: fun(nat,real),Xb: real] :
      ( aa(real,$o,sums(real,G),Xb)
     => aa(real,$o,sums(real,aTP_Lamp_wp(fun(nat,real),fun(nat,real),G)),Xb) ) ).

% sums_if'
tff(fact_3905_sums__if,axiom,
    ! [G: fun(nat,real),Xb: real,Fa: fun(nat,real),Y: real] :
      ( aa(real,$o,sums(real,G),Xb)
     => ( aa(real,$o,sums(real,Fa),Y)
       => aa(real,$o,sums(real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_wq(fun(nat,real),fun(fun(nat,real),fun(nat,real)),G),Fa)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)) ) ) ).

% sums_if
tff(fact_3906_sum__split__even__odd,axiom,
    ! [Fa: fun(nat,real),G: fun(nat,real),N: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_wr(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Fa),G)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ws(fun(nat,real),fun(nat,real),Fa)),set_ord_lessThan(nat,N))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_wt(fun(nat,real),fun(nat,real),G)),set_ord_lessThan(nat,N))) ).

% sum_split_even_odd
tff(fact_3907_monoseq__minus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A3: fun(nat,A)] :
          ( topological_monoseq(A,A3)
         => topological_monoseq(A,aTP_Lamp_wu(fun(nat,A),fun(nat,A),A3)) ) ) ).

% monoseq_minus
tff(fact_3908_Sum__Icc__int,axiom,
    ! [M: int,N: int] :
      ( aa(int,$o,ord_less_eq(int,M),N)
     => ( aa(set(int),int,groups7311177749621191930dd_sum(int,int,aTP_Lamp_wv(int,int)),set_or1337092689740270186AtMost(int,M,N)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(int,int,aa(int,fun(int,int),plus_plus(int),N),one_one(int)))),aa(int,int,aa(int,fun(int,int),times_times(int),M),aa(int,int,minus_minus(int,M),one_one(int))))),aa(num,int,numeral_numeral(int),bit0(one2))) ) ) ).

% Sum_Icc_int
tff(fact_3909_sum__pos__lt__pair,axiom,
    ! [Fa: fun(nat,real),K: nat] :
      ( summable(real,Fa)
     => ( ! [D3: nat] : aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,Fa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D3)))),aa(nat,real,Fa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D3)),one_one(nat))))))
       => aa(real,$o,ord_less(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,Fa),set_ord_lessThan(nat,K))),suminf(real,Fa)) ) ) ).

% sum_pos_lt_pair
tff(fact_3910_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( topological_monoseq(A,X5)
        <=> ( ! [N8: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,X5,N8)),aa(nat,A,X5,aa(nat,nat,suc,N8)))
            | ! [N8: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,X5,aa(nat,nat,suc,N8))),aa(nat,A,X5,N8)) ) ) ) ).

% monoseq_Suc
tff(fact_3911_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N3: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,X5,aa(nat,nat,suc,N3))),aa(nat,A,X5,N3))
         => topological_monoseq(A,X5) ) ) ).

% mono_SucI2
tff(fact_3912_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N3: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,X5,N3)),aa(nat,A,X5,aa(nat,nat,suc,N3)))
         => topological_monoseq(A,X5) ) ) ).

% mono_SucI1
tff(fact_3913_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [M3: nat,N3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,M3),N3)
             => aa(A,$o,ord_less_eq(A,aa(nat,A,X5,M3)),aa(nat,A,X5,N3)) )
         => topological_monoseq(A,X5) ) ) ).

% monoI1
tff(fact_3914_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [M3: nat,N3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,M3),N3)
             => aa(A,$o,ord_less_eq(A,aa(nat,A,X5,N3)),aa(nat,A,X5,M3)) )
         => topological_monoseq(A,X5) ) ) ).

% monoI2
tff(fact_3915_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Fa: fun(nat,A),Mm: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_uw(fun(nat,A),fun(nat,A),Fa)),set_ord_lessThan(nat,Mm)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_3916_sumr__cos__zero__one,axiom,
    ! [N: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ww(nat,real)),set_ord_lessThan(nat,aa(nat,nat,suc,N))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_3917_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C2: fun(nat,A),Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_wx(fun(nat,A),fun(A,fun(nat,A)),C2),Xb))
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_wy(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)),suminf(A,aa(A,fun(nat,A),aTP_Lamp_wx(fun(nat,A),fun(A,fun(nat,A)),C2),Xb))) ) ) ).

% diffs_equiv
tff(fact_3918_sin__paired,axiom,
    ! [Xb: real] : aa(real,$o,sums(real,aTP_Lamp_wz(real,fun(nat,real),Xb)),sin(real,Xb)) ).

% sin_paired
tff(fact_3919_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_3920_fact__1,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).

% fact_1
tff(fact_3921_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_3922_fact__Suc__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,aa(nat,nat,suc,zero_zero(nat))) = one_one(A) ) ) ).

% fact_Suc_0
tff(fact_3923_fact__2,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(num,A,numeral_numeral(A),bit0(one2)) ) ) ).

% fact_2
tff(fact_3924_diffs__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Fa: fun(nat,real),X3: nat] : aa(nat,A,diffs(A,aTP_Lamp_xa(fun(nat,real),fun(nat,A),Fa)),X3) = aa(real,A,real_Vector_of_real(A),aa(nat,real,diffs(real,Fa),X3)) ) ).

% diffs_of_real
tff(fact_3925_fact__nonzero,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semiri3467727345109120633visors(A) )
     => ! [N: nat] : semiring_char_0_fact(A,N) != zero_zero(A) ) ).

% fact_nonzero
tff(fact_3926_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),semiring_char_0_fact(A,N)) ) ).

% fact_ge_zero
tff(fact_3927_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : ~ aa(A,$o,ord_less(A,semiring_char_0_fact(A,N)),zero_zero(A)) ) ).

% fact_not_neg
tff(fact_3928_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : aa(A,$o,ord_less(A,zero_zero(A)),semiring_char_0_fact(A,N)) ) ).

% fact_gt_zero
tff(fact_3929_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : aa(A,$o,ord_less_eq(A,one_one(A)),semiring_char_0_fact(A,N)) ) ).

% fact_ge_1
tff(fact_3930_fact__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => aa(A,$o,ord_less_eq(A,semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N)) ) ) ).

% fact_mono
tff(fact_3931_diffs__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X3: nat] : aa(nat,A,diffs(A,aTP_Lamp_xb(fun(nat,A),fun(nat,A),C2)),X3) = aa(A,A,uminus_uminus(A),aa(nat,A,diffs(A,C2),X3)) ) ).

% diffs_minus
tff(fact_3932_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat,M: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N),M)
         => aa(A,$o,dvd_dvd(A,semiring_char_0_fact(A,N)),semiring_char_0_fact(A,M)) ) ) ).

% fact_dvd
tff(fact_3933_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,K),N)
         => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,N),K)))),semiring_char_0_fact(A,N)) ) ) ).

% choose_dvd
tff(fact_3934_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,N: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
         => ( aa(nat,$o,ord_less(nat,M),N)
           => aa(A,$o,ord_less(A,semiring_char_0_fact(A,M)),semiring_char_0_fact(A,N)) ) ) ) ).

% fact_less_mono
tff(fact_3935_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M: nat,N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),semiring_char_0_fact(A,N)),semiring_char_0_fact(A,M)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_3936_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [N: nat] : aa(A,$o,ord_less_eq(A,semiring_char_0_fact(A,N)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),N))) ) ).

% fact_le_power
tff(fact_3937_fact__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: num] : semiring_char_0_fact(A,aa(num,nat,numeral_numeral(nat),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),K)),semiring_char_0_fact(A,pred_numeral(K))) ) ).

% fact_numeral
tff(fact_3938_termdiff__converges__all,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xb: A] :
          ( ! [X2: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_xc(fun(nat,A),fun(A,fun(nat,A)),C2),X2))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_xd(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)) ) ) ).

% termdiff_converges_all
tff(fact_3939_square__fact__le__2__fact,axiom,
    ! [N: nat] : aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),times_times(real),semiring_char_0_fact(real,N)),semiring_char_0_fact(real,N))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% square_fact_le_2_fact
tff(fact_3940_cos__coeff__def,axiom,
    ! [X3: nat] :
      cos_coeff(X3) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X3),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),semiring_char_0_fact(real,X3)),zero_zero(real)) ).

% cos_coeff_def
tff(fact_3941_fact__num__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat] :
          semiring_char_0_fact(A,M) = $ite(M = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),M)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,M),one_one(nat))))) ) ).

% fact_num_eq_if
tff(fact_3942_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
         => ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,N),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_3943_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xb: real,N: nat,Diff: fun(nat,fun(A,real))] :
          ( ( Xb = zero_zero(real) )
         => ( ( N != zero_zero(nat) )
           => ( aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_xe(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Xb),Diff)),set_ord_lessThan(nat,N)) = aa(A,real,aa(nat,fun(A,real),Diff,zero_zero(nat)),zero_zero(A)) ) ) ) ) ).

% Maclaurin_zero
tff(fact_3944_Maclaurin__lemma,axiom,
    ! [H: real,Fa: fun(real,real),J: fun(nat,real),N: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),H)
     => ? [B9: real] : aa(real,real,Fa,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_xf(real,fun(fun(nat,real),fun(nat,real)),H),J)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),B9),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N)),semiring_char_0_fact(real,N)))) ) ).

% Maclaurin_lemma
tff(fact_3945_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,K5: real,C2: fun(nat,A)] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xb)),K5)
         => ( ! [X2: A] :
                ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X2)),K5)
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_xc(fun(nat,A),fun(A,fun(nat,A)),C2),X2)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_xg(A,fun(fun(nat,A),fun(nat,A)),Xb),C2)) ) ) ) ).

% termdiff_converges
tff(fact_3946_Maclaurin__cos__expansion,axiom,
    ! [Xb: real,N: nat] :
    ? [T2: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,T2)),abs_abs(real,Xb))
      & ( cos(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_xh(real,fun(nat,real),Xb)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_3947_Maclaurin__cos__expansion2,axiom,
    ! [Xb: real,N: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
       => ? [T2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),T2)
            & aa(real,$o,ord_less(real,T2),Xb)
            & ( cos(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_xh(real,fun(nat,real),Xb)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_3948_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,Xb),zero_zero(real))
       => ? [T2: real] :
            ( aa(real,$o,ord_less(real,Xb),T2)
            & aa(real,$o,ord_less(real,T2),zero_zero(real))
            & ( cos(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_xh(real,fun(nat,real),Xb)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_3949_cos__paired,axiom,
    ! [Xb: real] : aa(real,$o,sums(real,aTP_Lamp_xi(real,fun(nat,real),Xb)),cos(real,Xb)) ).

% cos_paired
tff(fact_3950_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ? [T2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),T2)
            & aa(real,$o,ord_less(real,T2),Xb)
            & ( sin(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_xj(real,fun(nat,real),Xb)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_3951_Maclaurin__sin__expansion4,axiom,
    ! [Xb: real,N: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ? [T2: real] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),T2)
          & aa(real,$o,ord_less_eq(real,T2),Xb)
          & ( sin(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_xj(real,fun(nat,real),Xb)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_3952_Maclaurin__sin__expansion2,axiom,
    ! [Xb: real,N: nat] :
    ? [T2: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,T2)),abs_abs(real,Xb))
      & ( sin(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_xj(real,fun(nat,real),Xb)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_3953_Maclaurin__sin__expansion,axiom,
    ! [Xb: real,N: nat] :
    ? [T2: real] : sin(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_xj(real,fun(nat,real),Xb)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,real,semiring_1_of_nat(real),N))),pi)))),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N))) ).

% Maclaurin_sin_expansion
tff(fact_3954_sin__coeff__0,axiom,
    sin_coeff(zero_zero(nat)) = zero_zero(real) ).

% sin_coeff_0
tff(fact_3955_fact__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => aa(nat,$o,ord_less_eq(nat,semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N)) ) ).

% fact_mono_nat
tff(fact_3956_fact__ge__self,axiom,
    ! [N: nat] : aa(nat,$o,ord_less_eq(nat,N),semiring_char_0_fact(nat,N)) ).

% fact_ge_self
tff(fact_3957_fact__less__mono__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(nat,$o,ord_less(nat,M),N)
       => aa(nat,$o,ord_less(nat,semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N)) ) ) ).

% fact_less_mono_nat
tff(fact_3958_fact__ge__Suc__0__nat,axiom,
    ! [N: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,N)) ).

% fact_ge_Suc_0_nat
tff(fact_3959_dvd__fact,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),M)
     => ( aa(nat,$o,ord_less_eq(nat,M),N)
       => aa(nat,$o,dvd_dvd(nat,M),semiring_char_0_fact(nat,N)) ) ) ).

% dvd_fact
tff(fact_3960_fact__diff__Suc,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,N),aa(nat,nat,suc,M))
     => ( semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),N)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),N)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,M),N))) ) ) ).

% fact_diff_Suc
tff(fact_3961_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,R2),N)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,N)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,N),R2)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),R2)) ) ).

% fact_div_fact_le_pow
tff(fact_3962_sin__coeff__def,axiom,
    ! [X3: nat] :
      sin_coeff(X3) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),X3),zero_zero(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,X3),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),semiring_char_0_fact(real,X3))) ).

% sin_coeff_def
tff(fact_3963_Maclaurin__exp__lt,axiom,
    ! [Xb: real,N: nat] :
      ( ( Xb != zero_zero(real) )
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
       => ? [T2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),abs_abs(real,T2))
            & aa(real,$o,ord_less(real,abs_abs(real,T2)),abs_abs(real,Xb))
            & ( aa(real,real,exp(real),Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_xk(real,fun(nat,real),Xb)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,exp(real),T2)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_3964_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A3: $o,B3: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Leaf((A3),(B3))) = zero_zero(nat) ).

% VEBT_internal.height.simps(1)
tff(fact_3965_pochhammer__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),comm_s3205402744901411588hammer(A,Z,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),N)) ) ).

% pochhammer_double
tff(fact_3966_exp__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,Xb),Y)
     => aa(real,$o,ord_less(real,aa(real,real,exp(real),Xb)),aa(real,real,exp(real),Y)) ) ).

% exp_less_mono
tff(fact_3967_exp__less__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,exp(real),Xb)),aa(real,real,exp(real),Y))
    <=> aa(real,$o,ord_less(real,Xb),Y) ) ).

% exp_less_cancel_iff
tff(fact_3968_pochhammer__1,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,one_one(nat)) = A3 ) ).

% pochhammer_1
tff(fact_3969_exp__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( aa(A,A,exp(A),zero_zero(A)) = one_one(A) ) ) ).

% exp_zero
tff(fact_3970_pochhammer__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,zero_zero(nat)) = one_one(A) ) ).

% pochhammer_0
tff(fact_3971_pochhammer__Suc0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% pochhammer_Suc0
tff(fact_3972_exp__eq__one__iff,axiom,
    ! [Xb: real] :
      ( ( aa(real,real,exp(real),Xb) = one_one(real) )
    <=> ( Xb = zero_zero(real) ) ) ).

% exp_eq_one_iff
tff(fact_3973_exp__less__one__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,exp(real),Xb)),one_one(real))
    <=> aa(real,$o,ord_less(real,Xb),zero_zero(real)) ) ).

% exp_less_one_iff
tff(fact_3974_one__less__exp__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),aa(real,real,exp(real),Xb))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),Xb) ) ).

% one_less_exp_iff
tff(fact_3975_exp__le__one__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),Xb)),one_one(real))
    <=> aa(real,$o,ord_less_eq(real,Xb),zero_zero(real)) ) ).

% exp_le_one_iff
tff(fact_3976_one__le__exp__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),aa(real,real,exp(real),Xb))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb) ) ).

% one_le_exp_iff
tff(fact_3977_exp__ln__iff,axiom,
    ! [Xb: real] :
      ( ( aa(real,real,exp(real),aa(real,real,ln_ln(real),Xb)) = Xb )
    <=> aa(real,$o,ord_less(real,zero_zero(real)),Xb) ) ).

% exp_ln_iff
tff(fact_3978_exp__ln,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,real,exp(real),aa(real,real,ln_ln(real),Xb)) = Xb ) ) ).

% exp_ln
tff(fact_3979_exp__less__cancel,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,exp(real),Xb)),aa(real,real,exp(real),Y))
     => aa(real,$o,ord_less(real,Xb),Y) ) ).

% exp_less_cancel
tff(fact_3980_exp__not__eq__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,exp(A),Xb) != zero_zero(A) ) ).

% exp_not_eq_zero
tff(fact_3981_exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
     => ? [X2: real] : aa(real,real,exp(real),X2) = Y ) ).

% exp_total
tff(fact_3982_exp__gt__zero,axiom,
    ! [Xb: real] : aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,exp(real),Xb)) ).

% exp_gt_zero
tff(fact_3983_not__exp__less__zero,axiom,
    ! [Xb: real] : ~ aa(real,$o,ord_less(real,aa(real,real,exp(real),Xb)),zero_zero(real)) ).

% not_exp_less_zero
tff(fact_3984_not__exp__le__zero,axiom,
    ! [Xb: real] : ~ aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),Xb)),zero_zero(real)) ).

% not_exp_le_zero
tff(fact_3985_exp__ge__zero,axiom,
    ! [Xb: real] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,exp(real),Xb)) ).

% exp_ge_zero
tff(fact_3986_exp__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : aa(A,A,exp(A),aa(A,A,minus_minus(A,Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,exp(A),Xb)),aa(A,A,exp(A),Y)) ) ).

% exp_diff
tff(fact_3987_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,N: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xb)
         => aa(A,$o,ord_less(A,zero_zero(A)),comm_s3205402744901411588hammer(A,Xb,N)) ) ) ).

% pochhammer_pos
tff(fact_3988_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M: nat,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,M) != zero_zero(A) )
         => ( aa(nat,$o,ord_less_eq(nat,N),M)
           => ( comm_s3205402744901411588hammer(A,A3,N) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_3989_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N: nat,M: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,N) = zero_zero(A) )
         => ( aa(nat,$o,ord_less_eq(nat,N),M)
           => ( comm_s3205402744901411588hammer(A,A3,M) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_3990_pochhammer__fact,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_semiring_1(A) )
     => ! [N: nat] : semiring_char_0_fact(A,N) = comm_s3205402744901411588hammer(A,one_one(A),N) ) ).

% pochhammer_fact
tff(fact_3991_exp__gt__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => aa(real,$o,ord_less(real,one_one(real)),aa(real,real,exp(real),Xb)) ) ).

% exp_gt_one
tff(fact_3992_exp__minus__inverse,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),Xb)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xb))) = one_one(A) ) ).

% exp_minus_inverse
tff(fact_3993_exp__of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [N: nat,Xb: A] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),Xb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Xb)),N) ) ).

% exp_of_nat_mult
tff(fact_3994_exp__of__nat2__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,N: nat] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(nat,A,semiring_1_of_nat(A),N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Xb)),N) ) ).

% exp_of_nat2_mult
tff(fact_3995_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,N: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xb)
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),comm_s3205402744901411588hammer(A,Xb,N)) ) ) ).

% pochhammer_nonneg
tff(fact_3996_pochhammer__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [N: nat] :
          comm_s3205402744901411588hammer(A,zero_zero(A),N) = $ite(N = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% pochhammer_0_left
tff(fact_3997_exp__ge__add__one__self__aux,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb)),aa(real,real,exp(real),Xb)) ) ).

% exp_ge_add_one_self_aux
tff(fact_3998_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less_eq(real,one_one(real)),Y)
     => ? [X2: real] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),X2)
          & aa(real,$o,ord_less_eq(real,X2),aa(real,real,minus_minus(real,Y),one_one(real)))
          & ( aa(real,real,exp(real),X2) = Y ) ) ) ).

% lemma_exp_total
tff(fact_3999_ln__ge__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,Y),aa(real,real,ln_ln(real),Xb))
      <=> aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),Y)),Xb) ) ) ).

% ln_ge_iff
tff(fact_4000_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A)),N)) ) ).

% pochhammer_rec
tff(fact_4001_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,K: nat] :
          ( aa(nat,$o,ord_less(nat,N),K)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_4002_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [N: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) = zero_zero(A) )
        <=> aa(nat,$o,ord_less(nat,N),K) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_4003_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N: nat] :
          ( ( comm_s3205402744901411588hammer(A,A3,N) = zero_zero(A) )
        <=> ? [K3: nat] :
              ( aa(nat,$o,ord_less(nat,K3),N)
              & ( A3 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K3)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_4004_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,K),N)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_4005_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A,A3: A] :
          powr(A,Xb,A3) = $ite(Xb = zero_zero(A),zero_zero(A),aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,ln_ln(A),Xb)))) ) ).

% powr_def
tff(fact_4006_exp__le,axiom,
    aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),one_one(real))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))) ).

% exp_le
tff(fact_4007_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [N: nat,Xb: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),aa(nat,A,semiring_1_of_nat(A),N)))),N) = aa(A,A,exp(A),Xb) ) ) ) ).

% exp_divide_power_eq
tff(fact_4008_tanh__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,tanh(A),Xb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,exp(A),Xb)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xb)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),Xb)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xb)))) ) ).

% tanh_altdef
tff(fact_4009_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,N: nat,Z: A] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( comm_s3205402744901411588hammer(A,Z,N) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,M)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ).

% pochhammer_product
tff(fact_4010_exp__half__le2,axiom,
    aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% exp_half_le2
tff(fact_4011_exp__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] : aa(A,A,exp(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Z)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% exp_double
tff(fact_4012_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,R2),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R2),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R2)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_4013_pochhammer__same,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_ring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N)),N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),semiring_char_0_fact(A,N)) ) ).

% pochhammer_same
tff(fact_4014_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,Z)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Z))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% exp_bound_half
tff(fact_4015_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B3: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B3),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).

% pochhammer_minus
tff(fact_4016_pochhammer__minus_H,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B3: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B3),K)) ) ).

% pochhammer_minus'
tff(fact_4017_VEBT__internal_Oheight_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A2: $o,B2: $o] : Xb != vEBT_Leaf((A2),(B2))
     => ~ ! [Uu2: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb != vEBT_Node(Uu2,Deg,TreeList2,Summary) ) ).

% VEBT_internal.height.cases
tff(fact_4018_exp__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),one_one(real))
       => aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),Xb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% exp_bound
tff(fact_4019_real__exp__bound__lemma,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less_eq(real,aa(real,real,exp(real),Xb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),Xb))) ) ) ).

% real_exp_bound_lemma
tff(fact_4020_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),N))),Xb)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
       => aa(real,$o,ord_less_eq(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),aa(nat,real,semiring_1_of_nat(real),N)))),N)),aa(real,real,exp(real),Xb)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_4021_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xb: real,N: nat] :
      ( aa(real,$o,ord_less_eq(real,Xb),aa(nat,real,semiring_1_of_nat(real),N))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
       => aa(real,$o,ord_less_eq(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),aa(nat,real,semiring_1_of_nat(real),N)))),N)),aa(real,real,exp(real),aa(real,real,uminus_uminus(real),Xb))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_4022_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,Z)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2))))
         => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,exp(A),Z))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),real_V7770717601297561774m_norm(A,Z)))) ) ) ).

% exp_bound_lemma
tff(fact_4023_Maclaurin__exp__le,axiom,
    ! [Xb: real,N: nat] :
    ? [T2: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,T2)),abs_abs(real,Xb))
      & ( aa(real,real,exp(real),Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_xk(real,fun(nat,real),Xb)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,exp(real),T2)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N))) ) ) ).

% Maclaurin_exp_le
tff(fact_4024_exp__lower__Taylor__quadratic,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(num,real,numeral_numeral(real),bit0(one2))))),aa(real,real,exp(real),Xb)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_4025_log__base__10__eq2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),aa(real,real,exp(real),one_one(real)))),aa(real,real,ln_ln(real),Xb)) ) ) ).

% log_base_10_eq2
tff(fact_4026_tanh__real__altdef,axiom,
    ! [Xb: real] : aa(real,real,tanh(real),Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,one_one(real)),aa(real,real,exp(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),bit0(one2)))),Xb)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,exp(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),bit0(one2)))),Xb)))) ).

% tanh_real_altdef
tff(fact_4027_fact__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))),N))),semiring_char_0_fact(A,N)) ) ).

% fact_double
tff(fact_4028_log__base__10__eq1,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),aa(real,real,exp(real),one_one(real)))),aa(real,real,ln_ln(real),aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2))))))),aa(real,real,ln_ln(real),Xb)) ) ) ).

% log_base_10_eq1
tff(fact_4029_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,N))),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_xl(A,fun(nat,A),Z),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_4030_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] :
          comm_s3205402744901411588hammer(A,A3,N) = $ite(N = zero_zero(nat),one_one(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_xm(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,minus_minus(nat,N),one_one(nat)),one_one(A))) ) ).

% pochhammer_code
tff(fact_4031_of__nat__code,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),N) = semiri8178284476397505188at_aux(A,aTP_Lamp_xn(A,A),N,zero_zero(A)) ) ).

% of_nat_code
tff(fact_4032_floor__log__nat__eq__powr__iff,axiom,
    ! [B3: nat,K: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B3)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),N) )
        <=> ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),N)),K)
            & aa(nat,$o,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_4033_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A4: set(B)] : groups7121269368397514597t_prod(B,A,aTP_Lamp_xo(B,A),A4) = one_one(A) ) ).

% prod.neutral_const
tff(fact_4034_of__nat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Fa: fun(B,nat),A4: set(B)] : aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(B,nat,Fa,A4)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_xp(fun(B,nat),fun(B,A),Fa),A4) ) ).

% of_nat_prod
tff(fact_4035_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Fa: fun(B,int),A4: set(B)] : aa(int,A,ring_1_of_int(A),groups7121269368397514597t_prod(B,int,Fa,A4)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_xq(fun(B,int),fun(B,A),Fa),A4) ) ).

% of_int_prod
tff(fact_4036_of__real__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2191834092415804123ebra_1(A) )
     => ! [Fa: fun(B,real),S: set(B)] : aa(real,A,real_Vector_of_real(A),groups7121269368397514597t_prod(B,real,Fa,S)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_xr(fun(B,real),fun(B,A),Fa),S) ) ).

% of_real_prod
tff(fact_4037_prod__zero__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( semidom(B)
     => ! [A4: set(A),Fa: fun(A,B)] :
          ( finite_finite2(A,A4)
         => ( ( groups7121269368397514597t_prod(A,B,Fa,A4) = zero_zero(B) )
          <=> ? [X: A] :
                ( member(A,X,A4)
                & ( aa(A,B,Fa,X) = zero_zero(B) ) ) ) ) ) ).

% prod_zero_iff
tff(fact_4038_prod_Oempty,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A)] : groups7121269368397514597t_prod(B,A,G,bot_bot(set(B))) = one_one(A) ) ).

% prod.empty
tff(fact_4039_prod_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( ~ finite_finite2(A,A4)
         => ( groups7121269368397514597t_prod(A,B,G,A4) = one_one(B) ) ) ) ).

% prod.infinite
tff(fact_4040_floor__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,zero_zero(A)) = zero_zero(int) ) ) ).

% floor_zero
tff(fact_4041_floor__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archim6421214686448440834_floor(A,aa(num,A,numeral_numeral(A),V)) = aa(num,int,numeral_numeral(int),V) ) ).

% floor_numeral
tff(fact_4042_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_4043_prod_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A3: A,B3: fun(A,B)] :
          ( finite_finite2(A,S3)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xs(A,fun(fun(A,B),fun(A,B)),A3),B3),S3) = $ite(member(A,A3,S3),aa(A,B,B3,A3),one_one(B)) ) ) ) ).

% prod.delta
tff(fact_4044_prod_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A3: A,B3: fun(A,B)] :
          ( finite_finite2(A,S3)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xt(A,fun(fun(A,B),fun(A,B)),A3),B3),S3) = $ite(member(A,A3,S3),aa(A,B,B3,A3),one_one(B)) ) ) ) ).

% prod.delta'
tff(fact_4045_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb) ) ) ).

% zero_le_floor
tff(fact_4046_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,ord_less_eq(int,aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),V)),Xb) ) ) ).

% numeral_le_floor
tff(fact_4047_floor__less__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,Xb)),zero_zero(int))
        <=> aa(A,$o,ord_less(A,Xb),zero_zero(A)) ) ) ).

% floor_less_zero
tff(fact_4048_floor__less__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,Xb)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,ord_less(A,Xb),aa(num,A,numeral_numeral(A),V)) ) ) ).

% floor_less_numeral
tff(fact_4049_zero__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,ord_less(int,zero_zero(int)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,ord_less_eq(A,one_one(A)),Xb) ) ) ).

% zero_less_floor
tff(fact_4050_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,Xb)),zero_zero(int))
        <=> aa(A,$o,ord_less(A,Xb),one_one(A)) ) ) ).

% floor_le_zero
tff(fact_4051_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,ord_less_eq(int,one_one(int)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,ord_less_eq(A,one_one(A)),Xb) ) ) ).

% one_le_floor
tff(fact_4052_floor__less__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,Xb)),one_one(int))
        <=> aa(A,$o,ord_less(A,Xb),one_one(A)) ) ) ).

% floor_less_one
tff(fact_4053_floor__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num] : archim6421214686448440834_floor(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)) ) ).

% floor_neg_numeral
tff(fact_4054_floor__diff__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,Xb),aa(num,A,numeral_numeral(A),V))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,Xb)),aa(num,int,numeral_numeral(int),V)) ) ).

% floor_diff_numeral
tff(fact_4055_floor__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: num,N: nat] : archim6421214686448440834_floor(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),N)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),N) ) ).

% floor_numeral_power
tff(fact_4056_floor__diff__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,Xb),one_one(A))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,Xb)),one_one(int)) ) ).

% floor_diff_one
tff(fact_4057_floor__divide__eq__div__numeral,axiom,
    ! [A3: num,B3: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A3)),aa(num,real,numeral_numeral(real),B3))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),A3)),aa(num,int,numeral_numeral(int),B3)) ).

% floor_divide_eq_div_numeral
tff(fact_4058_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] :
          groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,ord_less(nat,aa(nat,nat,suc,N)),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N)))) ) ).

% prod.cl_ivl_Suc
tff(fact_4059_numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,ord_less(int,aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))),Xb) ) ) ).

% numeral_less_floor
tff(fact_4060_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,Xb)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,ord_less(A,Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),V)),one_one(A))) ) ) ).

% floor_le_numeral
tff(fact_4061_one__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,ord_less(int,one_one(int)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),bit0(one2))),Xb) ) ) ).

% one_less_floor
tff(fact_4062_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,Xb)),one_one(int))
        <=> aa(A,$o,ord_less(A,Xb),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% floor_le_one
tff(fact_4063_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),Xb) ) ) ).

% neg_numeral_le_floor
tff(fact_4064_floor__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,ord_less(A,Xb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ) ).

% floor_less_neg_numeral
tff(fact_4065_floor__one__divide__eq__div__numeral,axiom,
    ! [B3: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),B3))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(num,int,numeral_numeral(int),B3)) ).

% floor_one_divide_eq_div_numeral
tff(fact_4066_floor__minus__divide__eq__div__numeral,axiom,
    ! [A3: num,B3: num] : archim6421214686448440834_floor(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A3)),aa(num,real,numeral_numeral(real),B3)))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(num,int,numeral_numeral(int),B3)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_4067_neg__numeral__less__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,ord_less(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V))),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))),Xb) ) ) ).

% neg_numeral_less_floor
tff(fact_4068_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,Xb)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),V)))
        <=> aa(A,$o,ord_less(A,Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),one_one(A))) ) ) ).

% floor_le_neg_numeral
tff(fact_4069_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B3: num] : archim6421214686448440834_floor(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),B3)))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(num,int,numeral_numeral(int),B3)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_4070_prod_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A4)
             => ( aa(A,B,G,X2) = one_one(B) ) )
         => ( groups7121269368397514597t_prod(A,B,G,A4) = one_one(B) ) ) ) ).

% prod.neutral
tff(fact_4071_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A4: set(B)] :
          ( ( groups7121269368397514597t_prod(B,A,G,A4) != one_one(A) )
         => ~ ! [A2: B] :
                ( member(B,A2,A4)
               => ( aa(B,A,G,A2) = one_one(A) ) ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_4072_norm__prod__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Fa: fun(B,A),A4: set(B)] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,groups7121269368397514597t_prod(B,A,Fa,A4))),groups7121269368397514597t_prod(B,real,aTP_Lamp_xu(fun(B,A),fun(B,real),Fa),A4)) ) ).

% norm_prod_le
tff(fact_4073_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),H: fun(B,A),A4: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xv(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),A4) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G,A4)),groups7121269368397514597t_prod(B,A,H,A4)) ) ).

% prod.distrib
tff(fact_4074_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [Fa: fun(B,A),G: fun(B,A),A4: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xw(fun(B,A),fun(fun(B,A),fun(B,A)),Fa),G),A4) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(B,A,Fa,A4)),groups7121269368397514597t_prod(B,A,G,A4)) ) ).

% prod_dividef
tff(fact_4075_prod__power__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Fa: fun(B,A),A4: set(B),N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),groups7121269368397514597t_prod(B,A,Fa,A4)),N) = groups7121269368397514597t_prod(B,A,aa(nat,fun(B,A),aTP_Lamp_xx(fun(B,A),fun(nat,fun(B,A)),Fa),N),A4) ) ).

% prod_power_distrib
tff(fact_4076_prod_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(A),B6: set(B),G: fun(A,fun(B,C)),R: fun(A,fun(B,$o))] :
          ( finite_finite2(A,A4)
         => ( finite_finite2(B,B6)
           => ( groups7121269368397514597t_prod(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_xy(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B6),G),R),A4) = groups7121269368397514597t_prod(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_ya(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A4),G),R),B6) ) ) ) ) ).

% prod.swap_restrict
tff(fact_4077_prod__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [Fa: fun(A,B),A4: set(A)] : groups7121269368397514597t_prod(A,real,aTP_Lamp_yb(fun(A,B),fun(A,real),Fa),A4) = real_V7770717601297561774m_norm(B,groups7121269368397514597t_prod(A,B,Fa,A4)) ) ).

% prod_norm
tff(fact_4078_prod_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,fun(C,A)),B6: set(C),A4: set(B)] : groups7121269368397514597t_prod(B,A,aa(set(C),fun(B,A),aTP_Lamp_yc(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B6),A4) = groups7121269368397514597t_prod(C,A,aa(set(B),fun(C,A),aTP_Lamp_ye(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A4),B6) ) ).

% prod.swap
tff(fact_4079_abs__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [Fa: fun(B,A),A4: set(B)] : abs_abs(A,groups7121269368397514597t_prod(B,A,Fa,A4)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_yf(fun(B,A),fun(B,A),Fa),A4) ) ).

% abs_prod
tff(fact_4080_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Fa: fun(B,A),A3: A,A4: set(B)] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),groups7121269368397514597t_prod(B,A,aa(A,fun(B,A),aTP_Lamp_ur(fun(B,A),fun(A,fun(B,A)),Fa),A3),A4)),A3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),groups7121269368397514597t_prod(B,A,Fa,A4)),A3) ) ).

% mod_prod_eq
tff(fact_4081_prod__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A4: set(A),Fa: fun(A,B),G: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,A4)
             => ( aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,Fa,I2))
                & aa(B,$o,ord_less_eq(B,aa(A,B,Fa,I2)),aa(A,B,G,I2)) ) )
         => aa(B,$o,ord_less_eq(B,groups7121269368397514597t_prod(A,B,Fa,A4)),groups7121269368397514597t_prod(A,B,G,A4)) ) ) ).

% prod_mono
tff(fact_4082_prod__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A4: set(A),Fa: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A4)
             => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,Fa,X2)) )
         => aa(B,$o,ord_less_eq(B,zero_zero(B)),groups7121269368397514597t_prod(A,B,Fa,A4)) ) ) ).

% prod_nonneg
tff(fact_4083_prod__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A4: set(A),Fa: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A4)
             => aa(B,$o,ord_less(B,zero_zero(B)),aa(A,B,Fa,X2)) )
         => aa(B,$o,ord_less(B,zero_zero(B)),groups7121269368397514597t_prod(A,B,Fa,A4)) ) ) ).

% prod_pos
tff(fact_4084_prod__ge__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A4: set(A),Fa: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A4)
             => aa(B,$o,ord_less_eq(B,one_one(B)),aa(A,B,Fa,X2)) )
         => aa(B,$o,ord_less_eq(B,one_one(B)),groups7121269368397514597t_prod(A,B,Fa,A4)) ) ) ).

% prod_ge_1
tff(fact_4085_prod__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(B)
     => ! [A4: set(A),Fa: fun(A,B)] :
          ( finite_finite2(A,A4)
         => ( ? [X3: A] :
                ( member(A,X3,A4)
                & ( aa(A,B,Fa,X3) = zero_zero(B) ) )
           => ( groups7121269368397514597t_prod(A,B,Fa,A4) = zero_zero(B) ) ) ) ) ).

% prod_zero
tff(fact_4086_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xb))),Xb) ) ).

% of_int_floor_le
tff(fact_4087_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),Y)
         => aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Y)) ) ) ).

% floor_mono
tff(fact_4088_floor__less__cancel,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Y))
         => aa(A,$o,ord_less(A,Xb),Y) ) ) ).

% floor_less_cancel
tff(fact_4089_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Fa: fun(nat,A),A3: nat,B3: nat] : groups7121269368397514597t_prod(nat,A,Fa,set_or1337092689740270186AtMost(nat,A3,B3)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_yg(fun(nat,A),fun(nat,fun(A,A)),Fa),A3,B3,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_4090_floor__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,Xb)),archimedean_ceiling(A,Xb)) ) ).

% floor_le_ceiling
tff(fact_4091_floor__le__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,Xb)),archimedean_round(A,Xb)) ) ).

% floor_le_round
tff(fact_4092_prod_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( finite_finite2(A,A4)
         => ( groups7121269368397514597t_prod(A,B,G,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_uu(set(A),fun(fun(A,$o),fun(A,$o)),A4),P))) = groups7121269368397514597t_prod(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_yh(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P),A4) ) ) ) ).

% prod.inter_filter
tff(fact_4093_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_yi(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_4094_power__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C2: A,Fa: fun(B,nat),A4: set(B)] : aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,Fa),A4)) = groups7121269368397514597t_prod(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_yj(A,fun(fun(B,nat),fun(B,A)),C2),Fa),A4) ) ).

% power_sum
tff(fact_4095_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_yk(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_4096_prod__le__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A4: set(A),Fa: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A4)
             => ( aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,Fa,X2))
                & aa(B,$o,ord_less_eq(B,aa(A,B,Fa,X2)),one_one(B)) ) )
         => aa(B,$o,ord_less_eq(B,groups7121269368397514597t_prod(A,B,Fa,A4)),one_one(B)) ) ) ).

% prod_le_1
tff(fact_4097_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R: fun(A,fun(A,$o)),S3: set(B),H: fun(B,A),G: fun(B,A)] :
          ( aa(A,$o,aa(A,fun(A,$o),R,one_one(A)),one_one(A))
         => ( ! [X15: A,Y15: A,X23: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X15),X23)
                  & aa(A,$o,aa(A,fun(A,$o),R,Y15),Y23) )
               => aa(A,$o,aa(A,fun(A,$o),R,aa(A,A,aa(A,fun(A,A),times_times(A),X15),Y15)),aa(A,A,aa(A,fun(A,A),times_times(A),X23),Y23)) )
           => ( finite_finite2(B,S3)
             => ( ! [X2: B] :
                    ( member(B,X2,S3)
                   => aa(A,$o,aa(A,fun(A,$o),R,aa(B,A,H,X2)),aa(B,A,G,X2)) )
               => aa(A,$o,aa(A,fun(A,$o),R,groups7121269368397514597t_prod(B,A,H,S3)),groups7121269368397514597t_prod(B,A,G,S3)) ) ) ) ) ) ).

% prod.related
tff(fact_4098_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S6: set(A),T5: set(B),S3: set(A),I: fun(B,A),J: fun(A,B),T6: set(B),G: fun(A,C),H: fun(B,C)] :
          ( finite_finite2(A,S6)
         => ( finite_finite2(B,T5)
           => ( ! [A2: A] :
                  ( member(A,A2,aa(set(A),set(A),minus_minus(set(A),S3),S6))
                 => ( aa(B,A,I,aa(A,B,J,A2)) = A2 ) )
             => ( ! [A2: A] :
                    ( member(A,A2,aa(set(A),set(A),minus_minus(set(A),S3),S6))
                   => member(B,aa(A,B,J,A2),aa(set(B),set(B),minus_minus(set(B),T6),T5)) )
               => ( ! [B2: B] :
                      ( member(B,B2,aa(set(B),set(B),minus_minus(set(B),T6),T5))
                     => ( aa(A,B,J,aa(B,A,I,B2)) = B2 ) )
                 => ( ! [B2: B] :
                        ( member(B,B2,aa(set(B),set(B),minus_minus(set(B),T6),T5))
                       => member(A,aa(B,A,I,B2),aa(set(A),set(A),minus_minus(set(A),S3),S6)) )
                   => ( ! [A2: A] :
                          ( member(A,A2,S6)
                         => ( aa(A,C,G,A2) = one_one(C) ) )
                     => ( ! [B2: B] :
                            ( member(B,B2,T5)
                           => ( aa(B,C,H,B2) = one_one(C) ) )
                       => ( ! [A2: A] :
                              ( member(A,A2,S3)
                             => ( aa(B,C,H,aa(A,B,J,A2)) = aa(A,C,G,A2) ) )
                         => ( groups7121269368397514597t_prod(A,C,G,S3) = groups7121269368397514597t_prod(B,C,H,T6) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_4099_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,ord_less_eq(int,Z),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),Xb) ) ) ).

% le_floor_iff
tff(fact_4100_floor__less__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,ord_less(int,archim6421214686448440834_floor(A,Xb)),Z)
        <=> aa(A,$o,ord_less(A,Xb),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% floor_less_iff
tff(fact_4101_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] : aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Y))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))) ) ).

% le_floor_add
tff(fact_4102_floor__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,N: nat] :
          ( ( Xb = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xb)) )
         => ( archim6421214686448440834_floor(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),archim6421214686448440834_floor(A,Xb)),N) ) ) ) ).

% floor_power
tff(fact_4103_floor__divide__of__int__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [K: int,L: int] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) ) ).

% floor_divide_of_int_eq
tff(fact_4104_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),N: nat,I: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,N),I) = semiri8178284476397505188at_aux(A,Inc,N,aa(A,A,Inc,I)) ) ).

% of_nat_aux.simps(2)
tff(fact_4105_of__nat__aux_Osimps_I1_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),I: A] : semiri8178284476397505188at_aux(A,Inc,zero_zero(nat),I) = I ) ).

% of_nat_aux.simps(1)
tff(fact_4106_prod_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A4: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A4)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A4),collect(A,aTP_Lamp_yl(fun(A,B),fun(A,$o),G)))) = groups7121269368397514597t_prod(A,B,G,A4) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_4107_exp__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),Fa: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( aa(B,B,exp(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,Fa),I5)) = groups7121269368397514597t_prod(A,B,aTP_Lamp_ym(fun(A,B),fun(A,B),Fa),I5) ) ) ) ).

% exp_sum
tff(fact_4108_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_yn(fun(nat,A),fun(nat,fun(nat,A)),G),N),set_ord_lessThan(nat,N)) = groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,N)) ) ).

% prod.nat_diff_reindex
tff(fact_4109_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,N,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_yo(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M),set_or1337092689740270186AtMost(nat,N,M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_4110_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I: A,Fa: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,ord_less(B,one_one(B)),aa(A,B,Fa,I))
             => ( ! [I2: A] :
                    ( member(A,I2,I5)
                   => aa(B,$o,ord_less_eq(B,one_one(B)),aa(A,B,Fa,I2)) )
               => aa(B,$o,ord_less(B,one_one(B)),groups7121269368397514597t_prod(A,B,Fa,I5)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_4111_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),Fa: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I2: A] :
                  ( member(A,I2,I5)
                 => aa(B,$o,ord_less(B,one_one(B)),aa(A,B,Fa,I2)) )
             => aa(B,$o,ord_less(B,one_one(B)),groups7121269368397514597t_prod(A,B,Fa,I5)) ) ) ) ) ).

% less_1_prod
tff(fact_4112_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T6: set(A),S3: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,T6)
         => ( aa(set(A),$o,ord_less_eq(set(A),S3),T6)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T6),S3))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( ! [X2: A] :
                    ( member(A,X2,S3)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
               => ( groups7121269368397514597t_prod(A,B,G,T6) = groups7121269368397514597t_prod(A,B,H,S3) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_4113_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T6: set(A),S3: set(A),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,T6)
         => ( aa(set(A),$o,ord_less_eq(set(A),S3),T6)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T6),S3))
                 => ( aa(A,B,H,X2) = one_one(B) ) )
             => ( ! [X2: A] :
                    ( member(A,X2,S3)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
               => ( groups7121269368397514597t_prod(A,B,G,S3) = groups7121269368397514597t_prod(A,B,H,T6) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_4114_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T6: set(A),S3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,T6)
         => ( aa(set(A),$o,ord_less_eq(set(A),S3),T6)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T6),S3))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( groups7121269368397514597t_prod(A,B,G,T6) = groups7121269368397514597t_prod(A,B,G,S3) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_4115_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T6: set(A),S3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,T6)
         => ( aa(set(A),$o,ord_less_eq(set(A),S3),T6)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T6),S3))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( groups7121269368397514597t_prod(A,B,G,S3) = groups7121269368397514597t_prod(A,B,G,T6) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_4116_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(A),A4: set(A),B6: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,C5)
         => ( aa(set(A),$o,ord_less_eq(set(A),A4),C5)
           => ( aa(set(A),$o,ord_less_eq(set(A),B6),C5)
             => ( ! [A2: A] :
                    ( member(A,A2,aa(set(A),set(A),minus_minus(set(A),C5),A4))
                   => ( aa(A,B,G,A2) = one_one(B) ) )
               => ( ! [B2: A] :
                      ( member(A,B2,aa(set(A),set(A),minus_minus(set(A),C5),B6))
                     => ( aa(A,B,H,B2) = one_one(B) ) )
                 => ( ( groups7121269368397514597t_prod(A,B,G,C5) = groups7121269368397514597t_prod(A,B,H,C5) )
                   => ( groups7121269368397514597t_prod(A,B,G,A4) = groups7121269368397514597t_prod(A,B,H,B6) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_4117_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(A),A4: set(A),B6: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,C5)
         => ( aa(set(A),$o,ord_less_eq(set(A),A4),C5)
           => ( aa(set(A),$o,ord_less_eq(set(A),B6),C5)
             => ( ! [A2: A] :
                    ( member(A,A2,aa(set(A),set(A),minus_minus(set(A),C5),A4))
                   => ( aa(A,B,G,A2) = one_one(B) ) )
               => ( ! [B2: A] :
                      ( member(A,B2,aa(set(A),set(A),minus_minus(set(A),C5),B6))
                     => ( aa(A,B,H,B2) = one_one(B) ) )
                 => ( ( groups7121269368397514597t_prod(A,B,G,A4) = groups7121269368397514597t_prod(A,B,H,B6) )
                  <=> ( groups7121269368397514597t_prod(A,B,G,C5) = groups7121269368397514597t_prod(A,B,H,C5) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_4118_one__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),one_one(int)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),one_one(A))) ) ).

% one_add_floor
tff(fact_4119_floor__divide__of__nat__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [M: nat,N: nat] : archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)) ) ).

% floor_divide_of_nat_eq
tff(fact_4120_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),N))),aa(nat,A,G,aa(nat,nat,suc,N))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_4121_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,N))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,N))),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_4122_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_4123_ceiling__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          archimedean_ceiling(A,Xb) = $ite(Xb = aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Xb),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),one_one(int))) ) ).

% ceiling_altdef
tff(fact_4124_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(int,$o,ord_less_eq(int,aa(int,int,minus_minus(int,archimedean_ceiling(A,Xb)),archim6421214686448440834_floor(A,Xb))),one_one(int)) ) ).

% ceiling_diff_floor_le_1
tff(fact_4125_floor__eq,axiom,
    ! [N: int,Xb: real] :
      ( aa(real,$o,ord_less(real,aa(int,real,ring_1_of_int(real),N)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),N)),one_one(real)))
       => ( archim6421214686448440834_floor(real,Xb) = N ) ) ) ).

% floor_eq
tff(fact_4126_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : aa(real,$o,ord_less(real,R2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))),one_one(real))) ).

% real_of_int_floor_add_one_gt
tff(fact_4127_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : aa(real,$o,ord_less(real,aa(real,real,minus_minus(real,R2),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R2))) ).

% real_of_int_floor_gt_diff_one
tff(fact_4128_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_yi(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,N))) ) ).

% prod.lessThan_Suc_shift
tff(fact_4129_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N))),aa(nat,A,G,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,aTP_Lamp_yi(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,N))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_4130_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_yi(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,N)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_4131_fact__prod,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,aTP_Lamp_vj(nat,nat),set_or1337092689740270186AtMost(nat,one_one(nat),N))) ) ).

% fact_prod
tff(fact_4132_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A4: set(A),Fa: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A4)
         => ( ! [I2: A] :
                ( member(A,I2,A4)
               => ( aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,Fa,I2))
                  & aa(B,$o,ord_less(B,aa(A,B,Fa,I2)),aa(A,B,G,I2)) ) )
           => ( ( A4 != bot_bot(set(A)) )
             => aa(B,$o,ord_less(B,groups7121269368397514597t_prod(A,B,Fa,A4)),groups7121269368397514597t_prod(A,B,G,A4)) ) ) ) ) ).

% prod_mono_strict
tff(fact_4133_even__prod__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A4: set(A),Fa: fun(A,B)] :
          ( finite_finite2(A,A4)
         => ( aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),groups7121269368397514597t_prod(A,B,Fa,A4))
          <=> ? [X: A] :
                ( member(A,X,A4)
                & aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),aa(A,B,Fa,X)) ) ) ) ) ).

% even_prod_iff
tff(fact_4134_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Tb: A] :
          ( aa(int,$o,P,archim6421214686448440834_floor(A,Tb))
        <=> ! [I4: int] :
              ( ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),I4)),Tb)
                & aa(A,$o,ord_less(A,Tb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I4)),one_one(A))) )
             => aa(int,$o,P,I4) ) ) ) ).

% floor_split
tff(fact_4135_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A3: int] :
          ( ( archim6421214686448440834_floor(A,Xb) = A3 )
        <=> ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),A3)),Xb)
            & aa(A,$o,ord_less(A,Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A3)),one_one(A))) ) ) ) ).

% floor_eq_iff
tff(fact_4136_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),Z)),Xb)
         => ( aa(A,$o,ord_less(A,Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A)))
           => ( archim6421214686448440834_floor(A,Xb) = Z ) ) ) ) ).

% floor_unique
tff(fact_4137_less__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,ord_less(int,Z),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))),Xb) ) ) ).

% less_floor_iff
tff(fact_4138_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,ord_less_eq(int,archim6421214686448440834_floor(A,Xb)),Z)
        <=> aa(A,$o,ord_less(A,Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))) ) ) ).

% floor_le_iff
tff(fact_4139_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),B3)
           => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A3)),archim6421214686448440834_floor(A,B3))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))) ) ) ) ).

% le_mult_floor
tff(fact_4140_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(A,$o,ord_less_eq(A,aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xb))),Xb)
          & aa(A,$o,ord_less(A,Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),one_one(int)))) ) ) ).

% floor_correct
tff(fact_4141_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A),P2: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N))),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),P2)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_4142_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [A: $tType,Xb: fun(nat,fun(A,A)),Xaa: nat,Xba: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xb,Xaa,Xba,Xc) = Y )
     => ( Y = $ite(aa(nat,$o,ord_less(nat,Xba),Xaa),Xc,set_fo6178422350223883121st_nat(A,Xb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xaa),one_one(nat)),Xba,aa(A,A,aa(nat,fun(A,A),Xb,Xaa),Xc))) ) ) ).

% fold_atLeastAtMost_nat.elims
tff(fact_4143_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,Fa: fun(nat,fun(A,A)),A3: nat,B3: nat,Acc2: A] :
      set_fo6178422350223883121st_nat(A,Fa,A3,B3,Acc2) = $ite(aa(nat,$o,ord_less(nat,B3),A3),Acc2,set_fo6178422350223883121st_nat(A,Fa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),one_one(nat)),B3,aa(A,A,aa(nat,fun(A,A),Fa,A3),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_4144_floor__eq2,axiom,
    ! [N: int,Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(int,real,ring_1_of_int(real),N)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),N)),one_one(real)))
       => ( archim6421214686448440834_floor(real,Xb) = N ) ) ) ).

% floor_eq2
tff(fact_4145_prod_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),A3: A,B3: fun(A,B),C2: fun(A,B)] :
          ( finite_finite2(A,S3)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_yp(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A3),B3),C2),S3) = $ite(member(A,A3,S3),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B3,A3)),groups7121269368397514597t_prod(A,B,C2,aa(set(A),set(A),minus_minus(set(A),S3),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))))),groups7121269368397514597t_prod(A,B,C2,aa(set(A),set(A),minus_minus(set(A),S3),aa(set(A),set(A),insert(A,A3),bot_bot(set(A)))))) ) ) ) ).

% prod.delta_remove
tff(fact_4146_floor__divide__real__eq__div,axiom,
    ! [B3: int,A3: real] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B3)
     => ( archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),A3),aa(int,real,ring_1_of_int(real),B3))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),archim6421214686448440834_floor(real,A3)),B3) ) ) ).

% floor_divide_real_eq_div
tff(fact_4147_norm__prod__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),Z: fun(A,B),W: fun(A,B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,Z,I2))),one_one(real)) )
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,W,I2))),one_one(real)) )
           => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,groups7121269368397514597t_prod(A,B,Z,I5)),groups7121269368397514597t_prod(A,B,W,I5)))),aa(set(A),real,groups7311177749621191930dd_sum(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_yq(fun(A,B),fun(fun(A,B),fun(A,real)),Z),W)),I5)) ) ) ) ).

% norm_prod_diff
tff(fact_4148_fact__eq__fact__times,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,N),M)
     => ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,N)),groups7121269368397514597t_prod(nat,nat,aTP_Lamp_vj(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M))) ) ) ).

% fact_eq_fact_times
tff(fact_4149_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Q2)
         => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q2)))),Q2)),P2) ) ) ).

% floor_divide_lower
tff(fact_4150_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B6: set(A),A4: set(A),Fa: fun(A,B)] :
          ( finite_finite2(A,B6)
         => ( aa(set(A),$o,ord_less_eq(set(A),A4),B6)
           => ( ! [B2: A] :
                  ( member(A,B2,aa(set(A),set(A),minus_minus(set(A),B6),A4))
                 => aa(B,$o,ord_less_eq(B,one_one(B)),aa(A,B,Fa,B2)) )
             => ( ! [A2: A] :
                    ( member(A,A2,A4)
                   => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(A,B,Fa,A2)) )
               => aa(B,$o,ord_less_eq(B,groups7121269368397514597t_prod(A,B,Fa,A4)),groups7121269368397514597t_prod(A,B,Fa,B6)) ) ) ) ) ) ).

% prod_mono2
tff(fact_4151_prod__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom_divide(B)
     => ! [A4: set(A),Fa: fun(A,B),A3: A] :
          ( finite_finite2(A,A4)
         => ( ( aa(A,B,Fa,A3) != zero_zero(B) )
           => ( groups7121269368397514597t_prod(A,B,Fa,aa(set(A),set(A),minus_minus(set(A),A4),aa(set(A),set(A),insert(A,A3),bot_bot(set(A))))) = $ite(member(A,A3,A4),aa(B,B,aa(B,fun(B,B),divide_divide(B),groups7121269368397514597t_prod(A,B,Fa,A4)),aa(A,B,Fa,A3)),groups7121269368397514597t_prod(A,B,Fa,A4)) ) ) ) ) ).

% prod_diff1
tff(fact_4152_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_yr(A,fun(nat,A),A3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod
tff(fact_4153_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,N) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ys(A,fun(nat,fun(nat,A)),A3),N),set_or1337092689740270186AtMost(nat,one_one(nat),N)) ) ).

% pochhammer_prod_rev
tff(fact_4154_floor__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Q2)
         => aa(A,$o,ord_less(A,P2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),P2),Q2)))),one_one(A))),Q2)) ) ) ).

% floor_divide_upper
tff(fact_4155_fact__div__fact,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,N),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,N)) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_vj(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)),M)) ) ) ).

% fact_div_fact
tff(fact_4156_round__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : archimedean_round(A,Xb) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% round_def
tff(fact_4157_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_yt(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.in_pairs
tff(fact_4158_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Fa: fun(nat,A),A3: nat,B3: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_or1337092689740270186AtMost(nat,A3,B3)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_yu(fun(nat,A),fun(nat,fun(A,A)),Fa),A3,B3,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_4159_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,aa(nat,nat,suc,N)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ys(A,fun(nat,fun(nat,A)),A3),N),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_4160_floor__log__eq__powr__iff,axiom,
    ! [Xb: real,B3: real,K: int] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,one_one(real)),B3)
       => ( ( archim6421214686448440834_floor(real,aa(real,real,log(B3),Xb)) = K )
        <=> ( aa(real,$o,ord_less_eq(real,powr(real,B3,aa(int,real,ring_1_of_int(real),K))),Xb)
            & aa(real,$o,ord_less(real,Xb),powr(real,B3,aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))))) ) ) ) ) ).

% floor_log_eq_powr_iff
tff(fact_4161_floor__log2__div2,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
     => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_4162_floor__log__nat__eq__if,axiom,
    ! [B3: nat,N: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),N)),K)
     => ( aa(nat,$o,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))))
       => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),B3)
         => ( archim6421214686448440834_floor(real,aa(real,real,log(aa(nat,real,semiring_1_of_nat(real),B3)),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),N) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_4163_fact__code,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)),N,one_one(nat))) ) ).

% fact_code
tff(fact_4164_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          archimedean_round(A,Xb) = $ite(aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(one2)))),archimedean_frac(A,Xb)),archimedean_ceiling(A,Xb),archim6421214686448440834_floor(A,Xb)) ) ).

% round_altdef
tff(fact_4165_Maclaurin__sin__bound,axiom,
    ! [Xb: real,N: nat] : aa(real,$o,ord_less_eq(real,abs_abs(real,aa(real,real,minus_minus(real,sin(real,Xb)),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_xj(real,fun(nat,real),Xb)),set_ord_lessThan(nat,N))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),abs_abs(real,Xb)),N))) ).

% Maclaurin_sin_bound
tff(fact_4166_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R2: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_yv(A,fun(nat,A),R2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,M))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,R2),aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_4167_TBOUND__fi_H__adm,axiom,
    ! [D2: $tType,C: $tType,B: $tType,A: $tType,Foo: fun(A,fun(B,fun(C,nat)))] : comple1908693960933563346ssible(fun(product_prod(product_prod(A,B),C),heap_Time_Heap(D2)),partial_fun_lub(heap_Time_Heap(D2),heap_Time_Heap(D2),product_prod(product_prod(A,B),C),heap_Time_Heap_lub(D2)),partial_fun_ord(heap_Time_Heap(D2),heap_Time_Heap(D2),product_prod(product_prod(A,B),C),heap_Time_Heap_ord(D2)),aTP_Lamp_yw(fun(A,fun(B,fun(C,nat))),fun(fun(product_prod(product_prod(A,B),C),heap_Time_Heap(D2)),$o),Foo)) ).

% TBOUND_fi'_adm
tff(fact_4168_inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] : aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A3)) = A3 ) ).

% inverse_inverse_eq
tff(fact_4169_inverse__eq__iff__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,inverse_inverse(A),B3) )
        <=> ( A3 = B3 ) ) ) ).

% inverse_eq_iff_eq
tff(fact_4170_inverse__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% inverse_zero
tff(fact_4171_inverse__nonzero__iff__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% inverse_nonzero_iff_nonzero
tff(fact_4172_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_4173_inverse__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A] :
          ( ( aa(A,A,inverse_inverse(A),Xb) = one_one(A) )
        <=> ( Xb = one_one(A) ) ) ) ).

% inverse_eq_1_iff
tff(fact_4174_inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3)) ) ).

% inverse_mult_distrib
tff(fact_4175_inverse__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3) ) ).

% inverse_divide
tff(fact_4176_inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A3)) ) ).

% inverse_minus_eq
tff(fact_4177_abs__inverse,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A3: A] : abs_abs(A,aa(A,A,inverse_inverse(A),A3)) = aa(A,A,inverse_inverse(A),abs_abs(A,A3)) ) ).

% abs_inverse
tff(fact_4178_gbinomial__1,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : aa(nat,A,gbinomial(A,A3),one_one(nat)) = A3 ) ).

% gbinomial_1
tff(fact_4179_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less_eq(A,A3),zero_zero(A)) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_4180_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,inverse_inverse(A),A3))
        <=> aa(A,$o,ord_less_eq(A,zero_zero(A)),A3) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_4181_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,inverse_inverse(A),A3))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ).

% inverse_positive_iff_positive
tff(fact_4182_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% inverse_negative_iff_negative
tff(fact_4183_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
            <=> aa(A,$o,ord_less(A,B3),A3) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_4184_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
           => ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
            <=> aa(A,$o,ord_less(A,B3),A3) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_4185_gbinomial__0_I2_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [K: nat] : aa(nat,A,gbinomial(A,zero_zero(A)),aa(nat,nat,suc,K)) = zero_zero(A) ) ).

% gbinomial_0(2)
tff(fact_4186_gbinomial__0_I1_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : aa(nat,A,gbinomial(A,A3),zero_zero(nat)) = one_one(A) ) ).

% gbinomial_0(1)
tff(fact_4187_gbinomial__Suc0,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,zero_zero(nat))) = A3 ) ).

% gbinomial_Suc0
tff(fact_4188_frac__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int] : archimedean_frac(A,aa(int,A,ring_1_of_int(A),Z)) = zero_zero(A) ) ).

% frac_of_int
tff(fact_4189_prod__eq__1__iff,axiom,
    ! [A: $tType,A4: set(A),Fa: fun(A,nat)] :
      ( finite_finite2(A,A4)
     => ( ( groups7121269368397514597t_prod(A,nat,Fa,A4) = one_one(nat) )
      <=> ! [X: A] :
            ( member(A,X,A4)
           => ( aa(A,nat,Fa,X) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_4190_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => ( aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
            <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_4191_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B3)
           => ( aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
            <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_4192_right__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),A3)) = one_one(A) ) ) ) ).

% right_inverse
tff(fact_4193_left__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),A3) = one_one(A) ) ) ) ).

% left_inverse
tff(fact_4194_inverse__eq__divide__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num] : aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),W)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),W)) ) ).

% inverse_eq_divide_numeral
tff(fact_4195__092_060open_062heap_Oadmissible_A_I_092_060lambda_062vebt__predi_H_O_A_092_060forall_062x_Axa_Axb_O_Arefines_A_Ivebt__predi_Axa_Axb_J_A_Icurry_A_Icurry_Avebt__predi_H_J_Ax_Axa_Axb_J_J_092_060close_062,axiom,
    comple1908693960933563346ssible(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),partial_fun_lub(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_lub(option(nat))),partial_fun_ord(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_ord(option(nat))),aTP_Lamp_yx(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),$o)) ).

% \<open>heap.admissible (\<lambda>vebt_predi'. \<forall>x xa xb. refines (vebt_predi xa xb) (curry (curry vebt_predi') x xa xb))\<close>
tff(fact_4196_prod__pos__nat__iff,axiom,
    ! [A: $tType,A4: set(A),Fa: fun(A,nat)] :
      ( finite_finite2(A,A4)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),groups7121269368397514597t_prod(A,nat,Fa,A4))
      <=> ! [X: A] :
            ( member(A,X,A4)
           => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(A,nat,Fa,X)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_4197_inverse__eq__divide__neg__numeral,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num] : aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) ) ).

% inverse_eq_divide_neg_numeral
tff(fact_4198_int__prod,axiom,
    ! [A: $tType,Fa: fun(A,nat),A4: set(A)] : aa(nat,int,semiring_1_of_nat(int),groups7121269368397514597t_prod(A,nat,Fa,A4)) = groups7121269368397514597t_prod(A,int,aTP_Lamp_vr(fun(A,nat),fun(A,int),Fa),A4) ).

% int_prod
tff(fact_4199_power__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),A3)),N) = aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) ) ).

% power_inverse
tff(fact_4200_inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,inverse_inverse(A),B3) )
         => ( A3 = B3 ) ) ) ).

% inverse_eq_imp_eq
tff(fact_4201_mult__commute__imp__mult__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Y: A,Xb: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Y)),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),Y)) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
tff(fact_4202_field__class_Ofield__inverse__zero,axiom,
    ! [A: $tType] :
      ( field(A)
     => ( aa(A,A,inverse_inverse(A),zero_zero(A)) = zero_zero(A) ) ) ).

% field_class.field_inverse_zero
tff(fact_4203_inverse__zero__imp__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = zero_zero(A) )
         => ( A3 = zero_zero(A) ) ) ) ).

% inverse_zero_imp_zero
tff(fact_4204_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,inverse_inverse(A),B3) )
         => ( ( A3 != zero_zero(A) )
           => ( ( B3 != zero_zero(A) )
             => ( A3 = B3 ) ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
tff(fact_4205_nonzero__inverse__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,inverse_inverse(A),A3)) = A3 ) ) ) ).

% nonzero_inverse_inverse_eq
tff(fact_4206_nonzero__imp__inverse__nonzero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A3) != zero_zero(A) ) ) ) ).

% nonzero_imp_inverse_nonzero
tff(fact_4207_nonzero__norm__inverse,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),A3)) = aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,A3)) ) ) ) ).

% nonzero_norm_inverse
tff(fact_4208_nonzero__of__real__inverse,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [Xb: real] :
          ( ( Xb != zero_zero(real) )
         => ( aa(real,A,real_Vector_of_real(A),aa(real,real,inverse_inverse(real),Xb)) = aa(A,A,inverse_inverse(A),aa(real,A,real_Vector_of_real(A),Xb)) ) ) ) ).

% nonzero_of_real_inverse
tff(fact_4209_finite__set__of__finite__funs,axiom,
    ! [A: $tType,B: $tType,A4: set(A),B6: set(B),D: B] :
      ( finite_finite2(A,A4)
     => ( finite_finite2(B,B6)
       => finite_finite2(fun(A,B),collect(fun(A,B),aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_yy(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),A4),B6),D))) ) ) ).

% finite_set_of_finite_funs
tff(fact_4210_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R2: real,Xb: A] :
          ( aa(real,$o,ord_less_eq(real,R2),real_V7770717601297561774m_norm(A,Xb))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
           => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),Xb))),aa(real,real,inverse_inverse(real),R2)) ) ) ) ).

% norm_inverse_le_norm
tff(fact_4211_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ).

% positive_imp_inverse_positive
tff(fact_4212_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A3)),zero_zero(A)) ) ) ).

% negative_imp_inverse_negative
tff(fact_4213_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,inverse_inverse(A),A3))
         => ( ( A3 != zero_zero(A) )
           => aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_4214_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A3)),zero_zero(A))
         => ( ( A3 != zero_zero(A) )
           => aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_4215_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_4216_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less(A,B3),A3) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_4217_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
           => aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% less_imp_inverse_less
tff(fact_4218_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
           => aa(A,$o,ord_less(A,B3),A3) ) ) ) ).

% inverse_less_imp_less
tff(fact_4219_nonzero__inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_4220_inverse__numeral__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),aa(num,A,numeral_numeral(A),one2)) = aa(num,A,numeral_numeral(A),one2) ) ) ).

% inverse_numeral_1
tff(fact_4221_inverse__unique,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3) = one_one(A) )
         => ( aa(A,A,inverse_inverse(A),A3) = B3 ) ) ) ).

% inverse_unique
tff(fact_4222_nonzero__inverse__minus__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% nonzero_inverse_minus_eq
tff(fact_4223_inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] : aa(A,A,inverse_inverse(A),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ).

% inverse_eq_divide
tff(fact_4224_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),B3)) ) ).

% field_class.field_divide_inverse
tff(fact_4225_divide__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,inverse_inverse(A),B3)) ) ).

% divide_inverse
tff(fact_4226_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B3)),A3) ) ).

% divide_inverse_commute
tff(fact_4227_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xb)),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xb)),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),M)) ) ).

% power_mult_power_inverse_commute
tff(fact_4228_power__mult__inverse__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,M: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),M)),aa(A,A,inverse_inverse(A),Xb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),M)) ) ).

% power_mult_inverse_distrib
tff(fact_4229_mult__inverse__of__nat__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xaa: nat,Xb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xaa))),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Xaa))) ) ).

% mult_inverse_of_nat_commute
tff(fact_4230_nonzero__abs__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( abs_abs(A,aa(A,A,inverse_inverse(A),A3)) = aa(A,A,inverse_inverse(A),abs_abs(A,A3)) ) ) ) ).

% nonzero_abs_inverse
tff(fact_4231_admissible__heap,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,$o)))))] : comple1908693960933563346ssible(fun(A,heap_Time_Heap(B)),partial_fun_lub(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_lub(B)),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),aTP_Lamp_yz(fun(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,$o))))),fun(fun(A,heap_Time_Heap(B)),$o),P)) ).

% admissible_heap
tff(fact_4232_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,ord_less_eq(A,zero_zero(A)),archimedean_frac(A,Xb)) ) ).

% frac_ge_0
tff(fact_4233_frac__lt__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,ord_less(A,archimedean_frac(A,Xb)),one_one(A)) ) ).

% frac_lt_1
tff(fact_4234_frac__1__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),one_one(A))) = archimedean_frac(A,Xb) ) ).

% frac_1_eq
tff(fact_4235_exp__fdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X3: nat] : aa(nat,A,diffs(A,aTP_Lamp_za(nat,A)),X3) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,X3)) ) ).

% exp_fdiffs
tff(fact_4236_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_4237_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
         => ( aa(A,$o,ord_less(A,B3),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_4238_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),B3)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% le_imp_inverse_le
tff(fact_4239_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
           => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ).

% inverse_le_imp_le
tff(fact_4240_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),Xb)),one_one(A))
        <=> ( aa(A,$o,ord_less_eq(A,Xb),zero_zero(A))
            | aa(A,$o,ord_less_eq(A,one_one(A)),Xb) ) ) ) ).

% inverse_le_1_iff
tff(fact_4241_one__less__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,ord_less(A,one_one(A)),aa(A,A,inverse_inverse(A),Xb))
        <=> ( aa(A,$o,ord_less(A,zero_zero(A)),Xb)
            & aa(A,$o,ord_less(A,Xb),one_one(A)) ) ) ) ).

% one_less_inverse_iff
tff(fact_4242_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less(A,A3),one_one(A))
           => aa(A,$o,ord_less(A,one_one(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% one_less_inverse
tff(fact_4243_division__ring__inverse__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3))),aa(A,A,inverse_inverse(A),B3)) ) ) ) ) ).

% division_ring_inverse_add
tff(fact_4244_inverse__add,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),aa(A,A,inverse_inverse(A),A3))),aa(A,A,inverse_inverse(A),B3)) ) ) ) ) ).

% inverse_add
tff(fact_4245_field__class_Ofield__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),A3) = one_one(A) ) ) ) ).

% field_class.field_inverse
tff(fact_4246_division__ring__inverse__diff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,minus_minus(A,B3),A3))),aa(A,A,inverse_inverse(A),B3)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_4247_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A3) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_4248_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,A3),K)),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc_Suc
tff(fact_4249_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,K),N)
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),K) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,nat,minus_minus(nat,N),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_4250_inverse__powr,axiom,
    ! [Y: real,A3: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
     => ( powr(real,aa(real,real,inverse_inverse(real),Y),A3) = aa(real,real,inverse_inverse(real),powr(real,Y,A3)) ) ) ).

% inverse_powr
tff(fact_4251_prod__int__eq,axiom,
    ! [I: nat,J: nat] : groups7121269368397514597t_prod(nat,int,semiring_1_of_nat(int),set_or1337092689740270186AtMost(nat,I,J)) = groups7121269368397514597t_prod(int,int,aTP_Lamp_wv(int,int),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),J))) ).

% prod_int_eq
tff(fact_4252_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,inverse_inverse(A),Xb))
        <=> ( aa(A,$o,ord_less(A,zero_zero(A)),Xb)
            & aa(A,$o,ord_less_eq(A,Xb),one_one(A)) ) ) ) ).

% one_le_inverse_iff
tff(fact_4253_inverse__less__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),Xb)),one_one(A))
        <=> ( aa(A,$o,ord_less_eq(A,Xb),zero_zero(A))
            | aa(A,$o,ord_less(A,one_one(A)),Xb) ) ) ) ).

% inverse_less_1_iff
tff(fact_4254_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,$o,ord_less_eq(A,A3),one_one(A))
           => aa(A,$o,ord_less_eq(A,one_one(A)),aa(A,A,inverse_inverse(A),A3)) ) ) ) ).

% one_le_inverse
tff(fact_4255_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,ord_less(A,B3),A3) )
            & ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A))
             => aa(A,$o,ord_less(A,A3),B3) ) ) ) ) ).

% inverse_less_iff
tff(fact_4256_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3))
             => aa(A,$o,ord_less_eq(A,B3),A3) )
            & ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)),zero_zero(A))
             => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ) ).

% inverse_le_iff
tff(fact_4257_inverse__diff__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != zero_zero(A) )
         => ( ( B3 != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,inverse_inverse(A),A3)),aa(A,A,inverse_inverse(A),B3)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A3)),aa(A,A,minus_minus(A,A3),B3))),aa(A,A,inverse_inverse(A),B3))) ) ) ) ) ).

% inverse_diff_inverse
tff(fact_4258_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xb)
         => ? [N3: nat] : aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,N3)))),Xb) ) ) ).

% reals_Archimedean
tff(fact_4259_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A))),K)) ) ).

% gbinomial_addition_formula
tff(fact_4260_gbinomial__absorb__comp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,A3),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A))),K)) ) ).

% gbinomial_absorb_comp
tff(fact_4261_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),K)),A3)
         => aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,gbinomial(A,A3),K)) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_4262_forall__pos__mono__1,axiom,
    ! [P: fun(real,$o),E3: real] :
      ( ! [D3: real,E2: real] :
          ( aa(real,$o,ord_less(real,D3),E2)
         => ( aa(real,$o,P,D3)
           => aa(real,$o,P,E2) ) )
     => ( ! [N3: nat] : aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N3))))
       => ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
         => aa(real,$o,P,E3) ) ) ) ).

% forall_pos_mono_1
tff(fact_4263_forall__pos__mono,axiom,
    ! [P: fun(real,$o),E3: real] :
      ( ! [D3: real,E2: real] :
          ( aa(real,$o,ord_less(real,D3),E2)
         => ( aa(real,$o,P,D3)
           => aa(real,$o,P,E2) ) )
     => ( ! [N3: nat] :
            ( ( N3 != zero_zero(nat) )
           => aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N3))) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
         => aa(real,$o,P,E3) ) ) ) ).

% forall_pos_mono
tff(fact_4264_real__arch__inverse,axiom,
    ! [E3: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
    <=> ? [N8: nat] :
          ( ( N8 != zero_zero(nat) )
          & aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N8)))
          & aa(real,$o,ord_less(real,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),N8))),E3) ) ) ).

% real_arch_inverse
tff(fact_4265_sqrt__divide__self__eq,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,Xb)),Xb) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb)) ) ) ).

% sqrt_divide_self_eq
tff(fact_4266_ln__inverse,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,real,ln_ln(real),aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,ln_ln(real),Xb)) ) ) ).

% ln_inverse
tff(fact_4267_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] : groups7121269368397514597t_prod(nat,int,semiring_1_of_nat(int),set_or1337092689740270186AtMost(nat,I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J))) = groups7121269368397514597t_prod(int,int,aTP_Lamp_wv(int,int),set_or1337092689740270186AtMost(int,aa(nat,int,semiring_1_of_nat(int),I),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),J)))) ).

% prod_int_plus_eq
tff(fact_4268_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : summable(A,aTP_Lamp_zb(A,fun(nat,A),Xb)) ) ).

% summable_exp
tff(fact_4269_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xb)
         => ? [N3: nat] :
              ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N3)
              & aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N3))),Xb) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_4270_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,M: nat,N: nat] :
          ( ( Xb != zero_zero(A) )
         => ( aa(nat,$o,ord_less_eq(nat,M),N)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,minus_minus(nat,N),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xb)),M)) ) ) ) ) ).

% power_diff_conv_inverse
tff(fact_4271_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,gbinomial(A,A3),K)) ) ).

% Suc_times_gbinomial
tff(fact_4272_gbinomial__absorption,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A))),K)) ) ).

% gbinomial_absorption
tff(fact_4273_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M: nat,A3: A] :
          ( aa(nat,$o,ord_less_eq(nat,K),M)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),M)),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),M)),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_4274_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( ( archimedean_frac(A,Xb) = Xb )
        <=> ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
            & aa(A,$o,ord_less(A,Xb),one_one(A)) ) ) ) ).

% frac_eq
tff(fact_4275_log__inverse,axiom,
    ! [A3: real,Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
     => ( ( A3 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( aa(real,real,log(A3),aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,log(A3),Xb)) ) ) ) ) ).

% log_inverse
tff(fact_4276_frac__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          archimedean_frac(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = $ite(aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xb)),archimedean_frac(A,Y))),one_one(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xb)),archimedean_frac(A,Y)),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xb)),archimedean_frac(A,Y))),one_one(A))) ) ).

% frac_add
tff(fact_4277_ln__prod,axiom,
    ! [A: $tType,I5: set(A),Fa: fun(A,real)] :
      ( finite_finite2(A,I5)
     => ( ! [I2: A] :
            ( member(A,I2,I5)
           => aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,Fa,I2)) )
       => ( aa(real,real,ln_ln(real),groups7121269368397514597t_prod(A,real,Fa,I5)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_zc(fun(A,real),fun(A,real),Fa)),I5) ) ) ) ).

% ln_prod
tff(fact_4278_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))) ) ).

% gbinomial_rec
tff(fact_4279_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),aa(nat,A,gbinomial(A,A3),K)) ) ).

% gbinomial_factors
tff(fact_4280_gbinomial__negated__upper,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),K)),A3)),one_one(A))),K)) ) ).

% gbinomial_negated_upper
tff(fact_4281_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),N))),one_one(A))),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),N)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),N)) ) ).

% gbinomial_index_swap
tff(fact_4282_exp__plus__inverse__exp,axiom,
    ! [Xb: real] : aa(real,$o,ord_less_eq(real,aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,exp(real),Xb)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),Xb)))) ).

% exp_plus_inverse_exp
tff(fact_4283_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A3)),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),K)) ) ).

% gbinomial_minus
tff(fact_4284_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A))),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_4285_plus__inverse__ge__2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => aa(real,$o,ord_less_eq(real,aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,inverse_inverse(real),Xb))) ) ).

% plus_inverse_ge_2
tff(fact_4286_real__inv__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb))),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(real,real,inverse_inverse(real),Xb) ) ) ).

% real_inv_sqrt_pow2
tff(fact_4287_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),K)),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),A3),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer
tff(fact_4288_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A3),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer'
tff(fact_4289_tan__cot,axiom,
    ! [Xb: real] : aa(real,real,tan(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Xb)) = aa(real,real,inverse_inverse(real),aa(real,real,tan(real),Xb)) ).

% tan_cot
tff(fact_4290_TBOUND__adm,axiom,
    ! [B: $tType,A: $tType,Tb: fun(A,nat)] : comple1908693960933563346ssible(fun(A,heap_Time_Heap(B)),partial_fun_lub(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_lub(B)),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),aTP_Lamp_zd(fun(A,nat),fun(fun(A,heap_Time_Heap(B)),$o),Tb)) ).

% TBOUND_adm
tff(fact_4291_floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = $ite(aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),archimedean_frac(A,Xb)),archimedean_frac(A,Y))),one_one(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Y)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Y))),one_one(int))) ) ).

% floor_add
tff(fact_4292_refines__adm,axiom,
    ! [B: $tType,A: $tType,Tb: fun(A,heap_Time_Heap(B))] : comple1908693960933563346ssible(fun(A,heap_Time_Heap(B)),partial_fun_lub(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_lub(B)),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),aTP_Lamp_ze(fun(A,heap_Time_Heap(B)),fun(fun(A,heap_Time_Heap(B)),$o),Tb)) ).

% refines_adm
tff(fact_4293_real__le__x__sinh,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => aa(real,$o,ord_less_eq(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(real,real,exp(real),Xb)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),Xb)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% real_le_x_sinh
tff(fact_4294_real__le__abs__sinh,axiom,
    ! [Xb: real] : aa(real,$o,ord_less_eq(real,abs_abs(real,Xb)),abs_abs(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(real,real,exp(real),Xb)),aa(real,real,inverse_inverse(real),aa(real,real,exp(real),Xb)))),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% real_le_abs_sinh
tff(fact_4295_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_zf(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),N)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_4296_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_zg(A,fun(nat,A),A3),set_or1337092689740270186AtMost(nat,zero_zero(nat),K))),semiring_char_0_fact(A,aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc
tff(fact_4297_tan__sec,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tan(A),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),cos(A,Xb))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ) ).

% tan_sec
tff(fact_4298_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_4299_TBOUND__fi__adm,axiom,
    ! [C: $tType,B: $tType,A: $tType,Foo: fun(A,fun(B,nat))] : comple1908693960933563346ssible(fun(product_prod(A,B),heap_Time_Heap(C)),partial_fun_lub(heap_Time_Heap(C),heap_Time_Heap(C),product_prod(A,B),heap_Time_Heap_lub(C)),partial_fun_ord(heap_Time_Heap(C),heap_Time_Heap(C),product_prod(A,B),heap_Time_Heap_ord(C)),aTP_Lamp_zh(fun(A,fun(B,nat)),fun(fun(product_prod(A,B),heap_Time_Heap(C)),$o),Foo)) ).

% TBOUND_fi_adm
tff(fact_4300_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] :
          aa(nat,A,gbinomial(A,A3),K) = $ite(K = zero_zero(nat),one_one(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_zi(A,fun(nat,fun(A,A)),A3),zero_zero(nat),aa(nat,nat,minus_minus(nat,K),one_one(nat)),one_one(A))),semiring_char_0_fact(A,K))) ) ).

% gbinomial_code
tff(fact_4301_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_yv(A,fun(nat,A),A3)),set_ord_atMost(nat,M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),M)),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,gbinomial(A,A3),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_4302_binomial__code,axiom,
    ! [N: nat,K: nat] :
      aa(nat,nat,binomial(N),K) = $ite(
        aa(nat,$o,ord_less(nat,N),K),
        zero_zero(nat),
        $ite(aa(nat,$o,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),aa(nat,nat,binomial(N),aa(nat,nat,minus_minus(nat,N),K)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,N),K)),one_one(nat)),N,one_one(nat))),semiring_char_0_fact(nat,K))) ) ).

% binomial_code
tff(fact_4303_gbinomial__r__part__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),M))),one_one(A)))),set_ord_atMost(nat,M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ).

% gbinomial_r_part_sum
tff(fact_4304_exp__first__two__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,exp(A),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),Xb)),suminf(A,aTP_Lamp_zj(A,fun(nat,A),Xb))) ) ).

% exp_first_two_terms
tff(fact_4305_scaleR__zero__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real] : aa(A,A,real_V8093663219630862766scaleR(A,A3),zero_zero(A)) = zero_zero(A) ) ).

% scaleR_zero_right
tff(fact_4306_scaleR__cancel__right,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,Xb: A,B3: real] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb) = aa(A,A,real_V8093663219630862766scaleR(A,B3),Xb) )
        <=> ( ( A3 = B3 )
            | ( Xb = zero_zero(A) ) ) ) ) ).

% scaleR_cancel_right
tff(fact_4307_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_atMost(A,K))
        <=> aa(A,$o,ord_less_eq(A,I),K) ) ) ).

% atMost_iff
tff(fact_4308_scaleR__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,Xb: A,Y: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb) = aa(A,A,real_V8093663219630862766scaleR(A,A3),Y) )
        <=> ( ( Xb = Y )
            | ( A3 = zero_zero(real) ) ) ) ) ).

% scaleR_cancel_left
tff(fact_4309_binomial__n__n,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),N) = one_one(nat) ).

% binomial_n_n
tff(fact_4310_scaleR__zero__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,zero_zero(real)),Xb) = zero_zero(A) ) ).

% scaleR_zero_left
tff(fact_4311_scaleR__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,Xb: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(real) )
            | ( Xb = zero_zero(A) ) ) ) ) ).

% scaleR_eq_0_iff
tff(fact_4312_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_ord_atMost(A,Xb)),set_ord_atMost(A,Y))
        <=> aa(A,$o,ord_less_eq(A,Xb),Y) ) ) ).

% atMost_subset_iff
tff(fact_4313_binomial__0__Suc,axiom,
    ! [K: nat] : aa(nat,nat,binomial(zero_zero(nat)),aa(nat,nat,suc,K)) = zero_zero(nat) ).

% binomial_0_Suc
tff(fact_4314_binomial__1,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),aa(nat,nat,suc,zero_zero(nat))) = N ).

% binomial_1
tff(fact_4315_scaleR__power,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xb: real,Y: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,real_V8093663219630862766scaleR(A,Xb),Y)),N) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),N)) ) ).

% scaleR_power
tff(fact_4316_binomial__eq__0__iff,axiom,
    ! [N: nat,K: nat] :
      ( ( aa(nat,nat,binomial(N),K) = zero_zero(nat) )
    <=> aa(nat,$o,ord_less(nat,N),K) ) ).

% binomial_eq_0_iff
tff(fact_4317_binomial__n__0,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_4318_Icc__subset__Iic__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,H2: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or1337092689740270186AtMost(A,L,H)),set_ord_atMost(A,H2))
        <=> ( ~ aa(A,$o,ord_less_eq(A,L),H)
            | aa(A,$o,ord_less_eq(A,H),H2) ) ) ) ).

% Icc_subset_Iic_iff
tff(fact_4319_zero__less__binomial__iff,axiom,
    ! [N: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,binomial(N),K))
    <=> aa(nat,$o,ord_less_eq(nat,K),N) ) ).

% zero_less_binomial_iff
tff(fact_4320_atMost__0,axiom,
    set_ord_atMost(nat,zero_zero(nat)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))) ).

% atMost_0
tff(fact_4321_scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,W: num,A3: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),U)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A3)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),W))),A3) ) ).

% scaleR_times
tff(fact_4322_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [V: num,W: num,A3: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A3)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),W)),aa(num,real,numeral_numeral(real),V))),A3) ) ).

% inverse_scaleR_times
tff(fact_4323_fraction__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,V: num,W: num,A3: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),V))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A3)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),U)),aa(num,real,numeral_numeral(real),W))),aa(num,real,numeral_numeral(real),V))),A3) ) ).

% fraction_scaleR_times
tff(fact_4324_scaleR__half__double,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3)) = A3 ) ).

% scaleR_half_double
tff(fact_4325_scaleR__right_Osum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,G: fun(B,A),A4: set(B)] : aa(A,A,real_V8093663219630862766scaleR(A,A3),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A4)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_zk(real,fun(fun(B,A),fun(B,A)),A3),G)),A4) ) ).

% scaleR_right.sum
tff(fact_4326_scaleR__sum__right,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,Fa: fun(B,A),A4: set(B)] : aa(A,A,real_V8093663219630862766scaleR(A,A3),aa(set(B),A,groups7311177749621191930dd_sum(B,A,Fa),A4)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_zk(real,fun(fun(B,A),fun(B,A)),A3),Fa)),A4) ) ).

% scaleR_sum_right
tff(fact_4327_sum__choose__upper,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_zl(nat,fun(nat,nat),M)),set_ord_atMost(nat,N)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,M)) ).

% sum_choose_upper
tff(fact_4328_choose__one,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),one_one(nat)) = N ).

% choose_one
tff(fact_4329_scaleR__right__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,A3: real,B3: real] :
          ( ( Xb != zero_zero(A) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb) = aa(A,A,real_V8093663219630862766scaleR(A,B3),Xb) )
           => ( A3 = B3 ) ) ) ) ).

% scaleR_right_imp_eq
tff(fact_4330_scaleR__left__imp__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A3: real,Xb: A,Y: A] :
          ( ( A3 != zero_zero(real) )
         => ( ( aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb) = aa(A,A,real_V8093663219630862766scaleR(A,A3),Y) )
           => ( Xb = Y ) ) ) ) ).

% scaleR_left_imp_eq
tff(fact_4331_summable__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),R2: real] :
          ( summable(A,X5)
         => summable(A,aa(real,fun(nat,A),aTP_Lamp_zm(fun(nat,A),fun(real,fun(nat,A)),X5),R2)) ) ) ).

% summable_scaleR_right
tff(fact_4332_sums__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),A3: A,R2: real] :
          ( aa(A,$o,sums(A,X5),A3)
         => aa(A,$o,sums(A,aa(real,fun(nat,A),aTP_Lamp_zm(fun(nat,A),fun(real,fun(nat,A)),X5),R2)),aa(A,A,real_V8093663219630862766scaleR(A,R2),A3)) ) ) ).

% sums_scaleR_right
tff(fact_4333_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_atMost(A,U) = collect(A,aTP_Lamp_zn(A,fun(A,$o),U)) ) ).

% atMost_def
tff(fact_4334_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_zo(nat,fun(nat,nat),R2)),set_ord_atMost(nat,N)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R2),N))),N) ).

% sum_choose_lower
tff(fact_4335_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_zp(nat,fun(nat,nat),N)),set_ord_atMost(nat,M)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),one_one(nat))),M) ).

% choose_rising_sum(2)
tff(fact_4336_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_zp(nat,fun(nat,nat),N)),set_ord_atMost(nat,M)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ).

% choose_rising_sum(1)
tff(fact_4337_binomial__eq__0,axiom,
    ! [N: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,N),K)
     => ( aa(nat,nat,binomial(N),K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_4338_binomial__symmetric,axiom,
    ! [K: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),N)
     => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,binomial(N),aa(nat,nat,minus_minus(nat,N),K)) ) ) ).

% binomial_symmetric
tff(fact_4339_binomial__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,R2),N)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(N),R2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),N),R2)) ) ).

% binomial_le_pow
tff(fact_4340_atMost__atLeast0,axiom,
    ! [N: nat] : set_ord_atMost(nat,N) = set_or1337092689740270186AtMost(nat,zero_zero(nat),N) ).

% atMost_atLeast0
tff(fact_4341_sum__choose__diagonal,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => ( aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_zq(nat,fun(nat,fun(nat,nat)),M),N)),set_ord_atMost(nat,M)) = aa(nat,nat,binomial(aa(nat,nat,suc,N)),M) ) ) ).

% sum_choose_diagonal
tff(fact_4342_vandermonde,axiom,
    ! [M: nat,N: nat,R2: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_zr(nat,fun(nat,fun(nat,fun(nat,nat))),M),N),R2)),set_ord_atMost(nat,R2)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),R2) ).

% vandermonde
tff(fact_4343_of__real__def,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R2: real] : aa(real,A,real_Vector_of_real(A),R2) = aa(A,A,real_V8093663219630862766scaleR(A,R2),one_one(A)) ) ).

% of_real_def
tff(fact_4344_choose__row__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,binomial(N)),set_ord_atMost(nat,N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).

% choose_row_sum
tff(fact_4345_binomial,axiom,
    ! [A3: nat,B3: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),B3)),N) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_zs(nat,fun(nat,fun(nat,fun(nat,nat))),A3),B3),N)),set_ord_atMost(nat,N)) ).

% binomial
tff(fact_4346_scaleR__sum__left,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Fa: fun(B,real),A4: set(B),Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(set(B),real,groups7311177749621191930dd_sum(B,real,Fa),A4)),Xb) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_zt(fun(B,real),fun(A,fun(B,A)),Fa),Xb)),A4) ) ).

% scaleR_sum_left
tff(fact_4347_scaleR__left_Osum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [G: fun(B,real),A4: set(B),Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(set(B),real,groups7311177749621191930dd_sum(B,real,G),A4)),Xb) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_zt(fun(B,real),fun(A,fun(B,A)),G),Xb)),A4) ) ).

% scaleR_left.sum
tff(fact_4348_suminf__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),R2: real] :
          ( summable(A,X5)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,R2),suminf(A,X5)) = suminf(A,aa(real,fun(nat,A),aTP_Lamp_zm(fun(nat,A),fun(real,fun(nat,A)),X5),R2)) ) ) ) ).

% suminf_scaleR_right
tff(fact_4349_summable__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),Xb: A] :
          ( summable(real,X5)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_zu(fun(nat,real),fun(A,fun(nat,A)),X5),Xb)) ) ) ).

% summable_scaleR_left
tff(fact_4350_zero__less__binomial,axiom,
    ! [K: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),N)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,binomial(N),K)) ) ).

% zero_less_binomial
tff(fact_4351_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,N)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,N)),aa(nat,nat,binomial(N),K))),aa(nat,nat,suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_4352_choose__mult,axiom,
    ! [K: nat,M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),M)
     => ( aa(nat,$o,ord_less_eq(nat,M),N)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),M)),aa(nat,nat,binomial(M),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),K)),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_4353_binomial__absorb__comp,axiom,
    ! [N: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,N),K)),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),K)) ).

% binomial_absorb_comp
tff(fact_4354_sums__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),A3: real,Xb: A] :
          ( aa(real,$o,sums(real,X5),A3)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_zu(fun(nat,real),fun(A,fun(nat,A)),X5),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)) ) ) ).

% sums_scaleR_left
tff(fact_4355_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,B3: A,N: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),N) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_zv(A,fun(A,fun(nat,fun(nat,A))),A3),B3),N)),set_ord_atMost(nat,N)) ) ).

% binomial_ring
tff(fact_4356_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: real,Xb: A] :
          ( aa(real,$o,ord_less_eq(real,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B3),Xb)) ) ) ) ).

% scaleR_right_mono
tff(fact_4357_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B3: real,A3: real,C2: A] :
          ( aa(real,$o,ord_less_eq(real,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,C2),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B3),C2)) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_4358_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A3: A,B3: A,N: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3),N) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_zw(A,fun(A,fun(nat,fun(nat,A))),A3),B3),N)),set_ord_atMost(nat,N)) ) ).

% pochhammer_binomial_sum
tff(fact_4359_choose__square__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_zx(nat,fun(nat,nat),N)),set_ord_atMost(nat,N)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),N) ).

% choose_square_sum
tff(fact_4360_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B3))
        <=> ( ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
             => aa(A,$o,ord_less_eq(A,A3),B3) )
            & ( aa(real,$o,ord_less(real,C2),zero_zero(real))
             => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_4361_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B3))
          <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_4362_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B3))
          <=> aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_4363_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xb: A,Y: A,A3: real] :
          ( aa(A,$o,ord_less_eq(A,Xb),Y)
         => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A3),Y)) ) ) ) ).

% scaleR_left_mono
tff(fact_4364_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B3: A,A3: A,C2: real] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(real,$o,ord_less_eq(real,C2),zero_zero(real))
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B3)) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_4365_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,U: real,V: real,A3: A] :
          ( ( Xb = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V)),A3) )
        <=> $ite(V = zero_zero(real),Xb = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,V),Xb) = aa(A,A,real_V8093663219630862766scaleR(A,U),A3)) ) ) ).

% eq_vector_fraction_iff
tff(fact_4366_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,V: real,A3: A,Xb: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V)),A3) = Xb )
        <=> $ite(V = zero_zero(real),Xb = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,U),A3) = aa(A,A,real_V8093663219630862766scaleR(A,V),Xb)) ) ) ).

% vector_fraction_eq_iff
tff(fact_4367_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,E3: A,C2: A,B3: real,D: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B3),E3)),D))
        <=> aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,A3),B3)),E3)),C2)),D) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_4368_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,E3: A,C2: A,B3: real,D: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B3),E3)),D))
        <=> aa(A,$o,ord_less_eq(A,C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,B3),A3)),E3)),D)) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_4369_atMost__nat__numeral,axiom,
    ! [K: num] : set_ord_atMost(nat,aa(num,nat,numeral_numeral(nat),K)) = aa(set(nat),set(nat),insert(nat,aa(num,nat,numeral_numeral(nat),K)),set_ord_atMost(nat,pred_numeral(K))) ).

% atMost_nat_numeral
tff(fact_4370_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_ord_atMost(A,A3)),set_ord_lessThan(A,B3))
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% Iic_subset_Iio_iff
tff(fact_4371_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,Y: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb) )
         => ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_zy(A,fun(A,fun(nat,fun(nat,A))),Xb),Y),N)),set_ord_atMost(nat,N)) ) ) ) ).

% exp_series_add_commuting
tff(fact_4372_binomial__absorption,axiom,
    ! [K: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),K)) ).

% binomial_absorption
tff(fact_4373_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( ( N != one_one(nat) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_zz(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_4374_binomial__r__part__sum,axiom,
    ! [M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)),one_one(nat)))),set_ord_atMost(nat,M)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ).

% binomial_r_part_sum
tff(fact_4375_choose__linear__sum,axiom,
    ! [N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_aaa(nat,fun(nat,nat),N)),set_ord_atMost(nat,N)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,N),one_one(nat)))) ).

% choose_linear_sum
tff(fact_4376_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: A] :
          ( aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),B3)),zero_zero(A))
        <=> ( ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) )
            | ( aa(real,$o,ord_less(real,A3),zero_zero(real))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) )
            | ( A3 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_4377_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A3),B3))
        <=> ( ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) )
            | ( aa(real,$o,ord_less(real,A3),zero_zero(real))
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) )
            | ( A3 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_4378_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: real,Xb: A,Y: A] :
          ( aa(real,$o,ord_less_eq(real,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,Xb),Y)
           => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),B3)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
               => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B3),Y)) ) ) ) ) ) ).

% scaleR_mono
tff(fact_4379_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: real,C2: A,D: A] :
          ( aa(real,$o,ord_less_eq(real,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,C2),D)
           => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
             => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),C2)
               => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B3),D)) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_4380_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,Xb: A] :
          ( ( ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
              & aa(A,$o,ord_less_eq(A,Xb),zero_zero(A)) )
            | ( aa(real,$o,ord_less_eq(real,A3),zero_zero(real))
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb) ) )
         => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),zero_zero(A)) ) ) ).

% split_scaleR_neg_le
tff(fact_4381_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: A] :
          ( ( ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
              & aa(A,$o,ord_less_eq(A,zero_zero(A)),B3) )
            | ( aa(real,$o,ord_less_eq(real,A3),zero_zero(real))
              & aa(A,$o,ord_less_eq(A,B3),zero_zero(A)) ) )
         => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A3),B3)) ) ) ).

% split_scaleR_pos_le
tff(fact_4382_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,Xb: A] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_4383_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,Xb: A] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),A3)
         => ( aa(A,$o,ord_less_eq(A,Xb),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),zero_zero(A)) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_4384_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,Xb: A] :
          ( aa(real,$o,ord_less_eq(real,A3),zero_zero(real))
         => ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),zero_zero(A)) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_4385_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A3: real,B3: A] :
          ( aa(real,$o,ord_less_eq(real,A3),zero_zero(real))
         => ( aa(A,$o,ord_less_eq(A,B3),zero_zero(A))
           => aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A3),B3)) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_4386_suminf__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),Xb: A] :
          ( summable(real,X5)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,suminf(real,X5)),Xb) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_zu(fun(nat,real),fun(A,fun(nat,A)),X5),Xb)) ) ) ) ).

% suminf_scaleR_left
tff(fact_4387_scaleR__2,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(num,real,numeral_numeral(real),bit0(one2))),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Xb) ) ).

% scaleR_2
tff(fact_4388_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xb: A,A3: real] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),Xb)
         => ( aa(real,$o,ord_less_eq(real,A3),one_one(real))
           => aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)),Xb) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_4389_binomial__fact__lemma,axiom,
    ! [K: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),N)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,N),K)))),aa(nat,nat,binomial(N),K)) = semiring_char_0_fact(nat,N) ) ) ).

% binomial_fact_lemma
tff(fact_4390_real__vector__affinity__eq,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,Xb: A,C2: A,Y: A] :
          ( ( M != zero_zero(real) )
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,M),Xb)),C2) = Y )
          <=> ( Xb = aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),Y)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),C2)) ) ) ) ) ).

% real_vector_affinity_eq
tff(fact_4391_real__vector__eq__affinity,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [M: real,Y: A,Xb: A,C2: A] :
          ( ( M != zero_zero(real) )
         => ( ( Y = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,M),Xb)),C2) )
          <=> ( aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),Y)),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),M)),C2)) = Xb ) ) ) ) ).

% real_vector_eq_affinity
tff(fact_4392_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3))
          <=> aa(A,$o,ord_less_eq(A,B3),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% neg_le_divideR_eq
tff(fact_4393_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B3: A,A3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3)),A3)
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),B3) ) ) ) ).

% neg_divideR_le_eq
tff(fact_4394_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3))
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),B3) ) ) ) ).

% pos_le_divideR_eq
tff(fact_4395_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B3: A,A3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3)),A3)
          <=> aa(A,$o,ord_less_eq(A,B3),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% pos_divideR_le_eq
tff(fact_4396_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less(A,A3),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3))
          <=> aa(A,$o,ord_less(A,B3),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% neg_less_divideR_eq
tff(fact_4397_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B3: A,A3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3)),A3)
          <=> aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),B3) ) ) ) ).

% neg_divideR_less_eq
tff(fact_4398_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less(A,A3),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3))
          <=> aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),B3) ) ) ) ).

% pos_less_divideR_eq
tff(fact_4399_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B3: A,A3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3)),A3)
          <=> aa(A,$o,ord_less(A,B3),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% pos_divideR_less_eq
tff(fact_4400_nonzero__inverse__scaleR__distrib,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [A3: real,Xb: A] :
          ( ( A3 != zero_zero(real) )
         => ( ( Xb != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,real_V8093663219630862766scaleR(A,A3),Xb)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),A3)),aa(A,A,inverse_inverse(A),Xb)) ) ) ) ) ).

% nonzero_inverse_scaleR_distrib
tff(fact_4401_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_aab(nat,fun(nat,A),N)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_4402_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_uw(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N))) ) ).

% sum.atMost_Suc_shift
tff(fact_4403_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Fa: fun(nat,A),I: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_wh(fun(nat,A),fun(nat,A),Fa)),set_ord_atMost(nat,I)) = aa(A,A,minus_minus(A,aa(nat,A,Fa,zero_zero(nat))),aa(nat,A,Fa,aa(nat,nat,suc,I))) ) ).

% sum_telescope
tff(fact_4404_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat,D: fun(nat,A)] :
          ( ! [X: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_aac(fun(nat,A),fun(A,fun(nat,A)),C2),X)),set_ord_atMost(nat,N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_aac(fun(nat,A),fun(A,fun(nat,A)),D),X)),set_ord_atMost(nat,N))
        <=> ! [I4: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I4),N)
             => ( aa(nat,A,C2,I4) = aa(nat,A,D,I4) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_4405_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_yi(fun(nat,A),fun(nat,A),G),set_ord_atMost(nat,N))) ) ).

% prod.atMost_Suc_shift
tff(fact_4406_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A3: fun(nat,A),B6: A] :
          ( ! [N3: nat] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(nat,A,A3,N3))
         => ( ! [N3: nat] : aa(A,$o,ord_less_eq(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,A3),set_ord_atMost(nat,N3))),B6)
           => summable(A,A3) ) ) ) ).

% bounded_imp_summable
tff(fact_4407_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_aad(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_ord_atMost(nat,N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aaf(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_ord_lessThan(nat,N)) ) ).

% sum.nested_swap'
tff(fact_4408_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_aag(fun(nat,fun(nat,A)),fun(nat,A),A3),set_ord_atMost(nat,N)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aai(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N),set_ord_lessThan(nat,N)) ) ).

% prod.nested_swap'
tff(fact_4409_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,K),N)
         => aa(A,$o,ord_less_eq(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_4410_binomial__maximum,axiom,
    ! [N: nat,K: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% binomial_maximum
tff(fact_4411_binomial__antimono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),K4)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))),K)
       => ( aa(nat,$o,ord_less_eq(nat,K4),N)
         => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(N),K4)),aa(nat,nat,binomial(N),K)) ) ) ) ).

% binomial_antimono
tff(fact_4412_binomial__mono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),K4)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K4)),N)
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),K4)) ) ) ).

% binomial_mono
tff(fact_4413_binomial__maximum_H,axiom,
    ! [N: nat,K: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),N)) ).

% binomial_maximum'
tff(fact_4414_binomial__le__pow2,axiom,
    ! [N: nat,K: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,binomial(N),K)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% binomial_le_pow2
tff(fact_4415_choose__reduce__nat,axiom,
    ! [N: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_4416_times__binomial__minus1__eq,axiom,
    ! [K: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,binomial(N),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_4417_summable__exp__generic,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(A,aTP_Lamp_aaj(A,fun(nat,A),Xb)) ) ).

% summable_exp_generic
tff(fact_4418_sin__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_aak(A,fun(nat,A),Xb)),sin(A,Xb)) ) ).

% sin_converges
tff(fact_4419_sin__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X3: A] : sin(A,X3) = suminf(A,aTP_Lamp_aak(A,fun(nat,A),X3)) ) ).

% sin_def
tff(fact_4420_cos__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_aal(A,fun(nat,A),Xb)),cos(A,Xb)) ) ).

% cos_converges
tff(fact_4421_cos__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X3: A] : cos(A,X3) = suminf(A,aTP_Lamp_aal(A,fun(nat,A),X3)) ) ).

% cos_def
tff(fact_4422_summable__norm__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(real,aTP_Lamp_aam(A,fun(nat,real),Xb)) ) ).

% summable_norm_sin
tff(fact_4423_summable__norm__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(real,aTP_Lamp_aan(A,fun(nat,real),Xb)) ) ).

% summable_norm_cos
tff(fact_4424_binomial__altdef__nat,axiom,
    ! [K: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,K),N)
     => ( aa(nat,nat,binomial(N),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,K)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,N),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_4425_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B3: A,A3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3))),A3)
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_4426_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3)))
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_4427_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B3: A,A3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3))),A3)
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_4428_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less_eq(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3)))
          <=> aa(A,$o,ord_less_eq(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_4429_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3)))
          <=> aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_4430_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B3: A,A3: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3))),A3)
          <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_4431_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A3: A,B3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less(A,A3),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3)))
          <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B3)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_4432_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B3: A,A3: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B3))),A3)
          <=> aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A3)),aa(A,A,uminus_uminus(A),B3)) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_4433_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat] :
          ( ! [X: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_aac(fun(nat,A),fun(A,fun(nat,A)),C2),X)),set_ord_atMost(nat,N)) = zero_zero(A)
        <=> ! [I4: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I4),N)
             => ( aa(nat,A,C2,I4) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_4434_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C2: fun(nat,A),N: nat,K: nat] :
          ( ! [W2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_aao(fun(nat,A),fun(A,fun(nat,A)),C2),W2)),set_ord_atMost(nat,N)) = zero_zero(A)
         => ( aa(nat,$o,ord_less_eq(nat,K),N)
           => ( aa(nat,A,C2,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_4435_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_uw(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,N))) ) ).

% sum.atMost_shift
tff(fact_4436_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Fa: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_ord_atMost(nat,M))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)))) ) ).

% sum_up_index_split
tff(fact_4437_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_yi(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,N))) ) ).

% prod.atMost_shift
tff(fact_4438_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_aap(A,fun(nat,A),A3)),set_ord_atMost(nat,N)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(nat,A,semiring_1_of_nat(A),N))),one_one(A))),N) ) ).

% gbinomial_parallel_sum
tff(fact_4439_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,groups7311177749621191930dd_sum(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_aaq(nat,fun(nat,fun(nat,$o)),N)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_aas(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).

% sum.triangle_reindex_eq
tff(fact_4440_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : groups7121269368397514597t_prod(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_aaq(nat,fun(nat,fun(nat,$o)),N)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_aau(fun(nat,fun(nat,A)),fun(nat,A),G),set_ord_atMost(nat,N)) ) ).

% prod.triangle_reindex_eq
tff(fact_4441_binomial__less__binomial__Suc,axiom,
    ! [K: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))
     => aa(nat,$o,ord_less(nat,aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),aa(nat,nat,suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_4442_binomial__strict__mono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,K),K4)
     => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K4)),N)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,binomial(N),K)),aa(nat,nat,binomial(N),K4)) ) ) ).

% binomial_strict_mono
tff(fact_4443_binomial__strict__antimono,axiom,
    ! [K: nat,K4: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,K),K4)
     => ( aa(nat,$o,ord_less_eq(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
       => ( aa(nat,$o,ord_less_eq(nat,K4),N)
         => aa(nat,$o,ord_less(nat,aa(nat,nat,binomial(N),K4)),aa(nat,nat,binomial(N),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_4444_central__binomial__odd,axiom,
    ! [N: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
     => ( aa(nat,nat,binomial(N),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))) = aa(nat,nat,binomial(N),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% central_binomial_odd
tff(fact_4445_binomial__addition__formula,axiom,
    ! [N: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,nat,binomial(N),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,N),one_one(nat))),K)) ) ) ).

% binomial_addition_formula
tff(fact_4446_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_aav(nat,fun(nat,A),N)),set_ord_atMost(nat,N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ) ) ).

% choose_odd_sum
tff(fact_4447_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [N: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_aaw(nat,fun(nat,A),N)),set_ord_atMost(nat,N))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ) ) ).

% choose_even_sum
tff(fact_4448_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,K),N)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,N)),aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,N),K)))) ) ) ) ).

% binomial_fact
tff(fact_4449_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,K),N)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,N)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,N),K))) ) ) ) ).

% fact_binomial
tff(fact_4450_exp__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_aaj(A,fun(nat,A),Xb)),aa(A,A,exp(A),Xb)) ) ).

% exp_converges
tff(fact_4451_exp__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X3: A] : aa(A,A,exp(A),X3) = suminf(A,aTP_Lamp_aaj(A,fun(nat,A),X3)) ) ).

% exp_def
tff(fact_4452_summable__norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(real,aTP_Lamp_aax(A,fun(nat,real),Xb)) ) ).

% summable_norm_exp
tff(fact_4453_sin__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_aay(A,fun(nat,A),Xb)),sin(A,Xb)) ) ).

% sin_minus_converges
tff(fact_4454_cos__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_aaz(A,fun(nat,A),Xb)),cos(A,Xb)) ) ).

% cos_minus_converges
tff(fact_4455_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,N: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),Xb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_ord_atMost(nat,N))) = aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,N))) ) ).

% sum_gp_basic
tff(fact_4456_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,N: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,ord_less_eq(nat,K),N)
           => finite_finite2(A,collect(A,aa(nat,fun(A,$o),aTP_Lamp_aba(fun(nat,A),fun(nat,fun(A,$o)),C2),N))) ) ) ) ).

% polyfun_roots_finite
tff(fact_4457_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat] :
          ( finite_finite2(A,collect(A,aa(nat,fun(A,$o),aTP_Lamp_aba(fun(nat,A),fun(nat,fun(A,$o)),C2),N)))
        <=> ? [I4: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I4),N)
              & ( aa(nat,A,C2,I4) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_4458_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),A3: A,N: nat] :
          ( ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_abb(fun(nat,A),fun(A,fun(nat,A)),C2),A3)),set_ord_atMost(nat,N)) = zero_zero(A) )
         => ~ ! [B2: fun(nat,A)] :
                ~ ! [Z4: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_abb(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Z4),A3)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_abb(fun(nat,A),fun(A,fun(nat,A)),B2),Z4)),set_ord_lessThan(nat,N))) ) ) ).

% polyfun_linear_factor_root
tff(fact_4459_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),N: nat,A3: A] :
        ? [B2: fun(nat,A)] :
        ! [Z4: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_abb(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),set_ord_atMost(nat,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Z4),A3)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_abb(fun(nat,A),fun(A,fun(nat,A)),B2),Z4)),set_ord_lessThan(nat,N)))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_abb(fun(nat,A),fun(A,fun(nat,A)),C2),A3)),set_ord_atMost(nat,N))) ) ).

% polyfun_linear_factor
tff(fact_4460_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,N: nat,Xb: A] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_ord_atMost(nat,aa(nat,nat,minus_minus(nat,N),M)))) ) ) ) ).

% sum_power_shift
tff(fact_4461_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),minus_minus(set(nat),set_ord_atMost(nat,N)),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_4462_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : aa(set(product_prod(nat,nat)),A,groups7311177749621191930dd_sum(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_abc(nat,fun(nat,fun(nat,$o)),N)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_aas(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,N)) ) ).

% sum.triangle_reindex
tff(fact_4463_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),N: nat] : groups7121269368397514597t_prod(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_abc(nat,fun(nat,fun(nat,$o)),N)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_aau(fun(nat,fun(nat,A)),fun(nat,A),G),set_ord_lessThan(nat,N)) ) ).

% prod.triangle_reindex
tff(fact_4464_choose__two,axiom,
    ! [N: nat] : aa(nat,nat,binomial(N),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,minus_minus(nat,N),one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% choose_two
tff(fact_4465_summable__Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A3: fun(nat,A),B3: fun(nat,A)] :
          ( summable(real,aTP_Lamp_abd(fun(nat,A),fun(nat,real),A3))
         => ( summable(real,aTP_Lamp_abd(fun(nat,A),fun(nat,real),B3))
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_abf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B3)) ) ) ) ).

% summable_Cauchy_product
tff(fact_4466_cos__x__cos__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_abh(A,fun(A,fun(nat,A)),Xb),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Y))) ) ).

% cos_x_cos_y
tff(fact_4467_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A3: fun(nat,A),B3: fun(nat,A)] :
          ( summable(real,aTP_Lamp_abd(fun(nat,A),fun(nat,real),A3))
         => ( summable(real,aTP_Lamp_abd(fun(nat,A),fun(nat,real),B3))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A3)),suminf(A,B3)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_abf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B3)) ) ) ) ) ).

% Cauchy_product
tff(fact_4468_sums__cos__x__plus__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_abj(A,fun(A,fun(nat,A)),Xb),Y)),cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))) ) ).

% sums_cos_x_plus_y
tff(fact_4469_sin__x__sin__y,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_abl(A,fun(A,fun(nat,A)),Xb),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Y))) ) ).

% sin_x_sin_y
tff(fact_4470_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_vi(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,N)) ) ).

% sum.in_pairs_0
tff(fact_4471_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [M: nat,A3: fun(nat,A),N: nat,B3: fun(nat,A),Xb: A] :
          ( ! [I2: nat] :
              ( aa(nat,$o,ord_less(nat,M),I2)
             => ( aa(nat,A,A3,I2) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( aa(nat,$o,ord_less(nat,N),J2)
               => ( aa(nat,A,B3,J2) = zero_zero(A) ) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_abb(fun(nat,A),fun(A,fun(nat,A)),A3),Xb)),set_ord_atMost(nat,M))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_abb(fun(nat,A),fun(A,fun(nat,A)),B3),Xb)),set_ord_atMost(nat,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_abn(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A3),B3),Xb)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ) ) ).

% polynomial_product
tff(fact_4472_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_yt(fun(nat,A),fun(nat,A),G),set_ord_atMost(nat,N)) ) ).

% prod.in_pairs_0
tff(fact_4473_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),N: nat,K: A] :
          ( ! [X: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_aac(fun(nat,A),fun(A,fun(nat,A)),C2),X)),set_ord_atMost(nat,N)) = K
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X: nat] :
                ( member(nat,X,set_or1337092689740270186AtMost(nat,one_one(nat),N))
               => ( aa(nat,A,C2,X) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_4474_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_abo(A,fun(nat,A),A3)),set_ord_atMost(nat,M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),M)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A3),one_one(A))),M)) ) ).

% gbinomial_sum_lower_neg
tff(fact_4475_polynomial__product__nat,axiom,
    ! [M: nat,A3: fun(nat,nat),N: nat,B3: fun(nat,nat),Xb: nat] :
      ( ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,M),I2)
         => ( aa(nat,nat,A3,I2) = zero_zero(nat) ) )
     => ( ! [J2: nat] :
            ( aa(nat,$o,ord_less(nat,N),J2)
           => ( aa(nat,nat,B3,J2) = zero_zero(nat) ) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_abp(fun(nat,nat),fun(nat,fun(nat,nat)),A3),Xb)),set_ord_atMost(nat,M))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_abp(fun(nat,nat),fun(nat,fun(nat,nat)),B3),Xb)),set_ord_atMost(nat,N))) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_abr(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A3),B3),Xb)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))) ) ) ) ).

% polynomial_product_nat
tff(fact_4476_Cauchy__product__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A3: fun(nat,A),B3: fun(nat,A)] :
          ( summable(real,aTP_Lamp_abd(fun(nat,A),fun(nat,real),A3))
         => ( summable(real,aTP_Lamp_abd(fun(nat,A),fun(nat,real),B3))
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_abf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B3)),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A3)),suminf(A,B3))) ) ) ) ).

% Cauchy_product_sums
tff(fact_4477_sum_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),P2)
         => ( aa(nat,$o,ord_less_eq(nat,K),P2)
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_abs(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,P2)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_abt(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H)),set_ord_atMost(nat,aa(nat,nat,minus_minus(nat,P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% sum.zero_middle
tff(fact_4478_prod_Ozero__middle,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [P2: nat,K: nat,G: fun(nat,A),H: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),P2)
         => ( aa(nat,$o,ord_less_eq(nat,K),P2)
           => ( groups7121269368397514597t_prod(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_abu(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H),set_ord_atMost(nat,P2)) = groups7121269368397514597t_prod(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_abv(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),K),G),H),set_ord_atMost(nat,aa(nat,nat,minus_minus(nat,P2),aa(nat,nat,suc,zero_zero(nat))))) ) ) ) ) ).

% prod.zero_middle
tff(fact_4479_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A3: A,Xb: A,Y: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_abw(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),Xb),Y)),set_ord_atMost(nat,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_abx(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),Xb),Y)),set_ord_atMost(nat,M)) ) ).

% gbinomial_partial_sum_poly
tff(fact_4480_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,exp(A),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),suminf(A,aTP_Lamp_aby(A,fun(nat,A),Xb))) ) ).

% exp_first_term
tff(fact_4481_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,Z: A,A3: A] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),N)
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),N) = A3 )
          <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_abz(nat,fun(A,fun(A,fun(nat,A))),N),Z),A3)),set_ord_atMost(nat,N)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_4482_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,N: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_ord_atMost(nat,N)) = $ite(Xb = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,suc,N)))),aa(A,A,minus_minus(A,one_one(A)),Xb))) ) ).

% sum_gp0
tff(fact_4483_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_aca(nat,fun(nat,A),M)),set_ord_atMost(nat,M)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M) ) ).

% gbinomial_sum_nat_pow2
tff(fact_4484_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A3: A,Xb: A,Y: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_abw(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),Xb),Y)),set_ord_atMost(nat,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_acb(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A3),Xb),Y)),set_ord_atMost(nat,M)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_4485_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,A3: fun(nat,A),Xb: A,Y: A] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),N)
         => ( aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_abb(fun(nat,A),fun(A,fun(nat,A)),A3),Xb)),set_ord_atMost(nat,N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_abb(fun(nat,A),fun(A,fun(nat,A)),A3),Y)),set_ord_atMost(nat,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Xb),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_acd(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),N),A3),Xb),Y)),set_ord_lessThan(nat,N))) ) ) ) ).

% polyfun_diff_alt
tff(fact_4486_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,K: nat] : aa(A,A,exp(A),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_aaj(A,fun(nat,A),Xb)),set_ord_lessThan(nat,K))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_ace(A,fun(nat,fun(nat,A)),Xb),K))) ) ).

% exp_first_terms
tff(fact_4487_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E3: real,C2: fun(nat,A),N: nat] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
         => ? [M10: real] :
            ! [Z4: A] :
              ( aa(real,$o,ord_less_eq(real,M10),real_V7770717601297561774m_norm(A,Z4))
             => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_mz(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),set_ord_atMost(nat,N)))),aa(real,real,aa(real,fun(real,real),times_times(real),E3),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Z4)),aa(nat,nat,suc,N)))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_4488_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [N: nat,A3: fun(nat,A),Xb: A,Y: A] :
          ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),N)
         => ( aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_abb(fun(nat,A),fun(A,fun(nat,A)),A3),Xb)),set_ord_atMost(nat,N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_abb(fun(nat,A),fun(A,fun(nat,A)),A3),Y)),set_ord_atMost(nat,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Xb),Y)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_acg(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),N),A3),Xb),Y)),set_ord_lessThan(nat,N))) ) ) ) ).

% polyfun_diff
tff(fact_4489_central__binomial__lower__bound,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))),N)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N)))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),N))) ) ).

% central_binomial_lower_bound
tff(fact_4490_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_ach(A,fun(nat,A),Xb)),sinh(A,Xb)) ) ).

% sinh_converges
tff(fact_4491_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_aci(A,fun(nat,A),Xb)),cosh(A,Xb)) ) ).

% cosh_converges
tff(fact_4492_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T2: real] :
            ( aa(real,$o,ord_less_eq(real,zero_zero(real)),T2)
           => ( aa(real,$o,ord_less(real,T2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( Z != complex2(cos(real,T2),sin(real,T2)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_4493_sinh__real__zero__iff,axiom,
    ! [Xb: real] :
      ( ( sinh(real,Xb) = zero_zero(real) )
    <=> ( Xb = zero_zero(real) ) ) ).

% sinh_real_zero_iff
tff(fact_4494_sinh__real__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,sinh(real,Xb)),sinh(real,Y))
    <=> aa(real,$o,ord_less(real,Xb),Y) ) ).

% sinh_real_less_iff
tff(fact_4495_of__nat__id,axiom,
    ! [N: nat] : aa(nat,nat,semiring_1_of_nat(nat),N) = N ).

% of_nat_id
tff(fact_4496_sinh__real__neg__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,sinh(real,Xb)),zero_zero(real))
    <=> aa(real,$o,ord_less(real,Xb),zero_zero(real)) ) ).

% sinh_real_neg_iff
tff(fact_4497_sinh__real__pos__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),sinh(real,Xb))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),Xb) ) ).

% sinh_real_pos_iff
tff(fact_4498_sinh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),sinh(real,Xb))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb) ) ).

% sinh_real_nonneg_iff
tff(fact_4499_sinh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,sinh(real,Xb)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,Xb),zero_zero(real)) ) ).

% sinh_real_nonpos_iff
tff(fact_4500_sinh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( sinh(A,zero_zero(A)) = zero_zero(A) ) ) ).

% sinh_0
tff(fact_4501_cosh__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cosh(A,zero_zero(A)) = one_one(A) ) ) ).

% cosh_0
tff(fact_4502_sinh__less__cosh__real,axiom,
    ! [Xb: real] : aa(real,$o,ord_less(real,sinh(real,Xb)),cosh(real,Xb)) ).

% sinh_less_cosh_real
tff(fact_4503_tanh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,tanh(A),Xb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),sinh(A,Xb)),cosh(A,Xb)) ) ).

% tanh_def
tff(fact_4504_cosh__real__nonzero,axiom,
    ! [Xb: real] : cosh(real,Xb) != zero_zero(real) ).

% cosh_real_nonzero
tff(fact_4505_Complex__eq__numeral,axiom,
    ! [A3: real,B3: real,W: num] :
      ( ( complex2(A3,B3) = aa(num,complex,numeral_numeral(complex),W) )
    <=> ( ( A3 = aa(num,real,numeral_numeral(real),W) )
        & ( B3 = zero_zero(real) ) ) ) ).

% Complex_eq_numeral
tff(fact_4506_cosh__real__pos,axiom,
    ! [Xb: real] : aa(real,$o,ord_less(real,zero_zero(real)),cosh(real,Xb)) ).

% cosh_real_pos
tff(fact_4507_cosh__real__nonpos__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,Xb),zero_zero(real))
     => ( aa(real,$o,ord_less_eq(real,Y),zero_zero(real))
       => ( aa(real,$o,ord_less_eq(real,cosh(real,Xb)),cosh(real,Y))
        <=> aa(real,$o,ord_less_eq(real,Y),Xb) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_4508_cosh__real__nonneg__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less_eq(real,cosh(real,Xb)),cosh(real,Y))
        <=> aa(real,$o,ord_less_eq(real,Xb),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_4509_cosh__real__nonneg,axiom,
    ! [Xb: real] : aa(real,$o,ord_less_eq(real,zero_zero(real)),cosh(real,Xb)) ).

% cosh_real_nonneg
tff(fact_4510_sinh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : sinh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),sinh(A,Xb))),cosh(A,Xb)) ) ).

% sinh_double
tff(fact_4511_complex__eq__cancel__iff2,axiom,
    ! [Xb: real,Y: real,Xaa: real] :
      ( ( complex2(Xb,Y) = aa(real,complex,real_Vector_of_real(complex),Xaa) )
    <=> ( ( Xb = Xaa )
        & ( Y = zero_zero(real) ) ) ) ).

% complex_eq_cancel_iff2
tff(fact_4512_complex__of__real__code,axiom,
    ! [X3: real] : aa(real,complex,real_Vector_of_real(complex),X3) = complex2(X3,zero_zero(real)) ).

% complex_of_real_code
tff(fact_4513_complex__of__real__def,axiom,
    ! [R2: real] : aa(real,complex,real_Vector_of_real(complex),R2) = complex2(R2,zero_zero(real)) ).

% complex_of_real_def
tff(fact_4514_zero__complex_Ocode,axiom,
    zero_zero(complex) = complex2(zero_zero(real),zero_zero(real)) ).

% zero_complex.code
tff(fact_4515_Complex__eq__0,axiom,
    ! [A3: real,B3: real] :
      ( ( complex2(A3,B3) = zero_zero(complex) )
    <=> ( ( A3 = zero_zero(real) )
        & ( B3 = zero_zero(real) ) ) ) ).

% Complex_eq_0
tff(fact_4516_cosh__real__strict__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),Y)
       => aa(real,$o,ord_less(real,cosh(real,Xb)),cosh(real,Y)) ) ) ).

% cosh_real_strict_mono
tff(fact_4517_cosh__real__nonneg__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( aa(real,$o,ord_less(real,cosh(real,Xb)),cosh(real,Y))
        <=> aa(real,$o,ord_less(real,Xb),Y) ) ) ) ).

% cosh_real_nonneg_less_iff
tff(fact_4518_cosh__real__nonpos__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less_eq(real,Xb),zero_zero(real))
     => ( aa(real,$o,ord_less_eq(real,Y),zero_zero(real))
       => ( aa(real,$o,ord_less(real,cosh(real,Xb)),cosh(real,Y))
        <=> aa(real,$o,ord_less(real,Y),Xb) ) ) ) ).

% cosh_real_nonpos_less_iff
tff(fact_4519_Complex__eq__neg__numeral,axiom,
    ! [A3: real,B3: real,W: num] :
      ( ( complex2(A3,B3) = aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W)) )
    <=> ( ( A3 = aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),W)) )
        & ( B3 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_4520_cosh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ).

% cosh_square_eq
tff(fact_4521_hyperbolic__pythagoras,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = one_one(A) ) ).

% hyperbolic_pythagoras
tff(fact_4522_sinh__square__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(A)) ) ).

% sinh_square_eq
tff(fact_4523_one__complex_Ocode,axiom,
    one_one(complex) = complex2(one_one(real),zero_zero(real)) ).

% one_complex.code
tff(fact_4524_Complex__eq__1,axiom,
    ! [A3: real,B3: real] :
      ( ( complex2(A3,B3) = one_one(complex) )
    <=> ( ( A3 = one_one(real) )
        & ( B3 = zero_zero(real) ) ) ) ).

% Complex_eq_1
tff(fact_4525_arcosh__cosh__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,real,arcosh(real),cosh(real,Xb)) = Xb ) ) ).

% arcosh_cosh_real
tff(fact_4526_cosh__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : cosh(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cosh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),sinh(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% cosh_double
tff(fact_4527_Complex__sum_H,axiom,
    ! [A: $tType,Fa: fun(A,real),S: set(A)] : aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,aTP_Lamp_acj(fun(A,real),fun(A,complex),Fa)),S) = complex2(aa(set(A),real,groups7311177749621191930dd_sum(A,real,Fa),S),zero_zero(real)) ).

% Complex_sum'
tff(fact_4528_Complex__eq__neg__1,axiom,
    ! [A3: real,B3: real] :
      ( ( complex2(A3,B3) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) )
    <=> ( ( A3 = aa(real,real,uminus_uminus(real),one_one(real)) )
        & ( B3 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_1
tff(fact_4529_complex__norm,axiom,
    ! [Xb: real,Y: real] : real_V7770717601297561774m_norm(complex,complex2(Xb,Y)) = aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% complex_norm
tff(fact_4530_tanh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] :
          ( ( cosh(A,Xb) != zero_zero(A) )
         => ( ( cosh(A,Y) != zero_zero(A) )
           => ( aa(A,A,tanh(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,tanh(A),Xb)),aa(A,A,tanh(A),Y))),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,tanh(A),Xb)),aa(A,A,tanh(A),Y)))) ) ) ) ) ).

% tanh_add
tff(fact_4531_complex__inverse,axiom,
    ! [A3: real,B3: real] : aa(complex,complex,inverse_inverse(complex),complex2(A3,B3)) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),A3),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B3),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),B3)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A3),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B3),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% complex_inverse
tff(fact_4532_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sinh(A,Xb) = zero_zero(A) )
        <=> member(A,aa(A,A,exp(A),Xb),aa(set(A),set(A),insert(A,one_one(A)),aa(set(A),set(A),insert(A,aa(A,A,uminus_uminus(A),one_one(A))),bot_bot(set(A))))) ) ) ).

% sinh_zero_iff
tff(fact_4533_cosh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : cosh(A,Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),Z)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cosh_field_def
tff(fact_4534_sinh__field__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Z: A] : sinh(A,Z) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,exp(A),Z)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sinh_field_def
tff(fact_4535_cosh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cosh(A,Xb) = zero_zero(A) )
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,exp(A),Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% cosh_zero_iff
tff(fact_4536_cosh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : cosh(A,Xb) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,exp(A),Xb)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xb)))) ) ).

% cosh_def
tff(fact_4537_cosh__ln__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( cosh(real,aa(real,real,ln_ln(real),Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,inverse_inverse(real),Xb))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% cosh_ln_real
tff(fact_4538_sinh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : sinh(A,Xb) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(A,A,minus_minus(A,aa(A,A,exp(A),Xb)),aa(A,A,exp(A),aa(A,A,uminus_uminus(A),Xb)))) ) ).

% sinh_def
tff(fact_4539_sinh__ln__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( sinh(real,aa(real,real,ln_ln(real),Xb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,Xb),aa(real,real,inverse_inverse(real),Xb))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% sinh_ln_real
tff(fact_4540_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Xb: A,Y: B,A3: product_prod(A,B)] :
      ( aa(B,$o,aa(A,fun(B,$o),P,Xb),Y)
     => ( ( A3 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y) )
       => aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(A,B),A,product_fst(A,B),A3)),aa(product_prod(A,B),B,product_snd(A,B),A3)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_4541_cot__less__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),zero_zero(real))
       => aa(real,$o,ord_less(real,aa(real,real,cot(real),Xb)),zero_zero(real)) ) ) ).

% cot_less_zero
tff(fact_4542_sint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),aa(nat,nat,suc,zero_zero(nat)))))),ring_1_signed(A,int,W))
          & aa(int,$o,ord_less(int,ring_1_signed(A,int,W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% sint_range_size
tff(fact_4543_cot__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ( aa(A,A,cot(A),zero_zero(A)) = zero_zero(A) ) ) ).

% cot_zero
tff(fact_4544_signed__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ring_1(A)
        & type_len(B) )
     => ( ring_1_signed(B,A,zero_zero(word(B))) = zero_zero(A) ) ) ).

% signed_0
tff(fact_4545_More__Word_Osint__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( ring_1_signed(A,int,Xb) = zero_zero(int) )
        <=> ( Xb = zero_zero(word(A)) ) ) ) ).

% More_Word.sint_0
tff(fact_4546_cot__pi,axiom,
    aa(real,real,cot(real),pi) = zero_zero(real) ).

% cot_pi
tff(fact_4547_signed__minus__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ring_1(A)
        & type_len(B) )
     => ( ring_1_signed(B,A,aa(word(B),word(B),uminus_uminus(word(B)),one_one(word(B)))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% signed_minus_1
tff(fact_4548_sint__minus1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( ring_1_signed(A,int,Xb) = aa(int,int,uminus_uminus(int),one_one(int)) )
        <=> ( Xb = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% sint_minus1
tff(fact_4549_cot__npi,axiom,
    ! [N: nat] : aa(real,real,cot(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),pi)) = zero_zero(real) ).

% cot_npi
tff(fact_4550_cot__periodic,axiom,
    ! [Xb: real] : aa(real,real,cot(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) = aa(real,real,cot(real),Xb) ).

% cot_periodic
tff(fact_4551_Word_Osint__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,zero_zero(word(A))) = zero_zero(int) ) ) ).

% Word.sint_0
tff(fact_4552_signed__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_char_0(A) )
     => ! [W: word(B)] :
          ( ( ring_1_signed(B,A,W) = zero_zero(A) )
        <=> ( W = zero_zero(word(B)) ) ) ) ).

% signed_eq_0_iff
tff(fact_4553_sint__n1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% sint_n1
tff(fact_4554_cot__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X3: A] : aa(A,A,cot(A),X3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),cos(A,X3)),sin(A,X3)) ) ).

% cot_def
tff(fact_4555_sint__above__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xb: int] :
          ( aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),one_one(nat)))),Xb)
         => aa(int,$o,ord_less(int,ring_1_signed(A,int,W)),Xb) ) ) ).

% sint_above_size
tff(fact_4556_sint__below__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int,W: word(A)] :
          ( aa(int,$o,ord_less_eq(int,Xb),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),one_one(nat)))))
         => aa(int,$o,ord_less_eq(int,Xb),ring_1_signed(A,int,W)) ) ) ).

% sint_below_size
tff(fact_4557_cot__gt__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
       => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,cot(real),Xb)) ) ) ).

% cot_gt_zero
tff(fact_4558_tan__cot_H,axiom,
    ! [Xb: real] : aa(real,real,tan(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),Xb)) = aa(real,real,cot(real),Xb) ).

% tan_cot'
tff(fact_4559_divmod__BitM__2__eq,axiom,
    ! [M: num] : unique8689654367752047608divmod(int,bitM(M),bit0(one2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),M)),one_one(int))),one_one(int)) ).

% divmod_BitM_2_eq
tff(fact_4560_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [Fa: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,ord_less(fun(A,B),Fa),G)
        <=> ( aa(fun(A,B),$o,ord_less_eq(fun(A,B),Fa),G)
            & ~ aa(fun(A,B),$o,ord_less_eq(fun(A,B),G),Fa) ) ) ) ).

% less_fun_def
tff(fact_4561_i__even__power,axiom,
    ! [N: nat] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),imaginary_unit),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),aa(complex,complex,uminus_uminus(complex),one_one(complex))),N) ).

% i_even_power
tff(fact_4562_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,A3),A3) ) ).

% dual_order.refl
tff(fact_4563_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A] : aa(A,$o,ord_less_eq(A,Xb),Xb) ) ).

% order_refl
tff(fact_4564_dbl__dec__simps_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [K: num] : neg_numeral_dbl_dec(A,aa(num,A,numeral_numeral(A),K)) = aa(num,A,numeral_numeral(A),bitM(K)) ) ).

% dbl_dec_simps(5)
tff(fact_4565_pred__numeral__simps_I2_J,axiom,
    ! [K: num] : pred_numeral(bit0(K)) = aa(num,nat,numeral_numeral(nat),bitM(K)) ).

% pred_numeral_simps(2)
tff(fact_4566_power2__i,axiom,
    aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),imaginary_unit),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% power2_i
tff(fact_4567_exp__two__pi__i,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(num,complex,numeral_numeral(complex),bit0(one2))),aa(real,complex,real_Vector_of_real(complex),pi))),imaginary_unit)) = one_one(complex) ).

% exp_two_pi_i
tff(fact_4568_exp__two__pi__i_H,axiom,
    aa(complex,complex,exp(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),pi)),aa(num,complex,numeral_numeral(complex),bit0(one2))))) = one_one(complex) ).

% exp_two_pi_i'
tff(fact_4569_scast__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( ring_1_signed(B,word(A),zero_zero(word(B))) = zero_zero(word(A)) ) ) ).

% scast_0
tff(fact_4570_complex__i__not__zero,axiom,
    imaginary_unit != zero_zero(complex) ).

% complex_i_not_zero
tff(fact_4571_semiring__norm_I26_J,axiom,
    bitM(one2) = one2 ).

% semiring_norm(26)
tff(fact_4572_semiring__norm_I27_J,axiom,
    ! [N: num] : bitM(bit0(N)) = aa(num,num,bit1,bitM(N)) ).

% semiring_norm(27)
tff(fact_4573_semiring__norm_I28_J,axiom,
    ! [N: num] : bitM(aa(num,num,bit1,N)) = aa(num,num,bit1,bit0(N)) ).

% semiring_norm(28)
tff(fact_4574_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,ord_less_eq(A,Y),Xb)
         => ( aa(A,$o,ord_less_eq(A,Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_4575_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,ord_less_eq(A,Xb),Y)
         => aa(A,$o,ord_less_eq(A,Y),Xb) ) ) ).

% linorder_le_cases
tff(fact_4576_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B3: A,Fa: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( ( aa(A,B,Fa,B3) = C2 )
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,ord_less_eq(A,X2),Y2)
                 => aa(B,$o,ord_less_eq(B,aa(A,B,Fa,X2)),aa(A,B,Fa,Y2)) )
             => aa(B,$o,ord_less_eq(B,aa(A,B,Fa,A3)),C2) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_4577_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,Fa: fun(B,A),B3: B,C2: B] :
          ( ( A3 = aa(B,A,Fa,B3) )
         => ( aa(B,$o,ord_less_eq(B,B3),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,ord_less_eq(B,X2),Y2)
                 => aa(A,$o,ord_less_eq(A,aa(B,A,Fa,X2)),aa(B,A,Fa,Y2)) )
             => aa(A,$o,ord_less_eq(A,A3),aa(B,A,Fa,C2)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_4578_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),Y)
          | aa(A,$o,ord_less_eq(A,Y),Xb) ) ) ).

% linorder_linear
tff(fact_4579_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb = Y )
         => aa(A,$o,ord_less_eq(A,Xb),Y) ) ) ).

% order_eq_refl
tff(fact_4580_order__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B3: A,Fa: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(B,$o,ord_less_eq(B,aa(A,B,Fa,B3)),C2)
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,ord_less_eq(A,X2),Y2)
                 => aa(B,$o,ord_less_eq(B,aa(A,B,Fa,X2)),aa(A,B,Fa,Y2)) )
             => aa(B,$o,ord_less_eq(B,aa(A,B,Fa,A3)),C2) ) ) ) ) ).

% order_subst2
tff(fact_4581_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,Fa: fun(B,A),B3: B,C2: B] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(B,A,Fa,B3))
         => ( aa(B,$o,ord_less_eq(B,B3),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,ord_less_eq(B,X2),Y2)
                 => aa(A,$o,ord_less_eq(A,aa(B,A,Fa,X2)),aa(B,A,Fa,Y2)) )
             => aa(A,$o,ord_less_eq(A,A3),aa(B,A,Fa,C2)) ) ) ) ) ).

% order_subst1
tff(fact_4582_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
        <=> ( aa(A,$o,ord_less_eq(A,A3),B3)
            & aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_4583_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,B3),A3)
           => ( A3 = B3 ) ) ) ) ).

% antisym
tff(fact_4584_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,C2),B3)
           => aa(A,$o,ord_less_eq(A,C2),A3) ) ) ) ).

% dual_order.trans
tff(fact_4585_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,A3),B3)
           => ( A3 = B3 ) ) ) ) ).

% dual_order.antisym
tff(fact_4586_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
        <=> ( aa(A,$o,ord_less_eq(A,B3),A3)
            & aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% dual_order.eq_iff
tff(fact_4587_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A3: A,B3: A] :
          ( ! [A2: A,B2: A] :
              ( aa(A,$o,ord_less_eq(A,A2),B2)
             => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) )
         => ( ! [A2: A,B2: A] :
                ( aa(A,$o,aa(A,fun(A,$o),P,B2),A2)
               => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) )
           => aa(A,$o,aa(A,fun(A,$o),P,A3),B3) ) ) ) ).

% linorder_wlog
tff(fact_4588_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),Y)
         => ( aa(A,$o,ord_less_eq(A,Y),Z)
           => aa(A,$o,ord_less_eq(A,Xb),Z) ) ) ) ).

% order_trans
tff(fact_4589_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,B3),C2)
           => aa(A,$o,ord_less_eq(A,A3),C2) ) ) ) ).

% order.trans
tff(fact_4590_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),Y)
         => ( aa(A,$o,ord_less_eq(A,Y),Xb)
           => ( Xb = Y ) ) ) ) ).

% order_antisym
tff(fact_4591_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( ( B3 = C2 )
           => aa(A,$o,ord_less_eq(A,A3),C2) ) ) ) ).

% ord_le_eq_trans
tff(fact_4592_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 = B3 )
         => ( aa(A,$o,ord_less_eq(A,B3),C2)
           => aa(A,$o,ord_less_eq(A,A3),C2) ) ) ) ).

% ord_eq_le_trans
tff(fact_4593_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb = Y )
        <=> ( aa(A,$o,ord_less_eq(A,Xb),Y)
            & aa(A,$o,ord_less_eq(A,Y),Xb) ) ) ) ).

% order_class.order_eq_iff
tff(fact_4594_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( ( aa(A,$o,ord_less_eq(A,Xb),Y)
           => ~ aa(A,$o,ord_less_eq(A,Y),Z) )
         => ( ( aa(A,$o,ord_less_eq(A,Y),Xb)
             => ~ aa(A,$o,ord_less_eq(A,Xb),Z) )
           => ( ( aa(A,$o,ord_less_eq(A,Xb),Z)
               => ~ aa(A,$o,ord_less_eq(A,Z),Y) )
             => ( ( aa(A,$o,ord_less_eq(A,Z),Y)
                 => ~ aa(A,$o,ord_less_eq(A,Y),Xb) )
               => ( ( aa(A,$o,ord_less_eq(A,Y),Z)
                   => ~ aa(A,$o,ord_less_eq(A,Z),Xb) )
                 => ~ ( aa(A,$o,ord_less_eq(A,Z),Xb)
                     => ~ aa(A,$o,ord_less_eq(A,Xb),Y) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_4595_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(A,$o,ord_less_eq(A,A3),B3)
        <=> ( aa(A,$o,ord_less_eq(A,B3),A3)
            & ( B3 != A3 ) ) ) ) ).

% nle_le
tff(fact_4596_order__less__imp__not__less,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
         => ~ aa(A,$o,ord_less(A,Y),Xb) ) ) ).

% order_less_imp_not_less
tff(fact_4597_order__less__imp__not__eq2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
         => ( Y != Xb ) ) ) ).

% order_less_imp_not_eq2
tff(fact_4598_order__less__imp__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
         => ( Xb != Y ) ) ) ).

% order_less_imp_not_eq
tff(fact_4599_linorder__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
          | ( Xb = Y )
          | aa(A,$o,ord_less(A,Y),Xb) ) ) ).

% linorder_less_linear
tff(fact_4600_order__less__imp__triv,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,P: $o] :
          ( aa(A,$o,ord_less(A,Xb),Y)
         => ( aa(A,$o,ord_less(A,Y),Xb)
           => (P) ) ) ) ).

% order_less_imp_triv
tff(fact_4601_order__less__not__sym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
         => ~ aa(A,$o,ord_less(A,Y),Xb) ) ) ).

% order_less_not_sym
tff(fact_4602_order__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B3: A,Fa: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(B,$o,ord_less(B,aa(A,B,Fa,B3)),C2)
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,ord_less(A,X2),Y2)
                 => aa(B,$o,ord_less(B,aa(A,B,Fa,X2)),aa(A,B,Fa,Y2)) )
             => aa(B,$o,ord_less(B,aa(A,B,Fa,A3)),C2) ) ) ) ) ).

% order_less_subst2
tff(fact_4603_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,Fa: fun(B,A),B3: B,C2: B] :
          ( aa(A,$o,ord_less(A,A3),aa(B,A,Fa,B3))
         => ( aa(B,$o,ord_less(B,B3),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,ord_less(B,X2),Y2)
                 => aa(A,$o,ord_less(A,aa(B,A,Fa,X2)),aa(B,A,Fa,Y2)) )
             => aa(A,$o,ord_less(A,A3),aa(B,A,Fa,C2)) ) ) ) ) ).

% order_less_subst1
tff(fact_4604_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A] : ~ aa(A,$o,ord_less(A,Xb),Xb) ) ).

% order_less_irrefl
tff(fact_4605_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,B3: A,Fa: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( ( aa(A,B,Fa,B3) = C2 )
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,ord_less(A,X2),Y2)
                 => aa(B,$o,ord_less(B,aa(A,B,Fa,X2)),aa(A,B,Fa,Y2)) )
             => aa(B,$o,ord_less(B,aa(A,B,Fa,A3)),C2) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_4606_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A3: A,Fa: fun(B,A),B3: B,C2: B] :
          ( ( A3 = aa(B,A,Fa,B3) )
         => ( aa(B,$o,ord_less(B,B3),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,ord_less(B,X2),Y2)
                 => aa(A,$o,ord_less(A,aa(B,A,Fa,X2)),aa(B,A,Fa,Y2)) )
             => aa(A,$o,ord_less(A,A3),aa(B,A,Fa,C2)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_4607_order__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
         => ( aa(A,$o,ord_less(A,Y),Z)
           => aa(A,$o,ord_less(A,Xb),Z) ) ) ) ).

% order_less_trans
tff(fact_4608_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ~ aa(A,$o,ord_less(A,B3),A3) ) ) ).

% order_less_asym'
tff(fact_4609_linorder__neq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
        <=> ( aa(A,$o,ord_less(A,Xb),Y)
            | aa(A,$o,ord_less(A,Y),Xb) ) ) ) ).

% linorder_neq_iff
tff(fact_4610_order__less__asym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
         => ~ aa(A,$o,ord_less(A,Y),Xb) ) ) ).

% order_less_asym
tff(fact_4611_linorder__neqE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
         => ( ~ aa(A,$o,ord_less(A,Xb),Y)
           => aa(A,$o,ord_less(A,Y),Xb) ) ) ) ).

% linorder_neqE
tff(fact_4612_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( A3 != B3 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_4613_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( A3 != B3 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_4614_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( aa(A,$o,ord_less(A,C2),B3)
           => aa(A,$o,ord_less(A,C2),A3) ) ) ) ).

% dual_order.strict_trans
tff(fact_4615_not__less__iff__gr__or__eq,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,Xb),Y)
        <=> ( aa(A,$o,ord_less(A,Y),Xb)
            | ( Xb = Y ) ) ) ) ).

% not_less_iff_gr_or_eq
tff(fact_4616_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,B3),C2)
           => aa(A,$o,ord_less(A,A3),C2) ) ) ) ).

% order.strict_trans
tff(fact_4617_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A3: A,B3: A] :
          ( ! [A2: A,B2: A] :
              ( aa(A,$o,ord_less(A,A2),B2)
             => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) )
         => ( ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),P,A2),A2)
           => ( ! [A2: A,B2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),P,B2),A2)
                 => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) )
             => aa(A,$o,aa(A,fun(A,$o),P,A3),B3) ) ) ) ) ).

% linorder_less_wlog
tff(fact_4618_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_12: A] : aa(A,$o,P,X_12)
        <=> ? [N8: A] :
              ( aa(A,$o,P,N8)
              & ! [M6: A] :
                  ( aa(A,$o,ord_less(A,M6),N8)
                 => ~ aa(A,$o,P,M6) ) ) ) ) ).

% exists_least_iff
tff(fact_4619_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A] : ~ aa(A,$o,ord_less(A,A3),A3) ) ).

% dual_order.irrefl
tff(fact_4620_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ~ aa(A,$o,ord_less(A,A3),B3) ) ) ).

% dual_order.asym
tff(fact_4621_linorder__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,Xb),Y)
         => ( ( Xb != Y )
           => aa(A,$o,ord_less(A,Y),Xb) ) ) ) ).

% linorder_cases
tff(fact_4622_antisym__conv3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xb: A] :
          ( ~ aa(A,$o,ord_less(A,Y),Xb)
         => ( ~ aa(A,$o,ord_less(A,Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% antisym_conv3
tff(fact_4623_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A3: A] :
          ( ! [X2: A] :
              ( ! [Y4: A] :
                  ( aa(A,$o,ord_less(A,Y4),X2)
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X2) )
         => aa(A,$o,P,A3) ) ) ).

% less_induct
tff(fact_4624_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( ( B3 = C2 )
           => aa(A,$o,ord_less(A,A3),C2) ) ) ) ).

% ord_less_eq_trans
tff(fact_4625_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A,C2: A] :
          ( ( A3 = B3 )
         => ( aa(A,$o,ord_less(A,B3),C2)
           => aa(A,$o,ord_less(A,A3),C2) ) ) ) ).

% ord_eq_less_trans
tff(fact_4626_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ~ aa(A,$o,ord_less(A,B3),A3) ) ) ).

% order.asym
tff(fact_4627_less__imp__neq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
         => ( Xb != Y ) ) ) ).

% less_imp_neq
tff(fact_4628_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
         => ? [Z2: A] :
              ( aa(A,$o,ord_less(A,Xb),Z2)
              & aa(A,$o,ord_less(A,Z2),Y) ) ) ) ).

% dense
tff(fact_4629_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xb: A] :
        ? [X_1: A] : aa(A,$o,ord_less(A,Xb),X_1) ) ).

% gt_ex
tff(fact_4630_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xb: A] :
        ? [Y2: A] : aa(A,$o,ord_less(A,Y2),Xb) ) ).

% lt_ex
tff(fact_4631_eval__nat__numeral_I2_J,axiom,
    ! [N: num] : aa(num,nat,numeral_numeral(nat),bit0(N)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bitM(N))) ).

% eval_nat_numeral(2)
tff(fact_4632_one__plus__BitM,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bitM(N)) = bit0(N) ).

% one_plus_BitM
tff(fact_4633_BitM__plus__one,axiom,
    ! [N: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bitM(N)),one2) = bit0(N) ).

% BitM_plus_one
tff(fact_4634_imaginary__unit_Ocode,axiom,
    imaginary_unit = complex2(zero_zero(real),one_one(real)) ).

% imaginary_unit.code
tff(fact_4635_Complex__eq__i,axiom,
    ! [Xb: real,Y: real] :
      ( ( complex2(Xb,Y) = imaginary_unit )
    <=> ( ( Xb = zero_zero(real) )
        & ( Y = one_one(real) ) ) ) ).

% Complex_eq_i
tff(fact_4636_complex__of__real__i,axiom,
    ! [R2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),R2)),imaginary_unit) = complex2(zero_zero(real),R2) ).

% complex_of_real_i
tff(fact_4637_i__complex__of__real,axiom,
    ! [R2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),aa(real,complex,real_Vector_of_real(complex),R2)) = complex2(zero_zero(real),R2) ).

% i_complex_of_real
tff(fact_4638_numeral__BitM,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(num,A,numeral_numeral(A),bitM(N)) = aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),bit0(N))),one_one(A)) ) ).

% numeral_BitM
tff(fact_4639_odd__numeral__BitM,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [W: num] : ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,A,numeral_numeral(A),bitM(W))) ) ).

% odd_numeral_BitM
tff(fact_4640_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),Y)
         => ( aa(A,$o,ord_less(A,Xb),Y)
            | ( Xb = Y ) ) ) ) ).

% order_le_imp_less_or_eq
tff(fact_4641_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),Y)
          | aa(A,$o,ord_less(A,Y),Xb) ) ) ).

% linorder_le_less_linear
tff(fact_4642_order__less__le__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B3: A,Fa: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(B,$o,ord_less_eq(B,aa(A,B,Fa,B3)),C2)
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,ord_less(A,X2),Y2)
                 => aa(B,$o,ord_less(B,aa(A,B,Fa,X2)),aa(A,B,Fa,Y2)) )
             => aa(B,$o,ord_less(B,aa(A,B,Fa,A3)),C2) ) ) ) ) ).

% order_less_le_subst2
tff(fact_4643_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,Fa: fun(B,A),B3: B,C2: B] :
          ( aa(A,$o,ord_less(A,A3),aa(B,A,Fa,B3))
         => ( aa(B,$o,ord_less_eq(B,B3),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,ord_less_eq(B,X2),Y2)
                 => aa(A,$o,ord_less_eq(A,aa(B,A,Fa,X2)),aa(B,A,Fa,Y2)) )
             => aa(A,$o,ord_less(A,A3),aa(B,A,Fa,C2)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_4644_order__le__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,B3: A,Fa: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(B,$o,ord_less(B,aa(A,B,Fa,B3)),C2)
           => ( ! [X2: A,Y2: A] :
                  ( aa(A,$o,ord_less_eq(A,X2),Y2)
                 => aa(B,$o,ord_less_eq(B,aa(A,B,Fa,X2)),aa(A,B,Fa,Y2)) )
             => aa(B,$o,ord_less(B,aa(A,B,Fa,A3)),C2) ) ) ) ) ).

% order_le_less_subst2
tff(fact_4645_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A3: A,Fa: fun(B,A),B3: B,C2: B] :
          ( aa(A,$o,ord_less_eq(A,A3),aa(B,A,Fa,B3))
         => ( aa(B,$o,ord_less(B,B3),C2)
           => ( ! [X2: B,Y2: B] :
                  ( aa(B,$o,ord_less(B,X2),Y2)
                 => aa(A,$o,ord_less(A,aa(B,A,Fa,X2)),aa(B,A,Fa,Y2)) )
             => aa(A,$o,ord_less(A,A3),aa(B,A,Fa,C2)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_4646_order__less__le__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
         => ( aa(A,$o,ord_less_eq(A,Y),Z)
           => aa(A,$o,ord_less(A,Xb),Z) ) ) ) ).

% order_less_le_trans
tff(fact_4647_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),Y)
         => ( aa(A,$o,ord_less(A,Y),Z)
           => aa(A,$o,ord_less(A,Xb),Z) ) ) ) ).

% order_le_less_trans
tff(fact_4648_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ( A3 != B3 )
         => ( aa(A,$o,ord_less_eq(A,A3),B3)
           => aa(A,$o,ord_less(A,A3),B3) ) ) ) ).

% order_neq_le_trans
tff(fact_4649_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( ( A3 != B3 )
           => aa(A,$o,ord_less(A,A3),B3) ) ) ) ).

% order_le_neq_trans
tff(fact_4650_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
         => aa(A,$o,ord_less_eq(A,Xb),Y) ) ) ).

% order_less_imp_le
tff(fact_4651_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,Xb),Y)
        <=> aa(A,$o,ord_less_eq(A,Y),Xb) ) ) ).

% linorder_not_less
tff(fact_4652_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,ord_less_eq(A,Xb),Y)
        <=> aa(A,$o,ord_less(A,Y),Xb) ) ) ).

% linorder_not_le
tff(fact_4653_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
        <=> ( aa(A,$o,ord_less_eq(A,Xb),Y)
            & ( Xb != Y ) ) ) ) ).

% order_less_le
tff(fact_4654_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),Y)
        <=> ( aa(A,$o,ord_less(A,Xb),Y)
            | ( Xb = Y ) ) ) ) ).

% order_le_less
tff(fact_4655_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => aa(A,$o,ord_less_eq(A,B3),A3) ) ) ).

% dual_order.strict_implies_order
tff(fact_4656_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% order.strict_implies_order
tff(fact_4657_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
        <=> ( aa(A,$o,ord_less_eq(A,B3),A3)
            & ~ aa(A,$o,ord_less_eq(A,A3),B3) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_4658_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( aa(A,$o,ord_less_eq(A,C2),B3)
           => aa(A,$o,ord_less(A,C2),A3) ) ) ) ).

% dual_order.strict_trans2
tff(fact_4659_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B3: A,A3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( aa(A,$o,ord_less(A,C2),B3)
           => aa(A,$o,ord_less(A,C2),A3) ) ) ) ).

% dual_order.strict_trans1
tff(fact_4660_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
        <=> ( aa(A,$o,ord_less_eq(A,B3),A3)
            & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_4661_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
        <=> ( aa(A,$o,ord_less(A,B3),A3)
            | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_4662_dense__le__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
         => ( ! [W2: A] :
                ( aa(A,$o,ord_less(A,Xb),W2)
               => ( aa(A,$o,ord_less(A,W2),Y)
                 => aa(A,$o,ord_less_eq(A,W2),Z) ) )
           => aa(A,$o,ord_less_eq(A,Y),Z) ) ) ) ).

% dense_le_bounded
tff(fact_4663_dense__ge__bounded,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Z),Xb)
         => ( ! [W2: A] :
                ( aa(A,$o,ord_less(A,Z),W2)
               => ( aa(A,$o,ord_less(A,W2),Xb)
                 => aa(A,$o,ord_less_eq(A,Y),W2) ) )
           => aa(A,$o,ord_less_eq(A,Y),Z) ) ) ) ).

% dense_ge_bounded
tff(fact_4664_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
        <=> ( aa(A,$o,ord_less_eq(A,A3),B3)
            & ~ aa(A,$o,ord_less_eq(A,B3),A3) ) ) ) ).

% order.strict_iff_not
tff(fact_4665_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,B3),C2)
           => aa(A,$o,ord_less(A,A3),C2) ) ) ) ).

% order.strict_trans2
tff(fact_4666_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less(A,B3),C2)
           => aa(A,$o,ord_less(A,A3),C2) ) ) ) ).

% order.strict_trans1
tff(fact_4667_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
        <=> ( aa(A,$o,ord_less_eq(A,A3),B3)
            & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
tff(fact_4668_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
        <=> ( aa(A,$o,ord_less(A,A3),B3)
            | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
tff(fact_4669_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xb: A] :
          ( ~ aa(A,$o,ord_less_eq(A,Y),Xb)
         => aa(A,$o,ord_less(A,Xb),Y) ) ) ).

% not_le_imp_less
tff(fact_4670_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
        <=> ( aa(A,$o,ord_less_eq(A,Xb),Y)
            & ~ aa(A,$o,ord_less_eq(A,Y),Xb) ) ) ) ).

% less_le_not_le
tff(fact_4671_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z: A] :
          ( ! [X2: A] :
              ( aa(A,$o,ord_less(A,X2),Y)
             => aa(A,$o,ord_less_eq(A,X2),Z) )
         => aa(A,$o,ord_less_eq(A,Y),Z) ) ) ).

% dense_le
tff(fact_4672_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Y: A] :
          ( ! [X2: A] :
              ( aa(A,$o,ord_less(A,Z),X2)
             => aa(A,$o,ord_less_eq(A,Y),X2) )
         => aa(A,$o,ord_less_eq(A,Y),Z) ) ) ).

% dense_ge
tff(fact_4673_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),Y)
         => ( ~ aa(A,$o,ord_less(A,Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% antisym_conv2
tff(fact_4674_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,Xb),Y)
         => ( aa(A,$o,ord_less_eq(A,Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% antisym_conv1
tff(fact_4675_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( ~ aa(A,$o,ord_less(A,A3),B3)
        <=> ( ~ aa(A,$o,ord_less_eq(A,A3),B3)
            | ( A3 = B3 ) ) ) ) ).

% nless_le
tff(fact_4676_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,Xb),Y)
         => aa(A,$o,ord_less_eq(A,Y),Xb) ) ) ).

% leI
tff(fact_4677_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,ord_less_eq(A,Y),Xb)
         => ~ aa(A,$o,ord_less(A,Xb),Y) ) ) ).

% leD
tff(fact_4678_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),bot_bot(A))
         => ( A3 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_4679_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),bot_bot(A))
        <=> ( A3 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_4680_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,bot_bot(A)),A3) ) ).

% bot.extremum
tff(fact_4681_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] :
          ( ( A3 != bot_bot(A) )
        <=> aa(A,$o,ord_less(A,bot_bot(A)),A3) ) ) ).

% bot.not_eq_extremum
tff(fact_4682_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A3: A] : ~ aa(A,$o,ord_less(A,A3),bot_bot(A)) ) ).

% bot.extremum_strict
tff(fact_4683_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A] :
          aa(A,A,ord_max(A,A3),B3) = $ite(aa(A,$o,ord_less_eq(A,A3),B3),B3,A3) ) ).

% max_def
tff(fact_4684_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,ord_less_eq(A,Y),Xb)
         => ( aa(A,A,ord_max(A,Xb),Y) = Xb ) ) ) ).

% max_absorb1
tff(fact_4685_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),Y)
         => ( aa(A,A,ord_max(A,Xb),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_4686_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [Fa: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,ord_less_eq(fun(A,B),Fa),G)
        <=> ! [X: A] : aa(B,$o,ord_less_eq(B,aa(A,B,Fa,X)),aa(A,B,G,X)) ) ) ).

% le_fun_def
tff(fact_4687_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [Fa: fun(A,B),G: fun(A,B)] :
          ( ! [X2: A] : aa(B,$o,ord_less_eq(B,aa(A,B,Fa,X2)),aa(A,B,G,X2))
         => aa(fun(A,B),$o,ord_less_eq(fun(A,B),Fa),G) ) ) ).

% le_funI
tff(fact_4688_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [Fa: fun(A,B),G: fun(A,B),Xb: A] :
          ( aa(fun(A,B),$o,ord_less_eq(fun(A,B),Fa),G)
         => aa(B,$o,ord_less_eq(B,aa(A,B,Fa,Xb)),aa(A,B,G,Xb)) ) ) ).

% le_funE
tff(fact_4689_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [Fa: fun(A,B),G: fun(A,B),Xb: A] :
          ( aa(fun(A,B),$o,ord_less_eq(fun(A,B),Fa),G)
         => aa(B,$o,ord_less_eq(B,aa(A,B,Fa,Xb)),aa(A,B,G,Xb)) ) ) ).

% le_funD
tff(fact_4690_Arg__minus__ii,axiom,
    arg(aa(complex,complex,uminus_uminus(complex),imaginary_unit)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),pi)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% Arg_minus_ii
tff(fact_4691_csqrt__ii,axiom,
    csqrt(imaginary_unit) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),one_one(complex)),imaginary_unit)),aa(real,complex,real_Vector_of_real(complex),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% csqrt_ii
tff(fact_4692_Arg__ii,axiom,
    arg(imaginary_unit) = aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% Arg_ii
tff(fact_4693_csqrt__eq__0,axiom,
    ! [Z: complex] :
      ( ( csqrt(Z) = zero_zero(complex) )
    <=> ( Z = zero_zero(complex) ) ) ).

% csqrt_eq_0
tff(fact_4694_csqrt__0,axiom,
    csqrt(zero_zero(complex)) = zero_zero(complex) ).

% csqrt_0
tff(fact_4695_power2__csqrt,axiom,
    ! [Z: complex] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),csqrt(Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = Z ).

% power2_csqrt
tff(fact_4696_Arg__zero,axiom,
    arg(zero_zero(complex)) = zero_zero(real) ).

% Arg_zero
tff(fact_4697_of__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,complex,real_Vector_of_real(complex),aa(real,real,sqrt,Xb)) = csqrt(aa(real,complex,real_Vector_of_real(complex),Xb)) ) ) ).

% of_real_sqrt
tff(fact_4698_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),arg(Z))
      & aa(real,$o,ord_less_eq(real,arg(Z)),pi) ) ).

% Arg_bounded
tff(fact_4699_cis__minus__pi__half,axiom,
    cis(aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))) = aa(complex,complex,uminus_uminus(complex),imaginary_unit) ).

% cis_minus_pi_half
tff(fact_4700_int__ge__less__than__def,axiom,
    ! [D: int] : int_ge_less_than(D) = collect(product_prod(int,int),product_case_prod(int,int,$o,aTP_Lamp_ack(int,fun(int,fun(int,$o)),D))) ).

% int_ge_less_than_def
tff(fact_4701_int__ge__less__than2__def,axiom,
    ! [D: int] : int_ge_less_than2(D) = collect(product_prod(int,int),product_case_prod(int,int,$o,aTP_Lamp_acl(int,fun(int,fun(int,$o)),D))) ).

% int_ge_less_than2_def
tff(fact_4702_cis__zero,axiom,
    cis(zero_zero(real)) = one_one(complex) ).

% cis_zero
tff(fact_4703_cis__pi__half,axiom,
    cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) = imaginary_unit ).

% cis_pi_half
tff(fact_4704_cis__2pi,axiom,
    cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)) = one_one(complex) ).

% cis_2pi
tff(fact_4705_cis__neq__zero,axiom,
    ! [A3: real] : cis(A3) != zero_zero(complex) ).

% cis_neq_zero
tff(fact_4706_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => bij_betw(nat,complex,aTP_Lamp_acm(nat,fun(nat,complex),N),set_ord_lessThan(nat,N),collect(complex,aTP_Lamp_wd(nat,fun(complex,$o),N))) ) ).

% bij_betw_roots_unity
tff(fact_4707_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,nat),X22: A] : size_option(A,Xb,aa(A,option(A),some(A),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xb,X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_4708_set__decode__0,axiom,
    ! [Xb: nat] :
      ( member(nat,zero_zero(nat),nat_set_decode(Xb))
    <=> ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Xb) ) ).

% set_decode_0
tff(fact_4709_set__decode__zero,axiom,
    nat_set_decode(zero_zero(nat)) = bot_bot(set(nat)) ).

% set_decode_zero
tff(fact_4710_set__decode__Suc,axiom,
    ! [N: nat,Xb: nat] :
      ( member(nat,aa(nat,nat,suc,N),nat_set_decode(Xb))
    <=> member(nat,N,nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% set_decode_Suc
tff(fact_4711_prod_Oreindex__bij__betw,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [H: fun(A,B),S3: set(A),T6: set(B),G: fun(B,C)] :
          ( bij_betw(A,B,H,S3,T6)
         => ( groups7121269368397514597t_prod(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_acn(fun(A,B),fun(fun(B,C),fun(A,C)),H),G),S3) = groups7121269368397514597t_prod(B,C,G,T6) ) ) ) ).

% prod.reindex_bij_betw
tff(fact_4712_sum_Oreindex__bij__betw,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [H: fun(A,B),S3: set(A),T6: set(B),G: fun(B,C)] :
          ( bij_betw(A,B,H,S3,T6)
         => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aco(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S3) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),T6) ) ) ) ).

% sum.reindex_bij_betw
tff(fact_4713_subset__decode__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( aa(set(nat),$o,ord_less_eq(set(nat),nat_set_decode(M)),nat_set_decode(N))
     => aa(nat,$o,ord_less_eq(nat,M),N) ) ).

% subset_decode_imp_le
tff(fact_4714_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S6: set(A),T5: set(B),H: fun(A,B),S3: set(A),T6: set(B),G: fun(B,C)] :
          ( finite_finite2(A,S6)
         => ( finite_finite2(B,T5)
           => ( bij_betw(A,B,H,aa(set(A),set(A),minus_minus(set(A),S3),S6),aa(set(B),set(B),minus_minus(set(B),T6),T5))
             => ( ! [A2: A] :
                    ( member(A,A2,S6)
                   => ( aa(B,C,G,aa(A,B,H,A2)) = zero_zero(C) ) )
               => ( ! [B2: B] :
                      ( member(B,B2,T5)
                     => ( aa(B,C,G,B2) = zero_zero(C) ) )
                 => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aco(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S3) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),T6) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_4715_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S6: set(A),T5: set(B),H: fun(A,B),S3: set(A),T6: set(B),G: fun(B,C)] :
          ( finite_finite2(A,S6)
         => ( finite_finite2(B,T5)
           => ( bij_betw(A,B,H,aa(set(A),set(A),minus_minus(set(A),S3),S6),aa(set(B),set(B),minus_minus(set(B),T6),T5))
             => ( ! [A2: A] :
                    ( member(A,A2,S6)
                   => ( aa(B,C,G,aa(A,B,H,A2)) = one_one(C) ) )
               => ( ! [B2: B] :
                      ( member(B,B2,T5)
                     => ( aa(B,C,G,B2) = one_one(C) ) )
                 => ( groups7121269368397514597t_prod(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_acn(fun(A,B),fun(fun(B,C),fun(A,C)),H),G),S3) = groups7121269368397514597t_prod(B,C,G,T6) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_4716_option_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xb: fun(A,nat)] : size_option(A,Xb,none(A)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size_gen(1)
tff(fact_4717_set__decode__plus__power__2,axiom,
    ! [N: nat,Z: nat] :
      ( ~ member(nat,N,nat_set_decode(Z))
     => ( nat_set_decode(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),Z)) = aa(set(nat),set(nat),insert(nat,N),nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_4718_set__decode__def,axiom,
    ! [Xb: nat] : nat_set_decode(Xb) = collect(nat,aTP_Lamp_acp(nat,fun(nat,$o),Xb)) ).

% set_decode_def
tff(fact_4719_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% flip_bit_0
tff(fact_4720_forall__finite_I3_J,axiom,
    ! [Xb: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,suc,aa(nat,nat,suc,Xb)))
         => aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I4: nat] :
            ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,suc,Xb))
           => aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% forall_finite(3)
tff(fact_4721_forall__finite_I2_J,axiom,
    ! [P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,suc,zero_zero(nat)))
         => aa(nat,$o,P,I4) )
    <=> aa(nat,$o,P,zero_zero(nat)) ) ).

% forall_finite(2)
tff(fact_4722_of__bool__eq_I1_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa($o,A,zero_neq_one_of_bool(A),$false) = zero_zero(A) ) ) ).

% of_bool_eq(1)
tff(fact_4723_of__bool__eq__0__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P)) = zero_zero(A) )
        <=> ~ (P) ) ) ).

% of_bool_eq_0_iff
tff(fact_4724_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,ord_less_eq(A,aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q)))
        <=> ( (P)
           => (Q) ) ) ) ).

% of_bool_less_eq_iff
tff(fact_4725_of__bool__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,ord_less(A,aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q)))
        <=> ( ~ (P)
            & (Q) ) ) ) ).

% of_bool_less_iff
tff(fact_4726_of__bool__eq__1__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P)) = one_one(A) )
        <=> (P) ) ) ).

% of_bool_eq_1_iff
tff(fact_4727_of__bool__eq_I2_J,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( aa($o,A,zero_neq_one_of_bool(A),$true) = one_one(A) ) ) ).

% of_bool_eq(2)
tff(fact_4728_of__nat__of__bool,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: $o] : aa(nat,A,semiring_1_of_nat(A),aa($o,nat,zero_neq_one_of_bool(nat),(P))) = aa($o,A,zero_neq_one_of_bool(A),(P)) ) ).

% of_nat_of_bool
tff(fact_4729_abs__bool__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: $o] : abs_abs(A,aa($o,A,zero_neq_one_of_bool(A),(P))) = aa($o,A,zero_neq_one_of_bool(A),(P)) ) ).

% abs_bool_eq
tff(fact_4730_of__bool__or__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o,Q: $o] :
          aa($o,A,zero_neq_one_of_bool(A),
            ( (P)
            | (Q) )) = aa(A,A,ord_max(A,aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q))) ) ).

% of_bool_or_iff
tff(fact_4731_zero__less__of__bool__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P)))
        <=> (P) ) ) ).

% zero_less_of_bool_iff
tff(fact_4732_of__bool__less__one__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] :
          ( aa(A,$o,ord_less(A,aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A))
        <=> ~ (P) ) ) ).

% of_bool_less_one_iff
tff(fact_4733_Suc__0__mod__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa($o,nat,zero_neq_one_of_bool(nat),N != aa(nat,nat,suc,zero_zero(nat))) ).

% Suc_0_mod_eq
tff(fact_4734_of__bool__not__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [P: $o] : aa($o,A,zero_neq_one_of_bool(A),~ (P)) = aa(A,A,minus_minus(A,one_one(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).

% of_bool_not_iff
tff(fact_4735_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R2: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q2),aa($o,int,zero_neq_one_of_bool(int),R2 != zero_zero(int))) ).

% Divides.adjust_div_eq
tff(fact_4736_odd__of__bool__self,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [P2: $o] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa($o,A,zero_neq_one_of_bool(A),(P2)))
        <=> (P2) ) ) ).

% odd_of_bool_self
tff(fact_4737_mod__word__by__minus__1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),W),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa($o,word(A),zero_neq_one_of_bool(word(A)),aa(word(A),$o,ord_less(word(A),W),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))))) ) ).

% mod_word_by_minus_1_eq
tff(fact_4738_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B3: $o] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ).

% of_bool_half_eq_0
tff(fact_4739_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa($o,A,zero_neq_one_of_bool(A),N = zero_zero(nat)) ) ).

% one_div_2_pow_eq
tff(fact_4740_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa($o,A,zero_neq_one_of_bool(A),N = zero_zero(nat)) ) ).

% bits_1_div_exp
tff(fact_4741_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,ord_less(nat,zero_zero(nat)),N)) ) ).

% one_mod_2_pow_eq
tff(fact_4742_of__bool__conj,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [P: $o,Q: $o] :
          aa($o,A,zero_neq_one_of_bool(A),
            ( (P)
            & (Q) )) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),(P))),aa($o,A,zero_neq_one_of_bool(A),(Q))) ) ).

% of_bool_conj
tff(fact_4743_of__bool__eq__iff,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: $o,Q2: $o] :
          ( ( aa($o,A,zero_neq_one_of_bool(A),(P2)) = aa($o,A,zero_neq_one_of_bool(A),(Q2)) )
        <=> ( (P2)
          <=> (Q2) ) ) ) ).

% of_bool_eq_iff
tff(fact_4744_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa($o,A,zero_neq_one_of_bool(A),(P))) ) ).

% zero_less_eq_of_bool
tff(fact_4745_split__of__bool__asm,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,$o),P2: $o] :
          ( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P2)))
        <=> ~ ( ( (P2)
                & ~ aa(A,$o,P,one_one(A)) )
              | ( ~ (P2)
                & ~ aa(A,$o,P,zero_zero(A)) ) ) ) ) ).

% split_of_bool_asm
tff(fact_4746_split__of__bool,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P: fun(A,$o),P2: $o] :
          ( aa(A,$o,P,aa($o,A,zero_neq_one_of_bool(A),(P2)))
        <=> ( ( (P2)
             => aa(A,$o,P,one_one(A)) )
            & ( ~ (P2)
             => aa(A,$o,P,zero_zero(A)) ) ) ) ) ).

% split_of_bool
tff(fact_4747_of__bool__def,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ! [P2: $o] :
          aa($o,A,zero_neq_one_of_bool(A),(P2)) = $ite((P2),one_one(A),zero_zero(A)) ) ).

% of_bool_def
tff(fact_4748_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,ord_less_eq(A,aa($o,A,zero_neq_one_of_bool(A),(P))),one_one(A)) ) ).

% of_bool_less_eq_one
tff(fact_4749_Divides_Oadjust__div__def,axiom,
    ! [Qr: product_prod(int,int)] : adjust_div(Qr) = aa(product_prod(int,int),int,product_case_prod(int,int,int,aTP_Lamp_acq(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_4750_of__bool__odd__eq__mod__2,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A3: A] : aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% of_bool_odd_eq_mod_2
tff(fact_4751_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,$o),A3: A] :
          ( ! [A2: A] :
              ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
             => aa(A,$o,P,A2) )
         => ( ! [A2: A,B2: $o] :
                ( aa(A,$o,P,A2)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2))),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
                 => aa(A,$o,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) )
           => aa(A,$o,P,A3) ) ) ) ).

% bits_induct
tff(fact_4752_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,ord_less(nat,M),N))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) ) ).

% exp_mod_exp
tff(fact_4753_forall__finite_I1_J,axiom,
    ! [P: fun(nat,$o),I3: nat] :
      ( aa(nat,$o,ord_less(nat,I3),zero_zero(nat))
     => aa(nat,$o,P,I3) ) ).

% forall_finite(1)
tff(fact_4754_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M) != zero_zero(A) )
                & aa(nat,$o,ord_less_eq(nat,N),M) ))),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,M),N))) ) ).

% exp_div_exp_eq
tff(fact_4755_Comparator__Generator_OAll__less__Suc,axiom,
    ! [Xb: nat,P: fun(nat,$o)] :
      ( ! [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,suc,Xb))
         => aa(nat,$o,P,I4) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I4: nat] :
            ( aa(nat,$o,ord_less(nat,I4),Xb)
           => aa(nat,$o,P,aa(nat,nat,suc,I4)) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_4756_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
       => bij_betw(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(real,complex,real_Vector_of_real(complex),aa(real,real,root(N),real_V7770717601297561774m_norm(complex,C2)))),cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),arg(C2)),aa(nat,real,semiring_1_of_nat(real),N))))),collect(complex,aTP_Lamp_wd(nat,fun(complex,$o),N)),collect(complex,aa(nat,fun(complex,$o),aTP_Lamp_acr(complex,fun(nat,fun(complex,$o)),C2),N))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_4757_and__int_Osimps,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( member(int,K,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
        & member(int,L,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
        aa(int,int,uminus_uminus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
            & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
        aa(int,int,
          aa(int,fun(int,int),plus_plus(int),
            aa($o,int,zero_neq_one_of_bool(int),
              ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
              & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
          aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ).

% and_int.simps
tff(fact_4758_and__int_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Xaa) = Y )
     => ( Y = $ite(
            ( member(int,Xb,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & member(int,Xaa,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xb)
                & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xaa) ))),
            aa(int,int,
              aa(int,fun(int,int),plus_plus(int),
                aa($o,int,zero_neq_one_of_bool(int),
                  ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xb)
                  & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xaa) ))),
              aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xaa),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.elims
tff(fact_4759_and_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),A3) = A3 ) ).

% and.idem
tff(fact_4760_and_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B3) ) ).

% and.left_idem
tff(fact_4761_and_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B3)),B3) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B3) ) ).

% and.right_idem
tff(fact_4762_bit_Oconj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),zero_zero(A)) = zero_zero(A) ) ).

% bit.conj_zero_right
tff(fact_4763_bit_Oconj__zero__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),Xb) = zero_zero(A) ) ).

% bit.conj_zero_left
tff(fact_4764_zero__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),zero_zero(A)),A3) = zero_zero(A) ) ).

% zero_and_eq
tff(fact_4765_and__zero__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),zero_zero(A)) = zero_zero(A) ) ).

% and_zero_eq
tff(fact_4766_real__root__zero,axiom,
    ! [N: nat] : aa(real,real,root(N),zero_zero(real)) = zero_zero(real) ).

% real_root_zero
tff(fact_4767_bit_Oconj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),aa(A,A,uminus_uminus(A),one_one(A))) = Xb ) ).

% bit.conj_one_right
tff(fact_4768_and_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,uminus_uminus(A),one_one(A))) = A3 ) ).

% and.right_neutral
tff(fact_4769_and_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))),A3) = A3 ) ).

% and.left_neutral
tff(fact_4770_real__root__Suc__0,axiom,
    ! [Xb: real] : aa(real,real,root(aa(nat,nat,suc,zero_zero(nat))),Xb) = Xb ).

% real_root_Suc_0
tff(fact_4771_real__root__eq__iff,axiom,
    ! [N: nat,Xb: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( ( aa(real,real,root(N),Xb) = aa(real,real,root(N),Y) )
      <=> ( Xb = Y ) ) ) ).

% real_root_eq_iff
tff(fact_4772_root__0,axiom,
    ! [Xb: real] : aa(real,real,root(zero_zero(nat)),Xb) = zero_zero(real) ).

% root_0
tff(fact_4773_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L))
    <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
        | aa(int,$o,ord_less_eq(int,zero_zero(int)),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_4774_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,ord_less(int,K),zero_zero(int))
        & aa(int,$o,ord_less(int,L),zero_zero(int)) ) ) ).

% and_negative_int_iff
tff(fact_4775_and__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb))),one_one(A)) = one_one(A) ) ).

% and_numerals(8)
tff(fact_4776_and__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = one_one(A) ) ).

% and_numerals(2)
tff(fact_4777_real__root__eq__0__iff,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( ( aa(real,real,root(N),Xb) = zero_zero(real) )
      <=> ( Xb = zero_zero(real) ) ) ) ).

% real_root_eq_0_iff
tff(fact_4778_real__root__less__iff,axiom,
    ! [N: nat,Xb: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,aa(real,real,root(N),Xb)),aa(real,real,root(N),Y))
      <=> aa(real,$o,ord_less(real,Xb),Y) ) ) ).

% real_root_less_iff
tff(fact_4779_real__root__le__iff,axiom,
    ! [N: nat,Xb: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,root(N),Xb)),aa(real,real,root(N),Y))
      <=> aa(real,$o,ord_less_eq(real,Xb),Y) ) ) ).

% real_root_le_iff
tff(fact_4780_real__root__one,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,real,root(N),one_one(real)) = one_one(real) ) ) ).

% real_root_one
tff(fact_4781_real__root__eq__1__iff,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( ( aa(real,real,root(N),Xb) = one_one(real) )
      <=> ( Xb = one_one(real) ) ) ) ).

% real_root_eq_1_iff
tff(fact_4782_and__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(Xb))),one_one(A)) = zero_zero(A) ) ).

% and_numerals(5)
tff(fact_4783_and__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y))) = zero_zero(A) ) ).

% and_numerals(1)
tff(fact_4784_and__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(Xb))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(3)
tff(fact_4785_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,root(N),Y))
      <=> aa(real,$o,ord_less(real,zero_zero(real)),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_4786_real__root__lt__0__iff,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,aa(real,real,root(N),Xb)),zero_zero(real))
      <=> aa(real,$o,ord_less(real,Xb),zero_zero(real)) ) ) ).

% real_root_lt_0_iff
tff(fact_4787_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,root(N),Y))
      <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_4788_real__root__le__0__iff,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,root(N),Xb)),zero_zero(real))
      <=> aa(real,$o,ord_less_eq(real,Xb),zero_zero(real)) ) ) ).

% real_root_le_0_iff
tff(fact_4789_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,one_one(real)),aa(real,real,root(N),Y))
      <=> aa(real,$o,ord_less(real,one_one(real)),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_4790_real__root__lt__1__iff,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,aa(real,real,root(N),Xb)),one_one(real))
      <=> aa(real,$o,ord_less(real,Xb),one_one(real)) ) ) ).

% real_root_lt_1_iff
tff(fact_4791_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less_eq(real,one_one(real)),aa(real,real,root(N),Y))
      <=> aa(real,$o,ord_less_eq(real,one_one(real)),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_4792_real__root__le__1__iff,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less_eq(real,aa(real,real,root(N),Xb)),one_one(real))
      <=> aa(real,$o,ord_less_eq(real,Xb),one_one(real)) ) ) ).

% real_root_le_1_iff
tff(fact_4793_and__minus__numerals_I6_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),one_one(int)) = one_one(int) ).

% and_minus_numerals(6)
tff(fact_4794_and__minus__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = one_one(int) ).

% and_minus_numerals(2)
tff(fact_4795_and__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(6)
tff(fact_4796_and__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),bit0(Xb))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y))) ) ).

% and_numerals(4)
tff(fact_4797_real__root__pow__pos2,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),Xb)),N) = Xb ) ) ) ).

% real_root_pow_pos2
tff(fact_4798_and__minus__numerals_I1_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))) = zero_zero(int) ).

% and_minus_numerals(1)
tff(fact_4799_and__minus__numerals_I5_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))),one_one(int)) = zero_zero(int) ).

% and_minus_numerals(5)
tff(fact_4800_and__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% and_numerals(7)
tff(fact_4801_of__nat__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_and_eq
tff(fact_4802_of__int__and__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_and_eq
tff(fact_4803_and_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B3),C2)) ) ).

% and.assoc
tff(fact_4804_and_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B3) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B3),A3) ) ).

% and.commute
tff(fact_4805_and_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B3: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B3),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),B3),C2)) ) ).

% and.left_commute
tff(fact_4806_and__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B3) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( A3 = aa(A,A,uminus_uminus(A),one_one(A)) )
            & ( B3 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_4807_real__root__pos__pos__le,axiom,
    ! [Xb: real,N: nat] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,root(N),Xb)) ) ).

% real_root_pos_pos_le
tff(fact_4808_AND__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
     => aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Y)) ) ).

% AND_lower
tff(fact_4809_AND__upper1,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Y)),Xb) ) ).

% AND_upper1
tff(fact_4810_AND__upper2,axiom,
    ! [Y: int,Xb: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Y)),Y) ) ).

% AND_upper2
tff(fact_4811_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => ( aa(int,$o,ord_less_eq(int,Y),Z)
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_4812_AND__upper2_H,axiom,
    ! [Y: int,Z: int,Xb: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => ( aa(int,$o,ord_less_eq(int,Y),Z)
       => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_4813_real__root__less__mono,axiom,
    ! [N: nat,Xb: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,Xb),Y)
       => aa(real,$o,ord_less(real,aa(real,real,root(N),Xb)),aa(real,real,root(N),Y)) ) ) ).

% real_root_less_mono
tff(fact_4814_real__root__le__mono,axiom,
    ! [N: nat,Xb: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less_eq(real,Xb),Y)
       => aa(real,$o,ord_less_eq(real,aa(real,real,root(N),Xb)),aa(real,real,root(N),Y)) ) ) ).

% real_root_le_mono
tff(fact_4815_real__root__power,axiom,
    ! [N: nat,Xb: real,K: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),K)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),Xb)),K) ) ) ).

% real_root_power
tff(fact_4816_real__root__abs,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,real,root(N),abs_abs(real,Xb)) = abs_abs(real,aa(real,real,root(N),Xb)) ) ) ).

% real_root_abs
tff(fact_4817_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,L),zero_zero(int))
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),K) ) ).

% and_less_eq
tff(fact_4818_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => ( aa(int,$o,ord_less(int,Y),Z)
       => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z) ) ) ).

% AND_upper1''
tff(fact_4819_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xb: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => ( aa(int,$o,ord_less(int,Y),Z)
       => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Y)),Z) ) ) ).

% AND_upper2''
tff(fact_4820_even__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B3))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B3) ) ) ) ).

% even_and_iff
tff(fact_4821_real__root__gt__zero,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,root(N),Xb)) ) ) ).

% real_root_gt_zero
tff(fact_4822_real__root__strict__decreasing,axiom,
    ! [N: nat,N2: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,$o,ord_less(nat,N),N2)
       => ( aa(real,$o,ord_less(real,one_one(real)),Xb)
         => aa(real,$o,ord_less(real,aa(real,real,root(N2),Xb)),aa(real,real,root(N),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_4823_sqrt__def,axiom,
    sqrt = root(aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% sqrt_def
tff(fact_4824_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( abs_abs(real,aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),N))) = abs_abs(real,Y) ) ) ).

% root_abs_power
tff(fact_4825_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L))
    <=> ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
        | aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ) ) ).

% even_and_iff_int
tff(fact_4826_and__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),one_one(A)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% and_one_eq
tff(fact_4827_one__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),A3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% one_and_eq
tff(fact_4828_real__root__pos__pos,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,root(N),Xb)) ) ) ).

% real_root_pos_pos
tff(fact_4829_real__root__strict__increasing,axiom,
    ! [N: nat,N2: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,$o,ord_less(nat,N),N2)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( aa(real,$o,ord_less(real,Xb),one_one(real))
           => aa(real,$o,ord_less(real,aa(real,real,root(N),Xb)),aa(real,real,root(N2),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_4830_real__root__decreasing,axiom,
    ! [N: nat,N2: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,$o,ord_less_eq(nat,N),N2)
       => ( aa(real,$o,ord_less_eq(real,one_one(real)),Xb)
         => aa(real,$o,ord_less_eq(real,aa(real,real,root(N2),Xb)),aa(real,real,root(N),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_4831_odd__real__root__pow,axiom,
    ! [N: nat,Xb: real] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),Xb)),N) = Xb ) ) ).

% odd_real_root_pow
tff(fact_4832_odd__real__root__unique,axiom,
    ! [N: nat,Y: real,Xb: real] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
     => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),N) = Xb )
       => ( aa(real,real,root(N),Xb) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_4833_odd__real__root__power__cancel,axiom,
    ! [N: nat,Xb: real] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
     => ( aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N)) = Xb ) ) ).

% odd_real_root_power_cancel
tff(fact_4834_real__root__pow__pos,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),Xb)),N) = Xb ) ) ) ).

% real_root_pow_pos
tff(fact_4835_real__root__power__cancel,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
       => ( aa(real,real,root(N),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N)) = Xb ) ) ) ).

% real_root_power_cancel
tff(fact_4836_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Y)
       => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),N) = Xb )
         => ( aa(real,real,root(N),Xb) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_4837_real__root__increasing,axiom,
    ! [N: nat,N2: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,$o,ord_less_eq(nat,N),N2)
       => ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
         => ( aa(real,$o,ord_less_eq(real,Xb),one_one(real))
           => aa(real,$o,ord_less_eq(real,aa(real,real,root(N),Xb)),aa(real,real,root(N2),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_4838_log__root,axiom,
    ! [N: nat,A3: real,B3: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
       => ( aa(real,real,log(B3),aa(real,real,root(N),A3)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,log(B3),A3)),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ) ).

% log_root
tff(fact_4839_log__base__root,axiom,
    ! [N: nat,B3: real,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
       => ( aa(real,real,log(aa(real,real,root(N),B3)),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(real,real,log(B3),Xb)) ) ) ) ).

% log_base_root
tff(fact_4840_ln__root,axiom,
    ! [N: nat,B3: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
       => ( aa(real,real,ln_ln(real),aa(real,real,root(N),B3)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B3)),aa(nat,real,semiring_1_of_nat(real),N)) ) ) ) ).

% ln_root
tff(fact_4841_and__int__rec,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = aa(int,int,
        aa(int,fun(int,int),plus_plus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
            & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% and_int_rec
tff(fact_4842_root__powr__inverse,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,real,root(N),Xb) = powr(real,Xb,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),N))) ) ) ) ).

% root_powr_inverse
tff(fact_4843_and__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
        ( ( K = zero_zero(int) )
        | ( L = zero_zero(int) ) ),
        zero_zero(int),
        $ite(
          K = aa(int,int,uminus_uminus(int),one_one(int)),
          L,
          $ite(L = aa(int,int,uminus_uminus(int),one_one(int)),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),L),aa(num,int,numeral_numeral(int),bit0(one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ).

% and_int_unfold
tff(fact_4844_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K),L))
     => ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L) = $ite(
            ( member(int,K,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
            & member(int,L,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
            aa(int,int,uminus_uminus(int),
              aa($o,int,zero_neq_one_of_bool(int),
                ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
                & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
            aa(int,int,
              aa(int,fun(int,int),plus_plus(int),
                aa($o,int,zero_neq_one_of_bool(int),
                  ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
                  & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
              aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) ) ) ).

% and_int.psimps
tff(fact_4845_and__int_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: int] :
      ( ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Xaa) = Y )
     => ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xb),Xaa))
       => ~ ( ( Y = $ite(
                  ( member(int,Xb,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                  & member(int,Xaa,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) ),
                  aa(int,int,uminus_uminus(int),
                    aa($o,int,zero_neq_one_of_bool(int),
                      ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xb)
                      & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xaa) ))),
                  aa(int,int,
                    aa(int,fun(int,int),plus_plus(int),
                      aa($o,int,zero_neq_one_of_bool(int),
                        ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xb)
                        & ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xaa) ))),
                    aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xaa),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) )
           => ~ accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xb),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_4846_take__bit__word__Bit1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,M: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),N)),aa(num,word(A),numeral_numeral(word(A)),aa(num,num,bit1,M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),pred_numeral(N)),aa(num,word(A),numeral_numeral(word(A)),M)))) ) ).

% take_bit_word_Bit1_eq
tff(fact_4847_take__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),zero_zero(A)) = zero_zero(A) ) ).

% take_bit_of_0
tff(fact_4848_take__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A,B3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B3)) ) ).

% take_bit_and
tff(fact_4849_take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,zero_zero(nat)),A3) = zero_zero(A) ) ).

% take_bit_0
tff(fact_4850_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),one_one(A)) = one_one(A) ) ).

% take_bit_Suc_1
tff(fact_4851_take__bit__numeral__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),one_one(A)) = one_one(A) ) ).

% take_bit_numeral_1
tff(fact_4852_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),one_one(A)) = zero_zero(A) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_4853_word__no__log__defs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),A3)),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_no_log_defs(3)
tff(fact_4854_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,ord_less(nat,zero_zero(nat)),N)) ) ).

% take_bit_of_1
tff(fact_4855_and__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y))) = zero_zero(nat) ).

% and_nat_numerals(1)
tff(fact_4856_and__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),bit0(Xb))),aa(nat,nat,suc,zero_zero(nat))) = zero_zero(nat) ).

% and_nat_numerals(3)
tff(fact_4857_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3))
        <=> ( ( N = zero_zero(nat) )
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ) ).

% even_take_bit_eq
tff(fact_4858_and__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = one_one(nat) ).

% and_nat_numerals(2)
tff(fact_4859_and__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb))),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% and_nat_numerals(4)
tff(fact_4860_word__bitwise__1__simps_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),A3)),one_one(int))) ) ).

% word_bitwise_1_simps(4)
tff(fact_4861_word__bitwise__1__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_bitwise_1_simps(2)
tff(fact_4862_word__no__log__defs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),A3)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_no_log_defs(4)
tff(fact_4863_word__no__log__defs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_no_log_defs(5)
tff(fact_4864_word__no__log__defs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_no_log_defs(6)
tff(fact_4865_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,zero_zero(nat))),A3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_0
tff(fact_4866_and__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% and_Suc_0_eq
tff(fact_4867_Suc__0__and__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Suc_0_and_eq
tff(fact_4868_take__bit__word__Bit0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,M: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),N)),aa(num,word(A),numeral_numeral(word(A)),bit0(M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),pred_numeral(N)),aa(num,word(A),numeral_numeral(word(A)),M))) ) ).

% take_bit_word_Bit0_eq
tff(fact_4869_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,ord_less(nat,N),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% take_bit_of_exp
tff(fact_4870_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_of_2
tff(fact_4871_word__bitwise__1__simps_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),one_one(word(A))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_bitwise_1_simps(3)
tff(fact_4872_word__bitwise__1__simps_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),one_one(int))) ) ).

% word_bitwise_1_simps(5)
tff(fact_4873_take__bit__word__minus__Bit0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,M: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(M)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),pred_numeral(N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M)))) ) ).

% take_bit_word_minus_Bit0_eq
tff(fact_4874_word__log__esimps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),zero_zero(word(A))) = zero_zero(word(A)) ) ).

% word_log_esimps(1)
tff(fact_4875_word__log__esimps_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),zero_zero(word(A))),Xb) = zero_zero(word(A)) ) ).

% word_log_esimps(7)
tff(fact_4876_take__bit__add,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A3: A,B3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B3))) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) ) ).

% take_bit_add
tff(fact_4877_take__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,K: int] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(int,A,ring_1_of_int(A),K)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ).

% take_bit_of_int
tff(fact_4878_take__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)) ) ).

% take_bit_of_nat
tff(fact_4879_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A,B3: A,M: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),B3) )
         => ( aa(nat,$o,ord_less_eq(nat,M),N)
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),B3) ) ) ) ) ).

% take_bit_tightened
tff(fact_4880_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A,B3: A] :
          ( ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3) = aa(A,A,bit_ri4674362597316999326ke_bit(A,N),B3) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),B3) ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_4881_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat,A3: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)) = aa(A,A,
            $ite(aa(nat,$o,ord_less_eq(nat,N),M),bit_se2584673776208193580ke_bit(A,N),bit_ri4674362597316999326ke_bit(A,M)),
            A3) ) ).

% signed_take_bit_take_bit
tff(fact_4882_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A3: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A3)) = $ite(aa(nat,$o,ord_less_eq(nat,N),M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3))) ) ).

% take_bit_unset_bit_eq
tff(fact_4883_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A3: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A3)) = $ite(aa(nat,$o,ord_less_eq(nat,N),M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3))) ) ).

% take_bit_set_bit_eq
tff(fact_4884_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat,A3: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,N),bit_se8732182000553998342ip_bit(A,M,A3)) = $ite(aa(nat,$o,ord_less_eq(nat,N),M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3),bit_se8732182000553998342ip_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3))) ) ).

% take_bit_flip_bit_eq
tff(fact_4885_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat,A3: A] :
          ( aa(nat,$o,ord_less_eq(nat,M),aa(nat,nat,suc,N))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),A3) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_4886_even__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( aa(word(A),$o,even_word(A),A3)
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A3),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% even_word_iff
tff(fact_4887_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_bit0
tff(fact_4888_take__bit__eq__mod,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% take_bit_eq_mod
tff(fact_4889_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = zero_zero(A) )
        <=> aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),A3) ) ) ).

% take_bit_eq_0_iff
tff(fact_4890_bin__last__bintrunc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [L: nat,N: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_se2584673776208193580ke_bit(A,L),N))
        <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),L)
            & ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ) ) ).

% bin_last_bintrunc
tff(fact_4891_take__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_numeral_bit0
tff(fact_4892_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_4893_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,suc,N))),one_one(A)) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_4894_take__bit__numeral__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),K)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,nat,numeral_numeral(nat),K))),one_one(A)) ) ).

% take_bit_numeral_minus_1_eq
tff(fact_4895_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% take_bit_Suc
tff(fact_4896_and__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = $ite(
        ( ( M = zero_zero(nat) )
        | ( N = zero_zero(nat) ) ),
        zero_zero(nat),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% and_nat_unfold
tff(fact_4897_and__nat__rec,axiom,
    ! [M: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = aa(nat,nat,
        aa(nat,fun(nat,nat),plus_plus(nat),
          aa($o,nat,zero_neq_one_of_bool(nat),
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)
            & ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ))),
        aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% and_nat_rec
tff(fact_4898_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [K2: int,L2: int] :
            ( accp(product_prod(int,int),bit_and_int_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K2),L2))
           => ( ( ~ ( member(int,K2,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                    & member(int,L2,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),K2),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L2),aa(num,int,numeral_numeral(int),bit0(one2)))) )
             => aa(int,$o,aa(int,fun(int,$o),P,K2),L2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_4899_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = A3 )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = $ite(aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3),zero_zero(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A))) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_4900_take__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,pred_numeral(L)),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_numeral_bit1
tff(fact_4901_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = $ite(N = zero_zero(nat),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% take_bit_rec
tff(fact_4902_take__bit__word__minus__Bit1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,M: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),aa(num,num,bit1,M)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),pred_numeral(N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),inc(M)))))) ) ).

% take_bit_word_minus_Bit1_eq
tff(fact_4903_take__bit__numeral__minus__numeral__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,N: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),M)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),N))) = case_option(word(A),num,zero_zero(word(A)),aTP_Lamp_acs(num,fun(num,word(A)),M),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ) ).

% take_bit_numeral_minus_numeral_word
tff(fact_4904_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A0),A1))
     => ( ! [I2: int,J2: int] :
            ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I2),J2))
           => ( ( aa(int,$o,ord_less_eq(int,I2),J2)
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I2),one_one(int))),J2) )
             => aa(int,$o,aa(int,fun(int,$o),P,I2),J2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_4905_take__bit__num__simps_I1_J,axiom,
    ! [M: num] : bit_take_bit_num(zero_zero(nat),M) = none(num) ).

% take_bit_num_simps(1)
tff(fact_4906_take__bit__num__simps_I2_J,axiom,
    ! [N: nat] : bit_take_bit_num(aa(nat,nat,suc,N),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(2)
tff(fact_4907_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(5)
tff(fact_4908_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).

% pred_numeral_inc
tff(fact_4909_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),bit0(M)) = case_option(option(num),num,none(num),aTP_Lamp_act(num,option(num)),bit_take_bit_num(N,M)) ).

% take_bit_num_simps(3)
tff(fact_4910_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,N),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(N,M))) ).

% take_bit_num_simps(4)
tff(fact_4911_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),bit0(M)) = case_option(option(num),num,none(num),aTP_Lamp_act(num,option(num)),bit_take_bit_num(pred_numeral(R2),M)) ).

% take_bit_num_simps(6)
tff(fact_4912_add__neg__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(M))) ) ).

% add_neg_numeral_special(6)
tff(fact_4913_add__neg__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),one_one(A))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).

% add_neg_numeral_special(5)
tff(fact_4914_diff__numeral__special_I6_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num] : aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(num,A,numeral_numeral(A),inc(M)) ) ).

% diff_numeral_special(6)
tff(fact_4915_diff__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [N: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),N)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) ) ).

% diff_numeral_special(5)
tff(fact_4916_take__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,ord_less(nat,zero_zero(nat)),N)) ).

% take_bit_of_Suc_0
tff(fact_4917_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R2),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(pred_numeral(R2),M))) ).

% take_bit_num_simps(7)
tff(fact_4918_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),M)),aa(num,A,numeral_numeral(A),N)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ) ).

% take_bit_numeral_numeral
tff(fact_4919_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = case_option(int,num,zero_zero(int),aTP_Lamp_acu(num,fun(num,int),M),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),N)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_4920_take__bit__mult,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% take_bit_mult
tff(fact_4921_take__bit__minus,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),K)) ).

% take_bit_minus
tff(fact_4922_take__bit__diff,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,minus_minus(int,K),L)) ).

% take_bit_diff
tff(fact_4923_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M: nat] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_4924_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N: nat,Q2: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,bit_se2584673776208193580ke_bit(nat,M),Q2)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_4925_num__induct,axiom,
    ! [P: fun(num,$o),Xb: num] :
      ( aa(num,$o,P,one2)
     => ( ! [X2: num] :
            ( aa(num,$o,P,X2)
           => aa(num,$o,P,inc(X2)) )
       => aa(num,$o,P,Xb) ) ) ).

% num_induct
tff(fact_4926_add__inc,axiom,
    ! [Xb: num,Y: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),inc(Y)) = inc(aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),Y)) ).

% add_inc
tff(fact_4927_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N: nat,K: int] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => aa(int,$o,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_4928_take__bit__nonnegative,axiom,
    ! [N: nat,K: int] : aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ).

% take_bit_nonnegative
tff(fact_4929_take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),K)
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_4930_not__take__bit__negative,axiom,
    ! [N: nat,K: int] : ~ aa(int,$o,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),zero_zero(int)) ).

% not_take_bit_negative
tff(fact_4931_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,ord_less(int,K),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% take_bit_int_greater_self_iff
tff(fact_4932_inc_Osimps_I1_J,axiom,
    inc(one2) = bit0(one2) ).

% inc.simps(1)
tff(fact_4933_inc_Osimps_I3_J,axiom,
    ! [Xb: num] : inc(aa(num,num,bit1,Xb)) = bit0(inc(Xb)) ).

% inc.simps(3)
tff(fact_4934_inc_Osimps_I2_J,axiom,
    ! [Xb: num] : inc(bit0(Xb)) = aa(num,num,bit1,Xb) ).

% inc.simps(2)
tff(fact_4935_add__One,axiom,
    ! [Xb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),one2) = inc(Xb) ).

% add_One
tff(fact_4936_inc__BitM__eq,axiom,
    ! [N: num] : inc(bitM(N)) = bit0(N) ).

% inc_BitM_eq
tff(fact_4937_BitM__inc__eq,axiom,
    ! [N: num] : bitM(inc(N)) = aa(num,num,bit1,N) ).

% BitM_inc_eq
tff(fact_4938_take__bit__num__eq__Some__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: num,Q2: num] :
          ( ( bit_take_bit_num(M,N) = aa(num,option(num),some(num),Q2) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(num,A,numeral_numeral(A),N)) = aa(num,A,numeral_numeral(A),Q2) ) ) ) ).

% take_bit_num_eq_Some_imp
tff(fact_4939_mult__inc,axiom,
    ! [Xb: num,Y: num] : aa(num,num,aa(num,fun(num,num),times_times(num),Xb),inc(Y)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),times_times(num),Xb),Y)),Xb) ).

% mult_inc
tff(fact_4940_take__bit__decr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) != zero_zero(int) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,minus_minus(int,K),one_one(int))) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),one_one(int)) ) ) ).

% take_bit_decr_eq
tff(fact_4941_numeral__inc,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Xb: num] : aa(num,A,numeral_numeral(A),inc(Xb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Xb)),one_one(A)) ) ).

% numeral_inc
tff(fact_4942_take__bit__num__eq__None__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: num] :
          ( ( bit_take_bit_num(M,N) = none(num) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(num,A,numeral_numeral(A),N)) = zero_zero(A) ) ) ) ).

% take_bit_num_eq_None_imp
tff(fact_4943_take__bit__nat__eq__self,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_4944_take__bit__nat__less__exp,axiom,
    ! [N: nat,M: nat] : aa(nat,$o,ord_less(nat,aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% take_bit_nat_less_exp
tff(fact_4945_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M) = M )
    <=> aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_4946_take__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% take_bit_nat_def
tff(fact_4947_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% take_bit_Suc_minus_bit1
tff(fact_4948_take__bit__int__less__exp,axiom,
    ! [N: nat,K: int] : aa(int,$o,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).

% take_bit_int_less_exp
tff(fact_4949_take__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).

% take_bit_int_def
tff(fact_4950_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K))))),aa(num,int,numeral_numeral(int),bit0(one2)))),one_one(int)) ).

% take_bit_numeral_minus_bit1
tff(fact_4951_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),M)),M)
    <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_4952_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% take_bit_Suc_minus_bit0
tff(fact_4953_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,ord_less_eq(int,K),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))
    <=> aa(int,$o,ord_less(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_4954_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),K)
    <=> aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_4955_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = K )
    <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
        & aa(int,$o,ord_less(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_4956_take__bit__int__eq__self,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_4957_take__bit__incr__eq,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) != aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),one_one(int)) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ) ).

% take_bit_incr_eq
tff(fact_4958_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% take_bit_numeral_minus_bit0
tff(fact_4959_take__bit__int__less__eq,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),K)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
       => aa(int,$o,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,minus_minus(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ) ).

% take_bit_int_less_eq
tff(fact_4960_take__bit__int__greater__eq,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,ord_less(int,K),zero_zero(int))
     => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ).

% take_bit_int_greater_eq
tff(fact_4961_signed__take__bit__eq__take__bit__shift,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_4962_take__bit__minus__small__eq,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ( aa(int,$o,ord_less_eq(int,K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),K)) = aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_4963_and__mask__arith_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),N)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),N))) ) ) ) ).

% and_mask_arith'
tff(fact_4964_uint32_Osize__eq,axiom,
    ! [P2: uint32] : aa(uint32,nat,size_size(uint32),P2) = aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2)))))) ).

% uint32.size_eq
tff(fact_4965_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R2: A,A3: A,B3: A,C2: A,D: A] :
          ( ( R2 != zero_zero(A) )
         => ( ( ( A3 = B3 )
              & ( C2 != D ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),R2),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),aa(A,A,aa(A,fun(A,A),times_times(A),R2),D)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_4966_mask__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] :
          ( ( bit_se2239418461657761734s_mask(A,N) = zero_zero(A) )
        <=> ( N = zero_zero(nat) ) ) ) ).

% mask_eq_0_iff
tff(fact_4967_mask__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,zero_zero(nat)) = zero_zero(A) ) ) ).

% mask_0
tff(fact_4968_Word_Omask__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,zero_zero(nat))) = one_one(A) ) ) ).

% Word.mask_Suc_0
tff(fact_4969_take__bit__minus__one__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,uminus_uminus(A),one_one(A))) = bit_se2239418461657761734s_mask(A,N) ) ).

% take_bit_minus_one_eq_mask
tff(fact_4970_of__int__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(int,A,ring_1_of_int(A),bit_se2239418461657761734s_mask(int,N)) = bit_se2239418461657761734s_mask(A,N) ) ).

% of_int_mask_eq
tff(fact_4971_of__nat__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(nat,A,semiring_1_of_nat(A),bit_se2239418461657761734s_mask(nat,N)) = bit_se2239418461657761734s_mask(A,N) ) ).

% of_nat_mask_eq
tff(fact_4972_mask__twice2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,N),M)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),bit_se2239418461657761734s_mask(word(A),M))),bit_se2239418461657761734s_mask(word(A),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),bit_se2239418461657761734s_mask(word(A),N)) ) ) ) ).

% mask_twice2
tff(fact_4973_mask__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,one_one(nat)) = one_one(A) ) ) ).

% mask_1
tff(fact_4974_take__bit__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),bit_se2239418461657761734s_mask(A,N)) ) ).

% take_bit_eq_mask
tff(fact_4975_More__Word_Omask__Suc__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( bit_se2239418461657761734s_mask(word(A),aa(nat,nat,suc,zero_zero(nat))) = one_one(word(A)) ) ) ).

% More_Word.mask_Suc_0
tff(fact_4976_mask__bin,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(word(A),N) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,uminus_uminus(int),one_one(int)))) ) ).

% mask_bin
tff(fact_4977_word__1FF__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(num,word(A),numeral_numeral(word(A)),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))) = bit_se2239418461657761734s_mask(word(A),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))) ) ) ).

% word_1FF_is_mask
tff(fact_4978_word__FF__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(num,word(A),numeral_numeral(word(A)),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))))) = bit_se2239418461657761734s_mask(word(A),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))) ) ) ).

% word_FF_is_mask
tff(fact_4979_and__mask__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: num,N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(num,word(A),numeral_numeral(word(A)),I)),bit_se2239418461657761734s_mask(word(A),N)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(num,int,numeral_numeral(int),I))) ) ).

% and_mask_no
tff(fact_4980_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,N))
        <=> ( N = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_4981_mask__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),bit_se2239418461657761734s_mask(word(A),N)),one_one(word(A))) = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) ) ).

% mask_plus_1
tff(fact_4982_less__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: nat] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),bit_se2239418461657761734s_mask(word(A),N)) = Xb ) ) ) ).

% less_mask_eq
tff(fact_4983_mask__eq__decr__exp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(word(A),N) = aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),one_one(word(A))) ) ).

% mask_eq_decr_exp
tff(fact_4984_add__0__iff,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [B3: A,A3: A] :
          ( ( B3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B3),A3) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% add_0_iff
tff(fact_4985_mask__eq__exp__minus__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,N) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A)) ) ).

% mask_eq_exp_minus_1
tff(fact_4986_mask__Suc__rec,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(word(A),aa(nat,nat,suc,N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),bit_se2239418461657761734s_mask(word(A),N))),one_one(word(A))) ) ).

% mask_Suc_rec
tff(fact_4987_is__aligned__AND__less__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),N: nat,V: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),U),bit_se2239418461657761734s_mask(word(A),N)) = zero_zero(word(A)) )
         => ( aa(word(A),$o,ord_less(word(A),V),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),U),V) = zero_zero(word(A)) ) ) ) ) ).

% is_aligned_AND_less_0
tff(fact_4988_add__mask__fold,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: nat] : aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))),one_one(word(A))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),bit_se2239418461657761734s_mask(word(A),N)) ) ).

% add_mask_fold
tff(fact_4989_mask__eq__iff__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,N),aa(word(A),nat,size_size(word(A)),W))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)) = W )
          <=> aa(word(A),$o,ord_less(word(A),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ) ) ).

% mask_eq_iff_w2p
tff(fact_4990_and__mask__less__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,N),aa(word(A),nat,size_size(word(A)),Xb))
         => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),bit_se2239418461657761734s_mask(word(A),N))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ) ).

% and_mask_less_size
tff(fact_4991_word__and__mask__le__2pm1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] : aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N))),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),one_one(word(A)))) ) ).

% word_and_mask_le_2pm1
tff(fact_4992_word__mod__2p__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),bit_se2239418461657761734s_mask(word(A),N)) ) ) ) ).

% word_mod_2p_is_mask
tff(fact_4993_and__mask__arith,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),N)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),W)),N))) ) ).

% and_mask_arith
tff(fact_4994_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,pred_numeral(N)))) ) ).

% mask_numeral
tff(fact_4995_and__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(7)
tff(fact_4996_and__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(N))) ).

% and_minus_numerals(3)
tff(fact_4997_and__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bit0(N))) ).

% and_minus_numerals(4)
tff(fact_4998_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),bit_se2239418461657761734s_mask(nat,N))
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ).

% mask_nat_positive_iff
tff(fact_4999_and__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bit0(N))) ).

% and_minus_numerals(8)
tff(fact_5000_less__eq__mask,axiom,
    ! [N: nat] : aa(nat,$o,ord_less_eq(nat,N),bit_se2239418461657761734s_mask(nat,N)) ).

% less_eq_mask
tff(fact_5001_mask__nonnegative__int,axiom,
    ! [N: nat] : aa(int,$o,ord_less_eq(int,zero_zero(int)),bit_se2239418461657761734s_mask(int,N)) ).

% mask_nonnegative_int
tff(fact_5002_not__mask__negative__int,axiom,
    ! [N: nat] : ~ aa(int,$o,ord_less(int,bit_se2239418461657761734s_mask(int,N)),zero_zero(int)) ).

% not_mask_negative_int
tff(fact_5003_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one2,one2) = none(num) ).

% and_not_num.simps(1)
tff(fact_5004_less__mask,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),N)
     => aa(nat,$o,ord_less(nat,N),bit_se2239418461657761734s_mask(nat,N)) ) ).

% less_mask
tff(fact_5005_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] : bit_and_not_num(bit0(M),one2) = aa(num,option(num),some(num),bit0(M)) ).

% and_not_num.simps(4)
tff(fact_5006_and__not__num_Osimps_I2_J,axiom,
    ! [N: num] : bit_and_not_num(one2,bit0(N)) = aa(num,option(num),some(num),one2) ).

% and_not_num.simps(2)
tff(fact_5007_and__not__num_Osimps_I3_J,axiom,
    ! [N: num] : bit_and_not_num(one2,aa(num,num,bit1,N)) = none(num) ).

% and_not_num.simps(3)
tff(fact_5008_take__bit__eq__mask__iff,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = bit_se2239418461657761734s_mask(int,N) )
    <=> ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) = zero_zero(int) ) ) ).

% take_bit_eq_mask_iff
tff(fact_5009_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] : bit_and_not_num(aa(num,num,bit1,M),one2) = aa(num,option(num),some(num),bit0(M)) ).

% and_not_num.simps(7)
tff(fact_5010_Suc__mask__eq__exp,axiom,
    ! [N: nat] : aa(nat,nat,suc,bit_se2239418461657761734s_mask(nat,N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).

% Suc_mask_eq_exp
tff(fact_5011_mask__nat__less__exp,axiom,
    ! [N: nat] : aa(nat,$o,ord_less(nat,bit_se2239418461657761734s_mask(nat,N)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% mask_nat_less_exp
tff(fact_5012_mask__half__int,axiom,
    ! [N: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_se2239418461657761734s_mask(int,N)),aa(num,int,numeral_numeral(int),bit0(one2))) = bit_se2239418461657761734s_mask(int,aa(nat,nat,minus_minus(nat,N),one_one(nat))) ).

% mask_half_int
tff(fact_5013_mask__int__def,axiom,
    ! [N: nat] : bit_se2239418461657761734s_mask(int,N) = aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),one_one(int)) ).

% mask_int_def
tff(fact_5014_mask__nat__def,axiom,
    ! [N: nat] : bit_se2239418461657761734s_mask(nat,N) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),one_one(nat)) ).

% mask_nat_def
tff(fact_5015_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N: num] : bit_and_not_num(aa(num,num,bit1,M),bit0(N)) = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_acv(num,option(num)),bit_and_not_num(M,N)) ).

% and_not_num.simps(8)
tff(fact_5016_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = bit_se2239418461657761734s_mask(int,N) )
    <=> aa(int,$o,dvd_dvd(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_5017_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,aa(num,num,bit1,M)) = case_nat(option(num),none(num),aTP_Lamp_acw(num,fun(nat,option(num)),M),N) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_5018_slice__nth,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A),I: nat] :
      ( aa(nat,$o,ord_less(nat,From),To)
     => ( aa(nat,$o,ord_less_eq(nat,To),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(nat,$o,ord_less(nat,I),aa(nat,nat,minus_minus(nat,To),From))
         => ( aa(nat,A,nth(A,slice(A,From,To,Xs)),I) = aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),From),I)) ) ) ) ) ).

% slice_nth
tff(fact_5019_word__2p__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,N),aa(word(A),nat,size_size(word(A)),W))
         => ( aa(word(A),$o,ord_less(word(A),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))
          <=> aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ) ) ) ).

% word_2p_lem
tff(fact_5020_unsigned__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semiring_1(A)
        & type_len(B) )
     => ( aa(word(B),A,semiring_1_unsigned(B,A),zero_zero(word(B))) = zero_zero(A) ) ) ).

% unsigned_0
tff(fact_5021_unsigned__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semiring_1(A)
        & type_len(B) )
     => ( aa(word(B),A,semiring_1_unsigned(B,A),one_one(word(B))) = one_one(A) ) ) ).

% unsigned_1
tff(fact_5022_uint__nonnegative,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_nonnegative
tff(fact_5023_uint__le__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),Xb)),zero_zero(int))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),Xb) = zero_zero(int) ) ) ) ).

% uint_le_0_iff
tff(fact_5024_uint__ge__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),Xb)) ) ).

% uint_ge_0
tff(fact_5025_uint__lt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : ~ aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),Xb)),zero_zero(int)) ) ).

% uint_lt_0
tff(fact_5026_slice__complete,axiom,
    ! [A: $tType,Xs: list(A)] : slice(A,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs),Xs) = Xs ).

% slice_complete
tff(fact_5027_case__nat__numeral,axiom,
    ! [A: $tType,A3: A,Fa: fun(nat,A),V: num] : case_nat(A,A3,Fa,aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,Fa,pred_numeral(V)) ).

% case_nat_numeral
tff(fact_5028_word__le__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B3))) ) ) ).

% word_le_no
tff(fact_5029_word__less__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B3))) ) ) ).

% word_less_no
tff(fact_5030_slice__len,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A)] :
      ( aa(nat,$o,ord_less_eq(nat,From),To)
     => ( aa(nat,$o,ord_less_eq(nat,To),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),nat,size_size(list(A)),slice(A,From,To,Xs)) = aa(nat,nat,minus_minus(nat,To),From) ) ) ) ).

% slice_len
tff(fact_5031_case__nat__add__eq__if,axiom,
    ! [A: $tType,A3: A,Fa: fun(nat,A),V: num,N: nat] : case_nat(A,A3,Fa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),N)) = aa(nat,A,Fa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),N)) ).

% case_nat_add_eq_if
tff(fact_5032_word__div__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B3)))) ) ).

% word_div_no
tff(fact_5033_nat_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: B,F2: fun(nat,B),Nat: nat] : aa(B,A,H,case_nat(B,F1,F2,Nat)) = case_nat(A,aa(B,A,H,F1),aa(fun(nat,B),fun(nat,A),aTP_Lamp_acx(fun(B,A),fun(fun(nat,B),fun(nat,A)),H),F2),Nat) ).

% nat.case_distrib
tff(fact_5034_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F2: fun(nat,A),X22: nat] : case_nat(A,F1,F2,aa(nat,nat,suc,X22)) = aa(nat,A,F2,X22) ).

% old.nat.simps(5)
tff(fact_5035_old_Onat_Osimps_I4_J,axiom,
    ! [A: $tType,F1: A,F2: fun(nat,A)] : case_nat(A,F1,F2,zero_zero(nat)) = F1 ).

% old.nat.simps(4)
tff(fact_5036_uint__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( aa(word(A),int,semiring_1_unsigned(A,int),Xb) = zero_zero(int) )
        <=> ( Xb = zero_zero(word(A)) ) ) ) ).

% uint_0_iff
tff(fact_5037_uint__0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),zero_zero(word(A))) = zero_zero(int) ) ) ).

% uint_0_eq
tff(fact_5038_uint__1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),one_one(word(A))) = one_one(int) ) ) ).

% uint_1_eq
tff(fact_5039_word__le__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),A3),B3)
        <=> aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3)) ) ) ).

% word_le_def
tff(fact_5040_word__less__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A3),B3)
        <=> aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3)) ) ) ).

% word_less_def
tff(fact_5041_unsigned__greater__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & unique1627219031080169319umeral(A) )
     => ! [W: word(B)] : aa(A,$o,ord_less_eq(A,zero_zero(A)),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_greater_eq
tff(fact_5042_uint__div__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),V),W)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),V)),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_div_distrib
tff(fact_5043_uint__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xb),Y)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ).

% uint_div
tff(fact_5044_word__less__eq__iff__unsigned,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & linordered_semidom(B) )
     => ! [A3: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),A3),B3)
        <=> aa(B,$o,ord_less_eq(B,aa(word(A),B,semiring_1_unsigned(A,B),A3)),aa(word(A),B,semiring_1_unsigned(A,B),B3)) ) ) ).

% word_less_eq_iff_unsigned
tff(fact_5045_unsigned__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_char_0(A) )
     => ! [W: word(B)] :
          ( ( aa(word(B),A,semiring_1_unsigned(B,A),W) = zero_zero(A) )
        <=> ( W = zero_zero(word(B)) ) ) ) ).

% unsigned_eq_0_iff
tff(fact_5046_word__less__iff__unsigned,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & linordered_semidom(B) )
     => ! [A3: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A3),B3)
        <=> aa(B,$o,ord_less(B,aa(word(A),B,semiring_1_unsigned(A,B),A3)),aa(word(A),B,semiring_1_unsigned(A,B),B3)) ) ) ).

% word_less_iff_unsigned
tff(fact_5047_uint__add__ge0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xaa: word(A),Xb: word(B)] : aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xaa)),aa(word(B),int,semiring_1_unsigned(B,int),Xb))) ) ).

% uint_add_ge0
tff(fact_5048_uint__mult__ge0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xaa: word(A),Xb: word(B)] : aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),Xaa)),aa(word(B),int,semiring_1_unsigned(B,int),Xb))) ) ).

% uint_mult_ge0
tff(fact_5049_uint__add__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] : aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(A),int,semiring_1_unsigned(A,int),Y))) ) ).

% uint_add_le
tff(fact_5050_no__ulen__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),minus_minus(word(A),Xb),Y)),Xb)
        <=> aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),Xb)) ) ) ).

% no_ulen_sub
tff(fact_5051_uint__sub__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A)] :
          ( aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),Xb))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),Xb),Y)) = aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_sub_lem
tff(fact_5052_uint__sub__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] : aa(int,$o,ord_less_eq(int,aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),Xb),Y))) ) ).

% uint_sub_ge
tff(fact_5053_word__arith__power__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),N: nat] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),A3),N) = aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),N)) ) ).

% word_arith_power_alt
tff(fact_5054_word__div__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A3),B3) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))) ) ).

% word_div_def
tff(fact_5055_udvd__incr__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Up: int,Uq: int,Ua: int,N: int,K5: word(A),N5: int] :
          ( aa(int,$o,ord_less(int,Up),Uq)
         => ( ( Up = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(word(A),int,semiring_1_unsigned(A,int),K5))) )
           => ( ( Uq = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N5),aa(word(A),int,semiring_1_unsigned(A,int),K5))) )
             => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Up),aa(word(A),int,semiring_1_unsigned(A,int),K5))),Uq) ) ) ) ) ).

% udvd_incr_lem
tff(fact_5056_udvd__incr__lem0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Up: int,Uq: int,N: int,K5: word(A),N5: int] :
          ( aa(int,$o,ord_less(int,Up),Uq)
         => ( ( Up = aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(word(A),int,semiring_1_unsigned(A,int),K5)) )
           => ( ( Uq = aa(int,int,aa(int,fun(int,int),times_times(int),N5),aa(word(A),int,semiring_1_unsigned(A,int),K5)) )
             => aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Up),aa(word(A),int,semiring_1_unsigned(A,int),K5))),Uq) ) ) ) ) ).

% udvd_incr_lem0
tff(fact_5057_udvd__incr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q2: word(A),N: int,K5: word(A),N5: int] :
          ( aa(word(A),$o,ord_less(word(A),P2),Q2)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(word(A),int,semiring_1_unsigned(A,int),K5)) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),times_times(int),N5),aa(word(A),int,semiring_1_unsigned(A,int),K5)) )
             => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),K5)),Q2) ) ) ) ) ).

% udvd_incr0
tff(fact_5058_udvd__decr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q2: word(A),N: int,K5: word(A),N5: int] :
          ( aa(word(A),$o,ord_less(word(A),P2),Q2)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(word(A),int,semiring_1_unsigned(A,int),K5)) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),times_times(int),N5),aa(word(A),int,semiring_1_unsigned(A,int),K5)) )
             => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),times_times(int),N5),aa(word(A),int,semiring_1_unsigned(A,int),K5)) )
               => aa(word(A),$o,ord_less_eq(word(A),P2),aa(word(A),word(A),minus_minus(word(A),Q2),K5)) ) ) ) ) ) ).

% udvd_decr0
tff(fact_5059_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,Xb: A,Fa: fun(nat,A),N: nat] :
      case_nat(A,Xb,Fa,N) = $ite(N = zero_zero(nat),Xb,aa(nat,A,Fa,aa(nat,nat,minus_minus(nat,N),one_one(nat)))) ).

% Nitpick.case_nat_unfold
tff(fact_5060_udvd__incr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q2: word(A),Ua: int,N: int,K5: word(A),N5: int] :
          ( aa(word(A),$o,ord_less(word(A),P2),Q2)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(word(A),int,semiring_1_unsigned(A,int),K5))) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N5),aa(word(A),int,semiring_1_unsigned(A,int),K5))) )
             => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),K5)),Q2) ) ) ) ) ).

% udvd_incr'
tff(fact_5061_udvd__decr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q2: word(A),Ua: int,N: int,K5: word(A),N5: int] :
          ( aa(word(A),$o,ord_less(word(A),P2),Q2)
         => ( ( aa(word(A),int,semiring_1_unsigned(A,int),P2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N),aa(word(A),int,semiring_1_unsigned(A,int),K5))) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N5),aa(word(A),int,semiring_1_unsigned(A,int),K5))) )
             => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N5),aa(word(A),int,semiring_1_unsigned(A,int),K5))) )
               => aa(word(A),$o,ord_less_eq(word(A),P2),aa(word(A),word(A),minus_minus(word(A),Q2),K5)) ) ) ) ) ) ).

% udvd_decr'
tff(fact_5062_and__mask__lt__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] : aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ) ).

% and_mask_lt_2p
tff(fact_5063_mask__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)) = W )
        <=> aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ) ) ).

% mask_eq_iff
tff(fact_5064_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] : bit_take_bit_num(N,one2) = case_nat(option(num),none(num),aTP_Lamp_acy(nat,option(num)),N) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_5065_uint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),W))
          & aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),W))) ) ) ).

% uint_range_size
tff(fact_5066_uint__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))
         => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N) ) ) ) ).

% uint_2p
tff(fact_5067_and__mask__mod__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ).

% and_mask_mod_2p
tff(fact_5068_and__mask__dvd,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( aa(int,$o,dvd_dvd(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),aa(word(A),int,semiring_1_unsigned(A,int),W))
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)) = zero_zero(word(A)) ) ) ) ).

% and_mask_dvd
tff(fact_5069_no__plus__overflow__uint__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y))
        <=> aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xb))) ) ) ).

% no_plus_overflow_uint_size
tff(fact_5070_uint__plus__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y)) = $ite(aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xb))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xb)))) ) ).

% uint_plus_if_size
tff(fact_5071_uint__sub__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),Xb),Y)) = $ite(aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xb)))) ) ).

% uint_sub_if_size
tff(fact_5072_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,bit0(M)) = case_nat(option(num),none(num),aTP_Lamp_acz(num,fun(nat,option(num)),M),N) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_5073_Bit__Operations_Otake__bit__num__code,axiom,
    ! [N: nat,M: num] : bit_take_bit_num(N,M) = aa(product_prod(nat,num),option(num),product_case_prod(nat,num,option(num),aTP_Lamp_add(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),aa(nat,fun(num,product_prod(nat,num)),product_Pair(nat,num),N),M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_5074_powr__int,axiom,
    ! [Xb: real,I: int] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(int,real,ring_1_of_int(real),I)) = $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),I),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),nat2(I)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),nat2(aa(int,int,uminus_uminus(int),I))))) ) ) ).

% powr_int
tff(fact_5075_arctan__def,axiom,
    ! [Y: real] : aa(real,real,arctan,Y) = the(real,aTP_Lamp_ade(real,fun(real,$o),Y)) ).

% arctan_def
tff(fact_5076_nat__numeral,axiom,
    ! [K: num] : nat2(aa(num,int,numeral_numeral(int),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_numeral
tff(fact_5077_nat__1,axiom,
    nat2(one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_5078_nat__0__iff,axiom,
    ! [I: int] :
      ( ( nat2(I) = zero_zero(nat) )
    <=> aa(int,$o,ord_less_eq(int,I),zero_zero(int)) ) ).

% nat_0_iff
tff(fact_5079_nat__le__0,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less_eq(int,Z),zero_zero(int))
     => ( nat2(Z) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_5080_nat__neg__numeral,axiom,
    ! [K: num] : nat2(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) = zero_zero(nat) ).

% nat_neg_numeral
tff(fact_5081_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( aa(nat,$o,ord_less(nat,nat2(W)),nat2(Z))
    <=> ( aa(int,$o,ord_less(int,zero_zero(int)),Z)
        & aa(int,$o,ord_less(int,W),Z) ) ) ).

% zless_nat_conj
tff(fact_5082_of__nat__nat__take__bit__eq,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: nat,K: int] : aa(nat,A,semiring_1_of_nat(A),nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,N),K))) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ).

% of_nat_nat_take_bit_eq
tff(fact_5083_nat__zminus__int,axiom,
    ! [N: nat] : nat2(aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N))) = zero_zero(nat) ).

% nat_zminus_int
tff(fact_5084_int__nat__eq,axiom,
    ! [Z: int] :
      aa(nat,int,semiring_1_of_nat(int),nat2(Z)) = $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),Z),Z,zero_zero(int)) ).

% int_nat_eq
tff(fact_5085_Suc__unat__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( Xb != zero_zero(word(A)) )
         => ( aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),Xb),one_one(word(A))))) = aa(word(A),nat,semiring_1_unsigned(A,nat),Xb) ) ) ) ).

% Suc_unat_minus_one
tff(fact_5086_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),nat2(Z))
    <=> aa(int,$o,ord_less(int,zero_zero(int)),Z) ) ).

% zero_less_nat_eq
tff(fact_5087_diff__nat__numeral,axiom,
    ! [V: num,V4: num] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V)),aa(num,nat,numeral_numeral(nat),V4)) = nat2(aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),V)),aa(num,int,numeral_numeral(int),V4))) ).

% diff_nat_numeral
tff(fact_5088_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
         => ( aa(nat,A,semiring_1_of_nat(A),nat2(Z)) = aa(int,A,ring_1_of_int(A),Z) ) ) ) ).

% of_nat_nat
tff(fact_5089_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,N: nat] :
      ( ( nat2(Y) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),N) )
    <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),N) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_5090_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xb: num,N: nat,Y: int] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),N) = nat2(Y) )
    <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),N) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_5091_nat__ceiling__le__eq,axiom,
    ! [Xb: real,A3: nat] :
      ( aa(nat,$o,ord_less_eq(nat,nat2(archimedean_ceiling(real,Xb))),A3)
    <=> aa(real,$o,ord_less_eq(real,Xb),aa(nat,real,semiring_1_of_nat(real),A3)) ) ).

% nat_ceiling_le_eq
tff(fact_5092_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),nat2(Z))
    <=> aa(int,$o,ord_less(int,one_one(int)),Z) ) ).

% one_less_nat_eq
tff(fact_5093_nat__numeral__diff__1,axiom,
    ! [V: num] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V)),one_one(nat)) = nat2(aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),V)),one_one(int))) ).

% nat_numeral_diff_1
tff(fact_5094_nat__less__numeral__power__cancel__iff,axiom,
    ! [A3: int,Xb: num,N: nat] :
      ( aa(nat,$o,ord_less(nat,nat2(A3)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),N))
    <=> aa(int,$o,ord_less(int,A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),N)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_5095_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xb: num,N: nat,A3: int] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),N)),nat2(A3))
    <=> aa(int,$o,ord_less(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),N)),A3) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_5096_nat__le__numeral__power__cancel__iff,axiom,
    ! [A3: int,Xb: num,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,nat2(A3)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),N))
    <=> aa(int,$o,ord_less_eq(int,A3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),N)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_5097_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xb: num,N: nat,A3: int] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),N)),nat2(A3))
    <=> aa(int,$o,ord_less_eq(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),N)),A3) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_5098_ucast__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),zero_zero(word(B))) = zero_zero(word(A)) ) ) ).

% ucast_0
tff(fact_5099_ucast__0__I,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xb: word(A)] :
          ( ( Xb = zero_zero(word(A)) )
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb) = zero_zero(word(B)) ) ) ) ).

% ucast_0_I
tff(fact_5100_num_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: B,F2: fun(num,B),F32: fun(num,B),Num: num] : aa(B,A,H,case_num(B,F1,F2,F32,Num)) = case_num(A,aa(B,A,H,F1),aa(fun(num,B),fun(num,A),aTP_Lamp_adf(fun(B,A),fun(fun(num,B),fun(num,A)),H),F2),aa(fun(num,B),fun(num,A),aTP_Lamp_adf(fun(B,A),fun(fun(num,B),fun(num,A)),H),F32),Num) ).

% num.case_distrib
tff(fact_5101_nat__numeral__as__int,axiom,
    ! [X3: num] : aa(num,nat,numeral_numeral(nat),X3) = nat2(aa(num,int,numeral_numeral(int),X3)) ).

% nat_numeral_as_int
tff(fact_5102_nat__zero__as__int,axiom,
    zero_zero(nat) = nat2(zero_zero(int)) ).

% nat_zero_as_int
tff(fact_5103_nat__mono,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,Xb),Y)
     => aa(nat,$o,ord_less_eq(nat,nat2(Xb)),nat2(Y)) ) ).

% nat_mono
tff(fact_5104_nat__one__as__int,axiom,
    one_one(nat) = nat2(one_one(int)) ).

% nat_one_as_int
tff(fact_5105_ex__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ? [X_12: nat] : aa(nat,$o,P,X_12)
    <=> ? [X: int] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
          & aa(nat,$o,P,nat2(X)) ) ) ).

% ex_nat
tff(fact_5106_all__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ! [X_12: nat] : aa(nat,$o,P,X_12)
    <=> ! [X: int] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),X)
         => aa(nat,$o,P,nat2(X)) ) ) ).

% all_nat
tff(fact_5107_eq__nat__nat__iff,axiom,
    ! [Z: int,Z5: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z5)
       => ( ( nat2(Z) = nat2(Z5) )
        <=> ( Z = Z5 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_5108_unat__eq__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),Xb) = zero_zero(nat) )
        <=> ( Xb = zero_zero(word(A)) ) ) ) ).

% unat_eq_zero
tff(fact_5109_unat__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),zero_zero(word(A))) = zero_zero(nat) ) ) ).

% unat_0
tff(fact_5110_word__less__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A3),B3)
        <=> aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3)) ) ) ).

% word_less_nat_alt
tff(fact_5111_unat__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A3),B3)
         => aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3)) ) ) ).

% unat_mono
tff(fact_5112_word__le__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),A3),B3)
        <=> aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3)) ) ) ).

% word_le_nat_alt
tff(fact_5113_unat__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),one_one(word(A))) = one_one(nat) ) ) ).

% unat_1
tff(fact_5114_word__unat__and__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: nat,Y: word(A)] :
          ( ( aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),N)
            | aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),N) )
         => aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),Y))),N) ) ) ).

% word_unat_and_lt
tff(fact_5115_le__unat__uoi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: nat,Z: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,Y),aa(word(A),nat,semiring_1_unsigned(A,nat),Z))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)) = Y ) ) ) ).

% le_unat_uoi
tff(fact_5116_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> case_nat($o,$false,aTP_Lamp_adg(nat,$o),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_5117_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> case_nat($o,$true,aTP_Lamp_adh(nat,$o),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_5118_uno__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Z: word(A),N: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)),N))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)),N) ) ).

% uno_simps(2)
tff(fact_5119_max__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),ord_max(word(A),A3),B3)),C2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),ord_max(word(A),A3),B3))),aa(word(A),nat,semiring_1_unsigned(A,nat),C2)) ) ).

% max_lt
tff(fact_5120_unat__div__distrib,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),V),W)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),V)),aa(word(A),nat,semiring_1_unsigned(A,nat),W)) ) ).

% unat_div_distrib
tff(fact_5121_unat__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xb),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ).

% unat_div
tff(fact_5122_unat__of__nat__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A),C2: nat] :
          ( aa(word(A),$o,ord_less(word(A),A3),B3)
         => ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),B3) = C2 )
           => aa(word(A),$o,ord_less(word(A),A3),aa(nat,word(A),semiring_1_of_nat(word(A)),C2)) ) ) ) ).

% unat_of_nat_less
tff(fact_5123_verit__eq__simplify_I17_J,axiom,
    ! [A: $tType,F1: A,F2: fun(num,A),F32: fun(num,A),X22: num] : case_num(A,F1,F2,F32,bit0(X22)) = aa(num,A,F2,X22) ).

% verit_eq_simplify(17)
tff(fact_5124_verit__eq__simplify_I16_J,axiom,
    ! [A: $tType,F1: A,F2: fun(num,A),F32: fun(num,A)] : case_num(A,F1,F2,F32,one2) = F1 ).

% verit_eq_simplify(16)
tff(fact_5125_unset__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se2638667681897837118et_bit(nat),M),N) = nat2(aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),M),aa(nat,int,semiring_1_of_nat(int),N))) ).

% unset_bit_nat_def
tff(fact_5126_nat__mask__eq,axiom,
    ! [N: nat] : nat2(bit_se2239418461657761734s_mask(int,N)) = bit_se2239418461657761734s_mask(nat,N) ).

% nat_mask_eq
tff(fact_5127_ln__real__def,axiom,
    ! [Xb: real] : aa(real,real,ln_ln(real),Xb) = the(real,aTP_Lamp_adi(real,fun(real,$o),Xb)) ).

% ln_real_def
tff(fact_5128_suminf__def,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Fa: fun(nat,A)] : suminf(A,Fa) = the(A,sums(A,Fa)) ) ).

% suminf_def
tff(fact_5129_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),Z)
     => ( aa(nat,$o,ord_less(nat,nat2(W)),nat2(Z))
      <=> aa(int,$o,ord_less(int,W),Z) ) ) ).

% nat_mono_iff
tff(fact_5130_unat__eq__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),Xb) = aa(nat,nat,suc,zero_zero(nat)) )
        <=> ( Xb = one_one(word(A)) ) ) ) ).

% unat_eq_1
tff(fact_5131_unat__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb))
        <=> ( Xb != zero_zero(word(A)) ) ) ) ).

% unat_gt_0
tff(fact_5132_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( aa(nat,$o,ord_less(nat,M),nat2(Z))
    <=> aa(int,$o,ord_less(int,aa(nat,int,semiring_1_of_nat(int),M)),Z) ) ).

% zless_nat_eq_int_zless
tff(fact_5133_nat__le__iff,axiom,
    ! [Xb: int,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,nat2(Xb)),N)
    <=> aa(int,$o,ord_less_eq(int,Xb),aa(nat,int,semiring_1_of_nat(int),N)) ) ).

% nat_le_iff
tff(fact_5134_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] : aa(A,$o,ord_less_eq(A,R2),aa(nat,A,semiring_1_of_nat(A),nat2(archimedean_ceiling(A,R2)))) ) ).

% of_nat_ceiling
tff(fact_5135_nat__0__le,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( aa(nat,int,semiring_1_of_nat(int),nat2(Z)) = Z ) ) ).

% nat_0_le
tff(fact_5136_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = Z )
    <=> ( ( M = nat2(Z) )
        & aa(int,$o,ord_less_eq(int,zero_zero(int)),Z) ) ) ).

% int_eq_iff
tff(fact_5137_nat__uint__less__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: nat,Xb: word(A)] :
          ( ( nat2(aa(word(A),int,semiring_1_unsigned(A,int),Y)) = Z )
         => ( aa(word(A),$o,ord_less(word(A),Xb),Y)
           => aa(nat,$o,ord_less(nat,nat2(aa(word(A),int,semiring_1_unsigned(A,int),Xb))),Z) ) ) ) ).

% nat_uint_less_helper
tff(fact_5138_un__ui__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [A3: word(A),B3: word(B)] :
          ( aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(B),nat,semiring_1_unsigned(B,nat),B3))
        <=> aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(B),int,semiring_1_unsigned(B,int),B3)) ) ) ).

% un_ui_le
tff(fact_5139_word__of__nat__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,N),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb))
         => aa(word(A),$o,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),N)),Xb) ) ) ).

% word_of_nat_less
tff(fact_5140_unat__less__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: nat] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(nat,word(A),semiring_1_of_nat(word(A)),N))
         => aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),N) ) ) ).

% unat_less_helper
tff(fact_5141_word__of__nat__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,N),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb))
         => aa(word(A),$o,ord_less_eq(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),N)),Xb) ) ) ).

% word_of_nat_le
tff(fact_5142_word__unat__less__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),A3),aa(nat,word(A),semiring_1_of_nat(word(A)),B3))
         => aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),B3) ) ) ).

% word_unat_less_le
tff(fact_5143_word__arith__nat__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A3),B3) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))) ) ).

% word_arith_nat_div
tff(fact_5144_nat__plus__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),X3),Xa) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_plus_as_int
tff(fact_5145_and__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),N) = nat2(aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% and_nat_def
tff(fact_5146_nat__times__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),X3),Xa) = nat2(aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_times_as_int
tff(fact_5147_nat__minus__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa(nat,nat,minus_minus(nat,X3),Xa) = nat2(aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_minus_as_int
tff(fact_5148_nat__div__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),X3),Xa) = nat2(aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_div_as_int
tff(fact_5149_nat__mod__as__int,axiom,
    ! [X3: nat,Xa: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),X3),Xa) = nat2(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,semiring_1_of_nat(int),X3)),aa(nat,int,semiring_1_of_nat(int),Xa))) ).

% nat_mod_as_int
tff(fact_5150_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,M)),N)
    <=> case_nat($o,$false,ord_less_eq(nat,M),N) ) ).

% less_eq_nat.simps(2)
tff(fact_5151_ln__neg__is__const,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,Xb),zero_zero(real))
     => ( aa(real,real,ln_ln(real),Xb) = the(real,aTP_Lamp_adj(real,$o)) ) ) ).

% ln_neg_is_const
tff(fact_5152_max__Suc2,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,ord_max(nat,M),aa(nat,nat,suc,N)) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_adk(nat,fun(nat,nat),N),M) ).

% max_Suc2
tff(fact_5153_max__Suc1,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,ord_max(nat,aa(nat,nat,suc,N)),M) = case_nat(nat,aa(nat,nat,suc,N),aTP_Lamp_adl(nat,fun(nat,nat),N),M) ).

% max_Suc1
tff(fact_5154_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R2: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),R2)
         => aa(A,$o,ord_less_eq(A,aa(nat,A,semiring_1_of_nat(A),nat2(archim6421214686448440834_floor(A,R2)))),R2) ) ) ).

% of_nat_floor
tff(fact_5155_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),W)
     => ( aa(nat,$o,ord_less(nat,nat2(W)),nat2(Z))
      <=> aa(int,$o,ord_less(int,W),Z) ) ) ).

% nat_less_eq_zless
tff(fact_5156_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( nat2(W) = M )
    <=> $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),W),W = aa(nat,int,semiring_1_of_nat(int),M),M = zero_zero(nat)) ) ).

% nat_eq_iff
tff(fact_5157_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = nat2(W) )
    <=> $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),W),W = aa(nat,int,semiring_1_of_nat(int),M),M = zero_zero(nat)) ) ).

% nat_eq_iff2
tff(fact_5158_split__nat,axiom,
    ! [P: fun(nat,$o),I: int] :
      ( aa(nat,$o,P,nat2(I))
    <=> ( ! [N8: nat] :
            ( ( I = aa(nat,int,semiring_1_of_nat(int),N8) )
           => aa(nat,$o,P,N8) )
        & ( aa(int,$o,ord_less(int,I),zero_zero(int))
         => aa(nat,$o,P,zero_zero(nat)) ) ) ) ).

% split_nat
tff(fact_5159_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa(int,$o,ord_less(int,zero_zero(int)),W)
        | aa(int,$o,ord_less_eq(int,zero_zero(int)),Z) )
     => ( aa(nat,$o,ord_less_eq(nat,nat2(W)),nat2(Z))
      <=> aa(int,$o,ord_less_eq(int,W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_5160_nat__add__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z5)
       => ( nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat2(Z)),nat2(Z5)) ) ) ) ).

% nat_add_distrib
tff(fact_5161_le__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( aa(nat,$o,ord_less_eq(nat,N),nat2(K))
      <=> aa(int,$o,ord_less_eq(int,aa(nat,int,semiring_1_of_nat(int),N)),K) ) ) ).

% le_nat_iff
tff(fact_5162_Suc__as__int,axiom,
    ! [X3: nat] : aa(nat,nat,suc,X3) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),X3)),one_one(int))) ).

% Suc_as_int
tff(fact_5163_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B3: A] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(archim6421214686448440834_floor(A,A3))),nat2(archim6421214686448440834_floor(A,B3)))),nat2(archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)))) ) ).

% le_mult_nat_floor
tff(fact_5164_nat__mult__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( nat2(aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(Z)),nat2(Z5)) ) ) ).

% nat_mult_distrib
tff(fact_5165_unat__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),Xb)
        <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)) ) ) ).

% unat_1_0
tff(fact_5166_nat__diff__distrib,axiom,
    ! [Z5: int,Z: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z5)
     => ( aa(int,$o,ord_less_eq(int,Z5),Z)
       => ( nat2(aa(int,int,minus_minus(int,Z),Z5)) = aa(nat,nat,minus_minus(nat,nat2(Z)),nat2(Z5)) ) ) ) ).

% nat_diff_distrib
tff(fact_5167_nat__diff__distrib_H,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
       => ( nat2(aa(int,int,minus_minus(int,Xb),Y)) = aa(nat,nat,minus_minus(nat,nat2(Xb)),nat2(Y)) ) ) ) ).

% nat_diff_distrib'
tff(fact_5168_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa(nat,$o,ord_less_eq(nat,nat2(abs_abs(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat2(abs_abs(int,K))),nat2(abs_abs(int,L)))) ).

% nat_abs_triangle_ineq
tff(fact_5169_unat__max__word__pos,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))) ) ).

% unat_max_word_pos
tff(fact_5170_nat__div__distrib_H,axiom,
    ! [Y: int,Xb: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
     => ( nat2(aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(Xb)),nat2(Y)) ) ) ).

% nat_div_distrib'
tff(fact_5171_nat__div__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
     => ( nat2(aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(Xb)),nat2(Y)) ) ) ).

% nat_div_distrib
tff(fact_5172_nat__power__eq,axiom,
    ! [Z: int,N: nat] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( nat2(aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),nat2(Z)),N) ) ) ).

% nat_power_eq
tff(fact_5173_nat__mod__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
       => ( nat2(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xb),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),nat2(Xb)),nat2(Y)) ) ) ) ).

% nat_mod_distrib
tff(fact_5174_unatSuc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N) != zero_zero(word(A)) )
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N)) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),N)) ) ) ) ).

% unatSuc
tff(fact_5175_unatSuc2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N),one_one(word(A))) != zero_zero(word(A)) )
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N),one_one(word(A)))) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),N)) ) ) ) ).

% unatSuc2
tff(fact_5176_uno__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Z: word(A),M: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(word(A),nat,semiring_1_unsigned(A,nat),Z))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)) ) ) ) ).

% uno_simps(1)
tff(fact_5177_measure__unat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A)] :
          ( ( P2 != zero_zero(word(A)) )
         => aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),P2),one_one(word(A))))),aa(word(A),nat,semiring_1_unsigned(A,nat),P2)) ) ) ).

% measure_unat
tff(fact_5178_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),abs_abs(int,K)),abs_abs(int,L)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(abs_abs(int,K))),nat2(abs_abs(int,L)))) ).

% div_abs_eq_div_nat
tff(fact_5179_word__of__int__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
         => ( aa(int,word(A),ring_1_of_int(word(A)),Xb) = aa(nat,word(A),semiring_1_of_nat(word(A)),nat2(Xb)) ) ) ) ).

% word_of_int_nat
tff(fact_5180_nat__floor__neg,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,Xb),zero_zero(real))
     => ( nat2(archim6421214686448440834_floor(real,Xb)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_5181_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),abs_abs(int,K)),abs_abs(int,L)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),nat2(abs_abs(int,K))),nat2(abs_abs(int,L)))) ).

% mod_abs_eq_div_nat
tff(fact_5182_nat__take__bit__eq,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),nat2(K)) ) ) ).

% nat_take_bit_eq
tff(fact_5183_take__bit__nat__eq,axiom,
    ! [K: int,N: nat] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),nat2(K)) = nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ) ) ).

% take_bit_nat_eq
tff(fact_5184_floor__eq3,axiom,
    ! [N: nat,Xb: real] :
      ( aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),N)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N)))
       => ( nat2(archim6421214686448440834_floor(real,Xb)) = N ) ) ) ).

% floor_eq3
tff(fact_5185_le__nat__floor,axiom,
    ! [Xb: nat,A3: real] :
      ( aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),Xb)),A3)
     => aa(nat,$o,ord_less_eq(nat,Xb),nat2(archim6421214686448440834_floor(real,A3))) ) ).

% le_nat_floor
tff(fact_5186_nat__2,axiom,
    nat2(aa(num,int,numeral_numeral(int),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))) ).

% nat_2
tff(fact_5187_diff__Suc,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_vj(nat,nat),aa(nat,nat,minus_minus(nat,M),N)) ).

% diff_Suc
tff(fact_5188_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Z)
     => ( aa(nat,nat,suc,nat2(Z)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)) ) ) ).

% Suc_nat_eq_nat_zadd1
tff(fact_5189_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),W)
     => ( aa(nat,$o,ord_less(nat,nat2(W)),M)
      <=> aa(int,$o,ord_less(int,W),aa(nat,int,semiring_1_of_nat(int),M)) ) ) ).

% nat_less_iff
tff(fact_5190_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z5: int] :
      ( aa(int,$o,ord_less_eq(int,Z),zero_zero(int))
     => ( nat2(aa(int,int,aa(int,fun(int,int),times_times(int),Z),Z5)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(aa(int,int,uminus_uminus(int),Z))),nat2(aa(int,int,uminus_uminus(int),Z5))) ) ) ).

% nat_mult_distrib_neg
tff(fact_5191_word__overflow__unat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),one_one(word(A)))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),one_one(nat)) )
          | ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% word_overflow_unat
tff(fact_5192_nat__abs__int__diff,axiom,
    ! [A3: nat,B3: nat] :
      nat2(abs_abs(int,aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),A3)),aa(nat,int,semiring_1_of_nat(int),B3)))) = $ite(aa(nat,$o,ord_less_eq(nat,A3),B3),aa(nat,nat,minus_minus(nat,B3),A3),aa(nat,nat,minus_minus(nat,A3),B3)) ).

% nat_abs_int_diff
tff(fact_5193_unat__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),W),one_one(word(A)))) = aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),W)),one_one(nat)) ) ) ) ).

% unat_minus_one
tff(fact_5194_floor__eq4,axiom,
    ! [N: nat,Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(nat,real,semiring_1_of_nat(real),N)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N)))
       => ( nat2(archim6421214686448440834_floor(real,Xb)) = N ) ) ) ).

% floor_eq4
tff(fact_5195_diff__nat__eq__if,axiom,
    ! [Z: int,Z5: int] :
      aa(nat,nat,minus_minus(nat,nat2(Z)),nat2(Z5)) = $ite(
        aa(int,$o,ord_less(int,Z5),zero_zero(int)),
        nat2(Z),
        $let(
          d: int,
          d:= aa(int,int,minus_minus(int,Z),Z5),
          $ite(aa(int,$o,ord_less(int,d),zero_zero(int)),zero_zero(nat),nat2(d)) ) ) ).

% diff_nat_eq_if
tff(fact_5196_of__int__of__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: int] :
          aa(int,A,ring_1_of_int(A),K) = $ite(aa(int,$o,ord_less(int,K),zero_zero(int)),aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),nat2(aa(int,int,uminus_uminus(int),K)))),aa(nat,A,semiring_1_of_nat(A),nat2(K))) ) ).

% of_int_of_nat
tff(fact_5197_arccos__def,axiom,
    ! [Y: real] : aa(real,real,arccos,Y) = the(real,aTP_Lamp_adm(real,fun(real,$o),Y)) ).

% arccos_def
tff(fact_5198_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa(nat,$o,dvd_dvd(nat,nat2(Z)),M)
    <=> $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),Z),aa(int,$o,dvd_dvd(int,Z),aa(nat,int,semiring_1_of_nat(int),M)),M = zero_zero(nat)) ) ).

% nat_dvd_iff
tff(fact_5199_lt__plus__1__le__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,MaxBound: word(A),Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,N),aa(word(A),nat,semiring_1_unsigned(A,nat),MaxBound))
         => ( aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(nat,word(A),semiring_1_of_nat(word(A)),N)))
          <=> aa(word(A),$o,ord_less_eq(word(A),Xb),aa(nat,word(A),semiring_1_of_nat(word(A)),N)) ) ) ) ).

% lt_plus_1_le_word
tff(fact_5200_odd__word__imp__even__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),one_one(word(A))) = zero_zero(word(A)) )
            | aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),one_one(word(A))))) ) ) ) ).

% odd_word_imp_even_next
tff(fact_5201_even__word__imp__odd__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb))
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),one_one(word(A))) = zero_zero(word(A)) )
            | ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),one_one(word(A))))) ) ) ) ).

% even_word_imp_odd_next
tff(fact_5202_even__nat__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
     => ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),nat2(K))
      <=> aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K) ) ) ).

% even_nat_iff
tff(fact_5203_word__div__eq__1__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),M: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),N),M) = one_one(word(A)) )
        <=> ( aa(word(A),$o,ord_less_eq(word(A),M),N)
            & aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),M))) ) ) ) ).

% word_div_eq_1_iff
tff(fact_5204_and__mask__dvd__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(word(A),nat,semiring_1_unsigned(A,nat),W))
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)) = zero_zero(word(A)) ) ) ) ).

% and_mask_dvd_nat
tff(fact_5205_of__nat__eq__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N) = W )
        <=> ? [Q4: nat] : N = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),W)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),W)))) ) ) ).

% of_nat_eq_size
tff(fact_5206_pi__half,axiom,
    aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))) = the(real,aTP_Lamp_adn(real,$o)) ).

% pi_half
tff(fact_5207_pi__def,axiom,
    pi = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),the(real,aTP_Lamp_adn(real,$o))) ).

% pi_def
tff(fact_5208_unat__plus__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y)) = $ite(aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xb))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xb)))) ) ).

% unat_plus_if_size
tff(fact_5209_unat__sub__if__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),Xb),Y)) = $ite(aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xb)))),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))) ) ).

% unat_sub_if_size
tff(fact_5210_no__plus__overflow__unat__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y))
        <=> aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xb))) ) ) ).

% no_plus_overflow_unat_size
tff(fact_5211_word__unat__mask__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),aa(word(A),nat,size_size(word(A)),W))
         => aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),M)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)) ) ) ).

% word_unat_mask_lt
tff(fact_5212_powr__real__of__int,axiom,
    ! [Xb: real,N: int] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(int,real,ring_1_of_int(real),N)) = $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),N),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),nat2(N)),aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),nat2(aa(int,int,uminus_uminus(int),N))))) ) ) ).

% powr_real_of_int
tff(fact_5213_arcsin__def,axiom,
    ! [Y: real] : aa(real,real,arcsin,Y) = the(real,aTP_Lamp_ado(real,fun(real,$o),Y)) ).

% arcsin_def
tff(fact_5214_old_Orec__prod__def,axiom,
    ! [C: $tType,B: $tType,A: $tType,X3: fun(A,fun(B,C)),Xa: product_prod(A,B)] : product_rec_prod(A,B,C,X3,Xa) = the(C,product_rec_set_prod(A,B,C,X3,Xa)) ).

% old.rec_prod_def
tff(fact_5215_the__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( aa(A,$o,P,A3)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = A3 ) )
       => ( the(A,P) = A3 ) ) ) ).

% the_equality
tff(fact_5216_the__eq__trivial,axiom,
    ! [A: $tType,A3: A] : the(A,aTP_Lamp_gc(A,fun(A,$o),A3)) = A3 ).

% the_eq_trivial
tff(fact_5217_the__sym__eq__trivial,axiom,
    ! [A: $tType,Xb: A] : the(A,fequal(A,Xb)) = Xb ).

% the_sym_eq_trivial
tff(fact_5218_The__split__eq,axiom,
    ! [A: $tType,B: $tType,Xb: A,Y: B] : the(product_prod(A,B),product_case_prod(A,B,$o,aa(B,fun(A,fun(B,$o)),aTP_Lamp_adp(A,fun(B,fun(A,fun(B,$o))),Xb),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y) ).

% The_split_eq
tff(fact_5219_The__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : the(product_prod(A,B),product_case_prod(A,B,$o,P)) = the(product_prod(A,B),aTP_Lamp_adq(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P)) ).

% The_case_prod
tff(fact_5220_the1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y2: A] :
              ( aa(A,$o,P,Y2)
             => ( Y2 = X3 ) ) )
     => ( aa(A,$o,P,A3)
       => ( the(A,P) = A3 ) ) ) ).

% the1_equality
tff(fact_5221_the1I2,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y2: A] :
              ( aa(A,$o,P,Y2)
             => ( Y2 = X3 ) ) )
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,the(A,P)) ) ) ).

% the1I2
tff(fact_5222_If__def,axiom,
    ! [A: $tType,P: $o,Xb: A,Y: A] :
      $ite((P),Xb,Y) = the(A,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_adr($o,fun(A,fun(A,fun(A,$o))),(P)),Xb),Y)) ).

% If_def
tff(fact_5223_theI2,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,A3)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = A3 ) )
       => ( ! [X2: A] :
              ( aa(A,$o,P,X2)
             => aa(A,$o,Q,X2) )
         => aa(A,$o,Q,the(A,P)) ) ) ) ).

% theI2
tff(fact_5224_theI_H,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y2: A] :
              ( aa(A,$o,P,Y2)
             => ( Y2 = X3 ) ) )
     => aa(A,$o,P,the(A,P)) ) ).

% theI'
tff(fact_5225_theI,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( aa(A,$o,P,A3)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = A3 ) )
       => aa(A,$o,P,the(A,P)) ) ) ).

% theI
tff(fact_5226_floor__real__def,axiom,
    ! [Xb: real] : archim6421214686448440834_floor(real,Xb) = the(int,aTP_Lamp_ads(real,fun(int,$o),Xb)) ).

% floor_real_def
tff(fact_5227_htt__vebt__inserti__invar__vebt,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => time_htt(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Tb),Tib),aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),vEBT_vebt_inserti,Tib),Xb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(Tb,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),nat2(archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))))))) ) ).

% htt_vebt_inserti_invar_vebt
tff(fact_5228_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] : order_Greatest(A,P) = the(A,aTP_Lamp_adt(fun(A,$o),fun(A,$o),P)) ) ).

% Greatest_def
tff(fact_5229_modulo__int__def,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = $ite(
        L = zero_zero(int),
        K,
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),nat2(abs_abs(int,K))),nat2(abs_abs(int,L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),abs_abs(int,L)),aa($o,int,zero_neq_one_of_bool(int),~ aa(int,$o,dvd_dvd(int,L),K)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),nat2(abs_abs(int,K))),nat2(abs_abs(int,L))))))) ) ).

% modulo_int_def
tff(fact_5230_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,sgn_sgn(A),A3)) = aa(A,A,sgn_sgn(A),A3) ) ).

% sgn_sgn
tff(fact_5231_htt__vebt__buildupi_H,axiom,
    ! [N: nat] : time_htt(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(N),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(N)),vEBT_V441764108873111860ildupi(N)) ).

% htt_vebt_buildupi'
tff(fact_5232_htt__vebt__buildupi,axiom,
    ! [N: nat] : time_htt(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(N),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(N)),vEBT_V441764108873111860ildupi(N)) ).

% htt_vebt_buildupi
tff(fact_5233_sgn__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_zero
tff(fact_5234_sgn__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),zero_zero(A)) = zero_zero(A) ) ) ).

% sgn_0
tff(fact_5235_sgn__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),one_one(A)) = one_one(A) ) ) ).

% sgn_1
tff(fact_5236_sgn__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( aa(A,A,sgn_sgn(A),one_one(A)) = one_one(A) ) ) ).

% sgn_one
tff(fact_5237_sgn__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A3: A,B3: A] : aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),B3)) ) ).

% sgn_divide
tff(fact_5238_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),A3)) ) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_5239_power__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A,N: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sgn_sgn(A),A3)),N) ) ).

% power_sgn
tff(fact_5240_inverse__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A] : aa(A,A,inverse_inverse(A),aa(A,A,sgn_sgn(A),A3)) = aa(A,A,sgn_sgn(A),A3) ) ).

% inverse_sgn
tff(fact_5241_sgn__inverse,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A3: A] : aa(A,A,sgn_sgn(A),aa(A,A,inverse_inverse(A),A3)) = aa(A,A,inverse_inverse(A),aa(A,A,sgn_sgn(A),A3)) ) ).

% sgn_inverse
tff(fact_5242_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) )
     => time_htt(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(N),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),U)) ) ).

% htt_vebt_buildupi'_univ
tff(fact_5243_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,N: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) )
     => time_htt(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(N),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),U)) ) ).

% htt_vebt_buildupi_univ
tff(fact_5244_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,sgn_sgn(A),A3)),zero_zero(A))
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% sgn_less
tff(fact_5245_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,sgn_sgn(A),A3))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ).

% sgn_greater
tff(fact_5246_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,sgn_sgn(A),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(A,A,sgn_sgn(A),B3)) ) ).

% divide_sgn
tff(fact_5247_htt__vebt__inserti,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : time_htt(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Tb),Tib),aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),vEBT_vebt_inserti,Tib),Xb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(Tb,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb)))) ).

% htt_vebt_inserti
tff(fact_5248_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A3)
         => ( aa(A,A,sgn_sgn(A),A3) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_5249_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( A3 != zero_zero(A) )
         => ( abs_abs(A,aa(A,A,sgn_sgn(A),A3)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_5250_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),A3)) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ).

% sgn_mult_self_eq
tff(fact_5251_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : abs_abs(A,aa(A,A,sgn_sgn(A),A3)) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ).

% sgn_abs
tff(fact_5252_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,sgn_sgn(A),abs_abs(A,A3)) = aa($o,A,zero_neq_one_of_bool(A),A3 != zero_zero(A)) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_5253_sgn__mult__dvd__iff,axiom,
    ! [R2: int,L: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R2)),L)),K)
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        & ( ( R2 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_5254_mult__sgn__dvd__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),L),aa(int,int,sgn_sgn(int),R2))),K)
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        & ( ( R2 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_5255_dvd__sgn__mult__iff,axiom,
    ! [L: int,R2: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R2)),K))
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        | ( R2 = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_5256_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R2: int] :
      ( aa(int,$o,dvd_dvd(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(int,int,sgn_sgn(int),R2)))
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        | ( R2 = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_5257_sgn__uint__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,int,sgn_sgn(int),aa(word(A),int,semiring_1_unsigned(A,int),W)) = aa($o,int,zero_neq_one_of_bool(int),W != zero_zero(word(A))) ) ).

% sgn_uint_eq
tff(fact_5258_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less(A,A3),zero_zero(A))
         => ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_5259_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,semiring_1_of_nat(A),N)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,ord_less(nat,zero_zero(nat)),N)) ) ).

% sgn_of_nat
tff(fact_5260_sgn__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( ( aa(A,A,sgn_sgn(A),Xb) = zero_zero(A) )
        <=> ( Xb = zero_zero(A) ) ) ) ).

% sgn_zero_iff
tff(fact_5261_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_5262_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_5263_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A,B3: A] : aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),aa(A,A,sgn_sgn(A),B3)) ) ).

% sgn_mult
tff(fact_5264_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B3) = aa(A,A,sgn_sgn(A),A3) )
         => ( aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = aa(A,A,sgn_sgn(A),A3) ) ) ) ).

% same_sgn_sgn_add
tff(fact_5265_htt__refine,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Tb: nat,C6: heap_Time_Heap(A)] :
      ( time_htt(A,P,C2,Q,Tb)
     => ( refine_Imp_refines(A,C6,C2)
       => time_htt(A,P,C6,Q,Tb) ) ) ).

% htt_refine
tff(fact_5266_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B3) != aa(A,A,sgn_sgn(A),A3) )
         => ( ( aa(A,A,sgn_sgn(A),A3) != zero_zero(A) )
           => ( ( aa(A,A,sgn_sgn(A),B3) != zero_zero(A) )
             => ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),B3)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_5267_sgn__minus__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% sgn_minus_1
tff(fact_5268_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B3: A,A3: A] :
          ( ( aa(A,A,sgn_sgn(A),B3) = aa(A,A,sgn_sgn(A),A3) )
         => ( abs_abs(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),abs_abs(A,A3)),abs_abs(A,B3)) ) ) ) ).

% same_sgn_abs_add
tff(fact_5269_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V: option(product_prod(nat,nat)),Va: nat,Vb: list(vEBT_VEBT),Vc: vEBT_VEBT,Vd: $o,Ve: $o] : aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_Node(V,Va,Vb,Vc)),vEBT_Leafi((Vd),(Ve))) = bot_bot(assn) ).

% vebt_assn_raw.simps(3)
tff(fact_5270_mult__sgn__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),Xb)),abs_abs(A,Xb)) = Xb ) ).

% mult_sgn_abs
tff(fact_5271_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A3)),abs_abs(A,A3)) = A3 ) ).

% sgn_mult_abs
tff(fact_5272_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),abs_abs(A,A3)),aa(A,A,sgn_sgn(A),A3)) = A3 ) ).

% abs_mult_sgn
tff(fact_5273_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K: A] : abs_abs(A,K) = aa(A,A,aa(A,fun(A,A),times_times(A),K),aa(A,A,sgn_sgn(A),K)) ) ).

% linordered_idom_class.abs_sgn
tff(fact_5274_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),divide_divide(int),abs_abs(int,K)),abs_abs(int,L)) ) ) ).

% div_eq_sgn_abs
tff(fact_5275_GreatestI__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B3: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y2: nat] :
            ( aa(nat,$o,P,Y2)
           => aa(nat,$o,ord_less_eq(nat,Y2),B3) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_nat
tff(fact_5276_Greatest__le__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B3: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y2: nat] :
            ( aa(nat,$o,P,Y2)
           => aa(nat,$o,ord_less_eq(nat,Y2),B3) )
       => aa(nat,$o,ord_less_eq(nat,K),order_Greatest(nat,P)) ) ) ).

% Greatest_le_nat
tff(fact_5277_GreatestI__ex__nat,axiom,
    ! [P: fun(nat,$o),B3: nat] :
      ( ? [X_13: nat] : aa(nat,$o,P,X_13)
     => ( ! [Y2: nat] :
            ( aa(nat,$o,P,Y2)
           => aa(nat,$o,ord_less_eq(nat,Y2),B3) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_ex_nat
tff(fact_5278_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = one_one(A) )
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A3) ) ) ).

% sgn_1_pos
tff(fact_5279_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          abs_abs(A,aa(A,A,sgn_sgn(A),A3)) = $ite(A3 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% abs_sgn_eq
tff(fact_5280_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ~ aa(int,$o,dvd_dvd(int,L),K)
       => ( aa(int,int,sgn_sgn(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L)) = aa(int,int,sgn_sgn(int),L) ) ) ) ).

% sgn_mod
tff(fact_5281_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y2: A] :
                ( aa(A,$o,P,Y2)
               => aa(A,$o,ord_less_eq(A,Y2),Xb) )
           => ( ! [X2: A] :
                  ( aa(A,$o,P,X2)
                 => ( ! [Y4: A] :
                        ( aa(A,$o,P,Y4)
                       => aa(A,$o,ord_less_eq(A,Y4),X2) )
                   => aa(A,$o,Q,X2) ) )
             => aa(A,$o,Q,order_Greatest(A,P)) ) ) ) ) ).

% GreatestI2_order
tff(fact_5282_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y2: A] :
                ( aa(A,$o,P,Y2)
               => aa(A,$o,ord_less_eq(A,Y2),Xb) )
           => ( order_Greatest(A,P) = Xb ) ) ) ) ).

% Greatest_equality
tff(fact_5283_sgn__if,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          aa(A,A,sgn_sgn(A),Xb) = $ite(
            Xb = zero_zero(A),
            zero_zero(A),
            $ite(aa(A,$o,ord_less(A,zero_zero(A)),Xb),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% sgn_if
tff(fact_5284_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( ( aa(A,A,sgn_sgn(A),A3) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ).

% sgn_1_neg
tff(fact_5285_zsgn__def,axiom,
    ! [I: int] :
      aa(int,int,sgn_sgn(int),I) = $ite(
        I = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,$o,ord_less(int,zero_zero(int)),I),one_one(int),aa(int,int,uminus_uminus(int),one_one(int))) ) ).

% zsgn_def
tff(fact_5286_norm__sgn,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          real_V7770717601297561774m_norm(A,aa(A,A,sgn_sgn(A),Xb)) = $ite(Xb = zero_zero(A),zero_zero(real),one_one(real)) ) ).

% norm_sgn
tff(fact_5287_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V)),abs_abs(int,K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V)),abs_abs(int,L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),abs_abs(int,K)),abs_abs(int,L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_5288_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,L),K)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(int,int,sgn_sgn(int),L))),aa(int,int,aa(int,fun(int,int),divide_divide(int),abs_abs(int,K)),abs_abs(int,L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_5289_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q2: int] :
      ( ( aa(int,int,sgn_sgn(int),R2) = aa(int,int,sgn_sgn(int),L) )
     => ( aa(int,$o,ord_less(int,abs_abs(int,R2)),abs_abs(int,L))
       => ( ( K = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q2),L)),R2) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R2)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_5290_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => ( ( aa(int,int,sgn_sgn(int),K) != aa(int,int,sgn_sgn(int),L) )
       => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),abs_abs(int,K)),abs_abs(int,L)))),aa($o,int,zero_neq_one_of_bool(int),~ aa(int,$o,dvd_dvd(int,L),K))) ) ) ) ).

% div_noneq_sgn_abs
tff(fact_5291_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K3: int] :
            ( ( A1 = K3 )
            & ( A22 = zero_zero(int) )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),K3) ) )
        | ? [L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),zero_zero(int)) )
            & ( L4 != zero_zero(int) )
            & ( K3 = aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L4) ) )
        | ? [R6: int,L4: int,K3: int,Q4: int] :
            ( ( A1 = K3 )
            & ( A22 = L4 )
            & ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q4),R6) )
            & ( aa(int,int,sgn_sgn(int),R6) = aa(int,int,sgn_sgn(int),L4) )
            & aa(int,$o,ord_less(int,abs_abs(int,R6)),abs_abs(int,L4))
            & ( K3 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L4)),R6) ) ) ) ) ).

% eucl_rel_int.simps
tff(fact_5292_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
     => ( ( ( A22 = zero_zero(int) )
         => ( A32 != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A1) ) )
       => ( ! [Q3: int] :
              ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),zero_zero(int)) )
             => ( ( A22 != zero_zero(int) )
               => ( A1 != aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A22) ) ) )
         => ~ ! [R4: int,Q3: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R4) )
               => ( ( aa(int,int,sgn_sgn(int),R4) = aa(int,int,sgn_sgn(int),A22) )
                 => ( aa(int,$o,ord_less(int,abs_abs(int,R4)),abs_abs(int,A22))
                   => ( A1 != aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Q3),A22)),R4) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_5293_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,N: nat] :
      aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = $ite(
        ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
        | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
        | ( N = zero_zero(nat) ) ),
        zero_zero(int),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N)),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,dvd_dvd(nat,N),M)))))) ) ).

% divide_int_unfold
tff(fact_5294_modulo__int__unfold,axiom,
    ! [K: int,M: nat,L: int,N: nat] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),N))) = $ite(
        ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
        | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
        | ( N = zero_zero(nat) ) ),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(nat,int,semiring_1_of_nat(int),M)),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,dvd_dvd(nat,N),M))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),N))))) ) ).

% modulo_int_unfold
tff(fact_5295_divide__int__def,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = $ite(
        L = zero_zero(int),
        zero_zero(int),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(abs_abs(int,K))),nat2(abs_abs(int,L)))),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(abs_abs(int,K))),nat2(abs_abs(int,L)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(int,$o,dvd_dvd(int,L),K)))))) ) ).

% divide_int_def
tff(fact_5296_vebt__buildupi__rule,axiom,
    ! [N: nat] : time_htt(vEBT_VEBTi,pure_assn(aa(nat,$o,ord_less(nat,zero_zero(nat)),N)),vEBT_vebt_buildupi(N),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(N)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% vebt_buildupi_rule
tff(fact_5297_floor__rat__def,axiom,
    ! [Xb: rat] : archim6421214686448440834_floor(rat,Xb) = the(int,aTP_Lamp_adu(rat,fun(int,$o),Xb)) ).

% floor_rat_def
tff(fact_5298_sgn__div__eq__sgn__mult,axiom,
    ! [A3: int,B3: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3) != zero_zero(int) )
     => ( aa(int,int,sgn_sgn(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),B3)) = aa(int,int,sgn_sgn(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),B3)) ) ) ).

% sgn_div_eq_sgn_mult
tff(fact_5299_highi__hT,axiom,
    ! [Xb: nat,N: nat] : time_htt(nat,one_one(assn),vEBT_VEBT_highi(Xb,N),aa(nat,fun(nat,assn),aTP_Lamp_adv(nat,fun(nat,fun(nat,assn)),Xb),N),one_one(nat)) ).

% highi_hT
tff(fact_5300_lowi__hT,axiom,
    ! [Xb: nat,N: nat] : time_htt(nat,one_one(assn),vEBT_VEBT_lowi(Xb,N),aa(nat,fun(nat,assn),aTP_Lamp_adw(nat,fun(nat,fun(nat,assn)),Xb),N),one_one(nat)) ).

% lowi_hT
tff(fact_5301_vebt__minti__hT,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi] : time_htt(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Tb),Tib),vEBT_vebt_minti(Tib),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_adx(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Tb),Tib),one_one(nat)) ).

% vebt_minti_hT
tff(fact_5302_vebt__maxti__hT,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi] : time_htt(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Tb),Tib),vEBT_vebt_maxti(Tib),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_ady(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Tb),Tib),one_one(nat)) ).

% vebt_maxti_hT
tff(fact_5303_minNrulli__ruleT,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi] : time_htt($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Tb),Tib),vEBT_VEBT_minNulli(Tib),aa(vEBT_VEBTi,fun($o,assn),aTP_Lamp_adz(vEBT_VEBT,fun(vEBT_VEBTi,fun($o,assn)),Tb),Tib),one_one(nat)) ).

% minNrulli_ruleT
tff(fact_5304_zero__le__sgn__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,sgn_sgn(real),Xb))
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb) ) ).

% zero_le_sgn_iff
tff(fact_5305_sgn__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,aa(real,real,sgn_sgn(real),Xb)),zero_zero(real))
    <=> aa(real,$o,ord_less_eq(real,Xb),zero_zero(real)) ) ).

% sgn_le_0_iff
tff(fact_5306_htt__vebt__memberi,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : time_htt($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Tb),Tib),aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),vEBT_vebt_memberi,Tib),Xb),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_aea(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),Tb),Tib),Xb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb)))) ).

% htt_vebt_memberi
tff(fact_5307_htt__vebt__memberi__invar__vebt,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => time_htt($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Tb),Tib),aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),vEBT_vebt_memberi,Tib),Xb),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_aea(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),Tb),Tib),Xb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),nat2(archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))))))) ) ).

% htt_vebt_memberi_invar_vebt
tff(fact_5308_htt__vebt__succi,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => time_htt(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Tb),Tib),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_vebt_succi,Tib),Xb),aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_aeb(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),Tb),Tib),Xb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),nat2(archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))))))) ) ).

% htt_vebt_succi
tff(fact_5309_sgn__rat__def,axiom,
    ! [A3: rat] :
      aa(rat,rat,sgn_sgn(rat),A3) = $ite(
        A3 = zero_zero(rat),
        zero_zero(rat),
        $ite(aa(rat,$o,ord_less(rat,zero_zero(rat)),A3),one_one(rat),aa(rat,rat,uminus_uminus(rat),one_one(rat))) ) ).

% sgn_rat_def
tff(fact_5310_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( aa(rat,$o,ord_less(rat,zero_zero(rat)),R2)
     => ~ ! [S2: rat] :
            ( aa(rat,$o,ord_less(rat,zero_zero(rat)),S2)
           => ! [T2: rat] :
                ( aa(rat,$o,ord_less(rat,zero_zero(rat)),T2)
               => ( R2 != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S2),T2) ) ) ) ) ).

% obtain_pos_sum
tff(fact_5311_less__eq__rat__def,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa(rat,$o,ord_less_eq(rat,Xb),Y)
    <=> ( aa(rat,$o,ord_less(rat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_rat_def
tff(fact_5312_abs__rat__def,axiom,
    ! [A3: rat] :
      abs_abs(rat,A3) = $ite(aa(rat,$o,ord_less(rat,A3),zero_zero(rat)),aa(rat,rat,uminus_uminus(rat),A3),A3) ).

% abs_rat_def
tff(fact_5313_sgn__root,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,real,sgn_sgn(real),aa(real,real,root(N),Xb)) = aa(real,real,sgn_sgn(real),Xb) ) ) ).

% sgn_root
tff(fact_5314_cis__Arg,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( cis(arg(Z)) = aa(complex,complex,sgn_sgn(complex),Z) ) ) ).

% cis_Arg
tff(fact_5315_sgn__real__def,axiom,
    ! [A3: real] :
      aa(real,real,sgn_sgn(real),A3) = $ite(
        A3 = zero_zero(real),
        zero_zero(real),
        $ite(aa(real,$o,ord_less(real,zero_zero(real)),A3),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ) ).

% sgn_real_def
tff(fact_5316_sgn__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,code_integer,sgn_sgn(code_integer),K) = $ite(
        K = zero_zero(code_integer),
        zero_zero(code_integer),
        $ite(aa(code_integer,$o,ord_less(code_integer,K),zero_zero(code_integer)),aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),one_one(code_integer)) ) ).

% sgn_integer_code
tff(fact_5317_sgn__power__injE,axiom,
    ! [A3: real,N: nat,Xb: real,B3: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),A3)),aa(nat,real,aa(real,fun(nat,real),power_power(real),abs_abs(real,A3)),N)) = Xb )
     => ( ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),B3)),aa(nat,real,aa(real,fun(nat,real),power_power(real),abs_abs(real,B3)),N)) )
       => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
         => ( A3 = B3 ) ) ) ) ).

% sgn_power_injE
tff(fact_5318_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,real,root(N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y)),aa(nat,real,aa(real,fun(nat,real),power_power(real),abs_abs(real,Y)),N))) = Y ) ) ).

% root_sgn_power
tff(fact_5319_sgn__power__root,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),aa(real,real,root(N),Xb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),abs_abs(real,aa(real,real,root(N),Xb))),N)) = Xb ) ) ).

% sgn_power_root
tff(fact_5320_cis__Arg__unique,axiom,
    ! [Z: complex,Xb: real] :
      ( ( aa(complex,complex,sgn_sgn(complex),Z) = cis(Xb) )
     => ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),Xb)
       => ( aa(real,$o,ord_less_eq(real,Xb),pi)
         => ( arg(Z) = Xb ) ) ) ) ).

% cis_Arg_unique
tff(fact_5321_split__root,axiom,
    ! [P: fun(real,$o),N: nat,Xb: real] :
      ( aa(real,$o,P,aa(real,real,root(N),Xb))
    <=> ( ( ( N = zero_zero(nat) )
         => aa(real,$o,P,zero_zero(real)) )
        & ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
         => ! [Y3: real] :
              ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Y3)),aa(nat,real,aa(real,fun(nat,real),power_power(real),abs_abs(real,Y3)),N)) = Xb )
             => aa(real,$o,P,Y3) ) ) ) ) ).

% split_root
tff(fact_5322_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero(complex) )
     => ( ( aa(complex,complex,sgn_sgn(complex),Z) = cis(arg(Z)) )
        & aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),arg(Z))
        & aa(real,$o,ord_less_eq(real,arg(Z)),pi) ) ) ).

% Arg_correct
tff(fact_5323_arctan__inverse,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero(real) )
     => ( aa(real,real,arctan,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Xb)) = aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Xb)),pi)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(real,real,arctan,Xb)) ) ) ).

% arctan_inverse
tff(fact_5324_vebt__pred_H__rf__abstr,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Tb),Tib),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),vEBT_VEBT_vebt_predi,Tb),Tib),Xb),aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_aec(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),Tb),Tib),Xb)) ) ).

% vebt_pred'_rf_abstr
tff(fact_5325_vebt__succi_H__rf__abstr,axiom,
    ! [Tb: vEBT_VEBT,N: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( vEBT_invar_vebt(Tb,N)
     => hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Tb),Tib),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),vEBT_VEBT_vebt_succi,Tb),Tib),Xb),aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_aeb(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),Tb),Tib),Xb)) ) ).

% vebt_succi'_rf_abstr
tff(fact_5326_vebt__memberi_H__rf__abstr,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : hoare_hoare_triple($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Tb),Tib),aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),vEBT_V854960066525838166emberi,Tb),Tib),Xb),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_aea(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),Tb),Tib),Xb)) ).

% vebt_memberi'_rf_abstr
tff(fact_5327_vebt__minti__h,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi] : hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Tb),Tib),vEBT_vebt_minti(Tib),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_adx(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Tb),Tib)) ).

% vebt_minti_h
tff(fact_5328_vebt__maxti__h,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi] : hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Tb),Tib),vEBT_vebt_maxti(Tib),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_ady(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Tb),Tib)) ).

% vebt_maxti_h
tff(fact_5329_minNulli__rule,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi] : hoare_hoare_triple($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Tb),Tib),vEBT_VEBT_minNulli(Tib),aa(vEBT_VEBTi,fun($o,assn),aTP_Lamp_adz(vEBT_VEBT,fun(vEBT_VEBTi,fun($o,assn)),Tb),Tib)) ).

% minNulli_rule
tff(fact_5330_hoare__triple__refines,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),C6: heap_Time_Heap(A)] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( refine_Imp_refines(A,C6,C2)
       => hoare_hoare_triple(A,P,C6,Q) ) ) ).

% hoare_triple_refines
tff(fact_5331_vebt__maxtilist,axiom,
    ! [I: nat,Ts: list(vEBT_VEBT),Tsi: list(vEBT_VEBTi)] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Ts))
     => hoare_hoare_triple(option(nat),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Ts),Tsi),vEBT_vebt_maxti(aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tsi),I)),aa(list(vEBT_VEBTi),fun(option(nat),assn),aa(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn)),aTP_Lamp_aed(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),I),Ts),Tsi)) ) ).

% vebt_maxtilist
tff(fact_5332_vebt__mintilist,axiom,
    ! [I: nat,Ts: list(vEBT_VEBT),Tsi: list(vEBT_VEBTi)] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Ts))
     => hoare_hoare_triple(option(nat),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Ts),Tsi),vEBT_vebt_minti(aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tsi),I)),aa(list(vEBT_VEBTi),fun(option(nat),assn),aa(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn)),aTP_Lamp_aee(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),I),Ts),Tsi)) ) ).

% vebt_mintilist
tff(fact_5333_repli__emp,axiom,
    ! [A: $tType,B: $tType,Xb: heap_Time_Heap(A),A4: fun(B,fun(A,assn)),Y: B,N: nat] :
      ( hoare_hoare_triple(A,one_one(assn),Xb,aa(B,fun(A,assn),A4,Y))
     => hoare_hoare_triple(list(A),one_one(assn),vEBT_VEBT_replicatei(A,N,Xb),vEBT_List_list_assn(B,A,A4,replicate(B,N,Y))) ) ).

% repli_emp
tff(fact_5334_vebt__inserti_H__rf__abstr,axiom,
    ! [Tb: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : hoare_hoare_triple(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Tb),Tib),aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),vEBT_V3964819847710782039nserti,Tb),Tib),Xb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(Tb,Xb))) ).

% vebt_inserti'_rf_abstr
tff(fact_5335_highi__h,axiom,
    ! [Xb: nat,N: nat] : hoare_hoare_triple(nat,one_one(assn),vEBT_VEBT_highi(Xb,N),aa(nat,fun(nat,assn),aTP_Lamp_adv(nat,fun(nat,fun(nat,assn)),Xb),N)) ).

% highi_h
tff(fact_5336_lowi__h,axiom,
    ! [Xb: nat,N: nat] : hoare_hoare_triple(nat,one_one(assn),vEBT_VEBT_lowi(Xb,N),aa(nat,fun(nat,assn),aTP_Lamp_adw(nat,fun(nat,fun(nat,assn)),Xb),N)) ).

% lowi_h
tff(fact_5337_builupi_Hcorr,axiom,
    ! [N: nat] : hoare_hoare_triple(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(N),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(N))) ).

% builupi'corr
tff(fact_5338_builupicorr,axiom,
    ! [N: nat] : hoare_hoare_triple(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(N),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(N))) ).

% builupicorr
tff(fact_5339_repli__cons__repl,axiom,
    ! [B: $tType,A: $tType,Q: assn,Xb: heap_Time_Heap(A),A4: fun(B,fun(A,assn)),Y: B,N: nat] :
      ( hoare_hoare_triple(A,Q,Xb,aa(B,fun(A,assn),aa(fun(B,fun(A,assn)),fun(B,fun(A,assn)),aTP_Lamp_aef(assn,fun(fun(B,fun(A,assn)),fun(B,fun(A,assn))),Q),A4),Y))
     => hoare_hoare_triple(list(A),Q,vEBT_VEBT_replicatei(A,N,Xb),aa(nat,fun(list(A),assn),aa(B,fun(nat,fun(list(A),assn)),aa(fun(B,fun(A,assn)),fun(B,fun(nat,fun(list(A),assn))),aTP_Lamp_aeg(assn,fun(fun(B,fun(A,assn)),fun(B,fun(nat,fun(list(A),assn)))),Q),A4),Y),N)) ) ).

% repli_cons_repl
tff(fact_5340_return__sp__rule,axiom,
    ! [A: $tType,P: assn,Xb: A] : hoare_hoare_triple(A,P,aa(A,heap_Time_Heap(A),heap_Time_return(A),Xb),aa(A,fun(A,assn),aTP_Lamp_aeh(assn,fun(A,fun(A,assn)),P),Xb)) ).

% return_sp_rule
tff(fact_5341_case__option__rule,axiom,
    ! [A: $tType,B: $tType,V: option(A),P: assn,Fn: heap_Time_Heap(B),Q: fun(B,assn),Fs: fun(A,heap_Time_Heap(B))] :
      ( ( ( V = none(A) )
       => hoare_hoare_triple(B,P,Fn,Q) )
     => ( ! [X2: A] :
            ( ( V = aa(A,option(A),some(A),X2) )
           => hoare_hoare_triple(B,P,aa(A,heap_Time_Heap(B),Fs,X2),Q) )
       => hoare_hoare_triple(B,P,case_option(heap_Time_Heap(B),A,Fn,Fs,V),Q) ) ) ).

% case_option_rule
tff(fact_5342_case__prod__rule,axiom,
    ! [A: $tType,B: $tType,C: $tType,Xb: product_prod(A,B),P: assn,Fa: fun(A,fun(B,heap_Time_Heap(C))),Q: fun(C,assn)] :
      ( ! [A2: A,B2: B] :
          ( ( Xb = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) )
         => hoare_hoare_triple(C,P,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),Fa,A2),B2),Q) )
     => hoare_hoare_triple(C,P,aa(product_prod(A,B),heap_Time_Heap(C),product_case_prod(A,B,heap_Time_Heap(C),Fa),Xb),Q) ) ).

% case_prod_rule
tff(fact_5343_return__wp__rule,axiom,
    ! [A: $tType,Q: fun(A,assn),Xb: A] : hoare_hoare_triple(A,aa(A,assn,Q,Xb),aa(A,heap_Time_Heap(A),heap_Time_return(A),Xb),Q) ).

% return_wp_rule
tff(fact_5344_bind__rule,axiom,
    ! [A: $tType,B: $tType,P: assn,Fa: heap_Time_Heap(A),R: fun(A,assn),G: fun(A,heap_Time_Heap(B)),Q: fun(B,assn)] :
      ( hoare_hoare_triple(A,P,Fa,R)
     => ( ! [X2: A] : hoare_hoare_triple(B,aa(A,assn,R,X2),aa(A,heap_Time_Heap(B),G,X2),Q)
       => hoare_hoare_triple(B,P,heap_Time_bind(A,B,Fa,G),Q) ) ) ).

% bind_rule
tff(fact_5345_frame__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),R: assn] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),R),C2,aa(assn,fun(A,assn),aTP_Lamp_aei(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ) ).

% frame_rule
tff(fact_5346_bind__rule_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,Fa: heap_Time_Heap(B),R: fun(B,assn),G: fun(B,heap_Time_Heap(C)),Q: fun(C,assn)] :
      ( ! [R4: A] : hoare_hoare_triple(B,P,Fa,aa(fun(B,assn),fun(B,assn),aTP_Lamp_aej(assn,fun(fun(B,assn),fun(B,assn)),P),R))
     => ( ! [X2: B] : hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,R,X2)),P),aa(B,heap_Time_Heap(C),G,X2),Q)
       => hoare_hoare_triple(C,P,heap_Time_bind(B,C,Fa,G),Q) ) ) ).

% bind_rule'
tff(fact_5347_list__assn__aux__ineq__len,axiom,
    ! [B: $tType,A: $tType,L: list(A),Li2: list(B),A4: fun(A,fun(B,assn))] :
      ( ( aa(list(A),nat,size_size(list(A)),L) != aa(list(B),nat,size_size(list(B)),Li2) )
     => ( aa(list(B),assn,vEBT_List_list_assn(A,B,A4,L),Li2) = bot_bot(assn) ) ) ).

% list_assn_aux_ineq_len
tff(fact_5348_nth__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xs: list(A),A3: array(A)] :
          ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
         => hoare_hoare_triple(A,aa(list(A),assn,snga_assn(A,A3),Xs),aa(nat,heap_Time_Heap(A),array_nth(A,A3),I),aa(array(A),fun(A,assn),aa(list(A),fun(array(A),fun(A,assn)),aTP_Lamp_aek(nat,fun(list(A),fun(array(A),fun(A,assn))),I),Xs),A3)) ) ) ).

% nth_rule
tff(fact_5349_Arg__def,axiom,
    ! [Z: complex] :
      arg(Z) = $ite(Z = zero_zero(complex),zero_zero(real),fChoice(real,aTP_Lamp_ael(complex,fun(real,$o),Z))) ).

% Arg_def
tff(fact_5350_some__insert__self,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ( S3 != bot_bot(set(A)) )
     => ( aa(set(A),set(A),insert(A,fChoice(A,aTP_Lamp_cc(set(A),fun(A,$o),S3))),S3) = S3 ) ) ).

% some_insert_self
tff(fact_5351_verit__sko__ex_H,axiom,
    ! [A: $tType,P: fun(A,$o),A4: $o] :
      ( ( aa(A,$o,P,fChoice(A,P))
      <=> (A4) )
     => ( ? [X_12: A] : aa(A,$o,P,X_12)
      <=> (A4) ) ) ).

% verit_sko_ex'
tff(fact_5352_verit__sko__forall,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X_12: A] : aa(A,$o,P,X_12)
    <=> aa(A,$o,P,fChoice(A,aTP_Lamp_vu(fun(A,$o),fun(A,$o),P))) ) ).

% verit_sko_forall
tff(fact_5353_verit__sko__forall_H,axiom,
    ! [A: $tType,P: fun(A,$o),A4: $o] :
      ( ( aa(A,$o,P,fChoice(A,aTP_Lamp_vu(fun(A,$o),fun(A,$o),P)))
      <=> (A4) )
     => ( ! [X_12: A] : aa(A,$o,P,X_12)
      <=> (A4) ) ) ).

% verit_sko_forall'
tff(fact_5354_verit__sko__forall_H_H,axiom,
    ! [A: $tType,B6: A,A4: A,P: fun(A,$o)] :
      ( ( B6 = A4 )
     => ( ( fChoice(A,P) = A4 )
      <=> ( fChoice(A,P) = B6 ) ) ) ).

% verit_sko_forall''
tff(fact_5355_verit__sko__ex__indirect,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o)] :
      ( ( Xb = fChoice(A,P) )
     => ( ? [X_12: A] : aa(A,$o,P,X_12)
      <=> aa(A,$o,P,Xb) ) ) ).

% verit_sko_ex_indirect
tff(fact_5356_verit__sko__ex__indirect2,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),P3: fun(A,$o)] :
      ( ( Xb = fChoice(A,P) )
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
          <=> aa(A,$o,P3,X2) )
       => ( ? [X_12: A] : aa(A,$o,P3,X_12)
        <=> aa(A,$o,P,Xb) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_5357_verit__sko__forall__indirect,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o)] :
      ( ( Xb = fChoice(A,aTP_Lamp_vu(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X_12: A] : aa(A,$o,P,X_12)
      <=> aa(A,$o,P,Xb) ) ) ).

% verit_sko_forall_indirect
tff(fact_5358_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),P3: fun(A,$o)] :
      ( ( Xb = fChoice(A,aTP_Lamp_vu(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
          <=> aa(A,$o,P3,X2) )
       => ( ! [X_12: A] : aa(A,$o,P3,X_12)
        <=> aa(A,$o,P,Xb) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_5359_some__elem,axiom,
    ! [A: $tType,S3: set(A)] :
      ( ( S3 != bot_bot(set(A)) )
     => member(A,fChoice(A,aTP_Lamp_cc(set(A),fun(A,$o),S3)),S3) ) ).

% some_elem
tff(fact_5360_of__list__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A)] : hoare_hoare_triple(array(A),one_one(assn),array_of_list(A,Xs),aTP_Lamp_aem(list(A),fun(array(A),assn),Xs)) ) ).

% of_list_rule
tff(fact_5361_length__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),Xs: list(A)] : hoare_hoare_triple(nat,aa(list(A),assn,snga_assn(A,A3),Xs),array_len(A,A3),aa(list(A),fun(nat,assn),aTP_Lamp_aen(array(A),fun(list(A),fun(nat,assn)),A3),Xs)) ) ).

% length_rule
tff(fact_5362_upd__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xs: list(A),A3: array(A),Xb: A] :
          ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
         => hoare_hoare_triple(array(A),aa(list(A),assn,snga_assn(A,A3),Xs),array_upd(A,I,Xb,A3),aa(A,fun(array(A),assn),aa(array(A),fun(A,fun(array(A),assn)),aa(list(A),fun(array(A),fun(A,fun(array(A),assn))),aTP_Lamp_aeo(nat,fun(list(A),fun(array(A),fun(A,fun(array(A),assn)))),I),Xs),A3),Xb)) ) ) ).

% upd_rule
tff(fact_5363_heaphelp,axiom,
    ! [A: $tType,Xaa: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Xb: vEBT_VEBTi,N: nat,Xc: vEBT_VEBTi,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( rep_assn(aa(assn,assn,
            aa(assn,fun(assn,assn),times_times(assn),
              aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,Xaa),Tree_is)),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),Xb))),
                pure_assn(( ( none(A) = none(A) )
                  & ( N = N ) )))),
            pure_assn(Xc = vEBT_Nodei(none(product_prod(nat,nat)),N,Xaa,Xb))),H)
     => rep_assn(aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_Node(none(product_prod(nat,nat)),N,TreeList,Summary2)),Xc),H) ) ).

% heaphelp
tff(fact_5364_some__sym__eq__trivial,axiom,
    ! [A: $tType,Xb: A] : fChoice(A,fequal(A,Xb)) = Xb ).

% some_sym_eq_trivial
tff(fact_5365_some__eq__trivial,axiom,
    ! [A: $tType,Xb: A] : fChoice(A,aTP_Lamp_gc(A,fun(A,$o),Xb)) = Xb ).

% some_eq_trivial
tff(fact_5366_some__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( aa(A,$o,P,A3)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = A3 ) )
       => ( fChoice(A,P) = A3 ) ) ) ).

% some_equality
tff(fact_5367_Eps__case__prod__eq,axiom,
    ! [A: $tType,B: $tType,Xb: A,Y: B] : fChoice(product_prod(A,B),product_case_prod(A,B,$o,aa(B,fun(A,fun(B,$o)),aTP_Lamp_adp(A,fun(B,fun(A,fun(B,$o))),Xb),Y))) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y) ).

% Eps_case_prod_eq
tff(fact_5368_split__paired__Eps,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o)] : fChoice(product_prod(A,B),P) = fChoice(product_prod(A,B),product_case_prod(A,B,$o,aTP_Lamp_aep(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),P))) ).

% split_paired_Eps
tff(fact_5369_assert_H__rule,axiom,
    ! [P: assn,Phi: $o] :
      ( ! [H5: product_prod(heap_ext(product_unit),set(nat))] :
          ( rep_assn(P,H5)
         => (Phi) )
     => hoare_hoare_triple(product_unit,P,refine_Imp_assert((Phi)),aTP_Lamp_aeq(assn,fun(product_unit,assn),P)) ) ).

% assert'_rule
tff(fact_5370_extract__pre__list__assn__lengthD,axiom,
    ! [B: $tType,A: $tType,A4: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B),H: product_prod(heap_ext(product_unit),set(nat))] :
      ( rep_assn(aa(list(B),assn,vEBT_List_list_assn(A,B,A4,Xs),Xsi),H)
     => ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ).

% extract_pre_list_assn_lengthD
tff(fact_5371_Eps__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : fChoice(product_prod(A,B),product_case_prod(A,B,$o,P)) = fChoice(product_prod(A,B),aTP_Lamp_adq(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P)) ).

% Eps_case_prod
tff(fact_5372_assert_H__bind__rule,axiom,
    ! [A: $tType,P: assn,Phi: $o,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( ! [H5: product_prod(heap_ext(product_unit),set(nat))] :
          ( rep_assn(P,H5)
         => (Phi) )
     => ( ( (Phi)
         => hoare_hoare_triple(A,P,C2,Q) )
       => hoare_hoare_triple(A,P,heap_Time_bind(product_unit,A,refine_Imp_assert((Phi)),aTP_Lamp_gb(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),C2)),Q) ) ) ).

% assert'_bind_rule
tff(fact_5373_htt__def,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Tb: nat] :
      ( time_htt(A,P,C2,Q,Tb)
    <=> ( hoare_hoare_triple(A,P,C2,Q)
        & ! [H6: heap_ext(product_unit),As: set(nat)] :
            ( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H6),As))
           => aa(nat,$o,ord_less_eq(nat,time_time(A,C2,H6)),Tb) ) ) ) ).

% htt_def
tff(fact_5374_httI,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Tb: nat] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( ! [H5: heap_ext(product_unit),As2: set(nat)] :
            ( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H5),As2))
           => aa(nat,$o,ord_less_eq(nat,time_time(A,C2,H5)),Tb) )
       => time_htt(A,P,C2,Q,Tb) ) ) ).

% httI
tff(fact_5375_someI2,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,A3)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2
tff(fact_5376_someI__ex,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X_13: A] : aa(A,$o,P,X_13)
     => aa(A,$o,P,fChoice(A,P)) ) ).

% someI_ex
tff(fact_5377_someI2__ex,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X_13: A] : aa(A,$o,P,X_13)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2_ex
tff(fact_5378_someI2__bex,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,A4)
          & aa(A,$o,P,X3) )
     => ( ! [X2: A] :
            ( ( member(A,X2,A4)
              & aa(A,$o,P,X2) )
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,fChoice(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_uu(set(A),fun(fun(A,$o),fun(A,$o)),A4),P))) ) ) ).

% someI2_bex
tff(fact_5379_some__eq__ex,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
    <=> ? [X_12: A] : aa(A,$o,P,X_12) ) ).

% some_eq_ex
tff(fact_5380_some1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A3: A] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y2: A] :
              ( aa(A,$o,P,Y2)
             => ( Y2 = X3 ) ) )
     => ( aa(A,$o,P,A3)
       => ( fChoice(A,P) = A3 ) ) ) ).

% some1_equality
tff(fact_5381_dependent__nat__choice,axiom,
    ! [A: $tType,P: fun(nat,fun(A,$o)),Q: fun(nat,fun(A,fun(A,$o)))] :
      ( ? [X_13: A] : aa(A,$o,aa(nat,fun(A,$o),P,zero_zero(nat)),X_13)
     => ( ! [X2: A,N3: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N3),X2)
           => ? [Y4: A] :
                ( aa(A,$o,aa(nat,fun(A,$o),P,aa(nat,nat,suc,N3)),Y4)
                & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N3),X2),Y4) ) )
       => ? [F4: fun(nat,A)] :
          ! [N4: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N4),aa(nat,A,F4,N4))
            & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N4),aa(nat,A,F4,N4)),aa(nat,A,F4,aa(nat,nat,suc,N4))) ) ) ) ).

% dependent_nat_choice
tff(fact_5382_some__in__eq,axiom,
    ! [A: $tType,A4: set(A)] :
      ( member(A,fChoice(A,aTP_Lamp_cc(set(A),fun(A,$o),A4)),A4)
    <=> ( A4 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_5383_vebt__assn__raw_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: vEBT_VEBTi,Y: assn] :
      ( ( aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Xb),Xaa) = Y )
     => ( ! [A2: $o,B2: $o] :
            ( ( Xb = vEBT_Leaf((A2),(B2)) )
           => ! [Ai: $o,Bi: $o] :
                ( ( Xaa = vEBT_Leafi((Ai),(Bi)) )
               => ( Y != pure_assn(( ( (Ai) = (A2) )
                      & ( (Bi) = (B2) ) )) ) ) )
       => ( ! [Mmo: option(product_prod(nat,nat)),Deg: nat,Tree_list: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mmo,Deg,Tree_list,Summary) )
             => ! [Mmoi: option(product_prod(nat,nat)),Degi: nat,Tree_array: array(vEBT_VEBTi),Summaryi: vEBT_VEBTi] :
                  ( ( Xaa = vEBT_Nodei(Mmoi,Degi,Tree_array,Summaryi) )
                 => ( Y != aa(assn,assn,
                        aa(assn,fun(assn,assn),times_times(assn),
                          aa(assn,assn,
                            aa(assn,fun(assn,assn),times_times(assn),
                              pure_assn(( ( Mmoi = Mmo )
                                & ( Degi = Deg ) ))),
                            aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),Summaryi))),
                        ex_assn(list(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn),aTP_Lamp_aer(list(vEBT_VEBT),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn)),Tree_list),Tree_array))) ) ) )
         => ( ( ? [V3: option(product_prod(nat,nat)),Va2: nat,Vb3: list(vEBT_VEBT),Vc3: vEBT_VEBT] : Xb = vEBT_Node(V3,Va2,Vb3,Vc3)
             => ( ? [Vd3: $o,Ve3: $o] : Xaa = vEBT_Leafi((Vd3),(Ve3))
               => ( Y != bot_bot(assn) ) ) )
           => ~ ( ? [Vd3: $o,Ve3: $o] : Xb = vEBT_Leaf((Vd3),(Ve3))
               => ( ? [V3: option(product_prod(nat,nat)),Va2: nat,Vb3: array(vEBT_VEBTi),Vc3: vEBT_VEBTi] : Xaa = vEBT_Nodei(V3,Va2,Vb3,Vc3)
                 => ( Y != bot_bot(assn) ) ) ) ) ) ) ) ).

% vebt_assn_raw.elims
tff(fact_5384_freeze__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),Xs: list(A)] : hoare_hoare_triple(list(A),aa(list(A),assn,snga_assn(A,A3),Xs),array_freeze(A,A3),aa(list(A),fun(list(A),assn),aTP_Lamp_aes(array(A),fun(list(A),fun(list(A),assn)),A3),Xs)) ) ).

% freeze_rule
tff(fact_5385_fun__of__rel__def,axiom,
    ! [A: $tType,B: $tType,R: set(product_prod(B,A)),Xb: B] : fun_of_rel(B,A,R,Xb) = fChoice(A,aa(B,fun(A,$o),aTP_Lamp_aet(set(product_prod(B,A)),fun(B,fun(A,$o)),R),Xb)) ).

% fun_of_rel_def
tff(fact_5386_norm__pre__ex__rule,axiom,
    ! [A: $tType,B: $tType,P: fun(A,assn),Fa: heap_Time_Heap(B),Q: fun(B,assn)] :
      ( ! [X2: A] : hoare_hoare_triple(B,aa(A,assn,P,X2),Fa,Q)
     => hoare_hoare_triple(B,ex_assn(A,P),Fa,Q) ) ).

% norm_pre_ex_rule
tff(fact_5387_post__exI__rule,axiom,
    ! [B: $tType,A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,fun(B,assn)),Xb: B] :
      ( hoare_hoare_triple(A,P,C2,aa(B,fun(A,assn),aTP_Lamp_aeu(fun(A,fun(B,assn)),fun(B,fun(A,assn)),Q),Xb))
     => hoare_hoare_triple(A,P,C2,aTP_Lamp_aev(fun(A,fun(B,assn)),fun(A,assn),Q)) ) ).

% post_exI_rule
tff(fact_5388_norm__post__ex__rule__htt,axiom,
    ! [B: $tType,A: $tType,P: assn,Fa: heap_Time_Heap(A),Q: fun(B,fun(A,assn)),Xb: B,Tb: nat] :
      ( time_htt(A,P,Fa,aa(B,fun(A,assn),Q,Xb),Tb)
     => time_htt(A,P,Fa,aTP_Lamp_aex(fun(B,fun(A,assn)),fun(A,assn),Q),Tb) ) ).

% norm_post_ex_rule_htt
tff(fact_5389_norm__pre__ex__rule__htt,axiom,
    ! [B: $tType,A: $tType,P: fun(A,assn),Fa: heap_Time_Heap(B),Q: fun(B,assn),Tb: nat] :
      ( ! [X2: A] : time_htt(B,aa(A,assn,P,X2),Fa,Q,Tb)
     => time_htt(B,ex_assn(A,P),Fa,Q,Tb) ) ).

% norm_pre_ex_rule_htt
tff(fact_5390_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [Mmo2: option(product_prod(nat,nat)),Deg2: nat,Tree_list2: list(vEBT_VEBT),Summary2: vEBT_VEBT,Mmoi2: option(product_prod(nat,nat)),Degi2: nat,Tree_array2: array(vEBT_VEBTi),Summaryi2: vEBT_VEBTi] :
      aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_Node(Mmo2,Deg2,Tree_list2,Summary2)),vEBT_Nodei(Mmoi2,Degi2,Tree_array2,Summaryi2)) = aa(assn,assn,
        aa(assn,fun(assn,assn),times_times(assn),
          aa(assn,assn,
            aa(assn,fun(assn,assn),times_times(assn),
              pure_assn(( ( Mmoi2 = Mmo2 )
                & ( Degi2 = Deg2 ) ))),
            aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),Summaryi2))),
        ex_assn(list(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn),aTP_Lamp_aer(list(vEBT_VEBT),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn)),Tree_list2),Tree_array2))) ).

% vebt_assn_raw.simps(2)
tff(fact_5391_mod__h__bot__iff_I8_J,axiom,
    ! [A: $tType,R: fun(A,assn),H: heap_ext(product_unit)] :
      ( rep_assn(ex_assn(A,R),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))
    <=> ? [X: A] : rep_assn(aa(A,assn,R,X),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat)))) ) ).

% mod_h_bot_iff(8)
tff(fact_5392_mod__ex__dist,axiom,
    ! [A: $tType,P: fun(A,assn),H: product_prod(heap_ext(product_unit),set(nat))] :
      ( rep_assn(ex_assn(A,P),H)
    <=> ? [X: A] : rep_assn(aa(A,assn,P,X),H) ) ).

% mod_ex_dist
tff(fact_5393_norm__assertion__simps_I17_J,axiom,
    ! [A: $tType,R: assn,Q: fun(A,assn)] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R),ex_assn(A,Q)) = ex_assn(A,aa(fun(A,assn),fun(A,assn),aTP_Lamp_aey(assn,fun(fun(A,assn),fun(A,assn)),R),Q)) ).

% norm_assertion_simps(17)
tff(fact_5394_ex__assn__const,axiom,
    ! [A: $tType,C2: assn] : ex_assn(A,aTP_Lamp_aez(assn,fun(A,assn),C2)) = C2 ).

% ex_assn_const
tff(fact_5395_norm__assertion__simps_I16_J,axiom,
    ! [A: $tType,Q: fun(A,assn),R: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),ex_assn(A,Q)),R) = ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_aei(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ).

% norm_assertion_simps(16)
tff(fact_5396_frame__rule__left,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),R: assn] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R),P),C2,aa(assn,fun(A,assn),aTP_Lamp_afa(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ) ).

% frame_rule_left
tff(fact_5397_ex__distrib__star,axiom,
    ! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_aei(fun(A,assn),fun(assn,fun(A,assn)),P),Q)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),ex_assn(A,P)),Q) ).

% ex_distrib_star
tff(fact_5398_ex__one__point__gen,axiom,
    ! [A: $tType,P: fun(A,assn),V: A] :
      ( ! [H5: product_prod(heap_ext(product_unit),set(nat)),X2: A] :
          ( rep_assn(aa(A,assn,P,X2),H5)
         => ( X2 = V ) )
     => ( ex_assn(A,P) = aa(A,assn,P,V) ) ) ).

% ex_one_point_gen
tff(fact_5399_mod__exI,axiom,
    ! [A: $tType,P: fun(A,assn),H: product_prod(heap_ext(product_unit),set(nat))] :
      ( ? [X3: A] : rep_assn(aa(A,assn,P,X3),H)
     => rep_assn(ex_assn(A,P),H) ) ).

% mod_exI
tff(fact_5400_mod__exE,axiom,
    ! [A: $tType,P: fun(A,assn),H: product_prod(heap_ext(product_unit),set(nat))] :
      ( rep_assn(ex_assn(A,P),H)
     => ~ ! [X2: A] : ~ rep_assn(aa(A,assn,P,X2),H) ) ).

% mod_exE
tff(fact_5401_mod__h__bot__normalize,axiom,
    ! [A: $tType,H: heap_ext(product_unit),P: assn] :
      ( syntax7388354845996824322omatch(A,heap_ext(product_unit),undefined(A),H)
     => ( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),bot_bot(set(nat))))
      <=> rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),undefined(heap_ext(product_unit))),bot_bot(set(nat)))) ) ) ).

% mod_h_bot_normalize
tff(fact_5402_length__corresp,axiom,
    ! [B: $tType,A: $tType] :
      ( heap(A)
     => ! [Tree_array2: array(A),Tree_is: list(B)] :
          ( ( ex_assn(list(A),snga_assn(A,Tree_array2)) = top_top(assn) )
         => ( aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),aa(list(B),nat,size_size(list(B)),Tree_is)) = array_len(A,Tree_array2) ) ) ) ).

% length_corresp
tff(fact_5403_vebt__assn__raw_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: vEBT_VEBTi,Y: assn] :
      ( ( aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Xb),Xaa) = Y )
     => ( accp(product_prod(vEBT_VEBT,vEBT_VEBTi),vEBT_v8524038756793281170aw_rel,aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),Xb),Xaa))
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ! [Ai: $o,Bi: $o] :
                  ( ( Xaa = vEBT_Leafi((Ai),(Bi)) )
                 => ( ( Y = pure_assn(( ( (Ai) = (A2) )
                          & ( (Bi) = (B2) ) )) )
                   => ~ accp(product_prod(vEBT_VEBT,vEBT_VEBTi),vEBT_v8524038756793281170aw_rel,aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),vEBT_Leaf((A2),(B2))),vEBT_Leafi((Ai),(Bi)))) ) ) )
         => ( ! [Mmo: option(product_prod(nat,nat)),Deg: nat,Tree_list: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mmo,Deg,Tree_list,Summary) )
               => ! [Mmoi: option(product_prod(nat,nat)),Degi: nat,Tree_array: array(vEBT_VEBTi),Summaryi: vEBT_VEBTi] :
                    ( ( Xaa = vEBT_Nodei(Mmoi,Degi,Tree_array,Summaryi) )
                   => ( ( Y = aa(assn,assn,
                            aa(assn,fun(assn,assn),times_times(assn),
                              aa(assn,assn,
                                aa(assn,fun(assn,assn),times_times(assn),
                                  pure_assn(( ( Mmoi = Mmo )
                                    & ( Degi = Deg ) ))),
                                aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),Summaryi))),
                            ex_assn(list(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn),aTP_Lamp_aer(list(vEBT_VEBT),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn)),Tree_list),Tree_array))) )
                     => ~ accp(product_prod(vEBT_VEBT,vEBT_VEBTi),vEBT_v8524038756793281170aw_rel,aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),vEBT_Node(Mmo,Deg,Tree_list,Summary)),vEBT_Nodei(Mmoi,Degi,Tree_array,Summaryi))) ) ) )
           => ( ! [V3: option(product_prod(nat,nat)),Va2: nat,Vb3: list(vEBT_VEBT),Vc3: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(V3,Va2,Vb3,Vc3) )
                 => ! [Vd3: $o,Ve3: $o] :
                      ( ( Xaa = vEBT_Leafi((Vd3),(Ve3)) )
                     => ( ( Y = bot_bot(assn) )
                       => ~ accp(product_prod(vEBT_VEBT,vEBT_VEBTi),vEBT_v8524038756793281170aw_rel,aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),vEBT_Node(V3,Va2,Vb3,Vc3)),vEBT_Leafi((Vd3),(Ve3)))) ) ) )
             => ~ ! [Vd3: $o,Ve3: $o] :
                    ( ( Xb = vEBT_Leaf((Vd3),(Ve3)) )
                   => ! [V3: option(product_prod(nat,nat)),Va2: nat,Vb3: array(vEBT_VEBTi),Vc3: vEBT_VEBTi] :
                        ( ( Xaa = vEBT_Nodei(V3,Va2,Vb3,Vc3) )
                       => ( ( Y = bot_bot(assn) )
                         => ~ accp(product_prod(vEBT_VEBT,vEBT_VEBTi),vEBT_v8524038756793281170aw_rel,aa(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi),aa(vEBT_VEBT,fun(vEBT_VEBTi,product_prod(vEBT_VEBT,vEBT_VEBTi)),product_Pair(vEBT_VEBT,vEBT_VEBTi),vEBT_Leaf((Vd3),(Ve3))),vEBT_Nodei(V3,Va2,Vb3,Vc3))) ) ) ) ) ) ) ) ) ).

% vebt_assn_raw.pelims
tff(fact_5404_top__option__def,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ( top_top(option(A)) = aa(A,option(A),some(A),top_top(A)) ) ) ).

% top_option_def
tff(fact_5405_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( ( A3 != top_top(A) )
        <=> aa(A,$o,ord_less(A,A3),top_top(A)) ) ) ).

% top.not_eq_extremum
tff(fact_5406_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : ~ aa(A,$o,ord_less(A,top_top(A)),A3) ) ).

% top.extremum_strict
tff(fact_5407_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,top_top(A)),A3)
         => ( A3 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_5408_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] :
          ( aa(A,$o,ord_less_eq(A,top_top(A)),A3)
        <=> ( A3 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_5409_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A3: A] : aa(A,$o,ord_less_eq(A,A3),top_top(A)) ) ).

% top_greatest
tff(fact_5410_assnle,axiom,
    ! [TreeList: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),X13: array(vEBT_VEBTi),Summary2: vEBT_VEBT,X14: vEBT_VEBTi] : aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14)))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is))) ).

% assnle
tff(fact_5411_hoare__triple__effect,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),H: heap_ext(product_unit),As3: set(nat)] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H),As3))
       => ? [H8: heap_ext(product_unit),R4: A,T2: nat] :
            ( heap_Time_effect(A,C2,H,H8,R4,T2)
            & rep_assn(aa(A,assn,Q,R4),aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H8),hoare_new_addrs(H,As3,H8))) ) ) ) ).

% hoare_triple_effect
tff(fact_5412_TBOUND__assert_H__bind,axiom,
    ! [A: $tType,P: $o,P3: $o,M: heap_Time_Heap(A),Tb: nat] :
      ( time_EQ($o,(P),(P3))
     => ( ( (P)
         => time_TBOUND(A,M,Tb) )
       => time_TBOUND(A,heap_Time_bind(product_unit,A,refine_Imp_assert((P)),aTP_Lamp_gb(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),M)),
            $ite((P3),Tb,zero_zero(nat))) ) ) ).

% TBOUND_assert'_bind
tff(fact_5413_finite__option__UNIV,axiom,
    ! [A: $tType] :
      ( finite_finite2(option(A),top_top(set(option(A))))
    <=> finite_finite2(A,top_top(set(A))) ) ).

% finite_option_UNIV
tff(fact_5414_Collect__const,axiom,
    ! [A: $tType,P: $o] :
      collect(A,aTP_Lamp_afb($o,fun(A,$o),(P))) = $ite((P),top_top(set(A)),bot_bot(set(A))) ).

% Collect_const
tff(fact_5415_finite__Collect__not,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( finite_finite2(A,collect(A,P))
     => ( finite_finite2(A,collect(A,aTP_Lamp_vu(fun(A,$o),fun(A,$o),P)))
      <=> finite_finite2(A,top_top(set(A))) ) ) ).

% finite_Collect_not
tff(fact_5416_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P: $o] :
      collect(product_prod(A,B),product_case_prod(A,B,$o,aTP_Lamp_afc($o,fun(A,fun(B,$o)),(P)))) = $ite((P),top_top(set(product_prod(A,B))),bot_bot(set(product_prod(A,B)))) ).

% Collect_const_case_prod
tff(fact_5417_triv__exI,axiom,
    ! [A: $tType,Q: fun(A,assn),Xb: A] : aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(A,assn,Q,Xb)),ex_assn(A,Q)) ).

% triv_exI
tff(fact_5418_ent__ex__postI,axiom,
    ! [A: $tType,P: assn,Q: fun(A,assn),Xb: A] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(A,assn,Q,Xb))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,P),ex_assn(A,Q)) ) ).

% ent_ex_postI
tff(fact_5419_ent__ex__preI,axiom,
    ! [A: $tType,P: fun(A,assn),Q: assn] :
      ( ! [X2: A] : aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(A,assn,P,X2)),Q)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,ex_assn(A,P)),Q) ) ).

% ent_ex_preI
tff(fact_5420_enorm__exI_H,axiom,
    ! [A: $tType,Z6: fun(A,$o),P: assn,Q: fun(A,assn)] :
      ( ! [X2: A] :
          ( aa(A,$o,Z6,X2)
         => aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(A,assn,Q,X2)) )
     => ( ? [X_13: A] : aa(A,$o,Z6,X_13)
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,P),ex_assn(A,Q)) ) ) ).

% enorm_exI'
tff(fact_5421_top__empty__eq2,axiom,
    ! [B: $tType,A: $tType,X3: A,Xa: B] :
      ( aa(B,$o,aa(A,fun(B,$o),top_top(fun(A,fun(B,$o))),X3),Xa)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X3),Xa),top_top(set(product_prod(A,B)))) ) ).

% top_empty_eq2
tff(fact_5422_cons__pre__rule,axiom,
    ! [A: $tType,P: assn,P3: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),P3)
     => ( hoare_hoare_triple(A,P3,C2,Q)
       => hoare_hoare_triple(A,P,C2,Q) ) ) ).

% cons_pre_rule
tff(fact_5423_return__cons__rule,axiom,
    ! [A: $tType,P: assn,Q: fun(A,assn),Xb: A] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(A,assn,Q,Xb))
     => hoare_hoare_triple(A,P,aa(A,heap_Time_Heap(A),heap_Time_return(A),Xb),Q) ) ).

% return_cons_rule
tff(fact_5424_fi__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Ps: assn,F5: assn] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Ps),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),F5))
       => hoare_hoare_triple(A,Ps,C2,aa(assn,fun(A,assn),aTP_Lamp_aei(fun(A,assn),fun(assn,fun(A,assn)),Q),F5)) ) ) ).

% fi_rule
tff(fact_5425_htt__cons__rule,axiom,
    ! [A: $tType,P3: assn,C2: heap_Time_Heap(A),Q6: fun(A,assn),T3: nat,P: assn,Q: fun(A,assn),Tb: nat] :
      ( time_htt(A,P3,C2,Q6,T3)
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),P3)
       => ( ! [X2: A] : aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(A,assn,Q6,X2)),aa(A,assn,Q,X2))
         => ( aa(nat,$o,ord_less_eq(nat,T3),Tb)
           => time_htt(A,P,C2,Q,Tb) ) ) ) ) ).

% htt_cons_rule
tff(fact_5426_TBOUND__Let__strong,axiom,
    ! [B: $tType,A: $tType,V: A,V4: A,Fa: fun(A,heap_Time_Heap(B)),Bnd: fun(A,nat)] :
      ( time_EQ(A,V,V4)
     => ( time_TBOUND(B,aa(A,heap_Time_Heap(B),Fa,V),aa(A,nat,Bnd,V))
       => time_TBOUND(B,aa(A,heap_Time_Heap(B),Fa,V),aa(A,nat,Bnd,V4)) ) ) ).

% TBOUND_Let_strong
tff(fact_5427_listI__assn__wrap__insert,axiom,
    ! [A: $tType,P: assn,Uu: vEBT_VEBT,Uua: nat,Xi: vEBT_VEBTi,I5: set(nat),I: nat,Xs: list(vEBT_VEBT),Xsi: list(vEBT_VEBTi),F5: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(Uu,Uua)),Xi)),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,Xs,Xsi))),F5))
     => ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Xs))
       => ( member(nat,I,I5)
         => ( hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,I5,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,Xs,I,vEBT_vebt_insert(Uu,Uua)),list_update(vEBT_VEBTi,Xsi,I,Xi))),F5),C2,Q)
           => hoare_hoare_triple(A,P,C2,Q) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_5428_big__assn__simp,axiom,
    ! [H: nat,TreeList: list(vEBT_VEBT),L: nat,Xb: vEBT_VEBTi,Xaa: option(nat),X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,H),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L)),Xb)),pure_assn(Xaa = vEBT_vebt_mint(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H,Xb))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),insert(nat,H),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is)))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H,Xb))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),pure_assn(Xaa = vEBT_vebt_mint(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L))),list_update(vEBT_VEBTi,Tree_is,H,Xb)))) ) ).

% big_assn_simp
tff(fact_5429_big__assn__simp_H,axiom,
    ! [H: nat,TreeList: list(vEBT_VEBT),Xaa: vEBT_VEBT,L: nat,Xb: vEBT_VEBTi,Xba: option(nat),X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,H),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( ( Xaa = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),H),L) )
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Xaa),Xb)),pure_assn(Xba = vEBT_vebt_mint(Xaa)))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H,Xb))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),insert(nat,H),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is)))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,H,Xb))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),pure_assn(Xba = vEBT_vebt_mint(Xaa)))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,H,Xaa)),list_update(vEBT_VEBTi,Tree_is,H,Xb)))) ) ) ).

% big_assn_simp'
tff(fact_5430_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or7035219750837199246ssThan(A,L,U))
        <=> ( aa(A,$o,ord_less_eq(A,L),I)
            & aa(A,$o,ord_less(A,I),U) ) ) ) ).

% atLeastLessThan_iff
tff(fact_5431_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( set_or7035219750837199246ssThan(A,A3,B3) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_5432_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B3) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,ord_less(A,A3),B3) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_5433_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A3,B3) )
        <=> ~ aa(A,$o,ord_less(A,A3),B3) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5434_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ finite_finite2(A,set_or7035219750837199246ssThan(A,A3,B3))
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% infinite_Ico_iff
tff(fact_5435_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,J: A,M: A,N: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or7035219750837199246ssThan(A,I,J)),set_or7035219750837199246ssThan(A,M,N))
        <=> ( aa(A,$o,ord_less_eq(A,J),I)
            | ( aa(A,$o,ord_less_eq(A,M),I)
              & aa(A,$o,ord_less_eq(A,J),N) ) ) ) ) ).

% ivl_subset
tff(fact_5436_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,N: A,M: A] :
          ( aa(A,$o,ord_less_eq(A,I),N)
         => ( aa(set(A),set(A),minus_minus(set(A),set_or7035219750837199246ssThan(A,I,M)),set_or7035219750837199246ssThan(A,I,N)) = set_or7035219750837199246ssThan(A,N,M) ) ) ) ).

% ivl_diff
tff(fact_5437_local_Oext,axiom,
    ! [Y: nat,TreeList: list(vEBT_VEBT),X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,Y),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),Y)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),Y))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),insert(nat,Y),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),insert(nat,Y),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),Y)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),Y)))) ) ).

% local.ext
tff(fact_5438_recomp,axiom,
    ! [I: nat,TreeList: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),X13: array(vEBT_VEBTi),Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),I))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is))) ) ).

% recomp
tff(fact_5439_repack,axiom,
    ! [I: nat,TreeList: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),Rest: assn,X13: array(vEBT_VEBTi),Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),I))),Rest)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is)),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is)))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Rest),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is))) ) ).

% repack
tff(fact_5440_txe,axiom,
    ! [Y: nat,TreeList: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),X13: array(vEBT_VEBTi),Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,Y),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),Y)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),Y))),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),insert(nat,Y),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeList,Tree_is))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),Tree_is))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeList),Tree_is))) ) ).

% txe
tff(fact_5441_tcd,axiom,
    ! [A: $tType,I: nat,TreeList: list(vEBT_VEBT),TreeList4: list(A),Y: vEBT_VEBT,Xb: vEBT_VEBTi,X13: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summary2: vEBT_VEBT,X14: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(list(A),nat,size_size(list(A)),TreeList4) )
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Y),Xb)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,I,Xb)))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList))),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,I,Y),list_update(vEBT_VEBTi,Tree_is,I,Xb)))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13),list_update(vEBT_VEBTi,Tree_is,I,Xb))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary2),X14))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeList,I,Y)),list_update(vEBT_VEBTi,Tree_is,I,Xb)))) ) ) ).

% tcd
tff(fact_5442_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,ord_less(nat,N),M),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N))) ) ).

% sum.op_ivl_Suc
tff(fact_5443_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] :
          groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N))) = $ite(aa(nat,$o,ord_less(nat,N),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N))) ) ).

% prod.op_ivl_Suc
tff(fact_5444_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = collect(A,aTP_Lamp_do(A,$o)) ).

% UNIV_def
tff(fact_5445_ex__nat__less__eq,axiom,
    ! [N: nat,P: fun(nat,$o)] :
      ( ? [M6: nat] :
          ( aa(nat,$o,ord_less(nat,M6),N)
          & aa(nat,$o,P,M6) )
    <=> ? [X: nat] :
          ( member(nat,X,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))
          & aa(nat,$o,P,X) ) ) ).

% ex_nat_less_eq
tff(fact_5446_all__nat__less__eq,axiom,
    ! [N: nat,P: fun(nat,$o)] :
      ( ! [M6: nat] :
          ( aa(nat,$o,ord_less(nat,M6),N)
         => aa(nat,$o,P,M6) )
    <=> ! [X: nat] :
          ( member(nat,X,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))
         => aa(nat,$o,P,X) ) ) ).

% all_nat_less_eq
tff(fact_5447_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or7035219750837199246ssThan(A,A3,B3)),set_or7035219750837199246ssThan(A,C2,D))
         => ( aa(A,$o,ord_less_eq(A,B3),A3)
            | ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less_eq(A,B3),D) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_5448_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ~ finite_finite2(A,set_or7035219750837199246ssThan(A,A3,B3)) ) ) ).

% infinite_Ico
tff(fact_5449_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( aa(A,$o,ord_less(A,C2),D)
           => ( ( set_or7035219750837199246ssThan(A,A3,B3) = set_or7035219750837199246ssThan(A,C2,D) )
            <=> ( ( A3 = C2 )
                & ( B3 = D ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_5450_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B3) = set_or7035219750837199246ssThan(A,C2,D) )
         => ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less(A,C2),D)
             => ( A3 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_5451_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( ( set_or7035219750837199246ssThan(A,A3,B3) = set_or7035219750837199246ssThan(A,C2,D) )
         => ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less(A,C2),D)
             => ( B3 = D ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_5452_list__assn__conv__idx,axiom,
    ! [A: $tType,B: $tType,A4: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B)] : aa(list(B),assn,vEBT_List_list_assn(A,B,A4,Xs),Xsi) = vEBT_List_listI_assn(A,B,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs)),A4,Xs,Xsi) ).

% list_assn_conv_idx
tff(fact_5453_listI__assn__conv,axiom,
    ! [A: $tType,B: $tType,N: nat,Xs: list(A),A4: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( ( N = aa(list(A),nat,size_size(list(A)),Xs) )
     => ( vEBT_List_listI_assn(A,B,set_or7035219750837199246ssThan(nat,zero_zero(nat),N),A4,Xs,Xsi) = aa(list(B),assn,vEBT_List_list_assn(A,B,A4,Xs),Xsi) ) ) ).

% listI_assn_conv
tff(fact_5454_lessThan__atLeast0,axiom,
    ! [N: nat] : set_ord_lessThan(nat,N) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ).

% lessThan_atLeast0
tff(fact_5455_atLeastLessThan0,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,zero_zero(nat)) = bot_bot(set(nat)) ).

% atLeastLessThan0
tff(fact_5456_listI__assn__conv_H,axiom,
    ! [B: $tType,A: $tType,N: nat,Xs: list(A),A4: fun(A,fun(B,assn)),Xsi: list(B),F5: assn] :
      ( ( N = aa(list(A),nat,size_size(list(A)),Xs) )
     => ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,set_or7035219750837199246ssThan(nat,zero_zero(nat),N),A4,Xs,Xsi)),F5) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,A4,Xs),Xsi)),F5) ) ) ).

% listI_assn_conv'
tff(fact_5457_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_uw(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_5458_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ux(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_5459_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_yi(fun(nat,A),fun(nat,A),G),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_5460_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),K))) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_yk(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_5461_sum_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_add(B) )
     => ! [A3: A,C2: A,B3: A,D: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A3 = C2 )
         => ( ( B3 = D )
           => ( ! [X2: A] :
                  ( aa(A,$o,ord_less_eq(A,C2),X2)
                 => ( aa(A,$o,ord_less(A,X2),D)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),set_or7035219750837199246ssThan(A,A3,B3)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),set_or7035219750837199246ssThan(A,C2,D)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_5462_prod_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_mult(B) )
     => ! [A3: A,C2: A,B3: A,D: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A3 = C2 )
         => ( ( B3 = D )
           => ( ! [X2: A] :
                  ( aa(A,$o,ord_less_eq(A,C2),X2)
                 => ( aa(A,$o,ord_less(A,X2),D)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) ) )
             => ( groups7121269368397514597t_prod(A,B,G,set_or7035219750837199246ssThan(A,A3,B3)) = groups7121269368397514597t_prod(A,B,H,set_or7035219750837199246ssThan(A,C2,D)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_5463_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,P2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(nat,$o,ord_less_eq(nat,N),P2)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,N,P2))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,P2)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_5464_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,P2: nat,Fa: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(nat,$o,ord_less_eq(nat,N),P2)
           => ( aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_or7035219750837199246ssThan(nat,M,P2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_or7035219750837199246ssThan(nat,M,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_or7035219750837199246ssThan(nat,N,P2)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_5465_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,P2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(nat,$o,ord_less_eq(nat,N),P2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N))),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,N,P2))) = groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,P2)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_5466_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ).

% atLeast0_lessThan_Suc
tff(fact_5467_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N2: set(nat),N: nat] :
      ( aa(set(nat),$o,ord_less_eq(set(nat),N2),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))
     => finite_finite2(nat,N2) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_5468_listI__assn__weak__cong,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),I6: set(nat),A4: fun(A,fun(B,assn)),A9: fun(A,fun(B,assn)),Xs: list(A),Xs4: list(A),Xsi: list(B),Xsi2: list(B)] :
      ( ( I5 = I6 )
     => ( ( A4 = A9 )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Xs4) )
         => ( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(B),nat,size_size(list(B)),Xsi2) )
           => ( ! [I2: nat] :
                  ( member(nat,I2,I5)
                 => ( aa(nat,$o,ord_less(nat,I2),aa(list(A),nat,size_size(list(A)),Xs))
                   => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Xsi) )
                     => ( ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Xs4),I2) )
                        & ( aa(nat,B,nth(B,Xsi),I2) = aa(nat,B,nth(B,Xsi2),I2) ) ) ) ) )
             => ( vEBT_List_listI_assn(A,B,I5,A4,Xs,Xsi) = vEBT_List_listI_assn(A,B,I6,A9,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5469_listI__assn__cong,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),I6: set(nat),Xs: list(A),Xs4: list(A),Xsi: list(B),Xsi2: list(B),A4: fun(A,fun(B,assn)),A9: fun(A,fun(B,assn))] :
      ( ( I5 = I6 )
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Xs4) )
       => ( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(B),nat,size_size(list(B)),Xsi2) )
         => ( ! [I2: nat] :
                ( member(nat,I2,I5)
               => ( aa(nat,$o,ord_less(nat,I2),aa(list(A),nat,size_size(list(A)),Xs))
                 => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Xsi) )
                   => ( ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,nth(A,Xs4),I2) )
                      & ( aa(nat,B,nth(B,Xsi),I2) = aa(nat,B,nth(B,Xsi2),I2) )
                      & ( aa(B,assn,aa(A,fun(B,assn),A4,aa(nat,A,nth(A,Xs),I2)),aa(nat,B,nth(B,Xsi),I2)) = aa(B,assn,aa(A,fun(B,assn),A9,aa(nat,A,nth(A,Xs4),I2)),aa(nat,B,nth(B,Xsi2),I2)) ) ) ) ) )
           => ( vEBT_List_listI_assn(A,B,I5,A4,Xs,Xsi) = vEBT_List_listI_assn(A,B,I6,A9,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5470_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or7035219750837199246ssThan(A,A3,B3)),set_or1337092689740270186AtMost(A,C2,D))
        <=> ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less_eq(A,B3),D) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_5471_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or1337092689740270186AtMost(A,A3,B3)),set_or7035219750837199246ssThan(A,C2,D))
        <=> ( aa(A,$o,ord_less_eq(A,A3),B3)
           => ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less(A,B3),D) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_5472_subst__not__in,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A4: fun(A,fun(B,assn)),X1: A,Xsi: list(B),X22: B] :
      ( ~ member(nat,I,I5)
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,I5,A4,list_update(A,Xs,I,X1),list_update(B,Xsi,I,X22)) = vEBT_List_listI_assn(A,B,I5,A4,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5473_sum__shift__lb__Suc0__0__upt,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Fa: fun(nat,A),K: nat] :
          ( ( aa(nat,A,Fa,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_5474_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G,N)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_5475_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less(nat,M),N)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_5476_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: nat,B3: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,A3),B3)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B3))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,A3,B3))),aa(nat,A,G,B3)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_5477_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(nat,A,G,N)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_5478_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less(nat,M),N)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),N))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_5479_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: nat,B3: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,A3),B3)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,A3,aa(nat,nat,suc,B3))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,A3,B3))),aa(nat,A,G,B3)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_5480_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,N)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).

% sum.last_plus
tff(fact_5481_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,N)),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N))) ) ) ) ).

% prod.last_plus
tff(fact_5482_atLeastLessThanSuc,axiom,
    ! [M: nat,N: nat] :
      set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,N)) = $ite(aa(nat,$o,ord_less_eq(nat,M),N),aa(set(nat),set(nat),insert(nat,N),set_or7035219750837199246ssThan(nat,M,N)),bot_bot(set(nat))) ).

% atLeastLessThanSuc
tff(fact_5483_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,N: nat,Fa: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_vd(fun(nat,A),fun(nat,A),Fa)),set_or7035219750837199246ssThan(nat,M,N)) = aa(A,A,minus_minus(A,aa(nat,A,Fa,N)),aa(nat,A,Fa,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_5484_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_afd(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or7035219750837199246ssThan(nat,N,M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_5485_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_afe(fun(nat,fun(nat,A)),fun(nat,A),A3)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aaf(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% sum.nested_swap
tff(fact_5486_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,N,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_aff(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M),set_or7035219750837199246ssThan(nat,N,M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_5487_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: fun(nat,fun(nat,A)),N: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_afg(fun(nat,fun(nat,A)),fun(nat,A),A3),set_or1337092689740270186AtMost(nat,zero_zero(nat),N)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aai(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A3),N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% prod.nested_swap
tff(fact_5488_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),K: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afh(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_ord_lessThan(nat,N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).

% sum.nat_group
tff(fact_5489_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),K: nat,N: nat] : groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afi(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_ord_lessThan(nat,N)) = groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),K))) ) ).

% prod.nat_group
tff(fact_5490_prod__Suc__Suc__fact,axiom,
    ! [N: nat] : groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_Suc_fact
tff(fact_5491_prod__Suc__fact,axiom,
    ! [N: nat] : groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) = semiring_char_0_fact(nat,N) ).

% prod_Suc_fact
tff(fact_5492_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = $ite(aa(nat,$o,ord_less(nat,N),M),zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N))) ) ).

% sum.head_if
tff(fact_5493_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] :
          groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = $ite(aa(nat,$o,ord_less(nat,N),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N))),aa(nat,A,G,N))) ) ).

% prod.head_if
tff(fact_5494_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = $ite(aa(nat,$o,ord_less_eq(nat,M),pred_numeral(K)),aa(set(nat),set(nat),insert(nat,pred_numeral(K)),set_or7035219750837199246ssThan(nat,M,pred_numeral(K))),bot_bot(set(nat))) ).

% atLeastLessThan_nat_numeral
tff(fact_5495_fact__prod__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% fact_prod_Suc
tff(fact_5496_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,N,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_vc(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_5497_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,N,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_yo(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),N),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,N),M)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_5498_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A3: A,N: nat] : comm_s3205402744901411588hammer(A,A3,N) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_yr(A,fun(nat,A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% pochhammer_prod
tff(fact_5499_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [N: nat] : semiring_char_0_fact(A,N) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,minus_minus(nat,N),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% fact_prod_rev
tff(fact_5500_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Fa: fun(nat,A)] :
          ( summable(A,Fa)
        <=> ! [E5: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E5)
             => ? [N9: nat] :
                ! [M6: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,N9),M6)
                 => ! [N8: nat] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Fa),set_or7035219750837199246ssThan(nat,M6,N8)))),E5) ) ) ) ) ).

% summable_Cauchy
tff(fact_5501_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Fa: fun(nat,A),S: A,K: nat] :
          ( aa(A,$o,sums(A,Fa),S)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
           => aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_afj(fun(nat,A),fun(nat,fun(nat,A)),Fa),K)),S) ) ) ) ).

% sums_group
tff(fact_5502_listI__assn__insert,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A4: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( ~ member(nat,I,I5)
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),insert(nat,I),I5),A4,Xs,Xsi) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A4,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5503_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(set(nat),set(nat),minus_minus(set(nat),set_ord_lessThan(nat,N)),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_5504_listI__assn__subst,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A4: fun(A,fun(B,assn)),X1: A,Xsi: list(B),X22: B] :
      ( ~ member(nat,I,I5)
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),insert(nat,I),I5),A4,list_update(A,Xs,I,X1),list_update(B,Xsi,I,X22)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A4,X1),X22)),vEBT_List_listI_assn(A,B,I5,A4,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5505_rule__at__index,axiom,
    ! [B: $tType,A: $tType,C: $tType,P: assn,A4: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B),F5: assn,I: nat,C2: heap_Time_Heap(C),Q6: fun(C,assn),F6: fun(C,assn)] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,A4,Xs),Xsi)),F5))
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A4,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A4,Xs,Xsi))),F5),C2,Q6)
         => ( ! [R4: C] : aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(C,assn,Q6,R4)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A4,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A4,Xs,Xsi))),aa(C,assn,F6,R4)))
           => hoare_hoare_triple(C,P,C2,aa(fun(C,assn),fun(C,assn),aa(list(B),fun(fun(C,assn),fun(C,assn)),aa(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn))),aTP_Lamp_afk(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn)))),A4),Xs),Xsi),F6)) ) ) ) ) ).

% rule_at_index
tff(fact_5506_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,K),N)
         => ( semiring_char_0_fact(A,N) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,aa(nat,nat,minus_minus(nat,N),K),N)))),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,N),K))) ) ) ) ).

% fact_split
tff(fact_5507_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,K),N)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(N),K)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afl(nat,fun(nat,fun(nat,A)),K),N),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_5508_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afm(A,fun(nat,fun(nat,A)),A3),K),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_5509_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A3),K)),semiring_char_0_fact(A,K)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_afn(A,fun(nat,A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_5510_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,gbinomial(A,A3),K)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_afn(A,fun(nat,A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_5511_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A3: A,K: nat] : aa(nat,A,gbinomial(A,A3),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_zg(A,fun(nat,A),A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_5512_sum__power2,axiom,
    ! [K: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)))),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),one_one(nat)) ).

% sum_power2
tff(fact_5513_Sum__Ico__nat,axiom,
    ! [M: nat,N: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_vj(nat,nat)),set_or7035219750837199246ssThan(nat,M,N)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,minus_minus(nat,N),one_one(nat)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,minus_minus(nat,M),one_one(nat))))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Sum_Ico_nat
tff(fact_5514_listI__assn__extract,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A4: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( member(nat,I,I5)
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,I5,A4,Xs,Xsi) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A4,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A4,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5515_listI__assn__reinsert,axiom,
    ! [B: $tType,A: $tType,P: assn,A4: fun(A,fun(B,assn)),Xs: list(A),I: nat,Xsi: list(B),I5: set(nat),F5: assn,Q: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A4,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A4,Xs,Xsi))),F5))
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( member(nat,I,I5)
         => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A4,Xs,Xsi)),F5)),Q)
           => aa(assn,$o,aa(assn,fun(assn,$o),entails,P),Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5516_listI__assn__reinsert__upd,axiom,
    ! [B: $tType,A: $tType,P: assn,A4: fun(A,fun(B,assn)),Xb: A,Xi: B,I5: set(nat),I: nat,Xs: list(A),Xsi: list(B),F5: assn,Q: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A4,Xb),Xi)),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A4,Xs,Xsi))),F5))
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( member(nat,I,I5)
         => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A4,list_update(A,Xs,I,Xb),list_update(B,Xsi,I,Xi))),F5)),Q)
           => aa(assn,$o,aa(assn,fun(assn,$o),entails,P),Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5517_listI__assn__reinsert_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,A4: fun(A,fun(B,assn)),Xs: list(A),I: nat,Xsi: list(B),I5: set(nat),F5: assn,C2: heap_Time_Heap(C),Q: fun(C,assn)] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A4,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A4,Xs,Xsi))),F5))
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( member(nat,I,I5)
         => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A4,Xs,Xsi)),F5),C2,Q)
           => hoare_hoare_triple(C,P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5518_listI__assn__reinsert__upd_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,A4: fun(A,fun(B,assn)),Xb: A,Xi: B,I5: set(nat),I: nat,Xs: list(A),Xsi: list(B),F5: assn,C2: heap_Time_Heap(C),Q: fun(C,assn)] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A4,Xb),Xi)),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A4,Xs,Xsi))),F5))
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( member(nat,I,I5)
         => ( hoare_hoare_triple(C,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),vEBT_List_listI_assn(A,B,I5,A4,list_update(A,Xs,I,Xb),list_update(B,Xsi,I,Xi))),F5),C2,Q)
           => hoare_hoare_triple(C,P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5519_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [N: nat,A3: fun(nat,A),B3: fun(nat,A)] :
          ( ! [I2: nat,J2: nat] :
              ( aa(nat,$o,ord_less_eq(nat,I2),J2)
             => ( aa(nat,$o,ord_less(nat,J2),N)
               => aa(A,$o,ord_less_eq(A,aa(nat,A,A3,I2)),aa(nat,A,A3,J2)) ) )
         => ( ! [I2: nat,J2: nat] :
                ( aa(nat,$o,ord_less_eq(nat,I2),J2)
               => ( aa(nat,$o,ord_less(nat,J2),N)
                 => aa(A,$o,ord_less_eq(A,aa(nat,A,B3,J2)),aa(nat,A,B3,I2)) ) )
           => aa(A,$o,ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_afo(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A3),B3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,B3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_5520_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A3: fun(nat,nat),B3: fun(nat,nat)] :
      ( ! [I2: nat,J2: nat] :
          ( aa(nat,$o,ord_less_eq(nat,I2),J2)
         => ( aa(nat,$o,ord_less(nat,J2),N)
           => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,A3,I2)),aa(nat,nat,A3,J2)) ) )
     => ( ! [I2: nat,J2: nat] :
            ( aa(nat,$o,ord_less_eq(nat,I2),J2)
           => ( aa(nat,$o,ord_less(nat,J2),N)
             => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,B3,J2)),aa(nat,nat,B3,I2)) ) )
       => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_afp(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A3),B3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,A3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,B3),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_5521_root__def,axiom,
    ! [N: nat,Xb: real] :
      aa(real,real,root(N),Xb) = $ite(N = zero_zero(nat),zero_zero(real),the_inv_into(real,real,top_top(set(real)),aTP_Lamp_afq(nat,fun(real,real),N),Xb)) ).

% root_def
tff(fact_5522_finite__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] : finite_finite2(code_integer,set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U)) ).

% finite_atLeastZeroLessThan_integer
tff(fact_5523_word__to__1__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( set_or7035219750837199246ssThan(word(A),zero_zero(word(A)),one_one(word(A))) = aa(set(word(A)),set(word(A)),insert(word(A),zero_zero(word(A))),bot_bot(set(word(A)))) ) ) ).

% word_to_1_set
tff(fact_5524_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : finite_finite2(int,set_or7035219750837199246ssThan(int,zero_zero(int),U)) ).

% finite_atLeastZeroLessThan_int
tff(fact_5525_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,L,aa(int,int,aa(int,fun(int,int),plus_plus(int),U),one_one(int))) = set_or1337092689740270186AtMost(int,L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
tff(fact_5526_word__range__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: word(A),A3: word(A)] :
          ( ( B3 != zero_zero(word(A)) )
         => ( set_or1337092689740270186AtMost(word(A),A3,aa(word(A),word(A),minus_minus(word(A),B3),one_one(word(A)))) = set_or7035219750837199246ssThan(word(A),A3,B3) ) ) ) ).

% word_range_minus_1
tff(fact_5527_listI__assn__def,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),A4: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B)] :
      vEBT_List_listI_assn(A,B,I5,A4,Xs,Xsi) = aa(assn,assn,
        aa(assn,fun(assn,assn),times_times(assn),
          pure_assn(( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(A),nat,size_size(list(A)),Xs) )
            & aa(set(nat),$o,ord_less_eq(set(nat),I5),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) ))),
        finite_fold(nat,assn,aa(list(B),fun(nat,fun(assn,assn)),aa(list(A),fun(list(B),fun(nat,fun(assn,assn))),aTP_Lamp_afr(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),A4),Xs),Xsi),one_one(assn),I5)) ).

% listI_assn_def
tff(fact_5528_the__inv__into__def,axiom,
    ! [B: $tType,A: $tType,A4: set(B),Fa: fun(B,A),X3: A] : the_inv_into(B,A,A4,Fa,X3) = the(B,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_afs(set(B),fun(fun(B,A),fun(A,fun(B,$o))),A4),Fa),X3)) ).

% the_inv_into_def
tff(fact_5529_neg__mask__is__div_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,N),aa(word(A),nat,size_size(word(A)),W))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ) ) ).

% neg_mask_is_div'
tff(fact_5530_bit_Ocompl__eq__compl__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,bit_ri4277139882892585799ns_not(A),Xb) = aa(A,A,bit_ri4277139882892585799ns_not(A),Y) )
        <=> ( Xb = Y ) ) ) ).

% bit.compl_eq_compl_iff
tff(fact_5531_bit_Odouble__compl,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)) = Xb ) ).

% bit.double_compl
tff(fact_5532_bit_Oconj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),Xb) = zero_zero(A) ) ).

% bit.conj_cancel_left
tff(fact_5533_bit_Oconj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)) = zero_zero(A) ) ).

% bit.conj_cancel_right
tff(fact_5534_and__and__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A3),B3)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),B3)) = zero_zero(word(A)) ) ).

% and_and_not
tff(fact_5535_word__and__not,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xb)) = zero_zero(word(A)) ) ).

% word_and_not
tff(fact_5536_bit_Ocompl__one,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ) ).

% bit.compl_one
tff(fact_5537_bit_Ocompl__zero,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( aa(A,A,bit_ri4277139882892585799ns_not(A),zero_zero(A)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% bit.compl_zero
tff(fact_5538_NOT__mask__AND__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [W: A,N: nat] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),W),bit_se2239418461657761734s_mask(A,N))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N))) = zero_zero(A) ) ).

% NOT_mask_AND_mask
tff(fact_5539_minus__not__numeral__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,uminus_uminus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),N))) = aa(num,A,numeral_numeral(A),inc(N)) ) ).

% minus_not_numeral_eq
tff(fact_5540_word__bitwise__m1__simps_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = zero_zero(word(A)) ) ) ).

% word_bitwise_m1_simps(1)
tff(fact_5541_even__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4277139882892585799ns_not(A),A3))
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ).

% even_not_iff
tff(fact_5542_not__one__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ( aa(A,A,bit_ri4277139882892585799ns_not(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ).

% not_one_eq
tff(fact_5543_compl__of__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),one_one(word(A))) = aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) ) ) ).

% compl_of_1
tff(fact_5544_mask__eq__0__eq__x,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),W: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),W) = zero_zero(word(A)) )
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),W)) = Xb ) ) ) ).

% mask_eq_0_eq_x
tff(fact_5545_mask__eq__x__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),W: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),W) = Xb )
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),W)) = zero_zero(word(A)) ) ) ) ).

% mask_eq_x_eq_0
tff(fact_5546_not__diff__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B3: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,minus_minus(A,A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),B3) ) ).

% not_diff_distrib
tff(fact_5547_not__add__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B3: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = aa(A,A,minus_minus(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),B3) ) ).

% not_add_distrib
tff(fact_5548_take__bit__not__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3))) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)) ) ).

% take_bit_not_take_bit
tff(fact_5549_take__bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A,B3: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),B3)) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),B3) ) ) ) ).

% take_bit_not_iff
tff(fact_5550_of__int__not__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int] : aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4277139882892585799ns_not(int),K)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(int,A,ring_1_of_int(A),K)) ) ).

% of_int_not_eq
tff(fact_5551_of__int__not__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: num] : aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),K))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),K)) ) ).

% of_int_not_numeral
tff(fact_5552_minus__eq__not__plus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),one_one(A)) ) ).

% minus_eq_not_plus_1
tff(fact_5553_not__eq__complement,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),A3) = aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A3)),one_one(A)) ) ).

% not_eq_complement
tff(fact_5554_minus__eq__not__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A] : aa(A,A,uminus_uminus(A),A3) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,minus_minus(A,A3),one_one(A))) ) ).

% minus_eq_not_minus_1
tff(fact_5555_take__bit__not__eq__mask__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)) = aa(A,A,minus_minus(A,bit_se2239418461657761734s_mask(A,N)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)) ) ).

% take_bit_not_eq_mask_diff
tff(fact_5556_mask__out__first__mask__some,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: nat,Y: word(A),M: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N))) = Y )
         => ( aa(nat,$o,ord_less_eq(nat,N),M)
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) ) ) ) ) ).

% mask_out_first_mask_some
tff(fact_5557_mask__lower__twice,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,N),M)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N)))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) ) ) ) ).

% mask_lower_twice
tff(fact_5558_mask__AND__NOT__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N)))),bit_se2239418461657761734s_mask(word(A),N)) = zero_zero(word(A)) ) ).

% mask_AND_NOT_mask
tff(fact_5559_minus__numeral__inc__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(N))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),N)) ) ).

% minus_numeral_inc_eq
tff(fact_5560_not__numeral__Bit0__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bit0(N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,N))) ) ).

% not_numeral_Bit0_eq
tff(fact_5561_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_5562_not__numeral__BitM__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bitM(N))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(N))) ) ).

% not_numeral_BitM_eq
tff(fact_5563_multiple__mask__trivia,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,N: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N)))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),bit_se2239418461657761734s_mask(word(A),N)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) ) ) ) ).

% multiple_mask_trivia
tff(fact_5564_and__mask__0__iff__le__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N))) = zero_zero(word(A)) )
        <=> aa(word(A),$o,ord_less_eq(word(A),W),bit_se2239418461657761734s_mask(word(A),N)) ) ) ).

% and_mask_0_iff_le_mask
tff(fact_5565_minus__exp__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)) ) ).

% minus_exp_eq_not_mask
tff(fact_5566_NOT__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N)) = aa(word(A),word(A),uminus_uminus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ).

% NOT_mask
tff(fact_5567_neg__mask__is__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ).

% neg_mask_is_div
tff(fact_5568_signed__take__bit__eq__take__bit__minus,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),K)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,suc,N))),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_5569_neg__mask__add__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N)))),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),one_one(word(A)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xb),bit_se2239418461657761734s_mask(word(A),N)) ) ).

% neg_mask_add_mask
tff(fact_5570_Heap_Osize__gen,axiom,
    ! [A: $tType,Xaa: fun(A,nat),Xb: fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] : heap_Time_size_Heap(A,Xaa,heap_Time_Heap2(A,Xb)) = aa(nat,nat,suc,zero_zero(nat)) ).

% Heap.size_gen
tff(fact_5571_or_Oidem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),A3) = A3 ) ).

% or.idem
tff(fact_5572_or_Oleft__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3) ) ).

% or.left_idem
tff(fact_5573_or_Oright__idem,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3)),B3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3) ) ).

% or.right_idem
tff(fact_5574_bit__0__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ( aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),zero_zero(A)) = bot_bot(fun(nat,$o)) ) ) ).

% bit_0_eq
tff(fact_5575_or_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),zero_zero(A)),A3) = A3 ) ).

% or.left_neutral
tff(fact_5576_or_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),zero_zero(A)) = A3 ) ).

% or.right_neutral
tff(fact_5577_take__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A,B3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B3)) ) ).

% take_bit_or
tff(fact_5578_bit_Odisj__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_right
tff(fact_5579_bit_Odisj__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,uminus_uminus(A),one_one(A))),Xb) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_one_left
tff(fact_5580_bit_Ode__Morgan__disj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) ) ).

% bit.de_Morgan_disj
tff(fact_5581_bit_Ode__Morgan__conj,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) ) ).

% bit.de_Morgan_conj
tff(fact_5582_bit__numeral__Bit0__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),bit0(M))),aa(nat,nat,suc,N))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),M)),N) ) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_5583_bit__numeral__Bit1__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),aa(nat,nat,suc,N))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),M)),N) ) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_5584_or__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb)) ) ).

% or_numerals(8)
tff(fact_5585_or__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% or_numerals(2)
tff(fact_5586_bit_Odisj__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_right
tff(fact_5587_bit_Odisj__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),Xb) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.disj_cancel_left
tff(fact_5588_not__negative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,bit_ri4277139882892585799ns_not(int),K)),zero_zero(int))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% not_negative_int_iff
tff(fact_5589_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),K))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% not_nonnegative_int_iff
tff(fact_5590_signed__take__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K))
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_5591_signed__take__bit__negative__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K)),zero_zero(int))
    <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) ) ).

% signed_take_bit_negative_iff
tff(fact_5592_or__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(Xb))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y))) ) ).

% or_numerals(3)
tff(fact_5593_or__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(Xb))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb)) ) ).

% or_numerals(5)
tff(fact_5594_or__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% or_numerals(1)
tff(fact_5595_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(W)))),aa(nat,nat,suc,N))
    <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),N) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_5596_bit__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: num] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),bit0(W))),aa(num,nat,numeral_numeral(nat),N))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),W)),pred_numeral(N)) ) ) ).

% bit_numeral_simps(2)
tff(fact_5597_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W)))),aa(nat,nat,suc,N))
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W)),N) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_5598_bit__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: num] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,W))),aa(num,nat,numeral_numeral(nat),N))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),W)),pred_numeral(N)) ) ) ).

% bit_numeral_simps(3)
tff(fact_5599_word__no__log__defs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),A3))) ) ).

% word_no_log_defs(1)
tff(fact_5600_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),zero_zero(nat))
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ).

% bit_0
tff(fact_5601_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N: num] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(W)))),aa(num,nat,numeral_numeral(nat),N))
    <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),pred_numeral(N)) ) ).

% bit_minus_numeral_int(1)
tff(fact_5602_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N: num] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W)))),aa(num,nat,numeral_numeral(nat),N))
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W)),pred_numeral(N)) ) ).

% bit_minus_numeral_int(2)
tff(fact_5603_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),N)
        <=> ( ( N = zero_zero(nat) )
            & ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ) ).

% bit_mod_2_iff
tff(fact_5604_bin__nth__minus__Bit0,axiom,
    ! [N: nat,W: num] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),bit0(W))),N)
      <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W)),aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ).

% bin_nth_minus_Bit0
tff(fact_5605_bin__nth__minus__Bit1,axiom,
    ! [N: nat,W: num] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W))),N)
      <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W)),aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ).

% bin_nth_minus_Bit1
tff(fact_5606_word__of__int__not__numeral__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),Bin))) = aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin))),one_one(word(A))) ) ).

% word_of_int_not_numeral_eq
tff(fact_5607_word__no__log__defs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))) ) ).

% word_no_log_defs(2)
tff(fact_5608_or__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(7)
tff(fact_5609_or__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(6)
tff(fact_5610_or__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),bit0(Xb))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% or_numerals(4)
tff(fact_5611_bit__minus__int__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),K)),N)
    <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,minus_minus(int,K),one_one(int)))),N) ) ).

% bit_minus_int_iff
tff(fact_5612_signed__take__bit__eq__if__negative,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N))) ) ) ) ).

% signed_take_bit_eq_if_negative
tff(fact_5613_and__eq__not__not__or,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B3) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),aa(A,A,bit_ri4277139882892585799ns_not(A),B3))) ) ).

% and_eq_not_not_or
tff(fact_5614_or__eq__not__not__and,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),aa(A,A,bit_ri4277139882892585799ns_not(A),B3))) ) ).

% or_eq_not_not_and
tff(fact_5615_bit__not__int__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),N)
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) ) ).

% bit_not_int_iff
tff(fact_5616_bit__of__bool__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [B3: $o,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),N)
        <=> ( (B3)
            & ( N = zero_zero(nat) ) ) ) ) ).

% bit_of_bool_iff
tff(fact_5617_of__int__or__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_or_eq
tff(fact_5618_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,bit_se2584673776208193580ke_bit(A,M),A3)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),M)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N) ) ) ) ).

% bit_take_bit_iff
tff(fact_5619_bit__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),M),A3)),N)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
            & ( M != N ) ) ) ) ).

% bit_unset_bit_iff
tff(fact_5620_of__nat__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_or_eq
tff(fact_5621_bit__of__nat__iff__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(nat,A,semiring_1_of_nat(A),M)),N)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),M),N) ) ) ).

% bit_of_nat_iff_bit
tff(fact_5622_or_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B3),C2)) ) ).

% or.assoc
tff(fact_5623_bit__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3)),N)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
            | aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B3),N) ) ) ) ).

% bit_or_iff
tff(fact_5624_or_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B3),A3) ) ).

% or.commute
tff(fact_5625_or_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B3: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B3),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),B3),C2)) ) ).

% or.left_commute
tff(fact_5626_bit__numeral__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),M)),N)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(num,nat,numeral_numeral(nat),M)),N) ) ) ).

% bit_numeral_iff
tff(fact_5627_bit__and__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L)),N)
    <=> ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N)
        & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),L),N) ) ) ).

% bit_and_int_iff
tff(fact_5628_bit_Odisj__conj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),Z)),Xb) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),Xb)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Z),Xb)) ) ).

% bit.disj_conj_distrib2
tff(fact_5629_bit_Oconj__disj__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),Z)),Xb) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),Xb)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z),Xb)) ) ).

% bit.conj_disj_distrib2
tff(fact_5630_bit_Odisj__conj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),Z)) ) ).

% bit.disj_conj_distrib
tff(fact_5631_bit_Oconj__disj__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),Z)) ) ).

% bit.conj_disj_distrib
tff(fact_5632_bit__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B3)),N)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B3),N) ) ) ) ).

% bit_and_iff
tff(fact_5633_disjunctive__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] :
          ( ! [N3: nat] :
              ( ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N3)
              | ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B3),N3) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3) ) ) ) ).

% disjunctive_add
tff(fact_5634_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B3: A,N: nat] :
          ( ! [N3: nat] :
              ( ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N3)
              | ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B3),N3) )
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)),N)
          <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
              | aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B3),N) ) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_5635_word__log__esimps_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),zero_zero(word(A))),Xb) = Xb ) ).

% word_log_esimps(9)
tff(fact_5636_word__log__esimps_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xb),zero_zero(word(A))) = Xb ) ).

% word_log_esimps(3)
tff(fact_5637_word__or__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),A3),B3) = zero_zero(word(A)) )
        <=> ( ( A3 = zero_zero(word(A)) )
            & ( B3 = zero_zero(word(A)) ) ) ) ) ).

% word_or_zero
tff(fact_5638_or__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & ( B3 = zero_zero(A) ) ) ) ) ).

% or_eq_0_iff
tff(fact_5639_bit_Odisj__zero__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),zero_zero(A)) = Xb ) ).

% bit.disj_zero_right
tff(fact_5640_not__bit__1__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),one_one(A)),aa(nat,nat,suc,N)) ) ).

% not_bit_1_Suc
tff(fact_5641_bit__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: num] : ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),one_one(A)),aa(num,nat,numeral_numeral(nat),N)) ) ).

% bit_numeral_simps(1)
tff(fact_5642_bit__1__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),one_one(A)),N)
        <=> ( N = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_5643_signed__take__bit__eq__if__positive,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,N: nat] :
          ( ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
         => ( aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) ) ) ) ).

% signed_take_bit_eq_if_positive
tff(fact_5644_signed__take__bit__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)),aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)))) ) ).

% signed_take_bit_def
tff(fact_5645_disjunctive__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [B3: A,A3: A] :
          ( ! [N3: nat] :
              ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B3),N3)
             => aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N3) )
         => ( aa(A,A,minus_minus(A,A3),B3) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,bit_ri4277139882892585799ns_not(A),B3)) ) ) ) ).

% disjunctive_diff
tff(fact_5646_not__int__def,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),K)),one_one(int)) ).

% not_int_def
tff(fact_5647_and__not__numerals_I1_J,axiom,
    aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = zero_zero(int) ).

% and_not_numerals(1)
tff(fact_5648_word__plus__and__or__coroll,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),Y) = zero_zero(word(A)) )
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xb),Y) ) ) ) ).

% word_plus_and_or_coroll
tff(fact_5649_bit__not__int__iff_H,axiom,
    ! [K: int,N: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),K)),one_one(int))),N)
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) ) ).

% bit_not_int_iff'
tff(fact_5650_flip__bit__eq__if,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          bit_se8732182000553998342ip_bit(A,N,A3) = aa(A,A,
            aa(nat,fun(A,A),
              $ite(aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N),bit_se2638667681897837118et_bit(A),bit_se5668285175392031749et_bit(A)),
              N),
            A3) ) ).

% flip_bit_eq_if
tff(fact_5651_even__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
            & aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B3) ) ) ) ).

% even_or_iff
tff(fact_5652_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),Xb) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),Xb) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),Y) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( Xb = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_5653_bit__numeral__rec_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),bit0(W))),N)
        <=> case_nat($o,$false,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),W)),N) ) ) ).

% bit_numeral_rec(1)
tff(fact_5654_bit__numeral__rec_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,W))),N)
        <=> case_nat($o,$true,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),W)),N) ) ) ).

% bit_numeral_rec(2)
tff(fact_5655_not__int__div__2,axiom,
    ! [K: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),aa(num,int,numeral_numeral(int),bit0(one2))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% not_int_div_2
tff(fact_5656_even__not__iff__int,axiom,
    ! [K: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,bit_ri4277139882892585799ns_not(int),K))
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K) ) ).

% even_not_iff_int
tff(fact_5657_and__not__numerals_I4_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),bit0(M)) ).

% and_not_numerals(4)
tff(fact_5658_and__not__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = one_one(int) ).

% and_not_numerals(2)
tff(fact_5659_bit__imp__take__bit__positive,axiom,
    ! [N: nat,M: nat,K: int] :
      ( aa(nat,$o,ord_less(nat,N),M)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N)
       => aa(int,$o,ord_less(int,zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_5660_mask__subsume,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat,Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,N),M)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Y),bit_se2239418461657761734s_mask(word(A),N)))),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),M))) ) ) ) ).

% mask_subsume
tff(fact_5661_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N: num,Q2: num] :
      ( ( bit_and_not_num(M,N) = aa(num,option(num),some(num),Q2) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = aa(num,int,numeral_numeral(int),Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_5662_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,A3: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
         => ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_5663_bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),aa(nat,nat,suc,N))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),N) ) ) ).

% bit_Suc
tff(fact_5664_bit__iff__idd__imp__stable,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A] :
          ( ! [N3: nat] :
              ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N3)
            <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = A3 ) ) ) ).

% bit_iff_idd_imp_stable
tff(fact_5665_stable__imp__bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = A3 )
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
          <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_5666_bit_Ocompl__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),Y) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( aa(A,A,bit_ri4277139882892585799ns_not(A),Xb) = Y ) ) ) ) ).

% bit.compl_unique
tff(fact_5667_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N3: nat] :
          ( ! [M2: nat] :
              ( aa(nat,$o,ord_less_eq(nat,N3),M2)
             => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),M2)
              <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N3) ) )
         => ~ ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N3)
             => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),aa(nat,nat,minus_minus(nat,N3),one_one(nat)))
              <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N3) ) ) ) ).

% int_bit_bound
tff(fact_5668_and__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% and_not_numerals(5)
tff(fact_5669_and__not__numerals_I7_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(num,int,numeral_numeral(int),bit0(M)) ).

% and_not_numerals(7)
tff(fact_5670_and__not__numerals_I3_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = zero_zero(int) ).

% and_not_numerals(3)
tff(fact_5671_and__not__num__eq__None__iff,axiom,
    ! [M: num,N: num] :
      ( ( bit_and_not_num(M,N) = none(num) )
    <=> ( aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = zero_zero(int) ) ) ).

% and_not_num_eq_None_iff
tff(fact_5672_bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N))) ) ) ).

% bit_iff_odd
tff(fact_5673_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = zero_zero(A) )
        <=> ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_5674_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),N)
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) )
            & ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N) ) ) ) ).

% bit_not_iff_eq
tff(fact_5675_mask__Suc__exp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),bit_se2239418461657761734s_mask(A,N)) ) ).

% mask_Suc_exp
tff(fact_5676_and__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% and_not_numerals(9)
tff(fact_5677_and__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% and_not_numerals(6)
tff(fact_5678_int__numeral__not__and__num,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(N,M)) ).

% int_numeral_not_and_num
tff(fact_5679_int__numeral__and__not__num,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,N)) ).

% int_numeral_and_not_num
tff(fact_5680_bit__int__def,axiom,
    ! [K: int,N: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N)
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ) ).

% bit_int_def
tff(fact_5681_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),N)
          <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_5682_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
          <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,minus_minus(A,A3),one_one(A))),N)
              | ( N = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_5683_mask__Suc__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,N)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,N))) ) ).

% mask_Suc_double
tff(fact_5684_one__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3))) ) ).

% one_or_eq
tff(fact_5685_or__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),one_one(A)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3))) ) ).

% or_one_eq
tff(fact_5686_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A,N: nat] :
          ( ! [J2: nat] : ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),aa(nat,nat,suc,J2))
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3))),N)
          <=> $ite(N = zero_zero(nat),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3)),N)) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_5687_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
        <=> $ite(N = zero_zero(nat),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,minus_minus(nat,N),one_one(nat)))) ) ) ).

% bit_rec
tff(fact_5688_and__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% and_not_numerals(8)
tff(fact_5689_not__int__rec,axiom,
    ! [K: int] : aa(int,int,bit_ri4277139882892585799ns_not(int),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($o,int,zero_neq_one_of_bool(int),aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% not_int_rec
tff(fact_5690_Bit__Operations_Oset__bit__eq,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(int,int,aa(int,fun(int,int),times_times(int),aa($o,int,zero_neq_one_of_bool(int),~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ).

% Bit_Operations.set_bit_eq
tff(fact_5691_unset__bit__eq,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K) = aa(int,int,minus_minus(int,K),aa(int,int,aa(int,fun(int,int),times_times(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))) ).

% unset_bit_eq
tff(fact_5692_take__bit__Suc__from__most,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,N)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N)))),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)) ).

% take_bit_Suc_from_most
tff(fact_5693_int__not__code_I1_J,axiom,
    aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% int_not_code(1)
tff(fact_5694_xor__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = $ite(
        K = aa(int,int,uminus_uminus(int),one_one(int)),
        aa(int,int,bit_ri4277139882892585799ns_not(int),L),
        $ite(
          L = aa(int,int,uminus_uminus(int),one_one(int)),
          aa(int,int,bit_ri4277139882892585799ns_not(int),K),
          $ite(
            K = zero_zero(int),
            L,
            $ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),abs_abs(int,aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ) ).

% xor_int_unfold
tff(fact_5695_num_Osize__gen_I3_J,axiom,
    ! [X33: num] : size_num(aa(num,num,bit1,X33)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X33)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(3)
tff(fact_5696_bit_Oxor__left__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),Y)) = Y ) ).

% bit.xor_left_self
tff(fact_5697_bit_Oxor__self,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),Xb) = zero_zero(A) ) ).

% bit.xor_self
tff(fact_5698_xor__self__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),A3) = zero_zero(A) ) ).

% xor_self_eq
tff(fact_5699_xor_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),zero_zero(A)),A3) = A3 ) ).

% xor.left_neutral
tff(fact_5700_xor_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),zero_zero(A)) = A3 ) ).

% xor.right_neutral
tff(fact_5701_test__bit__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),one_one(word(A))),N)
        <=> ( N = zero_zero(nat) ) ) ) ).

% test_bit_1
tff(fact_5702_take__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A,B3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)),aa(A,A,bit_se2584673776208193580ke_bit(A,N),B3)) ) ).

% take_bit_xor
tff(fact_5703_bit_Oxor__compl__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),aa(A,A,bit_ri4277139882892585799ns_not(A),Y)) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),Y)) ) ).

% bit.xor_compl_right
tff(fact_5704_bit_Oxor__compl__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),Y) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),Y)) ) ).

% bit.xor_compl_left
tff(fact_5705_map__nth__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(nat)] : map(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),zero_zero(word(A))),Xs) = replicate($o,aa(list(nat),nat,size_size(list(nat)),Xs),$false) ) ).

% map_nth_0
tff(fact_5706_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L))
    <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
        & aa(int,$o,ord_less_eq(int,zero_zero(int)),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_5707_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),zero_zero(int))
    <=> ( aa(int,$o,ord_less(int,K),zero_zero(int))
        | aa(int,$o,ord_less(int,L),zero_zero(int)) ) ) ).

% or_negative_int_iff
tff(fact_5708_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L))
    <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
      <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_5709_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),zero_zero(int))
    <=> ~ ( aa(int,$o,ord_less(int,K),zero_zero(int))
        <=> aa(int,$o,ord_less(int,L),zero_zero(int)) ) ) ).

% xor_negative_int_iff
tff(fact_5710_bit_Oxor__one__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,uminus_uminus(A),one_one(A))),Xb) = aa(A,A,bit_ri4277139882892585799ns_not(A),Xb) ) ).

% bit.xor_one_left
tff(fact_5711_bit_Oxor__one__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),Xb) ) ).

% bit.xor_one_right
tff(fact_5712_bit_Oxor__cancel__left,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),Xb) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_left
tff(fact_5713_bit_Oxor__cancel__right,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% bit.xor_cancel_right
tff(fact_5714_xor__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(Xb))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(3)
tff(fact_5715_xor__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y)) ) ).

% xor_numerals(1)
tff(fact_5716_xor__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(num,A,numeral_numeral(A),bit0(Y)) ) ).

% xor_numerals(2)
tff(fact_5717_xor__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(Xb))),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb)) ) ).

% xor_numerals(5)
tff(fact_5718_xor__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb))),one_one(A)) = aa(num,A,numeral_numeral(A),bit0(Xb)) ) ).

% xor_numerals(8)
tff(fact_5719_or__minus__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N))) ).

% or_minus_numerals(2)
tff(fact_5720_or__minus__numerals_I6_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N))) ).

% or_minus_numerals(6)
tff(fact_5721_or__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% or_nat_numerals(2)
tff(fact_5722_or__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb)) ).

% or_nat_numerals(4)
tff(fact_5723_word__no__log__defs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),A3)),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_no_log_defs(7)
tff(fact_5724_xor__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y))) ) ).

% xor_numerals(7)
tff(fact_5725_or__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% or_nat_numerals(1)
tff(fact_5726_or__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(num,nat,numeral_numeral(nat),bit0(Xb))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb)) ).

% or_nat_numerals(3)
tff(fact_5727_word__bitwise__1__simps_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),A3)),one_one(int))) ) ).

% word_bitwise_1_simps(8)
tff(fact_5728_word__bitwise__1__simps_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_bitwise_1_simps(6)
tff(fact_5729_word__no__log__defs_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),A3)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_no_log_defs(8)
tff(fact_5730_word__no__log__defs_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_no_log_defs(9)
tff(fact_5731_word__no__log__defs_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_no_log_defs(10)
tff(fact_5732_or__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),M)),one_one(int))),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),N)),one_one(int)))) ).

% or_minus_minus_numerals
tff(fact_5733_and__minus__minus__numerals,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),M)),one_one(int))),aa(int,int,minus_minus(int,aa(num,int,numeral_numeral(int),N)),one_one(int)))) ).

% and_minus_minus_numerals
tff(fact_5734_xor__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),bit0(Xb))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(4)
tff(fact_5735_xor__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Xb: num,Y: num] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Xb))),aa(num,A,numeral_numeral(A),bit0(Y))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(num,A,numeral_numeral(A),Xb)),aa(num,A,numeral_numeral(A),Y)))) ) ).

% xor_numerals(6)
tff(fact_5736_word__bitwise__1__simps_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),one_one(word(A))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_bitwise_1_simps(7)
tff(fact_5737_word__bitwise__1__simps_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),one_one(int))) ) ).

% word_bitwise_1_simps(9)
tff(fact_5738_xor__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),L))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),L)) ).

% xor_int_def
tff(fact_5739_finite__bit__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : finite_finite2(nat,collect(nat,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W))) ) ).

% finite_bit_word
tff(fact_5740_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),L)
     => aa(int,$o,ord_less_eq(int,K),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)) ) ).

% or_greater_eq
tff(fact_5741_OR__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
       => aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xb),Y)) ) ) ).

% OR_lower
tff(fact_5742_bit__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),B3)),N)
        <=> ~ ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
            <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B3),N) ) ) ) ).

% bit_xor_iff
tff(fact_5743_bit__or__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L)),N)
    <=> ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N)
        | aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),L),N) ) ) ).

% bit_or_int_iff
tff(fact_5744_bit__xor__int__iff,axiom,
    ! [K: int,L: int,N: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)),N)
    <=> ~ ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N)
        <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),L),N) ) ) ).

% bit_xor_int_iff
tff(fact_5745_or__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = nat2(aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% or_nat_def
tff(fact_5746_int__or__code_I1_J,axiom,
    ! [J: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),zero_zero(int)),J) = J ).

% int_or_code(1)
tff(fact_5747_int__or__code_I2_J,axiom,
    ! [I: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),I),zero_zero(int)) = I ).

% int_or_code(2)
tff(fact_5748_nth__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),zero_zero(word(A))),N) ) ).

% nth_0
tff(fact_5749_word__exists__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( W != zero_zero(word(A)) )
         => ? [X_1: nat] : aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),X_1) ) ) ).

% word_exists_nth
tff(fact_5750_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,suc,N)) ).

% not_bit_Suc_0_Suc
tff(fact_5751_bit__Suc__0__iff,axiom,
    ! [N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,suc,zero_zero(nat))),N)
    <=> ( N = zero_zero(nat) ) ) ).

% bit_Suc_0_iff
tff(fact_5752_XOR__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
       => aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xb),Y)) ) ) ).

% XOR_lower
tff(fact_5753_int__xor__code_I2_J,axiom,
    ! [I: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),I),zero_zero(int)) = I ).

% int_xor_code(2)
tff(fact_5754_int__xor__code_I1_J,axiom,
    ! [J: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),zero_zero(int)),J) = J ).

% int_xor_code(1)
tff(fact_5755_bit_Oconj__xor__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),Z)) ) ).

% bit.conj_xor_distrib
tff(fact_5756_bit_Oconj__xor__distrib2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Y: A,Z: A,Xb: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Y),Z)),Xb) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Y),Xb)),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Z),Xb)) ) ).

% bit.conj_xor_distrib2
tff(fact_5757_xor_Oleft__commute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [B3: A,A3: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B3),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),C2)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B3),C2)) ) ).

% xor.left_commute
tff(fact_5758_xor_Ocommute,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),B3) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B3),A3) ) ).

% xor.commute
tff(fact_5759_xor_Oassoc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A,C2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),B3)),C2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),B3),C2)) ) ).

% xor.assoc
tff(fact_5760_of__nat__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_xor_eq
tff(fact_5761_of__int__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,L: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(int,A,ring_1_of_int(A),K)),aa(int,A,ring_1_of_int(A),L)) ) ).

% of_int_xor_eq
tff(fact_5762_plus__and__or,axiom,
    ! [Xb: int,Y: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Y)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xb),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Y) ).

% plus_and_or
tff(fact_5763_test__bit__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),N)
         => aa(nat,$o,ord_less(nat,N),aa(word(A),nat,size_size(word(A)),W)) ) ) ).

% test_bit_size
tff(fact_5764_word__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),V: word(A)] :
          ( ! [N3: nat] :
              ( aa(nat,$o,ord_less(nat,N3),aa(word(A),nat,size_size(word(A)),U))
             => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),U),N3)
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),V),N3) ) )
         => ( U = V ) ) ) ).

% word_eqI
tff(fact_5765_test__bit__over,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,size_size(word(A)),Xb)),N)
         => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),N) ) ) ).

% test_bit_over
tff(fact_5766_or__int__def,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,bit_ri4277139882892585799ns_not(int),K)),aa(int,int,bit_ri4277139882892585799ns_not(int),L))) ).

% or_int_def
tff(fact_5767_not__bit__Suc__0__numeral,axiom,
    ! [N: num] : ~ aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),N)) ).

% not_bit_Suc_0_numeral
tff(fact_5768_or__not__numerals_I1_J,axiom,
    aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).

% or_not_numerals(1)
tff(fact_5769_lsb__this__or__next,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),one_one(word(A)))),zero_zero(nat))
         => aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),zero_zero(nat)) ) ) ).

% lsb_this_or_next
tff(fact_5770_word__leI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),V: word(A)] :
          ( ! [N3: nat] :
              ( aa(nat,$o,ord_less(nat,N3),aa(word(A),nat,size_size(word(A)),U))
             => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),U),N3)
               => aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),V),N3) ) )
         => aa(word(A),$o,ord_less_eq(word(A),U),V) ) ) ).

% word_leI
tff(fact_5771_nth__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,I: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_se2239418461657761734s_mask(word(A),N)),I)
        <=> ( aa(nat,$o,ord_less(nat,I),N)
            & aa(nat,$o,ord_less(nat,I),aa(word(A),nat,size_size(word(A)),bit_se2239418461657761734s_mask(word(A),N))) ) ) ) ).

% nth_mask
tff(fact_5772_bit_Oxor__def2,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),aa(A,A,bit_ri4277139882892585799ns_not(A),Y))) ) ).

% bit.xor_def2
tff(fact_5773_bit_Oxor__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),Xb),Y) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),Xb),aa(A,A,bit_ri4277139882892585799ns_not(A),Y))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,bit_ri4277139882892585799ns_not(A),Xb)),Y)) ) ).

% bit.xor_def
tff(fact_5774_or__not__numerals_I4_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int)) ).

% or_not_numerals(4)
tff(fact_5775_or__not__numerals_I2_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N))) ).

% or_not_numerals(2)
tff(fact_5776_even__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),B3))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)
          <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B3) ) ) ) ).

% even_xor_iff
tff(fact_5777_bit__nat__iff,axiom,
    ! [K: int,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),nat2(K)),N)
    <=> ( aa(int,$o,ord_less_eq(int,zero_zero(int)),K)
        & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) ) ) ).

% bit_nat_iff
tff(fact_5778_overflow__imp__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),one_one(word(A))) = zero_zero(word(A)) )
         => aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),zero_zero(nat)) ) ) ).

% overflow_imp_lsb
tff(fact_5779_word__and__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A)] :
          aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),N),one_one(word(A))) = $ite(aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),N),zero_zero(nat)),one_one(word(A)),zero_zero(word(A))) ) ).

% word_and_1
tff(fact_5780_test__bit__bin_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),N)
        <=> ( aa(nat,$o,ord_less(nat,N),aa(word(A),nat,size_size(word(A)),W))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),N) ) ) ) ).

% test_bit_bin'
tff(fact_5781_le__mask__high__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),W),bit_se2239418461657761734s_mask(word(A),N))
        <=> ! [X: nat] :
              ( member(nat,X,set_or7035219750837199246ssThan(nat,N,aa(word(A),nat,size_size(word(A)),W)))
             => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),X) ) ) ) ).

% le_mask_high_bits
tff(fact_5782_or__not__numerals_I3_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N))) ).

% or_not_numerals(3)
tff(fact_5783_or__not__numerals_I7_J,axiom,
    ! [M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),one_one(int))) = aa(int,int,bit_ri4277139882892585799ns_not(int),zero_zero(int)) ).

% or_not_numerals(7)
tff(fact_5784_bang__is__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),M)
         => aa(word(A),$o,ord_less_eq(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)),Xb) ) ) ).

% bang_is_le
tff(fact_5785_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_5786_bit__nat__def,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),M),N)
    <=> ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ) ).

% bit_nat_def
tff(fact_5787_or__not__numerals_I6_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% or_not_numerals(6)
tff(fact_5788_odd__iff__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb))
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),zero_zero(nat)) ) ) ).

% odd_iff_lsb
tff(fact_5789_and__neq__0__is__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),N: nat,Xb: word(A)] :
          ( ( Y = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) )
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),Y) != zero_zero(word(A)) )
          <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),N) ) ) ) ).

% and_neq_0_is_nth
tff(fact_5790_nth__is__and__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),N)
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) != zero_zero(word(A)) ) ) ) ).

% nth_is_and_neq_0
tff(fact_5791_XOR__upper,axiom,
    ! [Xb: int,N: nat,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
     => ( aa(int,$o,ord_less(int,Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))
       => ( aa(int,$o,ord_less(int,Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))
         => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),Xb),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ) ) ) ).

% XOR_upper
tff(fact_5792_OR__upper,axiom,
    ! [Xb: int,N: nat,Y: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
     => ( aa(int,$o,ord_less(int,Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))
       => ( aa(int,$o,ord_less(int,Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))
         => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),Xb),Y)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ) ) ) ).

% OR_upper
tff(fact_5793_or__not__numerals_I5_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),bit0(M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(5)
tff(fact_5794_test__bit__int__code_I1_J,axiom,
    ! [N: nat] : ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),zero_zero(int)),N) ).

% test_bit_int_code(1)
tff(fact_5795_int__and__code_I1_J,axiom,
    ! [J: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),zero_zero(int)),J) = zero_zero(int) ).

% int_and_code(1)
tff(fact_5796_int__and__code_I2_J,axiom,
    ! [I: int] : aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),zero_zero(int)) = zero_zero(int) ).

% int_and_code(2)
tff(fact_5797_or__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% or_Suc_0_eq
tff(fact_5798_Suc__0__or__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% Suc_0_or_eq
tff(fact_5799_or__nat__rec,axiom,
    ! [M: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = aa(nat,nat,
        aa(nat,fun(nat,nat),plus_plus(nat),
          aa($o,nat,zero_neq_one_of_bool(nat),
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M)
            | ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ))),
        aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% or_nat_rec
tff(fact_5800_or__not__numerals_I8_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(8)
tff(fact_5801_or__not__numerals_I9_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,M))),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))))) ).

% or_not_numerals(9)
tff(fact_5802_xor__int__rec,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L) = aa(int,int,
        aa(int,fun(int,int),plus_plus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K) != ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L))),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% xor_int_rec
tff(fact_5803_xor__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),one_one(A)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)))),aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3))) ) ).

% xor_one_eq
tff(fact_5804_one__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),A3) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3)))),aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3))) ) ).

% one_xor_eq
tff(fact_5805_or__int__rec,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = aa(int,int,
        aa(int,fun(int,int),plus_plus(int),
          aa($o,int,zero_neq_one_of_bool(int),
            ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),K)
            | ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),L) ))),
        aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2)))))) ).

% or_int_rec
tff(fact_5806_or__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),N) = $ite(
        M = zero_zero(nat),
        N,
        $ite(N = zero_zero(nat),M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,ord_max(nat,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% or_nat_unfold
tff(fact_5807_or__int__unfold,axiom,
    ! [K: int,L: int] :
      aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L) = $ite(
        ( ( K = aa(int,int,uminus_uminus(int),one_one(int)) )
        | ( L = aa(int,int,uminus_uminus(int),one_one(int)) ) ),
        aa(int,int,uminus_uminus(int),one_one(int)),
        $ite(
          K = zero_zero(int),
          L,
          $ite(L = zero_zero(int),K,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,ord_max(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),L),aa(num,int,numeral_numeral(int),bit0(one2))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L),aa(num,int,numeral_numeral(int),bit0(one2))))))) ) ) ).

% or_int_unfold
tff(fact_5808_num_Osize__gen_I2_J,axiom,
    ! [X22: num] : size_num(bit0(X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_5809_or__minus__numerals_I5_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(N)))) ).

% or_minus_numerals(5)
tff(fact_5810_or__minus__numerals_I1_J,axiom,
    ! [N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(N)))) ).

% or_minus_numerals(1)
tff(fact_5811_bit_Oabstract__boolean__algebra__sym__diff__axioms,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => boolea3799213064322606851m_diff(A,bit_se5824344872417868541ns_and(A),bit_se1065995026697491101ons_or(A),bit_ri4277139882892585799ns_not(A),zero_zero(A),aa(A,A,uminus_uminus(A),one_one(A)),bit_se5824344971392196577ns_xor(A)) ) ).

% bit.abstract_boolean_algebra_sym_diff_axioms
tff(fact_5812_word__no__log__defs_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(num,int,numeral_numeral(int),A3)),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_no_log_defs(11)
tff(fact_5813_xor__nat__numerals_I4_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),bit0(Xb)) ).

% xor_nat_numerals(4)
tff(fact_5814_xor__nat__numerals_I3_J,axiom,
    ! [Xb: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(num,nat,numeral_numeral(nat),bit0(Xb))),aa(nat,nat,suc,zero_zero(nat))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Xb)) ).

% xor_nat_numerals(3)
tff(fact_5815_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y))) = aa(num,nat,numeral_numeral(nat),bit0(Y)) ).

% xor_nat_numerals(2)
tff(fact_5816_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),aa(num,nat,numeral_numeral(nat),bit0(Y))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Y)) ).

% xor_nat_numerals(1)
tff(fact_5817_or__minus__numerals_I8_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))),aa(num,int,numeral_numeral(int),M)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bit0(N)))) ).

% or_minus_numerals(8)
tff(fact_5818_or__minus__numerals_I4_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bit0(N)))) ).

% or_minus_numerals(4)
tff(fact_5819_word__bitwise__1__simps_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),one_one(word(A))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),one_one(int)),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_bitwise_1_simps(10)
tff(fact_5820_word__bitwise__1__simps_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(num,int,numeral_numeral(int),A3)),one_one(int))) ) ).

% word_bitwise_1_simps(12)
tff(fact_5821_word__no__log__defs_I14_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_no_log_defs(14)
tff(fact_5822_word__no__log__defs_I13_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(num,int,numeral_numeral(int),B3))) ) ).

% word_no_log_defs(13)
tff(fact_5823_word__no__log__defs_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(num,int,numeral_numeral(int),A3)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_no_log_defs(12)
tff(fact_5824_or__minus__numerals_I7_J,axiom,
    ! [N: num,M: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))),aa(num,int,numeral_numeral(int),M)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(N)))) ).

% or_minus_numerals(7)
tff(fact_5825_or__minus__numerals_I3_J,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(N)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(N)))) ).

% or_minus_numerals(3)
tff(fact_5826_word__bitwise__1__simps_I13_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),one_one(int))) ) ).

% word_bitwise_1_simps(13)
tff(fact_5827_word__bitwise__1__simps_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),one_one(word(A))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ).

% word_bitwise_1_simps(11)
tff(fact_5828_word__log__esimps_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xb),zero_zero(word(A))) = Xb ) ).

% word_log_esimps(5)
tff(fact_5829_word__log__esimps_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),zero_zero(word(A))),Xb) = Xb ) ).

% word_log_esimps(11)
tff(fact_5830_word__bw__same_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xb),Xb) = zero_zero(word(A)) ) ).

% word_bw_same(3)
tff(fact_5831_or__not__num__neg_Osimps_I1_J,axiom,
    bit_or_not_num_neg(one2,one2) = one2 ).

% or_not_num_neg.simps(1)
tff(fact_5832_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N: num] : bit_or_not_num_neg(bit0(N),one2) = bit0(one2) ).

% or_not_num_neg.simps(4)
tff(fact_5833_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(bit0(N),aa(num,num,bit1,M)) = bit0(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(6)
tff(fact_5834_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N: num] : bit_or_not_num_neg(aa(num,num,bit1,N),one2) = one2 ).

% or_not_num_neg.simps(7)
tff(fact_5835_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one2,aa(num,num,bit1,M)) = aa(num,num,bit1,M) ).

% or_not_num_neg.simps(3)
tff(fact_5836_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(bit0(N),bit0(M)) = bitM(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(5)
tff(fact_5837_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(aa(num,num,bit1,N),aa(num,num,bit1,M)) = bitM(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(9)
tff(fact_5838_xor__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = nat2(aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),N))) ).

% xor_nat_def
tff(fact_5839_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] : bit_or_not_num_neg(one2,bit0(M)) = aa(num,num,bit1,M) ).

% or_not_num_neg.simps(2)
tff(fact_5840_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N: num,M: num] : bit_or_not_num_neg(aa(num,num,bit1,N),bit0(M)) = bitM(bit_or_not_num_neg(N,M)) ).

% or_not_num_neg.simps(8)
tff(fact_5841_or__not__num__neg_Oelims,axiom,
    ! [Xb: num,Xaa: num,Y: num] :
      ( ( bit_or_not_num_neg(Xb,Xaa) = Y )
     => ( ( ( Xb = one2 )
         => ( ( Xaa = one2 )
           => ( Y != one2 ) ) )
       => ( ( ( Xb = one2 )
           => ! [M3: num] :
                ( ( Xaa = bit0(M3) )
               => ( Y != aa(num,num,bit1,M3) ) ) )
         => ( ( ( Xb = one2 )
             => ! [M3: num] :
                  ( ( Xaa = aa(num,num,bit1,M3) )
                 => ( Y != aa(num,num,bit1,M3) ) ) )
           => ( ( ? [N3: num] : Xb = bit0(N3)
               => ( ( Xaa = one2 )
                 => ( Y != bit0(one2) ) ) )
             => ( ! [N3: num] :
                    ( ( Xb = bit0(N3) )
                   => ! [M3: num] :
                        ( ( Xaa = bit0(M3) )
                       => ( Y != bitM(bit_or_not_num_neg(N3,M3)) ) ) )
               => ( ! [N3: num] :
                      ( ( Xb = bit0(N3) )
                     => ! [M3: num] :
                          ( ( Xaa = aa(num,num,bit1,M3) )
                         => ( Y != bit0(bit_or_not_num_neg(N3,M3)) ) ) )
                 => ( ( ? [N3: num] : Xb = aa(num,num,bit1,N3)
                     => ( ( Xaa = one2 )
                       => ( Y != one2 ) ) )
                   => ( ! [N3: num] :
                          ( ( Xb = aa(num,num,bit1,N3) )
                         => ! [M3: num] :
                              ( ( Xaa = bit0(M3) )
                             => ( Y != bitM(bit_or_not_num_neg(N3,M3)) ) ) )
                     => ~ ! [N3: num] :
                            ( ( Xb = aa(num,num,bit1,N3) )
                           => ! [M3: num] :
                                ( ( Xaa = aa(num,num,bit1,M3) )
                               => ( Y != bitM(bit_or_not_num_neg(N3,M3)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_5842_word__ops__nth__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less(nat,N),aa(word(A),nat,size_size(word(A)),Xb))
         => ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xb),Y)),N)
            <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),N)
                | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),N) ) )
            & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),Y)),N)
            <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),N)
                & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),N) ) )
            & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xb),Y)),N)
            <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),N)
                <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),N) ) )
            & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xb)),N)
            <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),N) ) ) ) ) ).

% word_ops_nth_size
tff(fact_5843_numeral__or__not__num__eq,axiom,
    ! [M: num,N: num] : aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,N)) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N)))) ).

% numeral_or_not_num_eq
tff(fact_5844_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(N,M))) ).

% int_numeral_not_or_num_neg
tff(fact_5845_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N: num] : aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),N))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,N))) ).

% int_numeral_or_not_num_neg
tff(fact_5846_bit__twiddle__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xb),
            aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xb),Y)),
              $ite(aa(word(A),$o,ord_less(word(A),Xb),Y),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),zero_zero(word(A))))) = aa(word(A),word(A),ord_max(word(A),Xb),Y) ) ).

% bit_twiddle_max
tff(fact_5847_xor__nat__unfold,axiom,
    ! [M: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = $ite(
        M = zero_zero(nat),
        N,
        $ite(N = zero_zero(nat),M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% xor_nat_unfold
tff(fact_5848_xor__nat__rec,axiom,
    ! [M: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),N) = aa(nat,nat,
        aa(nat,fun(nat,nat),plus_plus(nat),
          aa($o,nat,zero_neq_one_of_bool(nat),
            ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),M) != ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))),
        aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% xor_nat_rec
tff(fact_5849_Suc__0__xor__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% Suc_0_xor_eq
tff(fact_5850_xor__Suc__0__eq,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N))) ).

% xor_Suc_0_eq
tff(fact_5851_word__ops__lsb,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xb),Y)),zero_zero(nat))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),zero_zero(nat))
              | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),zero_zero(nat)) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),Y)),zero_zero(nat))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),zero_zero(nat))
              & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),zero_zero(nat)) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xb),Y)),zero_zero(nat))
          <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),zero_zero(nat))
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),zero_zero(nat)) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xb)),zero_zero(nat))
          <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),zero_zero(nat)) ) ) ) ).

% word_ops_lsb
tff(fact_5852_new__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,Xb: A] : hoare_hoare_triple(array(A),one_one(assn),array_new(A,N,Xb),aa(A,fun(array(A),assn),aTP_Lamp_aft(nat,fun(A,fun(array(A),assn)),N),Xb)) ) ).

% new_rule
tff(fact_5853_aux,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn)),A3: A,As3: list(A),C2: B,Cs: list(B)] : finite_fold(nat,assn,aa(list(B),fun(nat,fun(assn,assn)),aa(B,fun(list(B),fun(nat,fun(assn,assn))),aa(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))),aa(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn))))),aTP_Lamp_afu(fun(A,fun(B,assn)),fun(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))))),P),A3),As3),C2),Cs),one_one(assn),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),As3)))) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),P,A3),C2)),finite_fold(nat,assn,aa(list(B),fun(nat,fun(assn,assn)),aa(list(A),fun(list(B),fun(nat,fun(assn,assn))),aTP_Lamp_afr(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),P),As3),Cs),one_one(assn),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),As3)))) ).

% aux
tff(fact_5854_length__nth__simps_I4_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xb),Xs)),aa(nat,nat,suc,N)) = aa(nat,A,nth(A,Xs),N) ).

% length_nth_simps(4)
tff(fact_5855_nth__Cons__Suc,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xb),Xs)),aa(nat,nat,suc,N)) = aa(nat,A,nth(A,Xs),N) ).

% nth_Cons_Suc
tff(fact_5856_length__nth__simps_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xb),Xs)),zero_zero(nat)) = Xb ).

% length_nth_simps(3)
tff(fact_5857_nth__Cons__0,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xb),Xs)),zero_zero(nat)) = Xb ).

% nth_Cons_0
tff(fact_5858_enumerate__simps_I2_J,axiom,
    ! [A: $tType,N: nat,Xb: A,Xs: list(A)] : enumerate(A,N,aa(list(A),list(A),cons(A,Xb),Xs)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),cons(product_prod(nat,A),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),N),Xb)),enumerate(A,aa(nat,nat,suc,N),Xs)) ).

% enumerate_simps(2)
tff(fact_5859_nth__Cons__numeral,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),V: num] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xb),Xs)),aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),V)),one_one(nat))) ).

% nth_Cons_numeral
tff(fact_5860_nth__Cons__pos,axiom,
    ! [A: $tType,N: nat,Xb: A,Xs: list(A)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xb),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,N),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_5861_list__update_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I: nat,V: A] : list_update(A,aa(list(A),list(A),cons(A,Xb),Xs),I,V) = case_nat(list(A),aa(list(A),list(A),cons(A,V),Xs),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_afv(A,fun(list(A),fun(A,fun(nat,list(A)))),Xb),Xs),V),I) ).

% list_update.simps(2)
tff(fact_5862_impossible__Cons,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Xb: A] :
      ( aa(nat,$o,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys))
     => ( Xs != aa(list(A),list(A),cons(A,Xb),Ys) ) ) ).

% impossible_Cons
tff(fact_5863_length__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),cons(A,Xb),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_Cons
tff(fact_5864_length__Suc__conv,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Y3: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),cons(A,Y3),Ys3) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = N ) ) ) ).

% length_Suc_conv
tff(fact_5865_Suc__length__conv,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( ( aa(nat,nat,suc,N) = aa(list(A),nat,size_size(list(A)),Xs) )
    <=> ? [Y3: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),cons(A,Y3),Ys3) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = N ) ) ) ).

% Suc_length_conv
tff(fact_5866_length__nth__simps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),cons(A,Xb),Xs)) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_nth_simps(2)
tff(fact_5867_list__update__code_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A] : list_update(A,aa(list(A),list(A),cons(A,Xb),Xs),zero_zero(nat),Y) = aa(list(A),list(A),cons(A,Y),Xs) ).

% list_update_code(2)
tff(fact_5868_VEBT__internal_Oreplicatei_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,Xb: heap_Time_Heap(A)] : vEBT_VEBT_replicatei(A,aa(nat,nat,suc,N),Xb) = heap_Time_bind(A,list(A),Xb,aa(heap_Time_Heap(A),fun(A,heap_Time_Heap(list(A))),aTP_Lamp_afx(nat,fun(heap_Time_Heap(A),fun(A,heap_Time_Heap(list(A)))),N),Xb)) ).

% VEBT_internal.replicatei.simps(2)
tff(fact_5869_nth__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),N: nat] : aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xb),Xs)),N) = case_nat(A,Xb,nth(A,Xs),N) ).

% nth_Cons
tff(fact_5870_Suc__le__length__iff,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,N)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ? [X: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),cons(A,X),Ys3) )
          & aa(nat,$o,ord_less_eq(nat,N),aa(list(A),nat,size_size(list(A)),Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_5871_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X222: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),cons(A,X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X222)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_5872_nth__Cons_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),N: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xb),Xs)),N) = $ite(N = zero_zero(nat),Xb,aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,N),one_one(nat)))) ).

% nth_Cons'
tff(fact_5873_nth__non__equal__first__eq,axiom,
    ! [A: $tType,Xb: A,Y: A,Xs: list(A),N: nat] :
      ( ( Xb != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xb),Xs)),N) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,N),one_one(nat))) = Y )
          & aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_5874_nth__equal__first__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),N: nat] :
      ( ~ member(A,Xb,set2(A,Xs))
     => ( aa(nat,$o,ord_less_eq(nat,N),aa(list(A),nat,size_size(list(A)),Xs))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Xb),Xs)),N) = Xb )
        <=> ( N = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_5875_Cons__replicate__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),N: nat,Y: A] :
      ( ( aa(list(A),list(A),cons(A,Xb),Xs) = replicate(A,N,Y) )
    <=> ( ( Xb = Y )
        & aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
        & ( Xs = replicate(A,aa(nat,nat,minus_minus(nat,N),one_one(nat)),Xb) ) ) ) ).

% Cons_replicate_eq
tff(fact_5876_slice__Cons,axiom,
    ! [A: $tType,Begin: nat,End: nat,Xb: A,Xs: list(A)] :
      slice(A,Begin,End,aa(list(A),list(A),cons(A,Xb),Xs)) = $ite(
        ( ( Begin = zero_zero(nat) )
        & aa(nat,$o,ord_less(nat,zero_zero(nat)),End) ),
        aa(list(A),list(A),cons(A,Xb),slice(A,Begin,aa(nat,nat,minus_minus(nat,End),one_one(nat)),Xs)),
        slice(A,aa(nat,nat,minus_minus(nat,Begin),one_one(nat)),aa(nat,nat,minus_minus(nat,End),one_one(nat)),Xs) ) ).

% slice_Cons
tff(fact_5877_TBOUND__new,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,Xb: A] : time_TBOUND(array(A),array_new(A,N,Xb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ).

% TBOUND_new
tff(fact_5878_time__array__new,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,Xb: A,H: heap_ext(product_unit)] : time_time(array(A),array_new(A,N,Xb),H) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) ) ).

% time_array_new
tff(fact_5879_word__lsb__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin)))
        <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin))) ) ) ).

% word_lsb_neg_numeral
tff(fact_5880_lsb__odd,axiom,
    ! [A: $tType] :
      ( least_6119777620449941438nt_lsb(A)
     => ! [X3: A] :
          ( aa(A,$o,least_8051144512741203767sb_lsb(A),X3)
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),X3) ) ) ).

% lsb_odd
tff(fact_5881_word__lsb__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin))
        <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(num,int,numeral_numeral(int),Bin)) ) ) ).

% word_lsb_numeral
tff(fact_5882_int__lsb__numeral_I1_J,axiom,
    ~ aa(int,$o,least_8051144512741203767sb_lsb(int),zero_zero(int)) ).

% int_lsb_numeral(1)
tff(fact_5883_int__lsb__numeral_I2_J,axiom,
    aa(int,$o,least_8051144512741203767sb_lsb(int),one_one(int)) ).

% int_lsb_numeral(2)
tff(fact_5884_int__lsb__numeral_I6_J,axiom,
    ! [W: num] : ~ aa(int,$o,least_8051144512741203767sb_lsb(int),aa(num,int,numeral_numeral(int),bit0(W))) ).

% int_lsb_numeral(6)
tff(fact_5885_int__lsb__numeral_I3_J,axiom,
    aa(int,$o,least_8051144512741203767sb_lsb(int),aa(num,int,numeral_numeral(int),one2)) ).

% int_lsb_numeral(3)
tff(fact_5886_int__lsb__numeral_I7_J,axiom,
    ! [W: num] : aa(int,$o,least_8051144512741203767sb_lsb(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W))) ).

% int_lsb_numeral(7)
tff(fact_5887_int__lsb__numeral_I4_J,axiom,
    aa(int,$o,least_8051144512741203767sb_lsb(int),aa(int,int,uminus_uminus(int),one_one(int))) ).

% int_lsb_numeral(4)
tff(fact_5888_int__lsb__numeral_I8_J,axiom,
    ! [W: num] : ~ aa(int,$o,least_8051144512741203767sb_lsb(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(W)))) ).

% int_lsb_numeral(8)
tff(fact_5889_int__lsb__numeral_I5_J,axiom,
    aa(int,$o,least_8051144512741203767sb_lsb(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),one2))) ).

% int_lsb_numeral(5)
tff(fact_5890_int__lsb__numeral_I9_J,axiom,
    ! [W: num] : aa(int,$o,least_8051144512741203767sb_lsb(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W)))) ).

% int_lsb_numeral(9)
tff(fact_5891_lsb__int__def,axiom,
    ! [I: int] :
      ( aa(int,$o,least_8051144512741203767sb_lsb(int),I)
    <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),I),zero_zero(nat)) ) ).

% lsb_int_def
tff(fact_5892_word__lsb__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),W)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),zero_zero(nat)) ) ) ).

% word_lsb_alt
tff(fact_5893_word__lsb__1__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),one_one(word(A)))
        & ~ aa(word(B),$o,least_8051144512741203767sb_lsb(word(B)),zero_zero(word(B))) ) ) ).

% word_lsb_1_0
tff(fact_5894_bin__last__conv__lsb,axiom,
    aTP_Lamp_afy(int,$o) = least_8051144512741203767sb_lsb(int) ).

% bin_last_conv_lsb
tff(fact_5895_lsb__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [X3: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),X3)
        <=> ~ aa(word(A),$o,dvd_dvd(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),X3) ) ) ).

% lsb_word_eq
tff(fact_5896_word__lsb__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),A3)
        <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),int,semiring_1_unsigned(A,int),A3)) ) ) ).

% word_lsb_def
tff(fact_5897_word__lsb__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),W)
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),W)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(nat) ) ) ) ).

% word_lsb_nat
tff(fact_5898_word__lsb__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),W)
        <=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) ) ) ) ).

% word_lsb_int
tff(fact_5899_VEBT__internal_Oreplicatei_Oelims,axiom,
    ! [A: $tType,Xb: nat,Xaa: heap_Time_Heap(A),Y: heap_Time_Heap(list(A))] :
      ( ( vEBT_VEBT_replicatei(A,Xb,Xaa) = Y )
     => ( ( ( Xb = zero_zero(nat) )
         => ( Y != aa(list(A),heap_Time_Heap(list(A)),heap_Time_return(list(A)),nil(A)) ) )
       => ~ ! [N3: nat] :
              ( ( Xb = aa(nat,nat,suc,N3) )
             => ( Y != heap_Time_bind(A,list(A),Xaa,aa(nat,fun(A,heap_Time_Heap(list(A))),aTP_Lamp_afz(heap_Time_Heap(A),fun(nat,fun(A,heap_Time_Heap(list(A)))),Xaa),N3)) ) ) ) ) ).

% VEBT_internal.replicatei.elims
tff(fact_5900_hash__code__prod__simps,axiom,
    ! [A: $tType,B: $tType,H_a2: fun(A,uint32),H_b: fun(B,uint32),Xb: A,Xaa: B] : hash_hash_code_prod(A,B,H_a2,H_b,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Xaa)) = aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(A,uint32,H_a2,Xb)),aa(num,uint32,numeral_numeral(uint32),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))))),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(B,uint32,H_b,Xaa)),aa(num,uint32,numeral_numeral(uint32),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))))))))))))))))))))))))))))),aa(num,uint32,numeral_numeral(uint32),bit0(bit0(bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,one2))))))))))))))))))))))))))))))))) ).

% hash_code_prod_simps
tff(fact_5901_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X5: set(A)] :
      ( aa(set(A),$o,ord_less_eq(set(A),set2(A,Xs)),X5)
     => ( finite_finite2(A,X5)
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,count_list(A,Xs)),X5) = aa(list(A),nat,size_size(list(A)),Xs) ) ) ) ).

% sum_count_set
tff(fact_5902_length__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = zero_zero(nat) )
    <=> ( Xs = nil(A) ) ) ).

% length_0_conv
tff(fact_5903_empty__replicate,axiom,
    ! [A: $tType,N: nat,Xb: A] :
      ( ( nil(A) = replicate(A,N,Xb) )
    <=> ( N = zero_zero(nat) ) ) ).

% empty_replicate
tff(fact_5904_replicate__empty,axiom,
    ! [A: $tType,N: nat,Xb: A] :
      ( ( replicate(A,N,Xb) = nil(A) )
    <=> ( N = zero_zero(nat) ) ) ).

% replicate_empty
tff(fact_5905_count__notin,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ~ member(A,Xb,set2(A,Xs))
     => ( aa(A,nat,count_list(A,Xs),Xb) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_5906_length__greater__0__conv,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ( Xs != nil(A) ) ) ).

% length_greater_0_conv
tff(fact_5907_length__ge__1__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,suc,zero_zero(nat))),aa(list(A),nat,size_size(list(A)),L))
    <=> ( L != nil(A) ) ) ).

% length_ge_1_conv
tff(fact_5908_length__nth__simps_I1_J,axiom,
    ! [A: $tType] : aa(list(A),nat,size_size(list(A)),nil(A)) = zero_zero(nat) ).

% length_nth_simps(1)
tff(fact_5909_len__greater__imp__nonempty,axiom,
    ! [A: $tType,Xb: nat,L: list(A)] :
      ( aa(nat,$o,ord_less(nat,Xb),aa(list(A),nat,size_size(list(A)),L))
     => ( L != nil(A) ) ) ).

% len_greater_imp_nonempty
tff(fact_5910_list__induct2,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(A),fun(list(B),$o))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),nil(B))
       => ( ! [X2: A,Xs2: list(A),Y2: B,Ys4: list(B)] :
              ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys4) )
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs2),Ys4)
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),cons(A,X2),Xs2)),aa(list(B),list(B),cons(B,Y2),Ys4)) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ).

% list_induct2
tff(fact_5911_list__induct3,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xs: list(A),Ys: list(B),Zs: list(C),P: fun(list(A),fun(list(B),fun(list(C),$o)))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs) )
       => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,nil(A)),nil(B)),nil(C))
         => ( ! [X2: A,Xs2: list(A),Y2: B,Ys4: list(B),Z2: C,Zs2: list(C)] :
                ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys4) )
               => ( ( aa(list(B),nat,size_size(list(B)),Ys4) = aa(list(C),nat,size_size(list(C)),Zs2) )
                 => ( aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,Xs2),Ys4),Zs2)
                   => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,aa(list(A),list(A),cons(A,X2),Xs2)),aa(list(B),list(B),cons(B,Y2),Ys4)),aa(list(C),list(C),cons(C,Z2),Zs2)) ) ) )
           => aa(list(C),$o,aa(list(B),fun(list(C),$o),aa(list(A),fun(list(B),fun(list(C),$o)),P,Xs),Ys),Zs) ) ) ) ) ).

% list_induct3
tff(fact_5912_list__induct4,axiom,
    ! [C: $tType,A: $tType,B: $tType,D2: $tType,Xs: list(A),Ys: list(B),Zs: list(C),Ws: list(D2),P: fun(list(A),fun(list(B),fun(list(C),fun(list(D2),$o))))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ( aa(list(B),nat,size_size(list(B)),Ys) = aa(list(C),nat,size_size(list(C)),Zs) )
       => ( ( aa(list(C),nat,size_size(list(C)),Zs) = aa(list(D2),nat,size_size(list(D2)),Ws) )
         => ( aa(list(D2),$o,aa(list(C),fun(list(D2),$o),aa(list(B),fun(list(C),fun(list(D2),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D2),$o))),P,nil(A)),nil(B)),nil(C)),nil(D2))
           => ( ! [X2: A,Xs2: list(A),Y2: B,Ys4: list(B),Z2: C,Zs2: list(C),W2: D2,Ws2: list(D2)] :
                  ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys4) )
                 => ( ( aa(list(B),nat,size_size(list(B)),Ys4) = aa(list(C),nat,size_size(list(C)),Zs2) )
                   => ( ( aa(list(C),nat,size_size(list(C)),Zs2) = aa(list(D2),nat,size_size(list(D2)),Ws2) )
                     => ( aa(list(D2),$o,aa(list(C),fun(list(D2),$o),aa(list(B),fun(list(C),fun(list(D2),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D2),$o))),P,Xs2),Ys4),Zs2),Ws2)
                       => aa(list(D2),$o,aa(list(C),fun(list(D2),$o),aa(list(B),fun(list(C),fun(list(D2),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D2),$o))),P,aa(list(A),list(A),cons(A,X2),Xs2)),aa(list(B),list(B),cons(B,Y2),Ys4)),aa(list(C),list(C),cons(C,Z2),Zs2)),aa(list(D2),list(D2),cons(D2,W2),Ws2)) ) ) ) )
             => aa(list(D2),$o,aa(list(C),fun(list(D2),$o),aa(list(B),fun(list(C),fun(list(D2),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D2),$o))),P,Xs),Ys),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_5913_count__list_Osimps_I1_J,axiom,
    ! [A: $tType,Y: A] : aa(A,nat,count_list(A,nil(A)),Y) = zero_zero(nat) ).

% count_list.simps(1)
tff(fact_5914_replicate__0,axiom,
    ! [A: $tType,Xb: A] : replicate(A,zero_zero(nat),Xb) = nil(A) ).

% replicate_0
tff(fact_5915_length__compl__induct,axiom,
    ! [A: $tType,P: fun(list(A),$o),L: list(A)] :
      ( aa(list(A),$o,P,nil(A))
     => ( ! [E2: A,L2: list(A)] :
            ( ! [Ll: list(A)] :
                ( aa(nat,$o,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L2))
               => aa(list(A),$o,P,Ll) )
           => aa(list(A),$o,P,aa(list(A),list(A),cons(A,E2),L2)) )
       => aa(list(A),$o,P,L) ) ) ).

% length_compl_induct
tff(fact_5916_list__decomp__1,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = one_one(nat) )
     => ? [A2: A] : L = aa(list(A),list(A),cons(A,A2),nil(A)) ) ).

% list_decomp_1
tff(fact_5917_count__le__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] : aa(nat,$o,ord_less_eq(nat,aa(A,nat,count_list(A,Xs),Xb)),aa(list(A),nat,size_size(list(A)),Xs)) ).

% count_le_length
tff(fact_5918_VEBT__internal_Oreplicatei_Osimps_I1_J,axiom,
    ! [A: $tType,Xb: heap_Time_Heap(A)] : vEBT_VEBT_replicatei(A,zero_zero(nat),Xb) = aa(list(A),heap_Time_Heap(list(A)),heap_Time_return(list(A)),nil(A)) ).

% VEBT_internal.replicatei.simps(1)
tff(fact_5919_list__decomp__2,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
     => ? [A2: A,B2: A] : L = aa(list(A),list(A),cons(A,A2),aa(list(A),list(A),cons(A,B2),nil(A))) ) ).

% list_decomp_2
tff(fact_5920_count__list_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A] :
      aa(A,nat,count_list(A,aa(list(A),list(A),cons(A,Xb),Xs)),Y) = $ite(Xb = Y,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,count_list(A,Xs),Y)),one_one(nat)),aa(A,nat,count_list(A,Xs),Y)) ).

% count_list.simps(2)
tff(fact_5921_hash__code__option__simps_I2_J,axiom,
    ! [A: $tType,H_a2: fun(A,uint32),Xb: A] : hash_h1887023736457453652option(A,H_a2,aa(A,option(A),some(A),Xb)) = aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(A,uint32,H_a2,Xb)),aa(num,uint32,numeral_numeral(uint32),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(one2)))))))))))))))))))))))))))))))),aa(num,uint32,numeral_numeral(uint32),bit0(bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(one2))))))))))))))))))))))))))))))) ).

% hash_code_option_simps(2)
tff(fact_5922_hash__code__list__simps_I2_J,axiom,
    ! [A: $tType,H_a2: fun(A,uint32),Xb: A,Xaa: list(A)] : hash_hash_code_list(A,H_a2,aa(list(A),list(A),cons(A,Xb),Xaa)) = aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),aa(A,uint32,H_a2,Xb)),aa(num,uint32,numeral_numeral(uint32),bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2))))))))))))))))))))))))))))))))),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),hash_hash_code_list(A,H_a2,Xaa)),aa(num,uint32,numeral_numeral(uint32),aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(bit0(one2))))))))))))))))))))))))))))))))),aa(num,uint32,numeral_numeral(uint32),bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))))) ).

% hash_code_list_simps(2)
tff(fact_5923_hash__code__option__simps_I1_J,axiom,
    ! [A: $tType,H_a2: fun(A,uint32)] : hash_h1887023736457453652option(A,H_a2,none(A)) = aa(num,uint32,numeral_numeral(uint32),aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(bit0(bit0(bit0(bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,one2))))))))))))))))))))))))))))))) ).

% hash_code_option_simps(1)
tff(fact_5924_hash__code__list__simps_I1_J,axiom,
    ! [A: $tType,H_a2: fun(A,uint32)] : hash_hash_code_list(A,H_a2,nil(A)) = aa(num,uint32,numeral_numeral(uint32),bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(aa(num,num,bit1,bit0(aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(one2))))))))))))))))))))))))))))))) ).

% hash_code_list_simps(1)
tff(fact_5925_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A)] : aa(nat,A,nth(A,append(A,Xs,append(A,aa(list(A),list(A),cons(A,Xb),nil(A)),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xb ).

% concat_inth
tff(fact_5926_VEBT__internal_Oreplicatei_Opelims,axiom,
    ! [A: $tType,Xb: nat,Xaa: heap_Time_Heap(A),Y: heap_Time_Heap(list(A))] :
      ( ( vEBT_VEBT_replicatei(A,Xb,Xaa) = Y )
     => ( accp(product_prod(nat,heap_Time_Heap(A)),vEBT_V8535202610096940344ei_rel(A),aa(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A)),aa(nat,fun(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A))),product_Pair(nat,heap_Time_Heap(A)),Xb),Xaa))
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = aa(list(A),heap_Time_Heap(list(A)),heap_Time_return(list(A)),nil(A)) )
             => ~ accp(product_prod(nat,heap_Time_Heap(A)),vEBT_V8535202610096940344ei_rel(A),aa(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A)),aa(nat,fun(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A))),product_Pair(nat,heap_Time_Heap(A)),zero_zero(nat)),Xaa)) ) )
         => ~ ! [N3: nat] :
                ( ( Xb = aa(nat,nat,suc,N3) )
               => ( ( Y = heap_Time_bind(A,list(A),Xaa,aa(nat,fun(A,heap_Time_Heap(list(A))),aTP_Lamp_afz(heap_Time_Heap(A),fun(nat,fun(A,heap_Time_Heap(list(A)))),Xaa),N3)) )
                 => ~ accp(product_prod(nat,heap_Time_Heap(A)),vEBT_V8535202610096940344ei_rel(A),aa(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A)),aa(nat,fun(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A))),product_Pair(nat,heap_Time_Heap(A)),aa(nat,nat,suc,N3)),Xaa)) ) ) ) ) ) ).

% VEBT_internal.replicatei.pelims
tff(fact_5927_append__eq__append__conv,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Us: list(A),Vs: list(A)] :
      ( ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        | ( aa(list(A),nat,size_size(list(A)),Us) = aa(list(A),nat,size_size(list(A)),Vs) ) )
     => ( ( append(A,Xs,Us) = append(A,Ys,Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_5928_length__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),append(A,Xs,Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),nat,size_size(list(A)),Ys)) ).

% length_append
tff(fact_5929_nth__append__first,axiom,
    ! [A: $tType,I: nat,L: list(A),L3: list(A)] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
     => ( aa(nat,A,nth(A,append(A,L,L3)),I) = aa(nat,A,nth(A,L),I) ) ) ).

% nth_append_first
tff(fact_5930_nth__append__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A)] : aa(nat,A,nth(A,append(A,Xs,aa(list(A),list(A),cons(A,Xb),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xb ).

% nth_append_length
tff(fact_5931_nth__append__length__plus,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),N: nat] : aa(nat,A,nth(A,append(A,Xs,Ys)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs)),N)) = aa(nat,A,nth(A,Ys),N) ).

% nth_append_length_plus
tff(fact_5932_list__update__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A),Y: A] : list_update(A,append(A,Xs,aa(list(A),list(A),cons(A,Xb),Ys)),aa(list(A),nat,size_size(list(A)),Xs),Y) = append(A,Xs,aa(list(A),list(A),cons(A,Y),Ys)) ).

% list_update_length
tff(fact_5933_list__assn__aux__append2,axiom,
    ! [A: $tType,B: $tType,L22: list(A),L23: list(B),P: fun(A,fun(B,assn)),L1: list(A),L12: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L22) = aa(list(B),nat,size_size(list(B)),L23) )
     => ( aa(list(B),assn,vEBT_List_list_assn(A,B,P,append(A,L1,L22)),append(B,L12,L23)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,P,L1),L12)),aa(list(B),assn,vEBT_List_list_assn(A,B,P,L22),L23)) ) ) ).

% list_assn_aux_append2
tff(fact_5934_list__assn__aux__append,axiom,
    ! [A: $tType,B: $tType,L1: list(A),L12: list(B),P: fun(A,fun(B,assn)),L22: list(A),L23: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L1) = aa(list(B),nat,size_size(list(B)),L12) )
     => ( aa(list(B),assn,vEBT_List_list_assn(A,B,P,append(A,L1,L22)),append(B,L12,L23)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,P,L1),L12)),aa(list(B),assn,vEBT_List_list_assn(A,B,P,L22),L23)) ) ) ).

% list_assn_aux_append
tff(fact_5935_list__rest__coinc,axiom,
    ! [A: $tType,S22: list(A),S1: list(A),R1: list(A),R22: list(A)] :
      ( aa(nat,$o,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),S22)),aa(list(A),nat,size_size(list(A)),S1))
     => ( ( append(A,S1,R1) = append(A,S22,R22) )
       => ? [R1p: list(A)] : R22 = append(A,R1p,R1) ) ) ).

% list_rest_coinc
tff(fact_5936_enumerate__append__eq,axiom,
    ! [A: $tType,N: nat,Xs: list(A),Ys: list(A)] : enumerate(A,N,append(A,Xs,Ys)) = append(product_prod(nat,A),enumerate(A,N,Xs),enumerate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% enumerate_append_eq
tff(fact_5937_same__length__different,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != Ys )
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
       => ? [Pre: list(A),X2: A,Xs5: list(A),Y2: A,Ys5: list(A)] :
            ( ( X2 != Y2 )
            & ( Xs = append(A,Pre,append(A,aa(list(A),list(A),cons(A,X2),nil(A)),Xs5)) )
            & ( Ys = append(A,Pre,append(A,aa(list(A),list(A),cons(A,Y2),nil(A)),Ys5)) ) ) ) ) ).

% same_length_different
tff(fact_5938_list__update__append1,axiom,
    ! [A: $tType,I: nat,Xs: list(A),Ys: list(A),Xb: A] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
     => ( list_update(A,append(A,Xs,Ys),I,Xb) = append(A,list_update(A,Xs,I,Xb),Ys) ) ) ).

% list_update_append1
tff(fact_5939_length__Suc__conv__rev,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Y3: A,Ys3: list(A)] :
          ( ( Xs = append(A,Ys3,aa(list(A),list(A),cons(A,Y3),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = N ) ) ) ).

% length_Suc_conv_rev
tff(fact_5940_length__Suc__rev__conv,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,N) )
    <=> ? [Ys3: list(A),Y3: A] :
          ( ( Xs = append(A,Ys3,aa(list(A),list(A),cons(A,Y3),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = N ) ) ) ).

% length_Suc_rev_conv
tff(fact_5941_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] : aa(list(A),nat,size_size(list(A)),append(A,Xs,aa(list(A),list(A),cons(A,Xb),nil(A)))) = aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_append_singleton
tff(fact_5942_length__compl__rev__induct,axiom,
    ! [A: $tType,P: fun(list(A),$o),L: list(A)] :
      ( aa(list(A),$o,P,nil(A))
     => ( ! [L2: list(A),E2: A] :
            ( ! [Ll: list(A)] :
                ( aa(nat,$o,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),Ll)),aa(list(A),nat,size_size(list(A)),L2))
               => aa(list(A),$o,P,Ll) )
           => aa(list(A),$o,P,append(A,L2,aa(list(A),list(A),cons(A,E2),nil(A)))) )
       => aa(list(A),$o,P,L) ) ) ).

% length_compl_rev_induct
tff(fact_5943_nth__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),N: nat] :
      aa(nat,A,nth(A,append(A,Xs,Ys)),N) = $ite(aa(nat,$o,ord_less(nat,N),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,A,nth(A,Xs),N),aa(nat,A,nth(A,Ys),aa(nat,nat,minus_minus(nat,N),aa(list(A),nat,size_size(list(A)),Xs)))) ).

% nth_append
tff(fact_5944_list__update__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),N: nat,Xb: A] :
      list_update(A,append(A,Xs,Ys),N,Xb) = $ite(aa(nat,$o,ord_less(nat,N),aa(list(A),nat,size_size(list(A)),Xs)),append(A,list_update(A,Xs,N,Xb),Ys),append(A,Xs,list_update(A,Ys,aa(nat,nat,minus_minus(nat,N),aa(list(A),nat,size_size(list(A)),Xs)),Xb))) ).

% list_update_append
tff(fact_5945_product_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,Xb: A,Xs: list(A),Ys: list(B)] : product(A,B,aa(list(A),list(A),cons(A,Xb),Xs),Ys) = append(product_prod(A,B),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Ys),product(A,B,Xs,Ys)) ).

% product.simps(2)
tff(fact_5946_slice__prepend,axiom,
    ! [A: $tType,I: nat,K: nat,Xs: list(A),Ys: list(A)] :
      ( aa(nat,$o,ord_less_eq(nat,I),K)
     => ( aa(nat,$o,ord_less_eq(nat,K),aa(list(A),nat,size_size(list(A)),Xs))
       => $let(
            p: nat,
            p:= aa(list(A),nat,size_size(list(A)),Ys),
            slice(A,I,K,Xs) = slice(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),p),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),p),append(A,Ys,Xs)) ) ) ) ).

% slice_prepend
tff(fact_5947_quickcheck__narrowing__samples_Onarrowing__samples_Oelims,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),Zero: A,Xb: code_integer,Y: list(A)] :
          ( ( code_T4080844693773952564amples(A,A_of_integer,Zero,Xb) = Y )
         => ( Y = $ite(aa(code_integer,$o,ord_less(code_integer,zero_zero(code_integer)),Xb),aa(product_prod(A,A),list(A),product_case_prod(A,A,list(A),aa(code_integer,fun(A,fun(A,list(A))),aa(A,fun(code_integer,fun(A,fun(A,list(A)))),aTP_Lamp_aga(fun(code_integer,product_prod(A,A)),fun(A,fun(code_integer,fun(A,fun(A,list(A))))),A_of_integer),Zero),Xb)),aa(code_integer,product_prod(A,A),A_of_integer,Xb)),aa(list(A),list(A),cons(A,Zero),nil(A))) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.elims
tff(fact_5948_quickcheck__narrowing__samples_Onarrowing__samples_Osimps,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),Zero: A,I: code_integer] :
          code_T4080844693773952564amples(A,A_of_integer,Zero,I) = $ite(aa(code_integer,$o,ord_less(code_integer,zero_zero(code_integer)),I),aa(product_prod(A,A),list(A),product_case_prod(A,A,list(A),aa(code_integer,fun(A,fun(A,list(A))),aa(A,fun(code_integer,fun(A,fun(A,list(A)))),aTP_Lamp_aga(fun(code_integer,product_prod(A,A)),fun(A,fun(code_integer,fun(A,fun(A,list(A))))),A_of_integer),Zero),I)),aa(code_integer,product_prod(A,A),A_of_integer,I)),aa(list(A),list(A),cons(A,Zero),nil(A))) ) ).

% quickcheck_narrowing_samples.narrowing_samples.simps
tff(fact_5949_call__mono,axiom,
    ! [B: $tType,A: $tType,Ord: fun(B,fun(B,$o)),Tb: A] : comple7038119648293358887notone(fun(A,B),B,partial_fun_ord(B,B,A,Ord),Ord,aTP_Lamp_agb(A,fun(fun(A,B),B),Tb)) ).

% call_mono
tff(fact_5950_power__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [K: num,L: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),K)),aa(num,nat,numeral_numeral(nat),L)) = aa(num,A,numeral_numeral(A),pow(K,L)) ) ).

% power_numeral
tff(fact_5951_pow_Osimps_I1_J,axiom,
    ! [Xb: num] : pow(Xb,one2) = Xb ).

% pow.simps(1)
tff(fact_5952_if__mono,axiom,
    ! [A: $tType,B: $tType,Orda: fun(A,fun(A,$o)),Ordb: fun(B,fun(B,$o)),F5: fun(A,B),G4: fun(A,B),C2: $o] :
      ( comple7038119648293358887notone(A,B,Orda,Ordb,F5)
     => ( comple7038119648293358887notone(A,B,Orda,Ordb,G4)
       => comple7038119648293358887notone(A,B,Orda,Ordb,aa($o,fun(A,B),aa(fun(A,B),fun($o,fun(A,B)),aTP_Lamp_agc(fun(A,B),fun(fun(A,B),fun($o,fun(A,B))),F5),G4),(C2))) ) ) ).

% if_mono
tff(fact_5953_let__mono,axiom,
    ! [B: $tType,C: $tType,A: $tType,Orda: fun(B,fun(B,$o)),Ordb: fun(C,fun(C,$o)),B3: fun(B,fun(A,C)),Tb: A] :
      ( ! [X2: A] : comple7038119648293358887notone(B,C,Orda,Ordb,aa(A,fun(B,C),aTP_Lamp_agd(fun(B,fun(A,C)),fun(A,fun(B,C)),B3),X2))
     => comple7038119648293358887notone(B,C,Orda,Ordb,aa(A,fun(B,C),aTP_Lamp_agd(fun(B,fun(A,C)),fun(A,fun(B,C)),B3),Tb)) ) ).

% let_mono
tff(fact_5954_partial__function__definitions_Ofixp__induct__uc,axiom,
    ! [B: $tType,A: $tType,C: $tType,Leq: fun(A,fun(A,$o)),Lub: fun(set(A),A),U2: fun(C,fun(B,A)),F5: fun(C,C),C5: fun(fun(B,A),C),Fa: C,P: fun(fun(B,A),$o)] :
      ( partia7178651479351089652itions(A,Leq,Lub)
     => ( ! [X2: B] : comple7038119648293358887notone(fun(B,A),A,partial_fun_ord(A,A,B,Leq),Leq,aa(B,fun(fun(B,A),A),aa(fun(fun(B,A),C),fun(B,fun(fun(B,A),A)),aa(fun(C,C),fun(fun(fun(B,A),C),fun(B,fun(fun(B,A),A))),aTP_Lamp_age(fun(C,fun(B,A)),fun(fun(C,C),fun(fun(fun(B,A),C),fun(B,fun(fun(B,A),A)))),U2),F5),C5),X2))
       => ( ( Fa = aa(fun(B,A),C,C5,comple187402453842119260l_fixp(fun(B,A),partial_fun_lub(A,A,B,Lub),partial_fun_ord(A,A,B,Leq),aa(fun(fun(B,A),C),fun(fun(B,A),fun(B,A)),aa(fun(C,C),fun(fun(fun(B,A),C),fun(fun(B,A),fun(B,A))),aTP_Lamp_agf(fun(C,fun(B,A)),fun(fun(C,C),fun(fun(fun(B,A),C),fun(fun(B,A),fun(B,A)))),U2),F5),C5))) )
         => ( ! [F4: fun(B,A)] : aa(C,fun(B,A),U2,aa(fun(B,A),C,C5,F4)) = F4
           => ( comple1908693960933563346ssible(fun(B,A),partial_fun_lub(A,A,B,Lub),partial_fun_ord(A,A,B,Leq),P)
             => ( aa(fun(B,A),$o,P,aTP_Lamp_agg(fun(set(A),A),fun(B,A),Lub))
               => ( ! [F4: C] :
                      ( aa(fun(B,A),$o,P,aa(C,fun(B,A),U2,F4))
                     => aa(fun(B,A),$o,P,aa(C,fun(B,A),U2,aa(C,C,F5,F4))) )
                 => aa(fun(B,A),$o,P,aa(C,fun(B,A),U2,Fa)) ) ) ) ) ) ) ) ).

% partial_function_definitions.fixp_induct_uc
tff(fact_5955_cis__multiple__2pi,axiom,
    ! [N: real] :
      ( member(real,N,ring_1_Ints(real))
     => ( cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),N)) = one_one(complex) ) ) ).

% cis_multiple_2pi
tff(fact_5956_frac__eq__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( ( archimedean_frac(A,Xb) = zero_zero(A) )
        <=> member(A,Xb,ring_1_Ints(A)) ) ) ).

% frac_eq_0_iff
tff(fact_5957_frac__gt__0__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),archimedean_frac(A,Xb))
        <=> ~ member(A,Xb,ring_1_Ints(A)) ) ) ).

% frac_gt_0_iff
tff(fact_5958_Ints__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [N: num] : member(A,aa(num,A,numeral_numeral(A),N),ring_1_Ints(A)) ) ).

% Ints_numeral
tff(fact_5959_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),A3) = zero_zero(A) )
          <=> ( A3 = zero_zero(A) ) ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_5960_Ints__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A3: A,N: nat] :
          ( member(A,A3,ring_1_Ints(A))
         => member(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N),ring_1_Ints(A)) ) ) ).

% Ints_power
tff(fact_5961_Ints__0,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,zero_zero(A),ring_1_Ints(A)) ) ).

% Ints_0
tff(fact_5962_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,one_one(A),ring_1_Ints(A)) ) ).

% Ints_1
tff(fact_5963_partial__function__definitions_Oconst__mono,axiom,
    ! [B: $tType,A: $tType,Leq: fun(A,fun(A,$o)),Lub: fun(set(A),A),Ord: fun(B,fun(B,$o)),C2: A] :
      ( partia7178651479351089652itions(A,Leq,Lub)
     => comple7038119648293358887notone(B,A,Ord,Leq,aa(A,fun(B,A),aTP_Lamp_cp(A,fun(B,A)),C2)) ) ).

% partial_function_definitions.const_mono
tff(fact_5964_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A,B3: A] : finite_finite2(A,collect(A,aa(A,fun(A,$o),aTP_Lamp_agh(A,fun(A,fun(A,$o)),A3),B3))) ) ).

% finite_int_segment
tff(fact_5965_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3) != zero_zero(A) ) ) ) ).

% Ints_odd_nonzero
tff(fact_5966_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B3: int,A3: int] :
          ( aa(int,$o,dvd_dvd(int,B3),A3)
         => member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A3)),aa(int,A,ring_1_of_int(A),B3)),ring_1_Ints(A)) ) ) ).

% of_int_divide_in_Ints
tff(fact_5967_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A3: A] : finite_finite2(A,collect(A,aTP_Lamp_agi(A,fun(A,$o),A3))) ) ).

% finite_abs_int_segment
tff(fact_5968_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A3: A] :
          ( member(A,A3,ring_1_Ints(A))
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),A3)),A3)),zero_zero(A))
          <=> aa(A,$o,ord_less(A,A3),zero_zero(A)) ) ) ) ).

% Ints_odd_less_0
tff(fact_5969_Ints__nonzero__abs__ge1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( member(A,Xb,ring_1_Ints(A))
         => ( ( Xb != zero_zero(A) )
           => aa(A,$o,ord_less_eq(A,one_one(A)),abs_abs(A,Xb)) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_5970_Ints__nonzero__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( member(A,Xb,ring_1_Ints(A))
         => ( aa(A,$o,ord_less(A,abs_abs(A,Xb)),one_one(A))
           => ( Xb = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_5971_Ints__eq__abs__less1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( member(A,Xb,ring_1_Ints(A))
         => ( member(A,Y,ring_1_Ints(A))
           => ( ( Xb = Y )
            <=> aa(A,$o,ord_less(A,abs_abs(A,aa(A,A,minus_minus(A,Xb),Y))),one_one(A)) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_5972_sin__times__pi__eq__0,axiom,
    ! [Xb: real] :
      ( ( sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),pi)) = zero_zero(real) )
    <=> member(real,Xb,ring_1_Ints(real)) ) ).

% sin_times_pi_eq_0
tff(fact_5973_frac__neg,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          archimedean_frac(A,aa(A,A,uminus_uminus(A),Xb)) = $ite(member(A,Xb,ring_1_Ints(A)),zero_zero(A),aa(A,A,minus_minus(A,one_one(A)),archimedean_frac(A,Xb))) ) ).

% frac_neg
tff(fact_5974_le__mult__floor__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( member(A,A3,ring_1_Ints(A))
           => aa(B,$o,ord_less_eq(B,aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A3)),archim6421214686448440834_floor(A,B3)))),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_5975_mult__ceiling__le__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
         => ( member(A,A3,ring_1_Ints(A))
           => aa(B,$o,ord_less_eq(B,aa(int,B,ring_1_of_int(B),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),B3)))),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A3)),archimedean_ceiling(A,B3)))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_5976_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A3: A] :
          ( ( archimedean_frac(A,Xb) = A3 )
        <=> ( member(A,aa(A,A,minus_minus(A,Xb),A3),ring_1_Ints(A))
            & aa(A,$o,ord_less_eq(A,zero_zero(A)),A3)
            & aa(A,$o,ord_less(A,A3),one_one(A)) ) ) ) ).

% frac_unique_iff
tff(fact_5977_partial__function__definitions_Ofixp__rule__uc,axiom,
    ! [B: $tType,A: $tType,C: $tType,Leq: fun(A,fun(A,$o)),Lub: fun(set(A),A),U2: fun(C,fun(B,A)),F5: fun(C,C),C5: fun(fun(B,A),C),Fa: C] :
      ( partia7178651479351089652itions(A,Leq,Lub)
     => ( ! [X2: B] : comple7038119648293358887notone(fun(B,A),A,partial_fun_ord(A,A,B,Leq),Leq,aa(B,fun(fun(B,A),A),aa(fun(fun(B,A),C),fun(B,fun(fun(B,A),A)),aa(fun(C,C),fun(fun(fun(B,A),C),fun(B,fun(fun(B,A),A))),aTP_Lamp_age(fun(C,fun(B,A)),fun(fun(C,C),fun(fun(fun(B,A),C),fun(B,fun(fun(B,A),A)))),U2),F5),C5),X2))
       => ( ( Fa = aa(fun(B,A),C,C5,comple187402453842119260l_fixp(fun(B,A),partial_fun_lub(A,A,B,Lub),partial_fun_ord(A,A,B,Leq),aa(fun(fun(B,A),C),fun(fun(B,A),fun(B,A)),aa(fun(C,C),fun(fun(fun(B,A),C),fun(fun(B,A),fun(B,A))),aTP_Lamp_agf(fun(C,fun(B,A)),fun(fun(C,C),fun(fun(fun(B,A),C),fun(fun(B,A),fun(B,A)))),U2),F5),C5))) )
         => ( ! [F4: C] : aa(fun(B,A),C,C5,aa(C,fun(B,A),U2,F4)) = F4
           => ( Fa = aa(C,C,F5,Fa) ) ) ) ) ) ).

% partial_function_definitions.fixp_rule_uc
tff(fact_5978_partial__function__definitions_Omono__body__fixp,axiom,
    ! [A: $tType,B: $tType,Leq: fun(A,fun(A,$o)),Lub: fun(set(A),A),F5: fun(fun(B,A),fun(B,A))] :
      ( partia7178651479351089652itions(A,Leq,Lub)
     => ( ! [X2: B] : comple7038119648293358887notone(fun(B,A),A,partial_fun_ord(A,A,B,Leq),Leq,aa(B,fun(fun(B,A),A),aTP_Lamp_agj(fun(fun(B,A),fun(B,A)),fun(B,fun(fun(B,A),A)),F5),X2))
       => ( comple187402453842119260l_fixp(fun(B,A),partial_fun_lub(A,A,B,Lub),partial_fun_ord(A,A,B,Leq),F5) = aa(fun(B,A),fun(B,A),F5,comple187402453842119260l_fixp(fun(B,A),partial_fun_lub(A,A,B,Lub),partial_fun_ord(A,A,B,Leq),F5)) ) ) ) ).

% partial_function_definitions.mono_body_fixp
tff(fact_5979_sin__integer__2pi,axiom,
    ! [N: real] :
      ( member(real,N,ring_1_Ints(real))
     => ( sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),N)) = zero_zero(real) ) ) ).

% sin_integer_2pi
tff(fact_5980_cos__integer__2pi,axiom,
    ! [N: real] :
      ( member(real,N,ring_1_Ints(real))
     => ( cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),N)) = one_one(real) ) ) ).

% cos_integer_2pi
tff(fact_5981_admissible__fun,axiom,
    ! [A: $tType,B: $tType,Le: fun(A,fun(A,$o)),Lub: fun(set(A),A),Q: fun(B,fun(A,$o))] :
      ( partia7178651479351089652itions(A,Le,Lub)
     => ( ! [X2: B] : comple1908693960933563346ssible(A,Lub,Le,aa(B,fun(A,$o),Q,X2))
       => comple1908693960933563346ssible(fun(B,A),partial_fun_lub(A,A,B,Lub),partial_fun_ord(A,A,B,Le),aTP_Lamp_agk(fun(B,fun(A,$o)),fun(fun(B,A),$o),Q)) ) ) ).

% admissible_fun
tff(fact_5982_ran__nth__set__encoding__conv,axiom,
    ! [A: $tType,L: list(A)] : ran(nat,A,aTP_Lamp_agl(list(A),fun(nat,option(A)),L)) = set2(A,L) ).

% ran_nth_set_encoding_conv
tff(fact_5983_rat__inverse__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,inverse_inverse(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_agm(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_5984_quotient__of__number_I3_J,axiom,
    ! [K: num] : quotient_of(aa(num,rat,numeral_numeral(rat),K)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int)) ).

% quotient_of_number(3)
tff(fact_5985_rat__one__code,axiom,
    quotient_of(one_one(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int)) ).

% rat_one_code
tff(fact_5986_map__update__eta__repair_I2_J,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),K: B,V: A] :
      ( ( aa(B,option(A),M,K) = none(A) )
     => ( ran(B,A,aa(A,fun(B,option(A)),aa(B,fun(A,fun(B,option(A))),aTP_Lamp_agn(fun(B,option(A)),fun(B,fun(A,fun(B,option(A)))),M),K),V)) = aa(set(A),set(A),insert(A,V),ran(B,A,M)) ) ) ).

% map_update_eta_repair(2)
tff(fact_5987_rat__zero__code,axiom,
    quotient_of(zero_zero(rat)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% rat_zero_code
tff(fact_5988_quotient__of__number_I5_J,axiom,
    ! [K: num] : quotient_of(aa(rat,rat,uminus_uminus(rat),aa(num,rat,numeral_numeral(rat),K))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))),one_one(int)) ).

% quotient_of_number(5)
tff(fact_5989_quotient__of__number_I4_J,axiom,
    quotient_of(aa(rat,rat,uminus_uminus(rat),one_one(rat))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),one_one(int))),one_one(int)) ).

% quotient_of_number(4)
tff(fact_5990_quotient__of__denom__pos,axiom,
    ! [R2: rat,P2: int,Q2: int] :
      ( ( quotient_of(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => aa(int,$o,ord_less(int,zero_zero(int)),Q2) ) ).

% quotient_of_denom_pos
tff(fact_5991_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : aa(int,$o,ord_less(int,zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),quotient_of(R2))) ).

% quotient_of_denom_pos'
tff(fact_5992_rat__uminus__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,uminus_uminus(rat),P2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_ago(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_5993_rat__less__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa(rat,$o,ord_less(rat,P2),Q2)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aTP_Lamp_agq(rat,fun(int,fun(int,$o)),Q2)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_5994_rat__floor__code,axiom,
    ! [P2: rat] : archim6421214686448440834_floor(rat,P2) = aa(product_prod(int,int),int,product_case_prod(int,int,int,divide_divide(int)),quotient_of(P2)) ).

% rat_floor_code
tff(fact_5995_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(abs_abs(rat,P2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_agr(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_abs_code
tff(fact_5996_rat__less__eq__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa(rat,$o,ord_less_eq(rat,P2),Q2)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aTP_Lamp_agt(rat,fun(int,fun(int,$o)),Q2)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_5997_rat__sgn__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,sgn_sgn(rat),P2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,sgn_sgn(int),aa(product_prod(int,int),int,product_fst(int,int),quotient_of(P2)))),one_one(int)) ).

% rat_sgn_code
tff(fact_5998_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_agu(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_5999_quotient__of__int,axiom,
    ! [A3: int] : quotient_of(of_int(A3)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),one_one(int)) ).

% quotient_of_int
tff(fact_6000_ranI,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A3: B,B3: A] :
      ( ( aa(B,option(A),M,A3) = aa(A,option(A),some(A),B3) )
     => member(A,B3,ran(B,A,M)) ) ).

% ranI
tff(fact_6001_rat__plus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_agw(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_6002_rat__minus__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,minus_minus(rat,P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_agy(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_6003_normalize__denom__zero,axiom,
    ! [P2: int] : normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),zero_zero(int))) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)) ).

% normalize_denom_zero
tff(fact_6004_normalize__negative,axiom,
    ! [Q2: int,P2: int] :
      ( aa(int,$o,ord_less(int,Q2),zero_zero(int))
     => ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),P2)),aa(int,int,uminus_uminus(int),Q2))) ) ) ).

% normalize_negative
tff(fact_6005_normalize__denom__pos,axiom,
    ! [R2: product_prod(int,int),P2: int,Q2: int] :
      ( ( normalize(R2) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2) )
     => aa(int,$o,ord_less(int,zero_zero(int)),Q2) ) ).

% normalize_denom_pos
tff(fact_6006_normalize__crossproduct,axiom,
    ! [Q2: int,S: int,P2: int,R2: int] :
      ( ( Q2 != zero_zero(int) )
     => ( ( S != zero_zero(int) )
       => ( ( normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),P2),Q2)) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),R2),S)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P2),S) = aa(int,int,aa(int,fun(int,int),times_times(int),R2),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_6007_rat__divide__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_aha(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_6008_rat__times__code,axiom,
    ! [P2: rat,Q2: rat] : quotient_of(aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),P2),Q2)) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aTP_Lamp_ahc(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_times_code
tff(fact_6009_normalize__def,axiom,
    ! [P2: product_prod(int,int)] :
      normalize(P2) = $ite(
        aa(int,$o,ord_less(int,zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),P2)),
        $let(
          a2: int,
          a2:= gcd_gcd(int,aa(product_prod(int,int),int,product_fst(int,int),P2),aa(product_prod(int,int),int,product_snd(int,int),P2)),
          aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),a2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P2)),a2)) ),
        $ite(
          aa(product_prod(int,int),int,product_snd(int,int),P2) = zero_zero(int),
          aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),
          $let(
            a2: int,
            a2:= aa(int,int,uminus_uminus(int),gcd_gcd(int,aa(product_prod(int,int),int,product_fst(int,int),P2),aa(product_prod(int,int),int,product_snd(int,int),P2))),
            aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_fst(int,int),P2)),a2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(product_prod(int,int),int,product_snd(int,int),P2)),a2)) ) ) ) ).

% normalize_def
tff(fact_6010_Frct__code__post_I5_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),aa(num,int,numeral_numeral(int),K))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),one_one(rat)),aa(num,rat,numeral_numeral(rat),K)) ).

% Frct_code_post(5)
tff(fact_6011_gcd__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A] :
          ( ( gcd_gcd(A,A3,B3) = zero_zero(A) )
        <=> ( ( A3 = zero_zero(A) )
            & ( B3 = zero_zero(A) ) ) ) ) ).

% gcd_eq_0_iff
tff(fact_6012_gcd_Obottom__right__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : gcd_gcd(A,A3,one_one(A)) = one_one(A) ) ).

% gcd.bottom_right_bottom
tff(fact_6013_gcd_Obottom__left__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A] : gcd_gcd(A,one_one(A),A3) = one_one(A) ) ).

% gcd.bottom_left_bottom
tff(fact_6014_gcd__exp,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A3: A,N: nat,B3: A] : gcd_gcd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),N),aa(nat,A,aa(A,fun(nat,A),power_power(A),B3),N)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),gcd_gcd(A,A3,B3)),N) ) ).

% gcd_exp
tff(fact_6015_gcd__1__int,axiom,
    ! [M: int] : gcd_gcd(int,M,one_one(int)) = one_one(int) ).

% gcd_1_int
tff(fact_6016_gcd__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [N: num,A3: A] : gcd_gcd(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N)),A3) = gcd_gcd(A,aa(num,A,numeral_numeral(A),N),A3) ) ).

% gcd_neg_numeral_1
tff(fact_6017_gcd__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A3: A,N: num] : gcd_gcd(A,A3,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),N))) = gcd_gcd(A,A3,aa(num,A,numeral_numeral(A),N)) ) ).

% gcd_neg_numeral_2
tff(fact_6018_is__unit__gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,dvd_dvd(A,gcd_gcd(A,A3,B3)),one_one(A))
        <=> ( gcd_gcd(A,A3,B3) = one_one(A) ) ) ) ).

% is_unit_gcd_iff
tff(fact_6019_gcd__pos__int,axiom,
    ! [M: int,N: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),gcd_gcd(int,M,N))
    <=> ( ( M != zero_zero(int) )
        | ( N != zero_zero(int) ) ) ) ).

% gcd_pos_int
tff(fact_6020_gcd__neg__numeral__2__int,axiom,
    ! [Xb: int,N: num] : gcd_gcd(int,Xb,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = gcd_gcd(int,Xb,aa(num,int,numeral_numeral(int),N)) ).

% gcd_neg_numeral_2_int
tff(fact_6021_gcd__neg__numeral__1__int,axiom,
    ! [N: num,Xb: int] : gcd_gcd(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)),Xb) = gcd_gcd(int,aa(num,int,numeral_numeral(int),N),Xb) ).

% gcd_neg_numeral_1_int
tff(fact_6022_gcd__0__left__int,axiom,
    ! [Xb: int] : gcd_gcd(int,zero_zero(int),Xb) = abs_abs(int,Xb) ).

% gcd_0_left_int
tff(fact_6023_gcd__0__int,axiom,
    ! [Xb: int] : gcd_gcd(int,Xb,zero_zero(int)) = abs_abs(int,Xb) ).

% gcd_0_int
tff(fact_6024_gcd__ge__0__int,axiom,
    ! [Xb: int,Y: int] : aa(int,$o,ord_less_eq(int,zero_zero(int)),gcd_gcd(int,Xb,Y)) ).

% gcd_ge_0_int
tff(fact_6025_gcd__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( gcd_gcd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B3),A3),C2) = gcd_gcd(A,B3,C2) ) ) ) ).

% gcd_mult_unit1
tff(fact_6026_gcd__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( gcd_gcd(A,B3,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A3)) = gcd_gcd(A,B3,C2) ) ) ) ).

% gcd_mult_unit2
tff(fact_6027_gcd__div__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( gcd_gcd(A,B3,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A3)) = gcd_gcd(A,B3,C2) ) ) ) ).

% gcd_div_unit2
tff(fact_6028_gcd__div__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A3),one_one(A))
         => ( gcd_gcd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),A3),C2) = gcd_gcd(A,B3,C2) ) ) ) ).

% gcd_div_unit1
tff(fact_6029_gcd__le1__int,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),A3)
     => aa(int,$o,ord_less_eq(int,gcd_gcd(int,A3,B3)),A3) ) ).

% gcd_le1_int
tff(fact_6030_gcd__le2__int,axiom,
    ! [B3: int,A3: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
     => aa(int,$o,ord_less_eq(int,gcd_gcd(int,A3,B3)),B3) ) ).

% gcd_le2_int
tff(fact_6031_gcd__cases__int,axiom,
    ! [Xb: int,Y: int,P: fun(int,$o)] :
      ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
       => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
         => aa(int,$o,P,gcd_gcd(int,Xb,Y)) ) )
     => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
         => ( aa(int,$o,ord_less_eq(int,Y),zero_zero(int))
           => aa(int,$o,P,gcd_gcd(int,Xb,aa(int,int,uminus_uminus(int),Y))) ) )
       => ( ( aa(int,$o,ord_less_eq(int,Xb),zero_zero(int))
           => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
             => aa(int,$o,P,gcd_gcd(int,aa(int,int,uminus_uminus(int),Xb),Y)) ) )
         => ( ( aa(int,$o,ord_less_eq(int,Xb),zero_zero(int))
             => ( aa(int,$o,ord_less_eq(int,Y),zero_zero(int))
               => aa(int,$o,P,gcd_gcd(int,aa(int,int,uminus_uminus(int),Xb),aa(int,int,uminus_uminus(int),Y))) ) )
           => aa(int,$o,P,gcd_gcd(int,Xb,Y)) ) ) ) ) ).

% gcd_cases_int
tff(fact_6032_gcd__non__0__int,axiom,
    ! [Y: int,Xb: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),Y)
     => ( gcd_gcd(int,Xb,Y) = gcd_gcd(int,Y,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xb),Y)) ) ) ).

% gcd_non_0_int
tff(fact_6033_gcd__unique__int,axiom,
    ! [D: int,A3: int,B3: int] :
      ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),D)
        & aa(int,$o,dvd_dvd(int,D),A3)
        & aa(int,$o,dvd_dvd(int,D),B3)
        & ! [E5: int] :
            ( ( aa(int,$o,dvd_dvd(int,E5),A3)
              & aa(int,$o,dvd_dvd(int,E5),B3) )
           => aa(int,$o,dvd_dvd(int,E5),D) ) )
    <=> ( D = gcd_gcd(int,A3,B3) ) ) ).

% gcd_unique_int
tff(fact_6034_gcd__code__int,axiom,
    ! [K: int,L: int] :
      gcd_gcd(int,K,L) = abs_abs(int,
        $ite(L = zero_zero(int),K,gcd_gcd(int,L,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),abs_abs(int,K)),abs_abs(int,L))))) ).

% gcd_code_int
tff(fact_6035_Frct__code__post_I1_J,axiom,
    ! [A3: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A3)) = zero_zero(rat) ).

% Frct_code_post(1)
tff(fact_6036_Frct__code__post_I2_J,axiom,
    ! [A3: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A3),zero_zero(int))) = zero_zero(rat) ).

% Frct_code_post(2)
tff(fact_6037_Frct__code__post_I3_J,axiom,
    frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),one_one(int))) = one_one(rat) ).

% Frct_code_post(3)
tff(fact_6038_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),aa(num,int,numeral_numeral(int),L))) = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(num,rat,numeral_numeral(rat),K)),aa(num,rat,numeral_numeral(rat),L)) ).

% Frct_code_post(6)
tff(fact_6039_Frct__code__post_I4_J,axiom,
    ! [K: num] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(num,int,numeral_numeral(int),K)),one_one(int))) = aa(num,rat,numeral_numeral(rat),K) ).

% Frct_code_post(4)
tff(fact_6040_VEBT__internal_Ospace_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_space_rel2,Xb)
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ accp(vEBT_VEBT,vEBT_VEBT_space_rel2,vEBT_Leaf((A2),(B2))) ) )
         => ~ ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info,Deg,TreeList2,Summary) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Summary))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space,TreeList2),zero_zero(nat))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_space_rel2,vEBT_Node(Info,Deg,TreeList2,Summary)) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_6041_mergesort__by__rel__split__length,axiom,
    ! [A: $tType,Xs1: list(A),Xs22: list(A),Xs: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs1)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
      & ( aa(list(A),nat,size_size(list(A)),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs22)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% mergesort_by_rel_split_length
tff(fact_6042_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( gcd_gcd(nat,A3,B3) = zero_zero(nat) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B3 = zero_zero(nat) ) ) ) ).

% gcd_nat.eq_neutr_iff
tff(fact_6043_gcd__nat_Oleft__neutral,axiom,
    ! [A3: nat] : gcd_gcd(nat,zero_zero(nat),A3) = A3 ).

% gcd_nat.left_neutral
tff(fact_6044_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A3: nat,B3: nat] :
      ( ( zero_zero(nat) = gcd_gcd(nat,A3,B3) )
    <=> ( ( A3 = zero_zero(nat) )
        & ( B3 = zero_zero(nat) ) ) ) ).

% gcd_nat.neutr_eq_iff
tff(fact_6045_gcd__nat_Oright__neutral,axiom,
    ! [A3: nat] : gcd_gcd(nat,A3,zero_zero(nat)) = A3 ).

% gcd_nat.right_neutral
tff(fact_6046_gcd__0__nat,axiom,
    ! [Xb: nat] : gcd_gcd(nat,Xb,zero_zero(nat)) = Xb ).

% gcd_0_nat
tff(fact_6047_gcd__0__left__nat,axiom,
    ! [Xb: nat] : gcd_gcd(nat,zero_zero(nat),Xb) = Xb ).

% gcd_0_left_nat
tff(fact_6048_gcd__1__nat,axiom,
    ! [M: nat] : gcd_gcd(nat,M,one_one(nat)) = one_one(nat) ).

% gcd_1_nat
tff(fact_6049_gcd__Suc__0,axiom,
    ! [M: nat] : gcd_gcd(nat,M,aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,suc,zero_zero(nat)) ).

% gcd_Suc_0
tff(fact_6050_gcd__pos__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),gcd_gcd(nat,M,N))
    <=> ( ( M != zero_zero(nat) )
        | ( N != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_6051_gcd__diff2__nat,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,M),N)
     => ( gcd_gcd(nat,aa(nat,nat,minus_minus(nat,N),M),N) = gcd_gcd(nat,M,N) ) ) ).

% gcd_diff2_nat
tff(fact_6052_gcd__diff1__nat,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less_eq(nat,N),M)
     => ( gcd_gcd(nat,aa(nat,nat,minus_minus(nat,M),N),N) = gcd_gcd(nat,M,N) ) ) ).

% gcd_diff1_nat
tff(fact_6053_gcd__le2__nat,axiom,
    ! [B3: nat,A3: nat] :
      ( ( B3 != zero_zero(nat) )
     => aa(nat,$o,ord_less_eq(nat,gcd_gcd(nat,A3,B3)),B3) ) ).

% gcd_le2_nat
tff(fact_6054_gcd__le1__nat,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 != zero_zero(nat) )
     => aa(nat,$o,ord_less_eq(nat,gcd_gcd(nat,A3,B3)),A3) ) ).

% gcd_le1_nat
tff(fact_6055_gcd__non__0__nat,axiom,
    ! [Y: nat,Xb: nat] :
      ( ( Y != zero_zero(nat) )
     => ( gcd_gcd(nat,Xb,Y) = gcd_gcd(nat,Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Y)) ) ) ).

% gcd_non_0_nat
tff(fact_6056_gcd__nat_Osimps,axiom,
    ! [Xb: nat,Y: nat] :
      gcd_gcd(nat,Xb,Y) = $ite(Y = zero_zero(nat),Xb,gcd_gcd(nat,Y,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Y))) ).

% gcd_nat.simps
tff(fact_6057_gcd__nat_Oelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( gcd_gcd(nat,Xb,Xaa) = Y )
     => ( Y = $ite(Xaa = zero_zero(nat),Xb,gcd_gcd(nat,Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Xaa))) ) ) ).

% gcd_nat.elims
tff(fact_6058_bezout__nat,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 != zero_zero(nat) )
     => ? [X2: nat,Y2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),Y2)),gcd_gcd(nat,A3,B3)) ) ).

% bezout_nat
tff(fact_6059_bezout__gcd__nat_H,axiom,
    ! [B3: nat,A3: nat] :
    ? [X2: nat,Y2: nat] :
      ( ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),Y2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X2))
        & ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),Y2)) = gcd_gcd(nat,A3,B3) ) )
      | ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),X2))
        & ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B3),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),Y2)) = gcd_gcd(nat,A3,B3) ) ) ) ).

% bezout_gcd_nat'
tff(fact_6060_gcd__code__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      gcd_gcd(code_integer,K,L) = abs_abs(code_integer,
        $ite(L = zero_zero(code_integer),K,gcd_gcd(code_integer,L,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),abs_abs(code_integer,K)),abs_abs(code_integer,L))))) ).

% gcd_code_integer
tff(fact_6061_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_space_rel,Xb)
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
               => ~ accp(vEBT_VEBT,vEBT_VEBT_space_rel,vEBT_Leaf((A2),(B2))) ) )
         => ~ ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info,Deg,TreeList2,Summary) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Summary))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2,TreeList2),zero_zero(nat))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_space_rel,vEBT_Node(Info,Deg,TreeList2,Summary)) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_6062_VEBT__internal_Ocnt_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: real] :
      ( ( aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2,Xb)
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = one_one(real) )
               => ~ accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2,vEBT_Leaf((A2),(B2))) ) )
         => ~ ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info,Deg,TreeList2,Summary) )
               => ( ( Y = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Summary))),foldr(real,real,plus_plus(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt,TreeList2),zero_zero(real))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2,vEBT_Node(Info,Deg,TreeList2,Summary)) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
tff(fact_6063_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_cnt_rel,Xb)
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = one_one(nat) )
               => ~ accp(vEBT_VEBT,vEBT_VEBT_cnt_rel,vEBT_Leaf((A2),(B2))) ) )
         => ~ ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info,Deg,TreeList2,Summary) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Summary))),foldr(nat,nat,plus_plus(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2,TreeList2),zero_zero(nat))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_cnt_rel,vEBT_Node(Info,Deg,TreeList2,Summary)) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
tff(fact_6064_vebt__maxt_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_vebt_maxt_rel,Xb)
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = $ite(
                      (B2),
                      aa(nat,option(nat),some(nat),one_one(nat)),
                      $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) )
               => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Leaf((A2),(B2))) ) )
         => ( ! [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw) )
               => ( ( Y = none(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Ma2) )
                   => ~ accp(vEBT_VEBT,vEBT_vebt_maxt_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_6065_vebt__mint_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_vebt_mint_rel,Xb)
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = $ite(
                      (A2),
                      aa(nat,option(nat),some(nat),zero_zero(nat)),
                      $ite((B2),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) )
               => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Leaf((A2),(B2))) ) )
         => ( ! [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw) )
               => ( ( Y = none(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Mi2) )
                   => ~ accp(vEBT_VEBT,vEBT_vebt_mint_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_6066_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_t(Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,Xb)
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite((A2),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,vEBT_Leaf((A2),(B2))) ) )
         => ( ! [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_6067_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_a_x_t(Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,Xb)
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite((B2),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,vEBT_Leaf((A2),(B2))) ) )
         => ( ! [Uu2: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_6068_gcd__nat_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( gcd_gcd(nat,Xb,Xaa) = Y )
     => ( accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa))
       => ~ ( ( Y = $ite(Xaa = zero_zero(nat),Xb,gcd_gcd(nat,Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Xaa))) )
           => ~ accp(product_prod(nat,nat),gcd_nat_rel,aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xb),Xaa)) ) ) ) ).

% gcd_nat.pelims
tff(fact_6069_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062N_092_060_094sub_062u_092_060_094sub_062l_092_060_094sub_062l_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( vEBT_T_m_i_n_N_u_l_l(Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( ( Y = one_one(nat) )
             => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv)) )
               => ( ( Y = one_one(nat) )
                 => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Leaf($true,(Uv))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ( Y = one_one(nat) )
                   => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uw,Ux2,Uy2) )
                   => ( ( Y = one_one(nat) )
                     => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Node(none(product_prod(nat,nat)),Uw,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) )
                     => ( ( Y = one_one(nat) )
                       => ~ accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_6070_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xb)
      <=> (Y) )
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv)) )
               => ( ~ (Y)
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($true,(Uv))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Y)
                   => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uw,Ux2,Uy2) )
                   => ( (Y)
                     => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) )
                     => ( ~ (Y)
                       => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_6071_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xb)
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,Xb)
       => ( ! [Uv: $o] :
              ( ( Xb = vEBT_Leaf($true,(Uv)) )
             => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($true,(Uv))) )
         => ( ! [Uu2: $o] :
                ( ( Xb = vEBT_Leaf((Uu2),$true) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf((Uu2),$true)) )
           => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: list(vEBT_VEBT),Vc2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) )
                 => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_6072_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xb)
     => ( accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Leaf($false,$false)) )
         => ~ ! [Uw: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uw,Ux2,Uy2) )
               => ~ accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel,vEBT_Node(none(product_prod(nat,nat)),Uw,Ux2,Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_6073_setceilmax,axiom,
    ! [S: vEBT_VEBT,M: nat,Listy: list(vEBT_VEBT),N: nat] :
      ( vEBT_invar_vebt(S,M)
     => ( ! [X2: vEBT_VEBT] :
            ( member(vEBT_VEBT,X2,set2(vEBT_VEBT,Listy))
           => vEBT_invar_vebt(X2,N) )
       => ( ( M = aa(nat,nat,suc,N) )
         => ( ! [X2: vEBT_VEBT] :
                ( member(vEBT_VEBT,X2,set2(vEBT_VEBT,Listy))
               => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X2)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),N))) ) )
           => ( ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,S)) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) )
             => ( aa(nat,int,semiring_1_of_nat(int),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,S),set2(vEBT_VEBT,Listy))))) = archimedean_ceiling(real,aa(real,real,log(aa(num,real,numeral_numeral(real),bit0(one2))),aa(nat,real,semiring_1_of_nat(real),M))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_6074_flat__lub__def,axiom,
    ! [A: $tType,B3: A,A4: set(A)] :
      aa(set(A),A,partial_flat_lub(A,B3),A4) = $ite(aa(set(A),$o,ord_less_eq(set(A),A4),aa(set(A),set(A),insert(A,B3),bot_bot(set(A)))),B3,the(A,aa(set(A),fun(A,$o),aTP_Lamp_ahd(A,fun(set(A),fun(A,$o)),B3),A4))) ).

% flat_lub_def
tff(fact_6075_image__ident,axiom,
    ! [A: $tType,Y6: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_ct(A,A)),Y6) = Y6 ).

% image_ident
tff(fact_6076_height__compose__list,axiom,
    ! [Tb: vEBT_VEBT,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] :
      ( member(vEBT_VEBT,Tb,set2(vEBT_VEBT,TreeList))
     => aa(nat,$o,ord_less_eq(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_height,Tb)),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,Summary2),set2(vEBT_VEBT,TreeList))))) ) ).

% height_compose_list
tff(fact_6077_max__ins__scaled,axiom,
    ! [N: nat,X14: vEBT_VEBT,M: nat,X13: list(vEBT_VEBT)] : aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),lattic643756798349783984er_Max(nat,aa(set(nat),set(nat),insert(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14)),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),set2(vEBT_VEBT,X13))))))) ).

% max_ins_scaled
tff(fact_6078_height__i__max,axiom,
    ! [I: nat,X13: list(vEBT_VEBT),Foo: nat] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13))
     => aa(nat,$o,ord_less_eq(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I))),aa(nat,nat,ord_max(nat,Foo),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),set2(vEBT_VEBT,X13))))) ) ).

% height_i_max
tff(fact_6079_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S3: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S3) = S3 ) ).

% image_add_0
tff(fact_6080_max__idx__list,axiom,
    ! [I: nat,X13: list(vEBT_VEBT),N: nat,X14: vEBT_VEBT] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13))
     => aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13),I)))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),aa(nat,nat,ord_max(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14)),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),set2(vEBT_VEBT,X13)))))))) ) ).

% max_idx_list
tff(fact_6081_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_ahe(A,fun(A,A),A3)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_diff_right
tff(fact_6082_image__add__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_ahf(A,fun(A,A),K)),set_or1337092689740270186AtMost(A,I,J)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastAtMost'
tff(fact_6083_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D: A,A3: A,B3: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_ahg(A,fun(A,A),D)),set_or1337092689740270186AtMost(A,A3,B3)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,A3),D),aa(A,A,minus_minus(A,B3),D)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_6084_image__add__atLeastLessThan_H,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_ahf(A,fun(A,A),K)),set_or7035219750837199246ssThan(A,I,J)) = set_or7035219750837199246ssThan(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K)) ) ).

% image_add_atLeastLessThan'
tff(fact_6085_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero(nat) )
     => ( lattic643756798349783984er_Max(nat,collect(nat,aTP_Lamp_su(nat,fun(nat,$o),N))) = N ) ) ).

% Max_divisors_self_nat
tff(fact_6086_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite2(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,ord_less_eq(A,lattic643756798349783984er_Max(A,A4)),Xb)
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(A,$o,ord_less_eq(A,X),Xb) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_6087_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite2(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,ord_less(A,lattic643756798349783984er_Max(A,A4)),Xb)
            <=> ! [X: A] :
                  ( member(A,X,A4)
                 => aa(A,$o,ord_less(A,X),Xb) ) ) ) ) ) ).

% Max_less_iff
tff(fact_6088_range__constant,axiom,
    ! [B: $tType,A: $tType,Xb: A] : aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_cp(A,fun(B,A)),Xb)),top_top(set(B))) = aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))) ).

% range_constant
tff(fact_6089_Max__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [A4: set(A),C2: B] :
          ( finite_finite2(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( lattic643756798349783984er_Max(B,aa(set(A),set(B),image(A,B,aTP_Lamp_ahh(B,fun(A,B),C2)),A4)) = C2 ) ) ) ) ).

% Max_const
tff(fact_6090_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),D)
         => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),D)),set_or1337092689740270186AtMost(A,A3,B3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D),A3),aa(A,A,aa(A,fun(A,A),times_times(A),D),B3)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_6091_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D: A,A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),D)
         => ( aa(set(A),set(A),image(A,A,aTP_Lamp_ahi(A,fun(A,A),D)),set_or1337092689740270186AtMost(A,A3,B3)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),D),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),D)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_6092_rangeE,axiom,
    ! [A: $tType,B: $tType,B3: A,Fa: fun(B,A)] :
      ( member(A,B3,aa(set(B),set(A),image(B,A,Fa),top_top(set(B))))
     => ~ ! [X2: B] : B3 != aa(B,A,Fa,X2) ) ).

% rangeE
tff(fact_6093_range__composition,axiom,
    ! [A: $tType,C: $tType,B: $tType,Fa: fun(C,A),G: fun(B,C)] : aa(set(B),set(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_ahj(fun(C,A),fun(fun(B,C),fun(B,A)),Fa),G)),top_top(set(B))) = aa(set(C),set(A),image(C,A,Fa),aa(set(B),set(C),image(B,C,G),top_top(set(B)))) ).

% range_composition
tff(fact_6094_imageE,axiom,
    ! [A: $tType,B: $tType,B3: A,Fa: fun(B,A),A4: set(B)] :
      ( member(A,B3,aa(set(B),set(A),image(B,A,Fa),A4))
     => ~ ! [X2: B] :
            ( ( B3 = aa(B,A,Fa,X2) )
           => ~ member(B,X2,A4) ) ) ).

% imageE
tff(fact_6095_image__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,Fa: fun(B,A),G: fun(C,B),A4: set(C)] : aa(set(B),set(A),image(B,A,Fa),aa(set(C),set(B),image(C,B,G),A4)) = aa(set(C),set(A),image(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_cn(fun(B,A),fun(fun(C,B),fun(C,A)),Fa),G)),A4) ).

% image_image
tff(fact_6096_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,Fa: fun(B,A),A4: set(B),P: fun(A,$o)] : collect(A,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_ahk(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),Fa),A4),P)) = aa(set(B),set(A),image(B,A,Fa),collect(B,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_ahl(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Fa),A4),P))) ).

% Compr_image_eq
tff(fact_6097_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A4: set(A),Fa: fun(A,B)] :
      ( ~ finite_finite2(A,A4)
     => ( finite_finite2(B,aa(set(A),set(B),image(A,B,Fa),A4))
       => ? [X2: A] :
            ( member(A,X2,A4)
            & ~ finite_finite2(A,collect(A,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_ahm(set(A),fun(fun(A,B),fun(A,fun(A,$o))),A4),Fa),X2))) ) ) ) ).

% pigeonhole_infinite
tff(fact_6098_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite2(A,A4)
         => ( member(A,Xb,A4)
           => aa(A,$o,ord_less_eq(A,Xb),lattic643756798349783984er_Max(A,A4)) ) ) ) ).

% Max_ge
tff(fact_6099_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite2(A,A4)
         => ( ! [Y2: A] :
                ( member(A,Y2,A4)
               => aa(A,$o,ord_less_eq(A,Y2),Xb) )
           => ( member(A,Xb,A4)
             => ( lattic643756798349783984er_Max(A,A4) = Xb ) ) ) ) ) ).

% Max_eqI
tff(fact_6100_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B6: set(A)] :
          ( finite_finite2(A,A4)
         => ( finite_finite2(A,B6)
           => ( ! [X2: A] :
                  ( member(A,X2,A4)
                 => ? [Xa: A] :
                      ( member(A,Xa,B6)
                      & aa(A,$o,ord_less_eq(A,X2),Xa) ) )
             => ( ! [X2: A] :
                    ( member(A,X2,B6)
                   => ? [Xa: A] :
                        ( member(A,Xa,A4)
                        & aa(A,$o,ord_less_eq(A,X2),Xa) ) )
               => ( lattic643756798349783984er_Max(A,A4) = lattic643756798349783984er_Max(A,B6) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_6101_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),A3: A] :
          ( finite_finite2(A,A4)
         => ( member(A,A3,A4)
           => aa(A,$o,ord_less_eq(A,A3),lattic643756798349783984er_Max(A,A4)) ) ) ) ).

% Max.coboundedI
tff(fact_6102_Max__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S3: set(A),Fa: fun(A,B),K: B] :
          ( finite_finite2(A,S3)
         => ( ( S3 != bot_bot(set(A)) )
           => ( lattic643756798349783984er_Max(B,aa(set(A),set(B),image(A,B,aa(B,fun(A,B),aTP_Lamp_ahn(fun(A,B),fun(B,fun(A,B)),Fa),K)),S3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),lattic643756798349783984er_Max(B,aa(set(A),set(B),image(A,B,Fa),S3))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_6103_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),Fa: fun(A,C)] :
      ( finite_finite2(A,aa(set(B),set(A),image(B,A,G),top_top(set(B))))
     => finite_finite2(C,aa(set(B),set(C),image(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_aho(fun(B,A),fun(fun(A,C),fun(B,C)),G),Fa)),top_top(set(B)))) ) ).

% finite_range_imageI
tff(fact_6104_image__constant,axiom,
    ! [A: $tType,B: $tType,Xb: A,A4: set(A),C2: B] :
      ( member(A,Xb,A4)
     => ( aa(set(A),set(B),image(A,B,aTP_Lamp_ahp(B,fun(A,B),C2)),A4) = aa(set(B),set(B),insert(B,C2),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_6105_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,C2: A,A4: set(B)] :
      aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_cp(A,fun(B,A)),C2)),A4) = $ite(A4 = bot_bot(set(B)),bot_bot(set(A)),aa(set(A),set(A),insert(A,C2),bot_bot(set(A)))) ).

% image_constant_conv
tff(fact_6106_sum_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),H: fun(A,B),G: fun(A,C)] :
          ( finite_finite2(A,S3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),S3) = aa(set(C),B,groups7311177749621191930dd_sum(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ahr(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S3),H),G)),aa(set(A),set(C),image(A,C,G),S3)) ) ) ) ).

% sum.image_gen
tff(fact_6107_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,S: set(A),Tb: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_ahe(A,fun(A,A),A3)),aa(set(A),set(A),minus_minus(set(A),S),Tb)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),image(A,A,aTP_Lamp_ahe(A,fun(A,A),A3)),S)),aa(set(A),set(A),image(A,A,aTP_Lamp_ahe(A,fun(A,A),A3)),Tb)) ) ).

% translation_subtract_diff
tff(fact_6108_prod_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S3: set(A),H: fun(A,B),G: fun(A,C)] :
          ( finite_finite2(A,S3)
         => ( groups7121269368397514597t_prod(A,B,H,S3) = groups7121269368397514597t_prod(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ahs(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S3),H),G),aa(set(A),set(C),image(A,C,G),S3)) ) ) ) ).

% prod.image_gen
tff(fact_6109_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,Tb: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_ahe(A,fun(A,A),A3)),aa(set(A),set(A),uminus_uminus(set(A)),Tb)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image(A,A,aTP_Lamp_ahe(A,fun(A,A),A3)),Tb)) ) ).

% translation_subtract_Compl
tff(fact_6110_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite2(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ! [A2: A] :
                  ( member(A,A2,A4)
                 => aa(A,$o,ord_less_eq(A,A2),Xb) )
             => aa(A,$o,ord_less_eq(A,lattic643756798349783984er_Max(A,A4)),Xb) ) ) ) ) ).

% Max.boundedI
tff(fact_6111_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite2(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,ord_less_eq(A,lattic643756798349783984er_Max(A,A4)),Xb)
             => ! [A6: A] :
                  ( member(A,A6,A4)
                 => aa(A,$o,ord_less_eq(A,A6),Xb) ) ) ) ) ) ).

% Max.boundedE
tff(fact_6112_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),M: A] :
          ( finite_finite2(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ( M = lattic643756798349783984er_Max(A,A4) )
            <=> ( member(A,M,A4)
                & ! [X: A] :
                    ( member(A,X,A4)
                   => aa(A,$o,ord_less_eq(A,X),M) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_6113_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite2(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,ord_less_eq(A,Xb),lattic643756798349783984er_Max(A,A4))
            <=> ? [X: A] :
                  ( member(A,X,A4)
                  & aa(A,$o,ord_less_eq(A,Xb),X) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_6114_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),M: A] :
          ( finite_finite2(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( ( lattic643756798349783984er_Max(A,A4) = M )
            <=> ( member(A,M,A4)
                & ! [X: A] :
                    ( member(A,X,A4)
                   => aa(A,$o,ord_less_eq(A,X),M) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_6115_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),Xb: A] :
          ( finite_finite2(A,A4)
         => ( ( A4 != bot_bot(set(A)) )
           => ( aa(A,$o,ord_less(A,Xb),lattic643756798349783984er_Max(A,A4))
            <=> ? [X: A] :
                  ( member(A,X,A4)
                  & aa(A,$o,ord_less(A,Xb),X) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_6116_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),A3: A] :
          ( finite_finite2(A,A4)
         => ( ! [B2: A] :
                ( member(A,B2,A4)
               => aa(A,$o,ord_less_eq(A,B2),A3) )
           => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),insert(A,A3),A4)) = A3 ) ) ) ) ).

% Max_insert2
tff(fact_6117_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Uu,Deg2,TreeList,Summary2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,Summary2),set2(vEBT_VEBT,TreeList))))) ).

% VEBT_internal.height.simps(2)
tff(fact_6118_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] :
          ( ~ finite_finite2(A,A4)
         => ( lattic643756798349783984er_Max(A,A4) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Max.infinite
tff(fact_6119_sum_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [S3: set(A),T6: set(B),G: fun(A,B),H: fun(A,C)] :
          ( finite_finite2(A,S3)
         => ( finite_finite2(B,T6)
           => ( aa(set(B),$o,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,G),S3)),T6)
             => ( aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_ahu(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S3),G),H)),T6) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,H),S3) ) ) ) ) ) ).

% sum.group
tff(fact_6120_prod_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [S3: set(A),T6: set(B),G: fun(A,B),H: fun(A,C)] :
          ( finite_finite2(A,S3)
         => ( finite_finite2(B,T6)
           => ( aa(set(B),$o,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,G),S3)),T6)
             => ( groups7121269368397514597t_prod(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_ahv(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S3),G),H),T6) = groups7121269368397514597t_prod(A,C,H,S3) ) ) ) ) ) ).

% prod.group
tff(fact_6121_set__image__eq__pointwiseI,axiom,
    ! [B: $tType,A: $tType,L: list(A),L3: list(A),Fa: fun(A,B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(list(A),nat,size_size(list(A)),L3) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,ord_less(nat,I2),aa(list(A),nat,size_size(list(A)),L))
           => ( aa(A,B,Fa,aa(nat,A,nth(A,L),I2)) = aa(A,B,Fa,aa(nat,A,nth(A,L3),I2)) ) )
       => ( aa(set(A),set(B),image(A,B,Fa),set2(A,L)) = aa(set(A),set(B),image(A,B,Fa),set2(A,L3)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_6122_in__set__image__conv__nth,axiom,
    ! [A: $tType,B: $tType,Fa: fun(B,A),Xb: B,L: list(B)] :
      ( member(A,aa(B,A,Fa,Xb),aa(set(B),set(A),image(B,A,Fa),set2(B,L)))
    <=> ? [I4: nat] :
          ( aa(nat,$o,ord_less(nat,I4),aa(list(B),nat,size_size(list(B)),L))
          & ( aa(B,A,Fa,aa(nat,B,nth(B,L),I4)) = aa(B,A,Fa,Xb) ) ) ) ).

% in_set_image_conv_nth
tff(fact_6123_scaleR__image__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,Xb: A,Y: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(set(A),set(A),image(A,A,real_V8093663219630862766scaleR(A,C2)),set_or1337092689740270186AtMost(A,Xb,Y)) = set_or1337092689740270186AtMost(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),Xb),aa(A,A,real_V8093663219630862766scaleR(A,C2),Y)) ) ) ) ).

% scaleR_image_atLeastAtMost
tff(fact_6124_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A),B6: set(A)] :
          ( aa(set(A),$o,ord_less_eq(set(A),A4),B6)
         => ( ( A4 != bot_bot(set(A)) )
           => ( finite_finite2(A,B6)
             => aa(A,$o,ord_less_eq(A,lattic643756798349783984er_Max(A,A4)),lattic643756798349783984er_Max(A,B6)) ) ) ) ) ).

% Max.subset_imp
tff(fact_6125_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M8: set(A),N2: set(A)] :
          ( aa(set(A),$o,ord_less_eq(set(A),M8),N2)
         => ( ( M8 != bot_bot(set(A)) )
           => ( finite_finite2(A,N2)
             => aa(A,$o,ord_less_eq(A,lattic643756798349783984er_Max(A,M8)),lattic643756798349783984er_Max(A,N2)) ) ) ) ) ).

% Max_mono
tff(fact_6126_VEBT__internal_Oheight_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,Xb) = Y )
     => ( ( ? [A2: $o,B2: $o] : Xb = vEBT_Leaf((A2),(B2))
         => ( Y != zero_zero(nat) ) )
       => ~ ! [Uu2: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Uu2,Deg,TreeList2,Summary) )
             => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,Summary),set2(vEBT_VEBT,TreeList2))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_6127_divide__nat__def,axiom,
    ! [M: nat,N: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),N) = $ite(N = zero_zero(nat),zero_zero(nat),lattic643756798349783984er_Max(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ahw(nat,fun(nat,fun(nat,$o)),M),N)))) ).

% divide_nat_def
tff(fact_6128_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( gcd_gcd(nat,M,N) = lattic643756798349783984er_Max(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ahx(nat,fun(nat,fun(nat,$o)),N),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_6129_image__mult__atLeastAtMost__if,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xb: A,Y: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),C2)),set_or1337092689740270186AtMost(A,Xb,Y)) = $ite(
            aa(A,$o,ord_less(A,zero_zero(A)),C2),
            set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y)),
            $ite(aa(A,$o,ord_less_eq(A,Xb),Y),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),Y),aa(A,A,aa(A,fun(A,A),times_times(A),C2),Xb)),bot_bot(set(A))) ) ) ).

% image_mult_atLeastAtMost_if
tff(fact_6130_image__mult__atLeastAtMost__if_H,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,Xb: A,Y: A] :
          aa(set(A),set(A),image(A,A,aTP_Lamp_ahy(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,Xb,Y)) = $ite(
            aa(A,$o,ord_less_eq(A,Xb),Y),
            $ite(aa(A,$o,ord_less(A,zero_zero(A)),C2),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),Y),C2),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),C2))),
            bot_bot(set(A)) ) ) ).

% image_mult_atLeastAtMost_if'
tff(fact_6131_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A3: A,B3: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_ahz(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B3)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B3)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B3)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2))) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_6132_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A3: A,B3: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_aia(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B3)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B3)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B3)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A3)),C2))) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_6133_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A3: A,B3: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_aib(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B3)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_6134_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A3: A,B3: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_aic(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A3,B3)) = $ite(
            set_or1337092689740270186AtMost(A,A3,B3) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),M),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B3),M)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_6135_VEBT__internal_Oheight_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,Xb) = Y )
     => ( accp(vEBT_VEBT,vEBT_VEBT_height_rel,Xb)
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leaf((A2),(B2)) )
             => ( ( Y = zero_zero(nat) )
               => ~ accp(vEBT_VEBT,vEBT_VEBT_height_rel,vEBT_Leaf((A2),(B2))) ) )
         => ~ ! [Uu2: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,Deg,TreeList2,Summary) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(set(vEBT_VEBT),set(vEBT_VEBT),insert(vEBT_VEBT,Summary),set2(vEBT_VEBT,TreeList2))))) )
                 => ~ accp(vEBT_VEBT,vEBT_VEBT_height_rel,vEBT_Node(Uu2,Deg,TreeList2,Summary)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_6136_tailrec_Ofixp__induct__uc,axiom,
    ! [A: $tType,B: $tType,C: $tType,U2: fun(C,fun(A,B)),F5: fun(C,C),C5: fun(fun(A,B),C),Fa: C,P: fun(fun(A,B),$o)] :
      ( ! [X2: A] : comple7038119648293358887notone(fun(A,B),B,partial_fun_ord(B,B,A,partial_flat_ord(B,undefined(B))),partial_flat_ord(B,undefined(B)),aa(A,fun(fun(A,B),B),aa(fun(fun(A,B),C),fun(A,fun(fun(A,B),B)),aa(fun(C,C),fun(fun(fun(A,B),C),fun(A,fun(fun(A,B),B))),aTP_Lamp_aid(fun(C,fun(A,B)),fun(fun(C,C),fun(fun(fun(A,B),C),fun(A,fun(fun(A,B),B)))),U2),F5),C5),X2))
     => ( ( Fa = aa(fun(A,B),C,C5,comple187402453842119260l_fixp(fun(A,B),partial_fun_lub(B,B,A,partial_flat_lub(B,undefined(B))),partial_fun_ord(B,B,A,partial_flat_ord(B,undefined(B))),aa(fun(fun(A,B),C),fun(fun(A,B),fun(A,B)),aa(fun(C,C),fun(fun(fun(A,B),C),fun(fun(A,B),fun(A,B))),aTP_Lamp_aie(fun(C,fun(A,B)),fun(fun(C,C),fun(fun(fun(A,B),C),fun(fun(A,B),fun(A,B)))),U2),F5),C5))) )
       => ( ! [F4: fun(A,B)] : aa(C,fun(A,B),U2,aa(fun(A,B),C,C5,F4)) = F4
         => ( comple1908693960933563346ssible(fun(A,B),partial_fun_lub(B,B,A,partial_flat_lub(B,undefined(B))),partial_fun_ord(B,B,A,partial_flat_ord(B,undefined(B))),P)
           => ( aa(fun(A,B),$o,P,aTP_Lamp_aif(A,B))
             => ( ! [F4: C] :
                    ( aa(fun(A,B),$o,P,aa(C,fun(A,B),U2,F4))
                   => aa(fun(A,B),$o,P,aa(C,fun(A,B),U2,aa(C,C,F5,F4))) )
               => aa(fun(A,B),$o,P,aa(C,fun(A,B),U2,Fa)) ) ) ) ) ) ) ).

% tailrec.fixp_induct_uc
tff(fact_6137_bij__betw__Suc,axiom,
    ! [M8: set(nat),N2: set(nat)] :
      ( bij_betw(nat,nat,suc,M8,N2)
    <=> ( aa(set(nat),set(nat),image(nat,nat,suc),M8) = N2 ) ) ).

% bij_betw_Suc
tff(fact_6138_img__fst,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,S3: set(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),S3)
     => member(A,A3,aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),S3)) ) ).

% img_fst
tff(fact_6139_img__snd,axiom,
    ! [B: $tType,A: $tType,A3: A,B3: B,S3: set(product_prod(A,B))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),S3)
     => member(B,B3,aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),S3)) ) ).

% img_snd
tff(fact_6140_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A3: A,B3: B,A4: set(product_prod(A,B)),Fa: fun(A,fun(B,C))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A3),B3),A4)
     => member(C,aa(B,C,aa(A,fun(B,C),Fa,A3),B3),aa(set(product_prod(A,B)),set(C),image(product_prod(A,B),C,product_case_prod(A,B,C,Fa)),A4)) ) ).

% pair_imageI
tff(fact_6141_range__mult,axiom,
    ! [A3: real] :
      aa(set(real),set(real),image(real,real,aa(real,fun(real,real),times_times(real),A3)),top_top(set(real))) = $ite(A3 = zero_zero(real),aa(set(real),set(real),insert(real,zero_zero(real)),bot_bot(set(real))),top_top(set(real))) ).

% range_mult
tff(fact_6142_range__fst,axiom,
    ! [B: $tType,A: $tType] : aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),top_top(set(product_prod(A,B)))) = top_top(set(A)) ).

% range_fst
tff(fact_6143_range__snd,axiom,
    ! [B: $tType,A: $tType] : aa(set(product_prod(B,A)),set(A),image(product_prod(B,A),A,product_snd(B,A)),top_top(set(product_prod(B,A)))) = top_top(set(A)) ).

% range_snd
tff(fact_6144_Max__divisors__self__int,axiom,
    ! [N: int] :
      ( ( N != zero_zero(int) )
     => ( lattic643756798349783984er_Max(int,collect(int,aTP_Lamp_tg(int,fun(int,$o),N))) = abs_abs(int,N) ) ) ).

% Max_divisors_self_int
tff(fact_6145_nth__image__indices,axiom,
    ! [A: $tType,L: list(A)] : aa(set(nat),set(A),image(nat,A,nth(A,L)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = set2(A,L) ).

% nth_image_indices
tff(fact_6146_option_Opartial__function__definitions__axioms,axiom,
    ! [A: $tType] : partia7178651479351089652itions(option(A),partial_flat_ord(option(A),none(A)),partial_flat_lub(option(A),none(A))) ).

% option.partial_function_definitions_axioms
tff(fact_6147_option_Omono__body__fixp,axiom,
    ! [B: $tType,A: $tType,F5: fun(fun(A,option(B)),fun(A,option(B)))] :
      ( ! [X2: A] : comple7038119648293358887notone(fun(A,option(B)),option(B),partial_fun_ord(option(B),option(B),A,partial_flat_ord(option(B),none(B))),partial_flat_ord(option(B),none(B)),aa(A,fun(fun(A,option(B)),option(B)),aTP_Lamp_aig(fun(fun(A,option(B)),fun(A,option(B))),fun(A,fun(fun(A,option(B)),option(B))),F5),X2))
     => ( comple187402453842119260l_fixp(fun(A,option(B)),partial_fun_lub(option(B),option(B),A,partial_flat_lub(option(B),none(B))),partial_fun_ord(option(B),option(B),A,partial_flat_ord(option(B),none(B))),F5) = aa(fun(A,option(B)),fun(A,option(B)),F5,comple187402453842119260l_fixp(fun(A,option(B)),partial_fun_lub(option(B),option(B),A,partial_flat_lub(option(B),none(B))),partial_fun_ord(option(B),option(B),A,partial_flat_ord(option(B),none(B))),F5)) ) ) ).

% option.mono_body_fixp
tff(fact_6148_option_Ofixp__rule__uc,axiom,
    ! [A: $tType,B: $tType,C: $tType,U2: fun(C,fun(A,option(B))),F5: fun(C,C),C5: fun(fun(A,option(B)),C),Fa: C] :
      ( ! [X2: A] : comple7038119648293358887notone(fun(A,option(B)),option(B),partial_fun_ord(option(B),option(B),A,partial_flat_ord(option(B),none(B))),partial_flat_ord(option(B),none(B)),aa(A,fun(fun(A,option(B)),option(B)),aa(fun(fun(A,option(B)),C),fun(A,fun(fun(A,option(B)),option(B))),aa(fun(C,C),fun(fun(fun(A,option(B)),C),fun(A,fun(fun(A,option(B)),option(B)))),aTP_Lamp_aih(fun(C,fun(A,option(B))),fun(fun(C,C),fun(fun(fun(A,option(B)),C),fun(A,fun(fun(A,option(B)),option(B))))),U2),F5),C5),X2))
     => ( ( Fa = aa(fun(A,option(B)),C,C5,comple187402453842119260l_fixp(fun(A,option(B)),partial_fun_lub(option(B),option(B),A,partial_flat_lub(option(B),none(B))),partial_fun_ord(option(B),option(B),A,partial_flat_ord(option(B),none(B))),aa(fun(fun(A,option(B)),C),fun(fun(A,option(B)),fun(A,option(B))),aa(fun(C,C),fun(fun(fun(A,option(B)),C),fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_aii(fun(C,fun(A,option(B))),fun(fun(C,C),fun(fun(fun(A,option(B)),C),fun(fun(A,option(B)),fun(A,option(B))))),U2),F5),C5))) )
       => ( ! [F4: C] : aa(fun(A,option(B)),C,C5,aa(C,fun(A,option(B)),U2,F4)) = F4
         => ( Fa = aa(C,C,F5,Fa) ) ) ) ) ).

% option.fixp_rule_uc
tff(fact_6149_zero__notin__Suc__image,axiom,
    ! [A4: set(nat)] : ~ member(nat,zero_zero(nat),aa(set(nat),set(nat),image(nat,nat,suc),A4)) ).

% zero_notin_Suc_image
tff(fact_6150_None__notin__image__Some,axiom,
    ! [A: $tType,A4: set(A)] : ~ member(option(A),none(A),aa(set(A),set(option(A)),image(A,option(A),some(A)),A4)) ).

% None_notin_image_Some
tff(fact_6151_option_Ofixp__induct__uc,axiom,
    ! [A: $tType,B: $tType,C: $tType,U2: fun(C,fun(A,option(B))),F5: fun(C,C),C5: fun(fun(A,option(B)),C),Fa: C,P: fun(fun(A,option(B)),$o)] :
      ( ! [X2: A] : comple7038119648293358887notone(fun(A,option(B)),option(B),partial_fun_ord(option(B),option(B),A,partial_flat_ord(option(B),none(B))),partial_flat_ord(option(B),none(B)),aa(A,fun(fun(A,option(B)),option(B)),aa(fun(fun(A,option(B)),C),fun(A,fun(fun(A,option(B)),option(B))),aa(fun(C,C),fun(fun(fun(A,option(B)),C),fun(A,fun(fun(A,option(B)),option(B)))),aTP_Lamp_aih(fun(C,fun(A,option(B))),fun(fun(C,C),fun(fun(fun(A,option(B)),C),fun(A,fun(fun(A,option(B)),option(B))))),U2),F5),C5),X2))
     => ( ( Fa = aa(fun(A,option(B)),C,C5,comple187402453842119260l_fixp(fun(A,option(B)),partial_fun_lub(option(B),option(B),A,partial_flat_lub(option(B),none(B))),partial_fun_ord(option(B),option(B),A,partial_flat_ord(option(B),none(B))),aa(fun(fun(A,option(B)),C),fun(fun(A,option(B)),fun(A,option(B))),aa(fun(C,C),fun(fun(fun(A,option(B)),C),fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_aii(fun(C,fun(A,option(B))),fun(fun(C,C),fun(fun(fun(A,option(B)),C),fun(fun(A,option(B)),fun(A,option(B))))),U2),F5),C5))) )
       => ( ! [F4: fun(A,option(B))] : aa(C,fun(A,option(B)),U2,aa(fun(A,option(B)),C,C5,F4)) = F4
         => ( comple1908693960933563346ssible(fun(A,option(B)),partial_fun_lub(option(B),option(B),A,partial_flat_lub(option(B),none(B))),partial_fun_ord(option(B),option(B),A,partial_flat_ord(option(B),none(B))),P)
           => ( aa(fun(A,option(B)),$o,P,aTP_Lamp_gi(A,option(B)))
             => ( ! [F4: C] :
                    ( aa(fun(A,option(B)),$o,P,aa(C,fun(A,option(B)),U2,F4))
                   => aa(fun(A,option(B)),$o,P,aa(C,fun(A,option(B)),U2,aa(C,C,F5,F4))) )
               => aa(fun(A,option(B)),$o,P,aa(C,fun(A,option(B)),U2,Fa)) ) ) ) ) ) ) ).

% option.fixp_induct_uc
tff(fact_6152_in__fst__imageE,axiom,
    ! [B: $tType,A: $tType,Xb: A,S3: set(product_prod(A,B))] :
      ( member(A,Xb,aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),S3))
     => ~ ! [Y2: B] : ~ member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y2),S3) ) ).

% in_fst_imageE
tff(fact_6153_in__snd__imageE,axiom,
    ! [A: $tType,B: $tType,Y: A,S3: set(product_prod(B,A))] :
      ( member(A,Y,aa(set(product_prod(B,A)),set(A),image(product_prod(B,A),A,product_snd(B,A)),S3))
     => ~ ! [X2: B] : ~ member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),X2),Y),S3) ) ).

% in_snd_imageE
tff(fact_6154_option_Oleq__antisym,axiom,
    ! [A: $tType,Xb: option(A),Y: option(A)] :
      ( aa(option(A),$o,aa(option(A),fun(option(A),$o),partial_flat_ord(option(A),none(A)),Xb),Y)
     => ( aa(option(A),$o,aa(option(A),fun(option(A),$o),partial_flat_ord(option(A),none(A)),Y),Xb)
       => ( Xb = Y ) ) ) ).

% option.leq_antisym
tff(fact_6155_option_Oleq__trans,axiom,
    ! [A: $tType,Xb: option(A),Y: option(A),Z: option(A)] :
      ( aa(option(A),$o,aa(option(A),fun(option(A),$o),partial_flat_ord(option(A),none(A)),Xb),Y)
     => ( aa(option(A),$o,aa(option(A),fun(option(A),$o),partial_flat_ord(option(A),none(A)),Y),Z)
       => aa(option(A),$o,aa(option(A),fun(option(A),$o),partial_flat_ord(option(A),none(A)),Xb),Z) ) ) ).

% option.leq_trans
tff(fact_6156_option_Oleq__refl,axiom,
    ! [A: $tType,Xb: option(A)] : aa(option(A),$o,aa(option(A),fun(option(A),$o),partial_flat_ord(option(A),none(A)),Xb),Xb) ).

% option.leq_refl
tff(fact_6157_tailrec_Oleq__antisym,axiom,
    ! [A: $tType,Xb: A,Y: A] :
      ( aa(A,$o,aa(A,fun(A,$o),partial_flat_ord(A,undefined(A)),Xb),Y)
     => ( aa(A,$o,aa(A,fun(A,$o),partial_flat_ord(A,undefined(A)),Y),Xb)
       => ( Xb = Y ) ) ) ).

% tailrec.leq_antisym
tff(fact_6158_tailrec_Oleq__trans,axiom,
    ! [A: $tType,Xb: A,Y: A,Z: A] :
      ( aa(A,$o,aa(A,fun(A,$o),partial_flat_ord(A,undefined(A)),Xb),Y)
     => ( aa(A,$o,aa(A,fun(A,$o),partial_flat_ord(A,undefined(A)),Y),Z)
       => aa(A,$o,aa(A,fun(A,$o),partial_flat_ord(A,undefined(A)),Xb),Z) ) ) ).

% tailrec.leq_trans
tff(fact_6159_tailrec_Oleq__refl,axiom,
    ! [A: $tType,Xb: A] : aa(A,$o,aa(A,fun(A,$o),partial_flat_ord(A,undefined(A)),Xb),Xb) ).

% tailrec.leq_refl
tff(fact_6160_option_Oconst__mono,axiom,
    ! [A: $tType,B: $tType,Ord: fun(A,fun(A,$o)),C2: option(B)] : comple7038119648293358887notone(A,option(B),Ord,partial_flat_ord(option(B),none(B)),aTP_Lamp_aij(option(B),fun(A,option(B)),C2)) ).

% option.const_mono
tff(fact_6161_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A4: set(A),Fa: fun(nat,A),N: nat] :
      ( ( A4 = aa(set(nat),set(A),image(nat,A,Fa),collect(nat,aTP_Lamp_sx(nat,fun(nat,$o),N))) )
     => finite_finite2(A,A4) ) ).

% nat_seg_image_imp_finite
tff(fact_6162_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A4: set(A)] :
      ( finite_finite2(A,A4)
    <=> ? [N8: nat,F7: fun(nat,A)] : A4 = aa(set(nat),set(A),image(nat,A,F7),collect(nat,aTP_Lamp_sx(nat,fun(nat,$o),N8))) ) ).

% finite_conv_nat_seg_image
tff(fact_6163_fixp__induct__option,axiom,
    ! [C: $tType,A: $tType,B: $tType,U2: fun(C,fun(A,option(B))),F5: fun(C,C),C5: fun(fun(A,option(B)),C),Fa: C,P: fun(A,fun(B,$o)),Xb: A,Y: B] :
      ( ! [X2: A] : comple7038119648293358887notone(fun(A,option(B)),option(B),partial_fun_ord(option(B),option(B),A,partial_flat_ord(option(B),none(B))),partial_flat_ord(option(B),none(B)),aa(A,fun(fun(A,option(B)),option(B)),aa(fun(fun(A,option(B)),C),fun(A,fun(fun(A,option(B)),option(B))),aa(fun(C,C),fun(fun(fun(A,option(B)),C),fun(A,fun(fun(A,option(B)),option(B)))),aTP_Lamp_aih(fun(C,fun(A,option(B))),fun(fun(C,C),fun(fun(fun(A,option(B)),C),fun(A,fun(fun(A,option(B)),option(B))))),U2),F5),C5),X2))
     => ( ( Fa = aa(fun(A,option(B)),C,C5,comple187402453842119260l_fixp(fun(A,option(B)),partial_fun_lub(option(B),option(B),A,partial_flat_lub(option(B),none(B))),partial_fun_ord(option(B),option(B),A,partial_flat_ord(option(B),none(B))),aa(fun(fun(A,option(B)),C),fun(fun(A,option(B)),fun(A,option(B))),aa(fun(C,C),fun(fun(fun(A,option(B)),C),fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_aii(fun(C,fun(A,option(B))),fun(fun(C,C),fun(fun(fun(A,option(B)),C),fun(fun(A,option(B)),fun(A,option(B))))),U2),F5),C5))) )
       => ( ! [F4: fun(A,option(B))] : aa(C,fun(A,option(B)),U2,aa(fun(A,option(B)),C,C5,F4)) = F4
         => ( ! [F4: C,X2: A,Y2: B] :
                ( ! [Xa: A,Ya2: B] :
                    ( ( aa(A,option(B),aa(C,fun(A,option(B)),U2,F4),Xa) = aa(B,option(B),some(B),Ya2) )
                   => aa(B,$o,aa(A,fun(B,$o),P,Xa),Ya2) )
               => ( ( aa(A,option(B),aa(C,fun(A,option(B)),U2,aa(C,C,F5,F4)),X2) = aa(B,option(B),some(B),Y2) )
                 => aa(B,$o,aa(A,fun(B,$o),P,X2),Y2) ) )
           => ( ( aa(A,option(B),aa(C,fun(A,option(B)),U2,Fa),Xb) = aa(B,option(B),some(B),Y) )
             => aa(B,$o,aa(A,fun(B,$o),P,Xb),Y) ) ) ) ) ) ).

% fixp_induct_option
tff(fact_6164_notin__range__Some,axiom,
    ! [A: $tType,Xb: option(A)] :
      ( ~ member(option(A),Xb,aa(set(A),set(option(A)),image(A,option(A),some(A)),top_top(set(A))))
    <=> ( Xb = none(A) ) ) ).

% notin_range_Some
tff(fact_6165_fst__image__mp,axiom,
    ! [B: $tType,A: $tType,A4: set(product_prod(A,B)),B6: set(A),Xb: A,Y: B] :
      ( aa(set(A),$o,ord_less_eq(set(A),aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),A4)),B6)
     => ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),Y),A4)
       => member(A,Xb,B6) ) ) ).

% fst_image_mp
tff(fact_6166_snd__image__mp,axiom,
    ! [B: $tType,A: $tType,A4: set(product_prod(B,A)),B6: set(A),Xb: B,Y: A] :
      ( aa(set(A),$o,ord_less_eq(set(A),aa(set(product_prod(B,A)),set(A),image(product_prod(B,A),A,product_snd(B,A)),A4)),B6)
     => ( member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Xb),Y),A4)
       => member(A,Y,B6) ) ) ).

% snd_image_mp
tff(fact_6167_tailrec_Oconst__mono,axiom,
    ! [A: $tType,B: $tType,Ord: fun(A,fun(A,$o)),C2: B] : comple7038119648293358887notone(A,B,Ord,partial_flat_ord(B,undefined(B)),aTP_Lamp_ahp(B,fun(A,B),C2)) ).

% tailrec.const_mono
tff(fact_6168_option__admissible,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : comple1908693960933563346ssible(fun(A,option(B)),partial_fun_lub(option(B),option(B),A,partial_flat_lub(option(B),none(B))),partial_fun_ord(option(B),option(B),A,partial_flat_ord(option(B),none(B))),aTP_Lamp_aik(fun(A,fun(B,$o)),fun(fun(A,option(B)),$o),P)) ).

% option_admissible
tff(fact_6169_gcd__is__Max__divisors__int,axiom,
    ! [N: int,M: int] :
      ( ( N != zero_zero(int) )
     => ( gcd_gcd(int,M,N) = lattic643756798349783984er_Max(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_ail(int,fun(int,fun(int,$o)),N),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_6170_image__Suc__lessThan,axiom,
    ! [N: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_ord_lessThan(nat,N)) = set_or1337092689740270186AtMost(nat,one_one(nat),N) ).

% image_Suc_lessThan
tff(fact_6171_image__Suc__atMost,axiom,
    ! [N: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_ord_atMost(nat,N)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,N)) ).

% image_Suc_atMost
tff(fact_6172_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_6173_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_6174_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_ord_lessThan(nat,N))) ).

% lessThan_Suc_eq_insert_0
tff(fact_6175_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] : set_ord_atMost(nat,aa(nat,nat,suc,N)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_ord_atMost(nat,N))) ).

% atMost_Suc_eq_insert_0
tff(fact_6176_Collect__split__mono__strong,axiom,
    ! [B: $tType,A: $tType,X5: set(A),A4: set(product_prod(A,B)),Y6: set(B),P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
      ( ( X5 = aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),A4) )
     => ( ( Y6 = aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),A4) )
       => ( ! [X2: A] :
              ( member(A,X2,X5)
             => ! [Xa2: B] :
                  ( member(B,Xa2,Y6)
                 => ( aa(B,$o,aa(A,fun(B,$o),P,X2),Xa2)
                   => aa(B,$o,aa(A,fun(B,$o),Q,X2),Xa2) ) ) )
         => ( aa(set(product_prod(A,B)),$o,ord_less_eq(set(product_prod(A,B)),A4),collect(product_prod(A,B),product_case_prod(A,B,$o,P)))
           => aa(set(product_prod(A,B)),$o,ord_less_eq(set(product_prod(A,B)),A4),collect(product_prod(A,B),product_case_prod(A,B,$o,Q))) ) ) ) ) ).

% Collect_split_mono_strong
tff(fact_6177_UNIV__option__conv,axiom,
    ! [A: $tType] : top_top(set(option(A))) = aa(set(option(A)),set(option(A)),insert(option(A),none(A)),aa(set(A),set(option(A)),image(A,option(A),some(A)),top_top(set(A)))) ).

% UNIV_option_conv
tff(fact_6178_range__mod,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_aim(nat,fun(nat,nat),N)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),N) ) ) ).

% range_mod
tff(fact_6179_tailrec_Opartial__function__definitions__axioms,axiom,
    ! [A: $tType] : partia7178651479351089652itions(A,partial_flat_ord(A,undefined(A)),partial_flat_lub(A,undefined(A))) ).

% tailrec.partial_function_definitions_axioms
tff(fact_6180_finite__range__Some,axiom,
    ! [A: $tType] :
      ( finite_finite2(option(A),aa(set(A),set(option(A)),image(A,option(A),some(A)),top_top(set(A))))
    <=> finite_finite2(A,top_top(set(A))) ) ).

% finite_range_Some
tff(fact_6181_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image(int,int,aTP_Lamp_ain(int,fun(int,int),L)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(int,int,minus_minus(int,U),L))) = set_or7035219750837199246ssThan(int,L,U) ).

% image_add_int_atLeastLessThan
tff(fact_6182_heap__step__admissible,axiom,
    ! [A: $tType,C: $tType,B: $tType,D2: $tType,E: $tType,P: fun(E,fun(A,fun(C,fun(B,fun(D2,$o))))),Xb: E] : comple1908693960933563346ssible(fun(A,option(product_prod(B,product_prod(C,D2)))),partial_fun_lub(option(product_prod(B,product_prod(C,D2))),option(product_prod(B,product_prod(C,D2))),A,partial_flat_lub(option(product_prod(B,product_prod(C,D2))),none(product_prod(B,product_prod(C,D2))))),partial_fun_ord(option(product_prod(B,product_prod(C,D2))),option(product_prod(B,product_prod(C,D2))),A,partial_flat_ord(option(product_prod(B,product_prod(C,D2))),none(product_prod(B,product_prod(C,D2))))),aa(E,fun(fun(A,option(product_prod(B,product_prod(C,D2)))),$o),aTP_Lamp_aio(fun(E,fun(A,fun(C,fun(B,fun(D2,$o))))),fun(E,fun(fun(A,option(product_prod(B,product_prod(C,D2)))),$o)),P),Xb)) ).

% heap_step_admissible
tff(fact_6183_image__add__integer__atLeastLessThan,axiom,
    ! [L: code_integer,U: code_integer] : aa(set(code_integer),set(code_integer),image(code_integer,code_integer,aTP_Lamp_aip(code_integer,fun(code_integer,code_integer),L)),set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),aa(code_integer,code_integer,minus_minus(code_integer,U),L))) = set_or7035219750837199246ssThan(code_integer,L,U) ).

% image_add_integer_atLeastLessThan
tff(fact_6184_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),U)
     => ( set_or7035219750837199246ssThan(int,zero_zero(int),U) = aa(set(nat),set(int),image(nat,int,semiring_1_of_nat(int)),set_ord_lessThan(nat,nat2(U))) ) ) ).

% image_atLeastZeroLessThan_int
tff(fact_6185_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Xb: nat,Y: nat] :
      aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_aiq(nat,fun(nat,nat),C2)),set_or7035219750837199246ssThan(nat,Xb,Y)) = $ite(
        aa(nat,$o,ord_less(nat,C2),Y),
        set_or7035219750837199246ssThan(nat,aa(nat,nat,minus_minus(nat,Xb),C2),aa(nat,nat,minus_minus(nat,Y),C2)),
        $ite(aa(nat,$o,ord_less(nat,Xb),Y),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat))),bot_bot(set(nat))) ) ).

% image_minus_const_atLeastLessThan_nat
tff(fact_6186_fixp__induct__tailrec,axiom,
    ! [C: $tType,A: $tType,B: $tType,C2: B,U2: fun(C,fun(A,B)),F5: fun(C,C),C5: fun(fun(A,B),C),Fa: C,P: fun(A,fun(B,$o)),Xb: A,Y: B] :
      ( ! [X2: A] : comple7038119648293358887notone(fun(A,B),B,partial_fun_ord(B,B,A,partial_flat_ord(B,C2)),partial_flat_ord(B,C2),aa(A,fun(fun(A,B),B),aa(fun(fun(A,B),C),fun(A,fun(fun(A,B),B)),aa(fun(C,C),fun(fun(fun(A,B),C),fun(A,fun(fun(A,B),B))),aTP_Lamp_aid(fun(C,fun(A,B)),fun(fun(C,C),fun(fun(fun(A,B),C),fun(A,fun(fun(A,B),B)))),U2),F5),C5),X2))
     => ( ( Fa = aa(fun(A,B),C,C5,comple187402453842119260l_fixp(fun(A,B),partial_fun_lub(B,B,A,partial_flat_lub(B,C2)),partial_fun_ord(B,B,A,partial_flat_ord(B,C2)),aa(fun(fun(A,B),C),fun(fun(A,B),fun(A,B)),aa(fun(C,C),fun(fun(fun(A,B),C),fun(fun(A,B),fun(A,B))),aTP_Lamp_aie(fun(C,fun(A,B)),fun(fun(C,C),fun(fun(fun(A,B),C),fun(fun(A,B),fun(A,B)))),U2),F5),C5))) )
       => ( ! [F4: fun(A,B)] : aa(C,fun(A,B),U2,aa(fun(A,B),C,C5,F4)) = F4
         => ( ! [F4: C,X2: A,Y2: B] :
                ( ! [Xa: A,Ya2: B] :
                    ( ( aa(A,B,aa(C,fun(A,B),U2,F4),Xa) = Ya2 )
                   => ( ( Ya2 != C2 )
                     => aa(B,$o,aa(A,fun(B,$o),P,Xa),Ya2) ) )
               => ( ( aa(A,B,aa(C,fun(A,B),U2,aa(C,C,F5,F4)),X2) = Y2 )
                 => ( ( Y2 != C2 )
                   => aa(B,$o,aa(A,fun(B,$o),P,X2),Y2) ) ) )
           => ( ( aa(A,B,aa(C,fun(A,B),U2,Fa),Xb) = Y )
             => ( ( Y != C2 )
               => aa(B,$o,aa(A,fun(B,$o),P,Xb),Y) ) ) ) ) ) ) ).

% fixp_induct_tailrec
tff(fact_6187_tailrec__admissible,axiom,
    ! [B: $tType,A: $tType,C2: B,P: fun(A,fun(B,$o))] : comple1908693960933563346ssible(fun(A,B),partial_fun_lub(B,B,A,partial_flat_lub(B,C2)),partial_fun_ord(B,B,A,partial_flat_ord(B,C2)),aa(fun(A,fun(B,$o)),fun(fun(A,B),$o),aTP_Lamp_air(B,fun(fun(A,fun(B,$o)),fun(fun(A,B),$o)),C2),P)) ).

% tailrec_admissible
tff(fact_6188_tailrec_Omono__body__fixp,axiom,
    ! [B: $tType,A: $tType,F5: fun(fun(A,B),fun(A,B))] :
      ( ! [X2: A] : comple7038119648293358887notone(fun(A,B),B,partial_fun_ord(B,B,A,partial_flat_ord(B,undefined(B))),partial_flat_ord(B,undefined(B)),aa(A,fun(fun(A,B),B),aTP_Lamp_ais(fun(fun(A,B),fun(A,B)),fun(A,fun(fun(A,B),B)),F5),X2))
     => ( comple187402453842119260l_fixp(fun(A,B),partial_fun_lub(B,B,A,partial_flat_lub(B,undefined(B))),partial_fun_ord(B,B,A,partial_flat_ord(B,undefined(B))),F5) = aa(fun(A,B),fun(A,B),F5,comple187402453842119260l_fixp(fun(A,B),partial_fun_lub(B,B,A,partial_flat_lub(B,undefined(B))),partial_fun_ord(B,B,A,partial_flat_ord(B,undefined(B))),F5)) ) ) ).

% tailrec.mono_body_fixp
tff(fact_6189_tailrec_Ofixp__rule__uc,axiom,
    ! [A: $tType,B: $tType,C: $tType,U2: fun(C,fun(A,B)),F5: fun(C,C),C5: fun(fun(A,B),C),Fa: C] :
      ( ! [X2: A] : comple7038119648293358887notone(fun(A,B),B,partial_fun_ord(B,B,A,partial_flat_ord(B,undefined(B))),partial_flat_ord(B,undefined(B)),aa(A,fun(fun(A,B),B),aa(fun(fun(A,B),C),fun(A,fun(fun(A,B),B)),aa(fun(C,C),fun(fun(fun(A,B),C),fun(A,fun(fun(A,B),B))),aTP_Lamp_aid(fun(C,fun(A,B)),fun(fun(C,C),fun(fun(fun(A,B),C),fun(A,fun(fun(A,B),B)))),U2),F5),C5),X2))
     => ( ( Fa = aa(fun(A,B),C,C5,comple187402453842119260l_fixp(fun(A,B),partial_fun_lub(B,B,A,partial_flat_lub(B,undefined(B))),partial_fun_ord(B,B,A,partial_flat_ord(B,undefined(B))),aa(fun(fun(A,B),C),fun(fun(A,B),fun(A,B)),aa(fun(C,C),fun(fun(fun(A,B),C),fun(fun(A,B),fun(A,B))),aTP_Lamp_aie(fun(C,fun(A,B)),fun(fun(C,C),fun(fun(fun(A,B),C),fun(fun(A,B),fun(A,B)))),U2),F5),C5))) )
       => ( ! [F4: C] : aa(fun(A,B),C,C5,aa(C,fun(A,B),U2,F4)) = F4
         => ( Fa = aa(C,C,F5,Fa) ) ) ) ) ).

% tailrec.fixp_rule_uc
tff(fact_6190_UNIV__nat__eq,axiom,
    top_top(set(nat)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),top_top(set(nat)))) ).

% UNIV_nat_eq
tff(fact_6191_bind__return,axiom,
    ! [A: $tType,Fa: heap_Time_Heap(A)] : heap_Time_bind(A,A,Fa,heap_Time_return(A)) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aTP_Lamp_gb(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),Fa)) ).

% bind_return
tff(fact_6192_wait__bind__decon,axiom,
    ! [A: $tType,P: assn,M: heap_Time_Heap(A),Q: fun(A,assn),N: nat] :
      ( hoare_hoare_triple(A,P,M,Q)
     => hoare_hoare_triple(A,P,heap_Time_bind(product_unit,A,heap_Time_wait(N),aTP_Lamp_gb(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),M)),Q) ) ).

% wait_bind_decon
tff(fact_6193_wait__rule,axiom,
    ! [N: nat] : hoare_hoare_triple(product_unit,one_one(assn),heap_Time_wait(N),aTP_Lamp_ait(product_unit,assn)) ).

% wait_rule
tff(fact_6194_return__bind,axiom,
    ! [A: $tType,B: $tType,Xb: B,Fa: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,aa(B,heap_Time_Heap(B),heap_Time_return(B),Xb),Fa) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aa(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_aiu(B,fun(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A))),Xb),Fa)) ).

% return_bind
tff(fact_6195_sofl__test,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),ring_1_signed(A,int,Xb)),ring_1_signed(A,int,Y)) = ring_1_signed(A,int,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y)) )
        <=> ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),Xb)),one_one(nat))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y)),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y)),Y))) = zero_zero(word(A)) ) ) ) ).

% sofl_test
tff(fact_6196_take__bit__num__def,axiom,
    ! [N: nat,M: num] :
      bit_take_bit_num(N,M) = $ite(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M)) = zero_zero(nat),none(num),aa(num,option(num),some(num),num_of_nat(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,N),aa(num,nat,numeral_numeral(nat),M))))) ).

% take_bit_num_def
tff(fact_6197_drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),zero_zero(A)) = zero_zero(A) ) ).

% drop_bit_of_0
tff(fact_6198_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A3) ) ).

% drop_bit_drop_bit
tff(fact_6199_drop__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A,B3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3)),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),B3)) ) ).

% drop_bit_and
tff(fact_6200_drop__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A,B3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3)),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),B3)) ) ).

% drop_bit_or
tff(fact_6201_drop__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A,B3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3)),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),B3)) ) ).

% drop_bit_xor
tff(fact_6202_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] : num_of_nat(aa(num,nat,numeral_numeral(nat),Q2)) = Q2 ).

% num_of_nat_numeral_eq
tff(fact_6203_drop__bit__of__bool,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,B3: $o] :
          aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),aa($o,A,zero_neq_one_of_bool(A),(B3))) = aa($o,A,zero_neq_one_of_bool(A),
            ( ( N = zero_zero(nat) )
            & (B3) )) ) ).

% drop_bit_of_bool
tff(fact_6204_drop__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit0
tff(fact_6205_drop__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit1
tff(fact_6206_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),N = zero_zero(nat)) ) ).

% drop_bit_of_1
tff(fact_6207_drop__bit__numeral__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),pred_numeral(L)),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit0
tff(fact_6208_drop__bit__numeral__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [L: num,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),pred_numeral(L)),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_numeral_bit1
tff(fact_6209_drop__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,N),M)),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),A3)) ) ).

% drop_bit_take_bit
tff(fact_6210_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A3)) ) ).

% take_bit_drop_bit
tff(fact_6211_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = A3 )
        <=> ( aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3) = zero_zero(A) ) ) ) ).

% take_bit_eq_self_iff_drop_bit_eq_0
tff(fact_6212_drop__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,M: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),N),M)) ) ).

% drop_bit_of_nat
tff(fact_6213_of__nat__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),M),N)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_drop_bit
tff(fact_6214_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_6215_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_6216_numeral__num__of__nat,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(num,nat,numeral_numeral(nat),num_of_nat(N)) = N ) ) ).

% numeral_num_of_nat
tff(fact_6217_num__of__nat__One,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less_eq(nat,N),one_one(nat))
     => ( num_of_nat(N) = one2 ) ) ).

% num_of_nat_One
tff(fact_6218_drop__bit__half,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% drop_bit_half
tff(fact_6219_stable__imp__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))) = A3 )
         => ( aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3) = A3 ) ) ) ).

% stable_imp_drop_bit_eq
tff(fact_6220_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,N)),A3) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% drop_bit_Suc
tff(fact_6221_drop__bit__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% drop_bit_eq_div
tff(fact_6222_bit__iff__odd__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3)) ) ) ).

% bit_iff_odd_drop_bit
tff(fact_6223_even__drop__bit__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3))
        <=> ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N) ) ) ).

% even_drop_bit_iff_not_bit
tff(fact_6224_numeral__num__of__nat__unfold,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          aa(num,A,numeral_numeral(A),num_of_nat(N)) = $ite(N = zero_zero(nat),one_one(A),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% numeral_num_of_nat_unfold
tff(fact_6225_num__of__nat__double,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N)) = bit0(num_of_nat(N)) ) ) ).

% num_of_nat_double
tff(fact_6226_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
       => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),num_of_nat(M)),num_of_nat(N)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_6227_num__of__nat_Osimps_I2_J,axiom,
    ! [N: nat] :
      num_of_nat(aa(nat,nat,suc,N)) = $ite(aa(nat,$o,ord_less(nat,zero_zero(nat)),N),inc(num_of_nat(N)),one2) ).

% num_of_nat.simps(2)
tff(fact_6228_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3) = $ite(N = zero_zero(nat),A3,aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,minus_minus(nat,N),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% drop_bit_rec
tff(fact_6229_div__half__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xb),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Xb),Y)) = $let(
                q: word(A),
                q:= aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),one_one(nat)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),one_one(nat)),Xb)),Y)),
                $let(
                  r: word(A),
                  r:= aa(word(A),word(A),minus_minus(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),q),Y)),
                  $ite(aa(word(A),$o,ord_less_eq(word(A),Y),r),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),q),one_one(word(A)))),aa(word(A),word(A),minus_minus(word(A),r),Y)),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),q),r)) ) ) ) ) ) ).

% div_half_word
tff(fact_6230_admissible__all,axiom,
    ! [A: $tType,B: $tType,Lub: fun(set(B),B),Ord: fun(B,fun(B,$o)),P: fun(B,fun(A,$o))] :
      ( ! [Y2: A] : comple1908693960933563346ssible(B,Lub,Ord,aa(A,fun(B,$o),aTP_Lamp_aiv(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),Y2))
     => comple1908693960933563346ssible(B,Lub,Ord,aTP_Lamp_aiw(fun(B,fun(A,$o)),fun(B,$o),P)) ) ).

% admissible_all
tff(fact_6231_push__bit__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),zero_zero(A)) = zero_zero(A) ) ).

% push_bit_of_0
tff(fact_6232_push__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A3: A] :
          ( ( aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3) = zero_zero(A) )
        <=> ( A3 = zero_zero(A) ) ) ) ).

% push_bit_eq_0_iff
tff(fact_6233_drop__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),N),K))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_6234_drop__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),N),K)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% drop_bit_negative_int_iff
tff(fact_6235_drop__bit__minus__one,axiom,
    ! [N: nat] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),N),aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% drop_bit_minus_one
tff(fact_6236_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),A3) ) ).

% push_bit_push_bit
tff(fact_6237_push__bit__and,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A,B3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),B3)) ) ).

% push_bit_and
tff(fact_6238_push__bit__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A,B3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),B3)) ) ).

% push_bit_or
tff(fact_6239_push__bit__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A,B3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),B3)) ) ).

% push_bit_xor
tff(fact_6240_push__bit__Suc__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),K)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(num,A,numeral_numeral(A),bit0(K))) ) ).

% push_bit_Suc_numeral
tff(fact_6241_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_6242_drop__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),N = zero_zero(nat)) ).

% drop_bit_of_Suc_0
tff(fact_6243_push__bit__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(K)))) ) ).

% push_bit_Suc_minus_numeral
tff(fact_6244_push__bit__minus__one__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)) ) ).

% push_bit_minus_one_eq_not_mask
tff(fact_6245_push__bit__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [L: num,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(num,nat,numeral_numeral(nat),L)),aa(num,A,numeral_numeral(A),K)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),pred_numeral(L)),aa(num,A,numeral_numeral(A),bit0(K))) ) ).

% push_bit_numeral
tff(fact_6246_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit0(K)))) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_numeral_minus_bit0
tff(fact_6247_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,N)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),N),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_6248_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,N)),A3) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% push_bit_Suc
tff(fact_6249_push__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),one_one(A)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ).

% push_bit_of_1
tff(fact_6250_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3))
        <=> ( ( N != zero_zero(nat) )
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A3) ) ) ) ).

% even_push_bit_iff
tff(fact_6251_push__bit__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [L: num,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(num,nat,numeral_numeral(nat),L)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),K))) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),pred_numeral(L)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(K)))) ) ).

% push_bit_minus_numeral
tff(fact_6252_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(num,nat,numeral_numeral(nat),L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,K)))) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),pred_numeral(L)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_numeral_minus_bit1
tff(fact_6253_push__bit__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(num,nat,numeral_numeral(nat),N)),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,nat,numeral_numeral(nat),N))) ) ).

% push_bit_numeral_minus_1
tff(fact_6254_drop__bit__int__code_I1_J,axiom,
    ! [I: int] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),zero_zero(nat)),I) = I ).

% drop_bit_int_code(1)
tff(fact_6255_drop__bit__int__code_I2_J,axiom,
    ! [N: nat] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,N)),zero_zero(int)) = zero_zero(int) ).

% drop_bit_int_code(2)
tff(fact_6256_drop__bit__nat__eq,axiom,
    ! [N: nat,K: int] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),N),nat2(K)) = nat2(aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),N),K)) ).

% drop_bit_nat_eq
tff(fact_6257_push__bit__minus,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,uminus_uminus(A),A3)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3)) ) ).

% push_bit_minus
tff(fact_6258_push__bit__of__nat,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,M: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(nat,A,semiring_1_of_nat(A),M)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),N),M)) ) ).

% push_bit_of_nat
tff(fact_6259_of__nat__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),M),N)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),aa(nat,A,semiring_1_of_nat(A),N)) ) ).

% of_nat_push_bit
tff(fact_6260_push__bit__of__int,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,K: int] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(int,A,ring_1_of_int(A),K)) = aa(int,A,ring_1_of_int(A),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),K)) ) ).

% push_bit_of_int
tff(fact_6261_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),A3)) ) ).

% push_bit_take_bit
tff(fact_6262_push__bit__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A,B3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B3)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),B3)) ) ).

% push_bit_add
tff(fact_6263_take__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,M),N)),A3)) ) ).

% take_bit_push_bit
tff(fact_6264_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A3),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),one_one(A))) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_6265_bits__ident,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3))),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)) = A3 ) ).

% bits_ident
tff(fact_6266_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),N),A3) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A3),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),one_one(A))) ) ).

% set_bit_eq_or
tff(fact_6267_flip__bit__eq__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : bit_se8732182000553998342ip_bit(A,N,A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A3),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),one_one(A))) ) ).

% flip_bit_eq_xor
tff(fact_6268_push__bit__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),bit0(one2)))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% push_bit_double
tff(fact_6269_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_6270_push__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),bit_se2239418461657761734s_mask(A,N)) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,M))) ) ).

% push_bit_mask_eq
tff(fact_6271_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se2638667681897837118et_bit(A),N),A3) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),one_one(A)))) ) ).

% unset_bit_eq_and_not
tff(fact_6272_bin__rest__code,axiom,
    ! [I: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),I),aa(num,int,numeral_numeral(int),bit0(one2))) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),one_one(nat)),I) ).

% bin_rest_code
tff(fact_6273_drop__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).

% drop_bit_int_def
tff(fact_6274_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% push_bit_eq_mult
tff(fact_6275_exp__dvdE,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),A3)
         => ~ ! [B2: A] : A3 != aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),B2) ) ) ).

% exp_dvdE
tff(fact_6276_drop__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),N),M) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% drop_bit_nat_def
tff(fact_6277_admissible__const,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,$o)),Tb: $o] : comple1908693960933563346ssible(A,Lub,Ord,aTP_Lamp_afb($o,fun(A,$o),(Tb))) ).

% admissible_const
tff(fact_6278_admissible__conj,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,$o)),P: fun(A,$o),Q: fun(A,$o)] :
      ( comple1908693960933563346ssible(A,Lub,Ord,P)
     => ( comple1908693960933563346ssible(A,Lub,Ord,Q)
       => comple1908693960933563346ssible(A,Lub,Ord,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ta(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) ) ) ).

% admissible_conj
tff(fact_6279_admissible__True,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,$o))] : comple1908693960933563346ssible(A,Lub,Ord,aTP_Lamp_do(A,$o)) ).

% admissible_True
tff(fact_6280_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,M: nat,A3: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3))) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A3),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),bit_se2239418461657761734s_mask(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,N)))) ) ).

% slice_eq_mask
tff(fact_6281_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_aix(A,fun(nat,A),A3)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N)) ) ).

% take_bit_sum
tff(fact_6282_word__and__mask__or__conv__and__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),Index: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),N),Index)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),N),bit_se2239418461657761734s_mask(word(A),Index))),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),Index),one_one(word(A)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),N),bit_se2239418461657761734s_mask(word(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Index),one_one(nat)))) ) ) ) ).

% word_and_mask_or_conv_and_mask
tff(fact_6283_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat,A3: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3) = $let(
            l: A,
            l:= aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,N)),A3),
            $ite(aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),l),N),aa(A,A,aa(A,fun(A,A),plus_plus(A),l),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),one_one(A)))),l) ) ) ).

% signed_take_bit_code
tff(fact_6284_shiftr__integer__conv__div__pow2,axiom,
    ! [N: nat,Xb: code_integer] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4197421643247451524op_bit(code_integer),N),Xb) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),Xb),aa(nat,code_integer,aa(code_integer,fun(nat,code_integer),power_power(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),N)) ).

% shiftr_integer_conv_div_pow2
tff(fact_6285_set__bits__aux__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Fa: fun(nat,$o),N: nat,W: word(A)] :
          code_T2661198915054445665ts_aux(A,Fa,aa(nat,nat,suc,N),W) = code_T2661198915054445665ts_aux(A,Fa,N,
            aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),one_one(nat)),W)),
              $ite(aa(nat,$o,Fa,N),one_one(word(A)),zero_zero(word(A))))) ) ).

% set_bits_aux_Suc
tff(fact_6286_set__bits__aux__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Fa: fun(nat,$o),W: word(A)] : code_T2661198915054445665ts_aux(A,Fa,zero_zero(nat),W) = W ) ).

% set_bits_aux_0
tff(fact_6287_push__bit__nonnegative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),K))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_6288_push__bit__negative__int__iff,axiom,
    ! [N: nat,K: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),K)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,K),zero_zero(int)) ) ).

% push_bit_negative_int_iff
tff(fact_6289_push__bit__of__Suc__0,axiom,
    ! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),N),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).

% push_bit_of_Suc_0
tff(fact_6290_drop__bit__push__bit__int,axiom,
    ! [M: nat,N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),M),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),K)) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,minus_minus(nat,M),N)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),aa(nat,nat,minus_minus(nat,N),M)),K)) ).

% drop_bit_push_bit_int
tff(fact_6291_flip__bit__nat__def,axiom,
    ! [M: nat,N: nat] : bit_se8732182000553998342ip_bit(nat,M,N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),M),one_one(nat))) ).

% flip_bit_nat_def
tff(fact_6292_set__bit__nat__def,axiom,
    ! [M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5668285175392031749et_bit(nat),M),N) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),N),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),M),one_one(nat))) ).

% set_bit_nat_def
tff(fact_6293_push__bit__nat__eq,axiom,
    ! [N: nat,K: int] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),N),nat2(K)) = nat2(aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),K)) ).

% push_bit_nat_eq
tff(fact_6294_push__bit__int__code_I1_J,axiom,
    ! [I: int] : aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),zero_zero(nat)),I) = I ).

% push_bit_int_code(1)
tff(fact_6295_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),M),K)),N)
    <=> ( aa(nat,$o,ord_less_eq(nat,M),N)
        & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),aa(nat,nat,minus_minus(nat,N),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_6296_shiftl__integer__conv__mult__pow2,axiom,
    ! [N: nat,Xb: code_integer] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),N),Xb) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),Xb),aa(nat,code_integer,aa(code_integer,fun(nat,code_integer),power_power(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),N)) ).

% shiftl_integer_conv_mult_pow2
tff(fact_6297_Bit__Operations_Oset__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se5668285175392031749et_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),one_one(int))) ).

% Bit_Operations.set_bit_int_def
tff(fact_6298_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),M),Q2)),N)
    <=> ( aa(nat,$o,ord_less_eq(nat,M),N)
        & aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),Q2),aa(nat,nat,minus_minus(nat,N),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_6299_lsb__integer__code,axiom,
    ! [Xb: code_integer] :
      ( aa(code_integer,$o,least_8051144512741203767sb_lsb(code_integer),Xb)
    <=> aa(nat,$o,aa(code_integer,fun(nat,$o),bit_se5641148757651400278ts_bit(code_integer),Xb),zero_zero(nat)) ) ).

% lsb_integer_code
tff(fact_6300_flip__bit__int__def,axiom,
    ! [N: nat,K: int] : bit_se8732182000553998342ip_bit(int,N,K) = aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),one_one(int))) ).

% flip_bit_int_def
tff(fact_6301_unset__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se2638667681897837118et_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),one_one(int)))) ).

% unset_bit_int_def
tff(fact_6302_push__bit__int__def,axiom,
    ! [N: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),K) = aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).

% push_bit_int_def
tff(fact_6303_push__bit__nat__def,axiom,
    ! [N: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),N),M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) ).

% push_bit_nat_def
tff(fact_6304_push__bit__minus__one,axiom,
    ! [N: nat] : aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ).

% push_bit_minus_one
tff(fact_6305_set__bits__aux__rec,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Fa: fun(nat,$o),N: nat,W: word(A)] :
          code_T2661198915054445665ts_aux(A,Fa,N,W) = $ite(
            N = zero_zero(nat),
            W,
            $let(
              n: nat,
              n:= aa(nat,nat,minus_minus(nat,N),one_one(nat)),
              code_T2661198915054445665ts_aux(A,Fa,n,
                aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),one_one(nat)),W)),
                  $ite(aa(nat,$o,Fa,n),one_one(word(A)),zero_zero(word(A))))) ) ) ) ).

% set_bits_aux_rec
tff(fact_6306_Bit__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: $o] : bits_Bit_integer(code_integer_of_int(Xaa),(Xb)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($o,int,zero_neq_one_of_bool(int),(Xb))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Xaa))) ).

% Bit_integer.abs_eq
tff(fact_6307_test__bit__split,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A3: word(A),B3: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A3),B3) )
         => ( ! [N4: nat] :
                ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),B3),N4)
              <=> ( aa(nat,$o,ord_less(nat,N4),aa(word(B),nat,size_size(word(B)),B3))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),N4) ) )
            & ! [M2: nat] :
                ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A3),M2)
              <=> ( aa(nat,$o,ord_less(nat,M2),aa(word(A),nat,size_size(word(A)),A3))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(word(B),nat,size_size(word(B)),B3))) ) ) ) ) ) ).

% test_bit_split
tff(fact_6308_Bit__integer__code_I1_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,$false) = aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),one_one(nat)),I) ).

% Bit_integer_code(1)
tff(fact_6309_Bit__integer__code_I2_J,axiom,
    ! [I: code_integer] : bits_Bit_integer(I,$true) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),one_one(nat)),I)),one_one(code_integer)) ).

% Bit_integer_code(2)
tff(fact_6310_test__bit__split__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A3: word(A),B3: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A3),B3) )
        <=> ( ! [N8: nat] :
                ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),B3),N8)
              <=> ( aa(nat,$o,ord_less(nat,N8),aa(word(B),nat,size_size(word(B)),B3))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),N8) ) )
            & ! [M6: nat] :
                ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A3),M6)
              <=> ( aa(nat,$o,ord_less(nat,M6),aa(word(A),nat,size_size(word(A)),A3))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M6),aa(word(B),nat,size_size(word(B)),B3))) ) ) ) ) ) ).

% test_bit_split_eq
tff(fact_6311_test__bit__split_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A3: word(A),B3: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A3),B3) )
         => ! [N4: nat,M2: nat] :
              ( ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),B3),N4)
              <=> ( aa(nat,$o,ord_less(nat,N4),aa(word(B),nat,size_size(word(B)),B3))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),N4) ) )
              & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A3),M2)
              <=> ( aa(nat,$o,ord_less(nat,M2),aa(word(A),nat,size_size(word(A)),A3))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M2),aa(word(B),nat,size_size(word(B)),B3))) ) ) ) ) ) ).

% test_bit_split'
tff(fact_6312_dup__1,axiom,
    code_dup(one_one(code_integer)) = aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)) ).

% dup_1
tff(fact_6313_bin__last__integer__nbe,axiom,
    ! [I: code_integer] :
      ( bits_b8758750999018896077nteger(I)
    <=> ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))) != zero_zero(code_integer) ) ) ).

% bin_last_integer_nbe
tff(fact_6314_Code__Numeral_Odup__code_I1_J,axiom,
    code_dup(zero_zero(code_integer)) = zero_zero(code_integer) ).

% Code_Numeral.dup_code(1)
tff(fact_6315_bin__last__integer__code,axiom,
    ! [I: code_integer] :
      ( bits_b8758750999018896077nteger(I)
    <=> ( aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),I),one_one(code_integer)) != zero_zero(code_integer) ) ) ).

% bin_last_integer_code
tff(fact_6316_bin__last__integer_Oabs__eq,axiom,
    ! [Xb: int] :
      ( bits_b8758750999018896077nteger(code_integer_of_int(Xb))
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xb) ) ).

% bin_last_integer.abs_eq
tff(fact_6317_bitAND__integer__unfold,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),Xb),Y) = $ite(
        Xb = zero_zero(code_integer),
        zero_zero(code_integer),
        $ite(
          Xb = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),
          Y,
          bits_Bit_integer(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),bits_b2549910563261871055nteger(Xb)),bits_b2549910563261871055nteger(Y)),
            ( bits_b8758750999018896077nteger(Xb)
            & bits_b8758750999018896077nteger(Y) )) ) ) ).

% bitAND_integer_unfold
tff(fact_6318_bitOR__integer__unfold,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),Xb),Y) = $ite(
        Xb = zero_zero(code_integer),
        Y,
        $ite(
          Xb = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),
          aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),
          bits_Bit_integer(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),bits_b2549910563261871055nteger(Xb)),bits_b2549910563261871055nteger(Y)),
            ( bits_b8758750999018896077nteger(Xb)
            | bits_b8758750999018896077nteger(Y) )) ) ) ).

% bitOR_integer_unfold
tff(fact_6319_bin__rest__integer__code,axiom,
    ! [I: code_integer] : bits_b2549910563261871055nteger(I) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))) ).

% bin_rest_integer_code
tff(fact_6320_bin__rest__integer_Oabs__eq,axiom,
    ! [Xb: int] : bits_b2549910563261871055nteger(code_integer_of_int(Xb)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% bin_rest_integer.abs_eq
tff(fact_6321_bitXOR__integer__unfold,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344971392196577ns_xor(code_integer),Xb),Y) = $ite(
        Xb = zero_zero(code_integer),
        Y,
        $ite(
          Xb = aa(code_integer,code_integer,uminus_uminus(code_integer),one_one(code_integer)),
          aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),Y),
          bits_Bit_integer(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344971392196577ns_xor(code_integer),bits_b2549910563261871055nteger(Xb)),bits_b2549910563261871055nteger(Y)),
            ~ bits_b8758750999018896077nteger(Xb) = bits_b8758750999018896077nteger(Y)) ) ) ).

% bitXOR_integer_unfold
tff(fact_6322_case__prod__app,axiom,
    ! [D2: $tType,A: $tType,C: $tType,B: $tType,Fa: fun(B,fun(C,fun(D2,A))),Xb: product_prod(B,C),Y: D2] : aa(D2,A,aa(product_prod(B,C),fun(D2,A),product_case_prod(B,C,fun(D2,A),Fa),Xb),Y) = aa(product_prod(B,C),A,product_case_prod(B,C,A,aa(D2,fun(B,fun(C,A)),aTP_Lamp_fm(fun(B,fun(C,fun(D2,A))),fun(D2,fun(B,fun(C,A))),Fa),Y)),Xb) ).

% case_prod_app
tff(fact_6323_Uint32__code,axiom,
    ! [I: code_integer] :
      aa(code_integer,uint32,uint322,I) = $let(
        i: code_integer,
        i:= aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),I),aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2))))))))))))))))))))))))))))))))),
        $ite(aa(nat,$o,aa(code_integer,fun(nat,$o),bit_se5641148757651400278ts_bit(code_integer),i),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),uint32_signed(aa(code_integer,code_integer,minus_minus(code_integer,i),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))))))),uint32_signed(i)) ) ).

% Uint32_code
tff(fact_6324_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [Fa: fun(fun(A,B),C),G: C] :
          ( ! [X2: fun(A,B)] : aa(fun(A,B),C,Fa,X2) = G
         => ( aa(fun(A,B),C,Fa,aTP_Lamp_aiy(A,B)) = G ) ) ) ).

% fun_cong_unused_0
tff(fact_6325_Uint32__signed__def,axiom,
    ! [I: code_integer] :
      uint32_signed(I) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,I),aa(code_integer,code_integer,uminus_uminus(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))))))
        | aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))))),I) ),
        aa(code_integer,uint32,aa(fun(code_integer,uint32),fun(code_integer,uint32),undefined(fun(fun(code_integer,uint32),fun(code_integer,uint32))),uint322),I),
        aa(code_integer,uint32,uint322,I) ) ).

% Uint32_signed_def
tff(fact_6326_case__prod__Pair__iden,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B)] : aa(product_prod(A,B),product_prod(A,B),product_case_prod(A,B,product_prod(A,B),product_Pair(A,B)),P2) = P2 ).

% case_prod_Pair_iden
tff(fact_6327_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A4: set(A)] : set_Cons(A,A4,aa(set(list(A)),set(list(A)),insert(list(A),nil(A)),bot_bot(set(list(A))))) = aa(set(A),set(list(A)),image(A,list(A),aTP_Lamp_aiz(A,list(A))),A4) ).

% set_Cons_sing_Nil
tff(fact_6328_the__elem__def,axiom,
    ! [A: $tType,X5: set(A)] : the_elem(A,X5) = the(A,aTP_Lamp_aja(set(A),fun(A,$o),X5)) ).

% the_elem_def
tff(fact_6329_vebt__minti_Opelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_Time_Heap(option(nat))] :
      ( ( vEBT_vebt_minti(Xb) = Y )
     => ( accp(vEBT_VEBTi,vEBT_vebt_minti_rel,Xb)
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leafi((A2),(B2)) )
             => ( ( Y = $ite(
                      (A2),
                      aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),zero_zero(nat))),
                      $ite((B2),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),one_one(nat))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))) ) )
               => ~ accp(vEBT_VEBTi,vEBT_vebt_minti_rel,vEBT_Leafi((A2),(B2))) ) )
         => ( ! [Uu2: nat,Uv: array(vEBT_VEBTi),Uw: vEBT_VEBTi] :
                ( ( Xb = vEBT_Nodei(none(product_prod(nat,nat)),Uu2,Uv,Uw) )
               => ( ( Y = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)) )
                 => ~ accp(vEBT_VEBTi,vEBT_vebt_minti_rel,vEBT_Nodei(none(product_prod(nat,nat)),Uu2,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array(vEBT_VEBTi),Uz2: vEBT_VEBTi] :
                  ( ( Xb = vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),Mi2)) )
                   => ~ accp(vEBT_VEBTi,vEBT_vebt_minti_rel,vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_minti.pelims
tff(fact_6330_vebt__maxti_Opelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_Time_Heap(option(nat))] :
      ( ( vEBT_vebt_maxti(Xb) = Y )
     => ( accp(vEBT_VEBTi,vEBT_vebt_maxti_rel,Xb)
       => ( ! [A2: $o,B2: $o] :
              ( ( Xb = vEBT_Leafi((A2),(B2)) )
             => ( ( Y = $ite(
                      (B2),
                      aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),one_one(nat))),
                      $ite((A2),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),zero_zero(nat))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))) ) )
               => ~ accp(vEBT_VEBTi,vEBT_vebt_maxti_rel,vEBT_Leafi((A2),(B2))) ) )
         => ( ! [Uu2: nat,Uv: array(vEBT_VEBTi),Uw: vEBT_VEBTi] :
                ( ( Xb = vEBT_Nodei(none(product_prod(nat,nat)),Uu2,Uv,Uw) )
               => ( ( Y = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)) )
                 => ~ accp(vEBT_VEBTi,vEBT_vebt_maxti_rel,vEBT_Nodei(none(product_prod(nat,nat)),Uu2,Uv,Uw)) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: array(vEBT_VEBTi),Uz2: vEBT_VEBTi] :
                  ( ( Xb = vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),Ma2)) )
                   => ~ accp(vEBT_VEBTi,vEBT_vebt_maxti_rel,vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_maxti.pelims
tff(fact_6331_VEBT__internal_OminNulli_Opelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_Time_Heap($o)] :
      ( ( vEBT_VEBT_minNulli(Xb) = Y )
     => ( accp(vEBT_VEBTi,vEBT_V5740978063120863272li_rel,Xb)
       => ( ( ( Xb = vEBT_Leafi($false,$false) )
           => ( ( Y = aa($o,heap_Time_Heap($o),heap_Time_return($o),$true) )
             => ~ accp(vEBT_VEBTi,vEBT_V5740978063120863272li_rel,vEBT_Leafi($false,$false)) ) )
         => ( ! [Uv: $o] :
                ( ( Xb = vEBT_Leafi($true,(Uv)) )
               => ( ( Y = aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) )
                 => ~ accp(vEBT_VEBTi,vEBT_V5740978063120863272li_rel,vEBT_Leafi($true,(Uv))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leafi((Uu2),$true) )
                 => ( ( Y = aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) )
                   => ~ accp(vEBT_VEBTi,vEBT_V5740978063120863272li_rel,vEBT_Leafi((Uu2),$true)) ) )
             => ( ! [Uw: nat,Ux2: array(vEBT_VEBTi),Uy2: vEBT_VEBTi] :
                    ( ( Xb = vEBT_Nodei(none(product_prod(nat,nat)),Uw,Ux2,Uy2) )
                   => ( ( Y = aa($o,heap_Time_Heap($o),heap_Time_return($o),$true) )
                     => ~ accp(vEBT_VEBTi,vEBT_V5740978063120863272li_rel,vEBT_Nodei(none(product_prod(nat,nat)),Uw,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va3: nat,Vb2: array(vEBT_VEBTi),Vc2: vEBT_VEBTi] :
                      ( ( Xb = vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2) )
                     => ( ( Y = aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) )
                       => ~ accp(vEBT_VEBTi,vEBT_V5740978063120863272li_rel,vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va3,Vb2,Vc2)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.pelims
tff(fact_6332_upto__aux__rec,axiom,
    ! [I: int,J: int,Js: list(int)] :
      upto_aux(I,J,Js) = $ite(aa(int,$o,ord_less(int,J),I),Js,upto_aux(I,aa(int,int,minus_minus(int,J),one_one(int)),aa(list(int),list(int),cons(int,J),Js))) ).

% upto_aux_rec
tff(fact_6333_upto_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xb,Xaa) = Y )
     => ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xb),Xaa))
       => ~ ( ( Y = $ite(aa(int,$o,ord_less_eq(int,Xb),Xaa),aa(list(int),list(int),cons(int,Xb),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),one_one(int)),Xaa)),nil(int)) )
           => ~ accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Xb),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_6334_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( accp(product_prod(int,int),upto_rel,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I),J))
     => ( upto(I,J) = $ite(aa(int,$o,ord_less_eq(int,I),J),aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ) ) ).

% upto.psimps
tff(fact_6335_upto__empty,axiom,
    ! [J: int,I: int] :
      ( aa(int,$o,ord_less(int,J),I)
     => ( upto(I,J) = nil(int) ) ) ).

% upto_empty
tff(fact_6336_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil(int) = upto(I,J) )
    <=> aa(int,$o,ord_less(int,J),I) ) ).

% upto_Nil2
tff(fact_6337_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( upto(I,J) = nil(int) )
    <=> aa(int,$o,ord_less(int,J),I) ) ).

% upto_Nil
tff(fact_6338_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K))),J)
     => ( aa(nat,int,nth(int,upto(I,J)),K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),I),aa(nat,int,semiring_1_of_nat(int),K)) ) ) ).

% nth_upto
tff(fact_6339_length__upto,axiom,
    ! [I: int,J: int] : aa(list(int),nat,size_size(list(int)),upto(I,J)) = nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,J),I)),one_one(int))) ).

% length_upto
tff(fact_6340_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N: num] :
      upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = $ite(aa(int,$o,ord_less_eq(int,aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)),aa(list(int),list(int),cons(int,aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(num,int,numeral_numeral(int),N))),nil(int)) ).

% upto_rec_numeral(1)
tff(fact_6341_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N: num] :
      upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = $ite(aa(int,$o,ord_less_eq(int,aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))),aa(list(int),list(int),cons(int,aa(num,int,numeral_numeral(int),M)),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),M)),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))),nil(int)) ).

% upto_rec_numeral(2)
tff(fact_6342_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) = $ite(aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(num,int,numeral_numeral(int),N)),aa(list(int),list(int),cons(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(num,int,numeral_numeral(int),N))),nil(int)) ).

% upto_rec_numeral(3)
tff(fact_6343_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) = $ite(aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))),aa(list(int),list(int),cons(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M))),one_one(int)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))),nil(int)) ).

% upto_rec_numeral(4)
tff(fact_6344_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,I),J)
     => ( aa(int,$o,ord_less_eq(int,J),K)
       => ( upto(I,K) = append(int,upto(I,J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K)) ) ) ) ).

% upto_split2
tff(fact_6345_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,I),J)
     => ( aa(int,$o,ord_less_eq(int,J),K)
       => ( upto(I,K) = append(int,upto(I,aa(int,int,minus_minus(int,J),one_one(int))),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_6346_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or7035219750837199246ssThan(int,I,J) = set2(int,upto(I,aa(int,int,minus_minus(int,J),one_one(int)))) ).

% atLeastLessThan_upto
tff(fact_6347_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,ord_less_eq(int,I),J)
     => ( upto(I,J) = aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)) ) ) ).

% upto_rec1
tff(fact_6348_upto_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xb,Xaa) = Y )
     => ( Y = $ite(aa(int,$o,ord_less_eq(int,Xb),Xaa),aa(list(int),list(int),cons(int,Xb),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),one_one(int)),Xaa)),nil(int)) ) ) ).

% upto.elims
tff(fact_6349_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa(int,$o,ord_less_eq(int,I),J),aa(list(int),list(int),cons(int,I),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),J)),nil(int)) ).

% upto.simps
tff(fact_6350_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,ord_less_eq(int,I),J)
     => ( upto(I,J) = append(int,upto(I,aa(int,int,minus_minus(int,J),one_one(int))),aa(list(int),list(int),cons(int,J),nil(int))) ) ) ).

% upto_rec2
tff(fact_6351_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,ord_less_eq(int,I),J)
     => ( aa(int,$o,ord_less_eq(int,J),K)
       => ( upto(I,K) = append(int,upto(I,aa(int,int,minus_minus(int,J),one_one(int))),aa(list(int),list(int),cons(int,J),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),J),one_one(int)),K))) ) ) ) ).

% upto_split3
tff(fact_6352_concat__bit__Suc,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_concat_bit(aa(nat,nat,suc,N),K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,bit_concat_bit(N,aa(int,int,aa(int,fun(int,int),divide_divide(int),K),aa(num,int,numeral_numeral(int),bit0(one2)))),L))) ).

% concat_bit_Suc
tff(fact_6353_word__cat__split__size,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(A)
        & type_len(B) )
     => ! [Tb: word(A),U: word(B),V: word(C)] :
          ( aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,size_size(word(A)),Tb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),V)))
         => ( ( aa(word(C),product_prod(word(B),word(C)),aa(word(B),fun(word(C),product_prod(word(B),word(C))),product_Pair(word(B),word(C)),U),V) = word_split(A,B,C,Tb) )
           => ( Tb = word_cat(B,C,A,U,V) ) ) ) ) ).

% word_cat_split_size
tff(fact_6354_concat__bit__0,axiom,
    ! [K: int,L: int] : aa(int,int,bit_concat_bit(zero_zero(nat),K),L) = L ).

% concat_bit_0
tff(fact_6355_concat__bit__of__zero__2,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_concat_bit(N,K),zero_zero(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) ).

% concat_bit_of_zero_2
tff(fact_6356_concat__bit__nonnegative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,bit_concat_bit(N,K),L))
    <=> aa(int,$o,ord_less_eq(int,zero_zero(int)),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_6357_concat__bit__negative__iff,axiom,
    ! [N: nat,K: int,L: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,bit_concat_bit(N,K),L)),zero_zero(int))
    <=> aa(int,$o,ord_less(int,L),zero_zero(int)) ) ).

% concat_bit_negative_iff
tff(fact_6358_concat__bit__of__zero__1,axiom,
    ! [N: nat,L: int] : aa(int,int,bit_concat_bit(N,zero_zero(int)),L) = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),L) ).

% concat_bit_of_zero_1
tff(fact_6359_concat__bit__take__bit__eq,axiom,
    ! [N: nat,B3: int] : bit_concat_bit(N,aa(int,int,bit_se2584673776208193580ke_bit(int,N),B3)) = bit_concat_bit(N,B3) ).

% concat_bit_take_bit_eq
tff(fact_6360_concat__bit__eq__iff,axiom,
    ! [N: nat,K: int,L: int,R2: int,S: int] :
      ( ( aa(int,int,bit_concat_bit(N,K),L) = aa(int,int,bit_concat_bit(N,R2),S) )
    <=> ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),R2) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
tff(fact_6361_concat__bit__assoc,axiom,
    ! [N: nat,K: int,M: nat,L: int,R2: int] : aa(int,int,bit_concat_bit(N,K),aa(int,int,bit_concat_bit(M,L),R2)) = aa(int,int,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N),aa(int,int,bit_concat_bit(N,K),L)),R2) ).

% concat_bit_assoc
tff(fact_6362_concat__bit__eq,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_concat_bit(N,K),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),L)) ).

% concat_bit_eq
tff(fact_6363_concat__bit__def,axiom,
    ! [N: nat,K: int,L: int] : aa(int,int,bit_concat_bit(N,K),L) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(int,int,bit_se2584673776208193580ke_bit(int,N),K)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),L)) ).

% concat_bit_def
tff(fact_6364_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,bit_concat_bit(M,K),L)),N)
    <=> ( ( aa(nat,$o,ord_less(nat,N),M)
          & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) )
        | ( aa(nat,$o,ord_less_eq(nat,M),N)
          & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),L),aa(nat,nat,minus_minus(nat,N),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_6365_signed__take__bit__eq__concat__bit,axiom,
    ! [N: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,N),K) = aa(int,int,bit_concat_bit(N,K),aa(int,int,uminus_uminus(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N)))) ).

% signed_take_bit_eq_concat_bit
tff(fact_6366_test__bit__cat,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [A3: word(B),B3: word(C),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_cat(B,C,A,A3,B3)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),aa(word(A),nat,size_size(word(A)),word_cat(B,C,A,A3,B3)))
            & $ite(aa(nat,$o,ord_less(nat,N),aa(word(C),nat,size_size(word(C)),B3)),aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),B3),N),aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),A3),aa(nat,nat,minus_minus(nat,N),aa(word(C),nat,size_size(word(C)),B3)))) ) ) ) ).

% test_bit_cat
tff(fact_6367_word__cat__split__alt,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B)
        & type_len(C) )
     => ! [W: word(A),U: word(B),V: word(C)] :
          ( aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,size_size(word(A)),W)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),V)))
         => ( ( word_split(A,B,C,W) = aa(word(C),product_prod(word(B),word(C)),aa(word(B),fun(word(C),product_prod(word(B),word(C))),product_Pair(word(B),word(C)),U),V) )
           => ( word_cat(B,C,A,U,V) = W ) ) ) ) ).

% word_cat_split_alt
tff(fact_6368_word__split__cat__alt,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [W: word(A),U: word(B),V: word(C)] :
          ( ( W = word_cat(B,C,A,U,V) )
         => ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(B),nat,size_size(word(B)),U)),aa(word(C),nat,size_size(word(C)),V))),aa(word(A),nat,size_size(word(A)),W))
           => ( word_split(A,B,C,W) = aa(word(C),product_prod(word(B),word(C)),aa(word(B),fun(word(C),product_prod(word(B),word(C))),product_Pair(word(B),word(C)),U),V) ) ) ) ) ).

% word_split_cat_alt
tff(fact_6369_cat__slices,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A)
        & type_len(C) )
     => ! [A3: word(A),N: nat,C2: word(B),B3: word(C)] :
          ( ( A3 = slice2(B,A,N,C2) )
         => ( ( B3 = slice2(B,C,zero_zero(nat),C2) )
           => ( ( N = aa(word(C),nat,size_size(word(C)),B3) )
             => ( aa(nat,$o,ord_less_eq(nat,aa(word(B),nat,size_size(word(B)),C2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,size_size(word(A)),A3)),aa(word(C),nat,size_size(word(C)),B3)))
               => ( word_cat(A,C,B,A3,B3) = C2 ) ) ) ) ) ) ).

% cat_slices
tff(fact_6370_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F5: set(A),I5: set(A),Fa: fun(A,B),I: A] :
          ( finite_finite2(A,F5)
         => ( aa(set(A),$o,ord_less_eq(set(A),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajb(set(A),fun(fun(A,B),fun(A,$o)),I5),Fa))),F5)
           => ( groups1027152243600224163dd_sum(A,B,Fa,aa(set(A),set(A),minus_minus(set(A),I5),aa(set(A),set(A),insert(A,I),bot_bot(set(A))))) = $ite(member(A,I,I5),aa(B,B,minus_minus(B,groups1027152243600224163dd_sum(A,B,Fa,I5)),aa(A,B,Fa,I)),groups1027152243600224163dd_sum(A,B,Fa,I5)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_6371_slice__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [N: nat] : slice2(B,A,N,zero_zero(word(B))) = zero_zero(word(A)) ) ).

% slice_0
tff(fact_6372_sum_Oempty_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: fun(B,A)] : groups1027152243600224163dd_sum(B,A,P2,bot_bot(set(B))) = zero_zero(A) ) ).

% sum.empty'
tff(fact_6373_sum_Oinsert_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),P2: fun(A,B),I: A] :
          ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_to(set(A),fun(fun(A,B),fun(A,$o)),I5),P2)))
         => ( groups1027152243600224163dd_sum(A,B,P2,aa(set(A),set(A),insert(A,I),I5)) = $ite(member(A,I,I5),groups1027152243600224163dd_sum(A,B,P2,I5),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,P2,I)),groups1027152243600224163dd_sum(A,B,P2,I5))) ) ) ) ).

% sum.insert'
tff(fact_6374_sum_Onon__neutral_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),I5: set(B)] : groups1027152243600224163dd_sum(B,A,G,collect(B,aa(set(B),fun(B,$o),aTP_Lamp_ajc(fun(B,A),fun(set(B),fun(B,$o)),G),I5))) = groups1027152243600224163dd_sum(B,A,G,I5) ) ).

% sum.non_neutral'
tff(fact_6375_slice__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Tb: word(A)] : slice2(A,A,zero_zero(nat),Tb) = Tb ) ).

% slice_id
tff(fact_6376_sum_Odistrib__triv_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ajd(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I5)),groups1027152243600224163dd_sum(A,B,H,I5)) ) ) ) ).

% sum.distrib_triv'
tff(fact_6377_ucast__slice,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W) = slice2(B,A,zero_zero(nat),W) ) ).

% ucast_slice
tff(fact_6378_sum_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T6: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,ord_less_eq(set(A),S3),T6)
         => ( ! [X2: A] :
                ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T6),S3))
               => ( aa(A,B,G,X2) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,S3) = groups1027152243600224163dd_sum(A,B,G,T6) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_6379_sum_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T6: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,ord_less_eq(set(A),S3),T6)
         => ( ! [X2: A] :
                ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T6),S3))
               => ( aa(A,B,G,X2) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,T6) = groups1027152243600224163dd_sum(A,B,G,S3) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_6380_sum_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T6: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,ord_less_eq(set(A),S3),T6)
         => ( ! [I2: A] :
                ( member(A,I2,aa(set(A),set(A),minus_minus(set(A),T6),S3))
               => ( aa(A,B,H,I2) = zero_zero(B) ) )
           => ( ! [X2: A] :
                  ( member(A,X2,S3)
                 => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,S3) = groups1027152243600224163dd_sum(A,B,H,T6) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_6381_sum_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S3: set(A),T6: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,ord_less_eq(set(A),S3),T6)
         => ( ! [X2: A] :
                ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T6),S3))
               => ( aa(A,B,G,X2) = zero_zero(B) ) )
           => ( ! [X2: A] :
                  ( member(A,X2,S3)
                 => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,T6) = groups1027152243600224163dd_sum(A,B,H,S3) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_6382_slice__cat2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [A3: word(B),Tb: word(A)] : slice2(A,A,zero_zero(nat),word_cat(B,A,A,A3,Tb)) = Tb ) ).

% slice_cat2
tff(fact_6383_sum_Odistrib_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [I5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_to(set(A),fun(fun(A,B),fun(A,$o)),I5),G)))
         => ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_to(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ajd(fun(A,B),fun(fun(A,B),fun(A,B)),G),H),I5) = aa(B,B,aa(B,fun(B,B),plus_plus(B),groups1027152243600224163dd_sum(A,B,G,I5)),groups1027152243600224163dd_sum(A,B,H,I5)) ) ) ) ) ).

% sum.distrib'
tff(fact_6384_sum_OG__def,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [P2: fun(B,A),I5: set(B)] :
          groups1027152243600224163dd_sum(B,A,P2,I5) = $ite(finite_finite2(B,collect(B,aa(set(B),fun(B,$o),aTP_Lamp_ajc(fun(B,A),fun(set(B),fun(B,$o)),P2),I5))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,P2),collect(B,aa(set(B),fun(B,$o),aTP_Lamp_ajc(fun(B,A),fun(set(B),fun(B,$o)),P2),I5))),zero_zero(A)) ) ).

% sum.G_def
tff(fact_6385_slice__cat1,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(C)
        & type_len(B) )
     => ! [A3: word(A),B3: word(B)] :
          ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,size_size(word(A)),A3)),aa(word(B),nat,size_size(word(B)),B3))),aa(word(C),nat,size_size(word(C)),word_cat(A,B,C,A3,B3)))
         => ( slice2(C,A,aa(word(B),nat,size_size(word(B)),B3),word_cat(A,B,C,A3,B3)) = A3 ) ) ) ).

% slice_cat1
tff(fact_6386_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),Fa: fun(A,B),I: A] :
          ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajb(set(A),fun(fun(A,B),fun(A,$o)),I5),Fa)))
         => ( groups1027152243600224163dd_sum(A,B,Fa,aa(set(A),set(A),minus_minus(set(A),I5),aa(set(A),set(A),insert(A,I),bot_bot(set(A))))) = $ite(member(A,I,I5),aa(B,B,minus_minus(B,groups1027152243600224163dd_sum(A,B,Fa,I5)),aa(A,B,Fa,I)),groups1027152243600224163dd_sum(A,B,Fa,I5)) ) ) ) ).

% sum_diff1'
tff(fact_6387_split__slices,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [W: word(C),U: word(A),V: word(B)] :
          ( ( word_split(C,A,B,W) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),U),V) )
         => ( ( U = slice2(C,A,aa(word(B),nat,size_size(word(B)),V),W) )
            & ( V = slice2(C,B,zero_zero(nat),W) ) ) ) ) ).

% split_slices
tff(fact_6388_int__set__bit__conv__ops,axiom,
    ! [I: int,N: nat,B3: $o] :
      aa($o,int,aa(nat,fun($o,int),aa(int,fun(nat,fun($o,int)),generi7602027413899671122et_bit(int),I),N),(B3)) = $ite((B3),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),I),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),one_one(int))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),one_one(int))))) ).

% int_set_bit_conv_ops
tff(fact_6389_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs: list($o),N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),Bs)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),aa(list($o),nat,size_size(list($o)),Bs))
            & aa(nat,$o,nth($o,Bs),N) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_6390_int__set__bit__True__conv__OR,axiom,
    ! [I: int,N: nat] : aa($o,int,aa(nat,fun($o,int),aa(int,fun(nat,fun($o,int)),generi7602027413899671122et_bit(int),I),N),$true) = aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),I),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),one_one(int))) ).

% int_set_bit_True_conv_OR
tff(fact_6391_int__set__bit__False__conv__NAND,axiom,
    ! [I: int,N: nat] : aa($o,int,aa(nat,fun($o,int),aa(int,fun(nat,fun($o,int)),generi7602027413899671122et_bit(int),I),N),$false) = aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),N),one_one(int)))) ).

% int_set_bit_False_conv_NAND
tff(fact_6392_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list($o)] : aa(int,$o,ord_less(int,groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),Bs)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(list($o),nat,size_size(list($o)),Bs))) ).

% horner_sum_of_bool_2_less
tff(fact_6393_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Fa: fun(B,A),A3: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,Fa,A3,Xs) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_aje(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Fa),A3),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_6394_horner__sum__simps_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [Fa: fun(B,A),A3: A] : groups4207007520872428315er_sum(B,A,Fa,A3,nil(B)) = zero_zero(A) ) ).

% horner_sum_simps(1)
tff(fact_6395_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [Fa: fun(B,A),A3: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,Fa,A3,Xs) = foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_ajf(fun(B,A),fun(A,fun(B,fun(A,A))),Fa),A3),Xs,zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_6396_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Fa: fun(B,A),A3: A,Xs: list(B),Ys: list(B)] : groups4207007520872428315er_sum(B,A,Fa,A3,append(B,Xs,Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups4207007520872428315er_sum(B,A,Fa,A3,Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A3),aa(list(B),nat,size_size(list(B)),Xs))),groups4207007520872428315er_sum(B,A,Fa,A3,Ys))) ) ).

% horner_sum_append
tff(fact_6397_bit__set__bit__iff__2n,axiom,
    ! [A: $tType] :
      ( generic_set_set_bit(A)
     => ! [A3: A,M: nat,B3: $o,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa($o,A,aa(nat,fun($o,A),aa(A,fun(nat,fun($o,A)),generi7602027413899671122et_bit(A),A3),M),(B3))),N)
        <=> ( $ite(M = N,(B3),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N))
            & ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) ) ) ) ) ).

% bit_set_bit_iff_2n
tff(fact_6398_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] : groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),map(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),upt(zero_zero(nat),N))) = aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_6399_upt__0__eq__Nil__conv,axiom,
    ! [J: nat] :
      ( ( upt(zero_zero(nat),J) = nil(nat) )
    <=> ( J = zero_zero(nat) ) ) ).

% upt_0_eq_Nil_conv
tff(fact_6400_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa(nat,$o,ord_less_eq(nat,J),I)
     => ( upt(I,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_6401_upt__merge,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( aa(nat,$o,ord_less_eq(nat,I),J)
        & aa(nat,$o,ord_less_eq(nat,J),K) )
     => ( append(nat,upt(I,J),upt(J,K)) = upt(I,K) ) ) ).

% upt_merge
tff(fact_6402_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | aa(nat,$o,ord_less_eq(nat,J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_6403_length__upt,axiom,
    ! [I: nat,J: nat] : aa(list(nat),nat,size_size(list(nat)),upt(I,J)) = aa(nat,nat,minus_minus(nat,J),I) ).

% length_upt
tff(fact_6404_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),N)) = $ite(aa(nat,$o,ord_less(nat,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N)),aa(list(nat),list(nat),cons(nat,aa(num,nat,numeral_numeral(nat),M)),upt(aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),N))),nil(nat)) ).

% upt_rec_numeral
tff(fact_6405_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)),J)
     => ( aa(nat,nat,nth(nat,upt(I,J)),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K) ) ) ).

% nth_upt
tff(fact_6406_map__fst__enumerate,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : map(product_prod(nat,A),nat,product_fst(nat,A),enumerate(A,N,Xs)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(list(A),nat,size_size(list(A)),Xs))) ).

% map_fst_enumerate
tff(fact_6407_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa(nat,$o,ord_less(nat,I),J),aa(list(nat),list(nat),cons(nat,I),upt(aa(nat,nat,suc,I),J)),nil(nat)) ).

% upt_rec
tff(fact_6408_upt__0,axiom,
    ! [I: nat] : upt(I,zero_zero(nat)) = nil(nat) ).

% upt_0
tff(fact_6409_map__add__upt_H,axiom,
    ! [Ofs: nat,A3: nat,B3: nat] : map(nat,nat,aTP_Lamp_ajg(nat,fun(nat,nat),Ofs),upt(A3,B3)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A3),Ofs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B3),Ofs)) ).

% map_add_upt'
tff(fact_6410_map__add__upt,axiom,
    ! [N: nat,M: nat] : map(nat,nat,aTP_Lamp_ajg(nat,fun(nat,nat),N),upt(zero_zero(nat),M)) = upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)) ).

% map_add_upt
tff(fact_6411_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( upt(I,J) = aa(list(nat),list(nat),cons(nat,I),upt(aa(nat,nat,suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_6412_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = append(nat,upt(I,J),upt(J,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K))) ) ) ).

% upt_add_eq_append
tff(fact_6413_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs: list(nat),Ys: list(nat)] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( ( upt(I,J) = append(nat,Xs,Ys) )
      <=> ? [K3: nat] :
            ( aa(nat,$o,ord_less_eq(nat,I),K3)
            & aa(nat,$o,ord_less_eq(nat,K3),J)
            & ( upt(I,K3) = Xs )
            & ( upt(K3,J) = Ys ) ) ) ) ).

% upt_eq_append_conv
tff(fact_6414_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( append(nat,upt(zero_zero(nat),I),upt(I,J)) = upt(zero_zero(nat),J) ) ) ).

% upt_append
tff(fact_6415_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,ord_less_eq(nat,I),J)
     => ( upt(I,aa(nat,nat,suc,J)) = append(nat,upt(I,J),aa(list(nat),list(nat),cons(nat,J),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_6416_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      upt(I,aa(nat,nat,suc,J)) = $ite(aa(nat,$o,ord_less_eq(nat,I),J),append(nat,upt(I,J),aa(list(nat),list(nat),cons(nat,J),nil(nat))),nil(nat)) ).

% upt_Suc
tff(fact_6417_set__bit__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: nat,B3: $o] :
          ( aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,size_size(word(A)),Xb)),N)
         => ( aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),Xb),N),(B3)) = Xb ) ) ) ).

% set_bit_beyond
tff(fact_6418_test__bit__set__gen,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat,Xb: $o,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),W),N),(Xb))),M)
        <=> $ite(
              M = N,
              ( aa(nat,$o,ord_less(nat,N),aa(word(A),nat,size_size(word(A)),W))
              & (Xb) ),
              aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),M) ) ) ) ).

% test_bit_set_gen
tff(fact_6419_test__bit__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat,Xb: $o] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),W),N),(Xb))),N)
        <=> ( aa(nat,$o,ord_less(nat,N),aa(word(A),nat,size_size(word(A)),W))
            & (Xb) ) ) ) ).

% test_bit_set
tff(fact_6420_word__set__nth__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat,B3: $o] :
          ( ( aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),W),N),(B3)) = W )
        <=> ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),N)
            <=> (B3) )
            | aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,size_size(word(A)),W)),N) ) ) ) ).

% word_set_nth_iff
tff(fact_6421_upt__eq__lel__conv,axiom,
    ! [L: nat,H: nat,Is1: list(nat),I: nat,Is2: list(nat)] :
      ( ( upt(L,H) = append(nat,Is1,aa(list(nat),list(nat),cons(nat,I),Is2)) )
    <=> ( ( Is1 = upt(L,I) )
        & ( Is2 = upt(aa(nat,nat,suc,I),H) )
        & aa(nat,$o,ord_less_eq(nat,L),I)
        & aa(nat,$o,ord_less(nat,I),H) ) ) ).

% upt_eq_lel_conv
tff(fact_6422_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xb: nat,Xs: list(nat)] :
      ( ( upt(I,J) = aa(list(nat),list(nat),cons(nat,Xb),Xs) )
    <=> ( aa(nat,$o,ord_less(nat,I),J)
        & ( I = Xb )
        & ( upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),one_one(nat)),J) = Xs ) ) ) ).

% upt_eq_Cons_conv
tff(fact_6423_map__decr__upt,axiom,
    ! [M: nat,N: nat] : map(nat,nat,aTP_Lamp_ajh(nat,nat),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = upt(M,N) ).

% map_decr_upt
tff(fact_6424_atLeast__upt,axiom,
    ! [N: nat] : set_ord_lessThan(nat,N) = set2(nat,upt(zero_zero(nat),N)) ).

% atLeast_upt
tff(fact_6425_map__replicate__trivial,axiom,
    ! [A: $tType,Xb: A,I: nat] : map(nat,A,aTP_Lamp_aji(A,fun(nat,A),Xb),upt(zero_zero(nat),I)) = replicate(A,I,Xb) ).

% map_replicate_trivial
tff(fact_6426_enumerate__map__upt,axiom,
    ! [A: $tType,N: nat,Fa: fun(nat,A),M: nat] : enumerate(A,N,map(nat,A,Fa,upt(N,M))) = map(nat,product_prod(nat,A),aTP_Lamp_ajj(fun(nat,A),fun(nat,product_prod(nat,A)),Fa),upt(N,M)) ).

% enumerate_map_upt
tff(fact_6427_atMost__upto,axiom,
    ! [N: nat] : set_ord_atMost(nat,N) = set2(nat,upt(zero_zero(nat),aa(nat,nat,suc,N))) ).

% atMost_upto
tff(fact_6428_map__upt__Suc,axiom,
    ! [A: $tType,Fa: fun(nat,A),N: nat] : map(nat,A,Fa,upt(zero_zero(nat),aa(nat,nat,suc,N))) = aa(list(A),list(A),cons(A,aa(nat,A,Fa,zero_zero(nat))),map(nat,A,aTP_Lamp_ajk(fun(nat,A),fun(nat,A),Fa),upt(zero_zero(nat),N))) ).

% map_upt_Suc
tff(fact_6429_map__bit__range__eq__if__take__bit__eq,axiom,
    ! [N: nat,K: int,L: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),K) = aa(int,int,bit_se2584673776208193580ke_bit(int,N),L) )
     => ( map(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),upt(zero_zero(nat),N)) = map(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),L),upt(zero_zero(nat),N)) ) ) ).

% map_bit_range_eq_if_take_bit_eq
tff(fact_6430_map__nth,axiom,
    ! [A: $tType,Xs: list(A)] : map(nat,A,nth(A,Xs),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs))) = Xs ).

% map_nth
tff(fact_6431_one__bit__pow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),zero_zero(word(A))),N),$true) = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) ) ).

% one_bit_pow
tff(fact_6432_enumerate__replicate__eq,axiom,
    ! [A: $tType,N: nat,M: nat,A3: A] : enumerate(A,N,replicate(A,M,A3)) = map(nat,product_prod(nat,A),aTP_Lamp_ajl(A,fun(nat,product_prod(nat,A)),A3),upt(N,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M))) ).

% enumerate_replicate_eq
tff(fact_6433_nth__map__upt,axiom,
    ! [A: $tType,I: nat,N: nat,M: nat,Fa: fun(nat,A)] :
      ( aa(nat,$o,ord_less(nat,I),aa(nat,nat,minus_minus(nat,N),M))
     => ( aa(nat,A,nth(A,map(nat,A,Fa,upt(M,N))),I) = aa(nat,A,Fa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)) ) ) ).

% nth_map_upt
tff(fact_6434_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),N: nat,M: nat,Fa: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,minus_minus(nat,N),M) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,ord_less(nat,I2),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I2) = aa(nat,A,Fa,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I2)) ) )
       => ( map(nat,A,Fa,upt(M,N)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_6435_make__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,Fa: fun(nat,A)] : hoare_hoare_triple(array(A),one_one(assn),array_make(A,N,Fa),aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_ajm(nat,fun(fun(nat,A),fun(array(A),assn)),N),Fa)) ) ).

% make_rule
tff(fact_6436_sdiv__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),A3)),one_one(nat))))),signed7115095781618012415divide(int,ring_1_signed(A,int,A3),ring_1_signed(A,int,B3))) ) ).

% sdiv_word_min
tff(fact_6437_int__sdiv__simps_I2_J,axiom,
    ! [A3: int] : signed7115095781618012415divide(int,A3,zero_zero(int)) = zero_zero(int) ).

% int_sdiv_simps(2)
tff(fact_6438_sdiv__int__0__div,axiom,
    ! [Xb: int] : signed7115095781618012415divide(int,zero_zero(int),Xb) = zero_zero(int) ).

% sdiv_int_0_div
tff(fact_6439_sdiv__int__div__0,axiom,
    ! [Xb: int] : signed7115095781618012415divide(int,Xb,zero_zero(int)) = zero_zero(int) ).

% sdiv_int_div_0
tff(fact_6440_int__sdiv__simps_I1_J,axiom,
    ! [A3: int] : signed7115095781618012415divide(int,A3,one_one(int)) = A3 ).

% int_sdiv_simps(1)
tff(fact_6441_int__sdiv__same__is__1,axiom,
    ! [A3: int,B3: int] :
      ( ( A3 != zero_zero(int) )
     => ( ( signed7115095781618012415divide(int,A3,B3) = A3 )
      <=> ( B3 = one_one(int) ) ) ) ).

% int_sdiv_same_is_1
tff(fact_6442_int__sdiv__simps_I3_J,axiom,
    ! [A3: int] : signed7115095781618012415divide(int,A3,aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),A3) ).

% int_sdiv_simps(3)
tff(fact_6443_sdiv__int__numeral__numeral,axiom,
    ! [M: num,N: num] : signed7115095781618012415divide(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) ).

% sdiv_int_numeral_numeral
tff(fact_6444_int__sdiv__negated__is__minus1,axiom,
    ! [A3: int,B3: int] :
      ( ( A3 != zero_zero(int) )
     => ( ( signed7115095781618012415divide(int,A3,B3) = aa(int,int,uminus_uminus(int),A3) )
      <=> ( B3 = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% int_sdiv_negated_is_minus1
tff(fact_6445_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A3: int,B3: int] :
      ( ( signed7115095781618012415divide(int,A3,B3) != zero_zero(int) )
     => ( aa(int,int,sgn_sgn(int),signed7115095781618012415divide(int,A3,B3)) = aa(int,int,sgn_sgn(int),aa(int,int,aa(int,fun(int,int),times_times(int),A3),B3)) ) ) ).

% sgn_sdiv_eq_sgn_mult
tff(fact_6446_signed__divide__int__def,axiom,
    ! [K: int,L: int] : signed7115095781618012415divide(int,K,L) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),K)),aa(int,int,sgn_sgn(int),L))),aa(int,int,aa(int,fun(int,int),divide_divide(int),abs_abs(int,K)),abs_abs(int,L))) ).

% signed_divide_int_def
tff(fact_6447_time__array__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,Fa: fun(nat,A),H: heap_ext(product_unit)] : time_time(array(A),array_make(A,N,Fa),H) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) ) ).

% time_array_make
tff(fact_6448_TBOUND__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,Fa: fun(nat,A)] : time_TBOUND(array(A),array_make(A,N,Fa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ).

% TBOUND_make
tff(fact_6449_sdiv__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(int,$o,ord_less_eq(int,signed7115095781618012415divide(int,ring_1_signed(A,int,A3),ring_1_signed(A,int,B3))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,aa(word(A),nat,size_size(word(A)),A3)),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% sdiv_word_max
tff(fact_6450_array__of__list__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A)] : array_of_list(A,Xs) = array_make(A,aa(list(A),nat,size_size(list(A)),Xs),nth(A,Xs)) ) ).

% array_of_list_make
tff(fact_6451_array__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,Xb: A] : array_new(A,N,Xb) = array_make(A,N,aTP_Lamp_ajn(A,fun(nat,A),Xb)) ) ).

% array_make
tff(fact_6452_effect__makeI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),H2: heap_ext(product_unit),Fa: fun(nat,A),N: nat,H: heap_ext(product_unit)] :
          ( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H2) = array_alloc(A,map(nat,A,Fa,upt(zero_zero(nat),N)),H) )
         => heap_Time_effect(array(A),array_make(A,N,Fa),H,H2,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ).

% effect_makeI
tff(fact_6453_filter__upt__last,axiom,
    ! [A: $tType,P: fun(A,$o),L: list(A),Js: list(nat),J: nat,I: nat] :
      ( ( filter2(nat,aa(list(A),fun(nat,$o),aTP_Lamp_ajo(fun(A,$o),fun(list(A),fun(nat,$o)),P),L),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = append(nat,Js,aa(list(nat),list(nat),cons(nat,J),nil(nat))) )
     => ( aa(nat,$o,ord_less(nat,J),I)
       => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
         => ~ aa(A,$o,P,aa(nat,A,nth(A,L),I)) ) ) ) ).

% filter_upt_last
tff(fact_6454_filter__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),Xs: list(A)] : filter2(A,P,filter2(A,Q,Xs)) = filter2(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ajp(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),Xs) ).

% filter_filter
tff(fact_6455_set__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : set2(A,filter2(A,P,Xs)) = collect(A,aa(list(A),fun(A,$o),aTP_Lamp_ajq(fun(A,$o),fun(list(A),fun(A,$o)),P),Xs)) ).

% set_filter
tff(fact_6456_replicate__length__filter,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : replicate(A,aa(list(A),nat,size_size(list(A)),filter2(A,fequal(A,Xb),Xs)),Xb) = filter2(A,fequal(A,Xb),Xs) ).

% replicate_length_filter
tff(fact_6457_sum__length__filter__compl,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),filter2(A,P,Xs))),aa(list(A),nat,size_size(list(A)),filter2(A,aTP_Lamp_vu(fun(A,$o),fun(A,$o),P),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_6458_length__filter__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,$o,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),filter2(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_filter_le
tff(fact_6459_filter__conv__foldr,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : filter2(A,P,Xs) = foldr(A,list(A),aTP_Lamp_ajr(fun(A,$o),fun(A,fun(list(A),list(A))),P),Xs,nil(A)) ).

% filter_conv_foldr
tff(fact_6460_length__filter__less,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o)] :
      ( member(A,Xb,set2(A,Xs))
     => ( ~ aa(A,$o,P,Xb)
       => aa(nat,$o,ord_less(nat,aa(list(A),nat,size_size(list(A)),filter2(A,P,Xs))),aa(list(A),nat,size_size(list(A)),Xs)) ) ) ).

% length_filter_less
tff(fact_6461_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y: A] : aa(set(A),set(A),minus_minus(set(A),set2(A,Xs)),aa(set(A),set(A),insert(A,Y),bot_bot(set(A)))) = set2(A,filter2(A,aTP_Lamp_ajs(A,fun(A,$o),Y),Xs)) ).

% set_minus_filter_out
tff(fact_6462_upt__filter__extend,axiom,
    ! [U: nat,U3: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less_eq(nat,U),U3)
     => ( ! [I2: nat] :
            ( ( aa(nat,$o,ord_less_eq(nat,U),I2)
              & aa(nat,$o,ord_less(nat,I2),U3) )
           => ~ aa(nat,$o,P,I2) )
       => ( filter2(nat,P,upt(zero_zero(nat),U)) = filter2(nat,P,upt(zero_zero(nat),U3)) ) ) ) ).

% upt_filter_extend
tff(fact_6463_effect__of__listI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),H2: heap_ext(product_unit),Xs: list(A),H: heap_ext(product_unit)] :
          ( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H2) = array_alloc(A,Xs,H) )
         => heap_Time_effect(array(A),array_of_list(A,Xs),H,H2,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Xs))) ) ) ).

% effect_of_listI
tff(fact_6464_effect__newI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A),H2: heap_ext(product_unit),N: nat,Xb: A,H: heap_ext(product_unit)] :
          ( ( aa(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),heap_ext(product_unit))),product_Pair(array(A),heap_ext(product_unit)),A3),H2) = array_alloc(A,replicate(A,N,Xb),H) )
         => heap_Time_effect(array(A),array_new(A,N,Xb),H,H2,A3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat))) ) ) ).

% effect_newI
tff(fact_6465_Array__Time_Omake__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,Fa: fun(nat,A)] : array_make(A,N,Fa) = heap_Time_heap(array(A),aa(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_aju(nat,fun(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N),Fa)) ) ).

% Array_Time.make_def
tff(fact_6466_effect__heapI,axiom,
    ! [A: $tType,N: nat,Fa: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: A] :
      ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),Fa,H))) )
     => ( ( H2 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),Fa,H))) )
       => ( ( R2 = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),Fa,H)) )
         => heap_Time_effect(A,heap_Time_heap(A,Fa),H,H2,R2,N) ) ) ) ).

% effect_heapI
tff(fact_6467_effect__heapE,axiom,
    ! [A: $tType,Fa: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: A,N: nat] :
      ( heap_Time_effect(A,heap_Time_heap(A,Fa),H,H2,R2,N)
     => ~ ( ( H2 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),Fa,H))) )
         => ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),Fa,H))) )
           => ( R2 != aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),Fa,H)) ) ) ) ) ).

% effect_heapE
tff(fact_6468_return__def,axiom,
    ! [A: $tType,Xb: A] : aa(A,heap_Time_Heap(A),heap_Time_return(A),Xb) = heap_Time_heap(A,aTP_Lamp_ajv(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Xb)) ).

% return_def
tff(fact_6469_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xss: list(list(B))] : aa(nat,nat,ord_max(nat,aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),foldr(list(B),nat,aTP_Lamp_ajw(list(B),fun(nat,nat)),Xss,zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,aa(list(A),nat,size_size(list(A)),Xs)),foldr(list(B),nat,aTP_Lamp_ajx(list(B),fun(nat,nat)),filter2(list(B),aTP_Lamp_ajy(list(B),$o),Xss),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_6470_of__list__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A)] : array_of_list(A,Xs) = heap_Time_heap(array(A),aTP_Lamp_aka(list(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),Xs)) ) ).

% of_list_def
tff(fact_6471_new__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,Xb: A] : array_new(A,N,Xb) = heap_Time_heap(array(A),aa(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_akb(nat,fun(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N),Xb)) ) ).

% new_def
tff(fact_6472_execute__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,Fa: fun(nat,A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_make(A,N,Fa)),H) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_ajt(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N)),array_alloc(A,map(nat,A,Fa,upt(zero_zero(nat),N)),H))) ) ).

% execute_make
tff(fact_6473_tap__def,axiom,
    ! [A: $tType,Fa: fun(heap_ext(product_unit),A)] : heap_Time_tap(A,Fa) = heap_Time_Heap2(A,aTP_Lamp_akc(fun(heap_ext(product_unit),A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Fa)) ).

% tap_def
tff(fact_6474_execute__tap,axiom,
    ! [A: $tType,Fa: fun(heap_ext(product_unit),A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_tap(A,Fa)),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),A,Fa,H)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),one_one(nat)))) ).

% execute_tap
tff(fact_6475_execute__bind_I2_J,axiom,
    ! [A: $tType,B: $tType,Fa: heap_Time_Heap(A),H: heap_ext(product_unit),G: fun(A,heap_Time_Heap(B))] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,Fa),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
     => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_bind(A,B,Fa,G)),H) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ) ) ).

% execute_bind(2)
tff(fact_6476_time__def,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit)] : time_time(A,C2,H) = case_option(nat,product_prod(A,product_prod(heap_ext(product_unit),nat)),zero_zero(nat),product_case_prod(A,product_prod(heap_ext(product_unit),nat),nat,aTP_Lamp_ake(A,fun(product_prod(heap_ext(product_unit),nat),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),H)) ).

% time_def
tff(fact_6477_effectI,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),R2: A,H2: heap_ext(product_unit),N: nat] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),N))) )
     => heap_Time_effect(A,C2,H,H2,R2,N) ) ).

% effectI
tff(fact_6478_effect__def,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: A,N: nat] :
      ( heap_Time_effect(A,C2,H,H2,R2,N)
    <=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),N))) ) ) ).

% effect_def
tff(fact_6479_refines__def,axiom,
    ! [A: $tType,P2: heap_Time_Heap(A),Q2: heap_Time_Heap(A)] :
      ( refine_Imp_refines(A,P2,Q2)
    <=> ! [H6: heap_ext(product_unit)] : case_option($o,product_prod(A,product_prod(heap_ext(product_unit),nat)),$true,product_case_prod(A,product_prod(heap_ext(product_unit),nat),$o,aa(heap_ext(product_unit),fun(A,fun(product_prod(heap_ext(product_unit),nat),$o)),aTP_Lamp_akg(heap_Time_Heap(A),fun(heap_ext(product_unit),fun(A,fun(product_prod(heap_ext(product_unit),nat),$o))),P2),H6)),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,Q2),H6)) ) ).

% refines_def
tff(fact_6480_execute__bind__eq__SomeI,axiom,
    ! [A: $tType,B: $tType,Fa: heap_Time_Heap(A),H: heap_ext(product_unit),Xb: A,H2: heap_ext(product_unit),N: nat,G: fun(A,heap_Time_Heap(B)),Y: B,H7: heap_ext(product_unit),N5: nat] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,Fa),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Xb),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),N))) )
     => ( ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,aa(A,heap_Time_Heap(B),G,Xb)),H2) = aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),some(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat)),aa(B,fun(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat))),product_Pair(B,product_prod(heap_ext(product_unit),nat)),Y),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H7),N5))) )
       => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_bind(A,B,Fa,G)),H) = aa(product_prod(B,product_prod(heap_ext(product_unit),nat)),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),some(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat)),aa(B,fun(product_prod(heap_ext(product_unit),nat),product_prod(B,product_prod(heap_ext(product_unit),nat))),product_Pair(B,product_prod(heap_ext(product_unit),nat)),Y),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H7),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),N5)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_6481_effect__tapE,axiom,
    ! [A: $tType,Fa: fun(heap_ext(product_unit),A),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: A,N: nat] :
      ( heap_Time_effect(A,heap_Time_tap(A,Fa),H,H2,R2,N)
     => ~ ( ( H2 = H )
         => ( ( R2 = aa(heap_ext(product_unit),A,Fa,H) )
           => ( N != one_one(nat) ) ) ) ) ).

% effect_tapE
tff(fact_6482_effect__tapI,axiom,
    ! [A: $tType,H2: heap_ext(product_unit),H: heap_ext(product_unit),R2: A,Fa: fun(heap_ext(product_unit),A)] :
      ( ( H2 = H )
     => ( ( R2 = aa(heap_ext(product_unit),A,Fa,H) )
       => heap_Time_effect(A,heap_Time_tap(A,Fa),H,H2,R2,one_one(nat)) ) ) ).

% effect_tapI
tff(fact_6483_execute__of__list,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_of_list(A,Xs)),H) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_ajz(list(A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Xs)),array_alloc(A,Xs,H))) ) ).

% execute_of_list
tff(fact_6484_execute__new,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [N: nat,Xb: A,H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),heap_Time_execute(array(A),array_new(A,N,Xb)),H) = aa(product_prod(array(A),product_prod(heap_ext(product_unit),nat)),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),some(product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_ajt(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),N)),array_alloc(A,replicate(A,N,Xb),H))) ) ).

% execute_new
tff(fact_6485_effect__guardE,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),$o),Fa: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: A,N: nat] :
      ( heap_Time_effect(A,heap_Time_guard(A,P,Fa),H,H2,R2,N)
     => ~ ( ( H2 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),Fa,H))) )
         => ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),Fa,H))) )
           => ( ( R2 = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),Fa,H)) )
             => ~ aa(heap_ext(product_unit),$o,P,H) ) ) ) ) ).

% effect_guardE
tff(fact_6486_effect__guardI,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),$o),H: heap_ext(product_unit),H2: heap_ext(product_unit),Fa: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),N: nat,R2: A] :
      ( aa(heap_ext(product_unit),$o,P,H)
     => ( ( H2 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),Fa,H))) )
       => ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),Fa,H))) )
         => ( ( R2 = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),Fa,H)) )
           => heap_Time_effect(A,heap_Time_guard(A,P,Fa),H,H2,R2,N) ) ) ) ) ).

% effect_guardI
tff(fact_6487_guard__def,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),$o),Fa: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_guard(A,P,Fa) = heap_Time_Heap2(A,aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_akh(fun(heap_ext(product_unit),$o),fun(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),P),Fa)) ).

% guard_def
tff(fact_6488_the__res__def,axiom,
    ! [A: $tType,M: heap_Time_Heap(A),H: heap_ext(product_unit)] : time_the_res(A,M,H) = case_option(A,product_prod(A,product_prod(heap_ext(product_unit),nat)),undefined(A),product_case_prod(A,product_prod(heap_ext(product_unit),nat),A,aTP_Lamp_akj(A,fun(product_prod(heap_ext(product_unit),nat),A))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,M),H)) ).

% the_res_def
tff(fact_6489_effectE,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: A,N: nat] :
      ( heap_Time_effect(A,C2,H,H2,R2,N)
     => ~ ( ( R2 = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),H))) )
         => ( ( H2 = aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),H)))) )
           => ( ( N = aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,C2),H)))) )
             => ~ heap_Time_success(A,C2,H) ) ) ) ) ).

% effectE
tff(fact_6490_success__LetI,axiom,
    ! [A: $tType,B: $tType,Xb: A,Tb: A,Fa: fun(A,heap_Time_Heap(B)),H: heap_ext(product_unit)] :
      ( ( Xb = Tb )
     => ( heap_Time_success(B,aa(A,heap_Time_Heap(B),Fa,Xb),H)
       => heap_Time_success(B,aa(A,heap_Time_Heap(B),Fa,Tb),H) ) ) ).

% success_LetI
tff(fact_6491_success__returnI,axiom,
    ! [A: $tType,Xb: A,H: heap_ext(product_unit)] : heap_Time_success(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),Xb),H) ).

% success_returnI
tff(fact_6492_success__effectE,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( heap_Time_success(A,C2,H)
     => ~ ! [R4: A,H8: heap_ext(product_unit),N3: nat] : ~ heap_Time_effect(A,C2,H,H8,R4,N3) ) ).

% success_effectE
tff(fact_6493_effect__success,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: A,N: nat] :
      ( heap_Time_effect(A,C2,H,H2,R2,N)
     => heap_Time_success(A,C2,H) ) ).

% effect_success
tff(fact_6494_success__bind__effectI,axiom,
    ! [A: $tType,B: $tType,Fa: heap_Time_Heap(A),H: heap_ext(product_unit),H2: heap_ext(product_unit),Xb: A,N: nat,G: fun(A,heap_Time_Heap(B))] :
      ( heap_Time_effect(A,Fa,H,H2,Xb,N)
     => ( heap_Time_success(B,aa(A,heap_Time_Heap(B),G,Xb),H2)
       => heap_Time_success(B,heap_Time_bind(A,B,Fa,G),H) ) ) ).

% success_bind_effectI
tff(fact_6495_success__bind__executeI,axiom,
    ! [A: $tType,B: $tType,Fa: heap_Time_Heap(A),H: heap_ext(product_unit),Xb: A,H2: heap_ext(product_unit),N: nat,G: fun(A,heap_Time_Heap(B))] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,Fa),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Xb),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),N))) )
     => ( heap_Time_success(B,aa(A,heap_Time_Heap(B),G,Xb),H2)
       => heap_Time_success(B,heap_Time_bind(A,B,Fa,G),H) ) ) ).

% success_bind_executeI
tff(fact_6496_execute__bind__success,axiom,
    ! [B: $tType,A: $tType,Fa: heap_Time_Heap(A),H: heap_ext(product_unit),G: fun(A,heap_Time_Heap(B))] :
      ( heap_Time_success(A,Fa,H)
     => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_bind(A,B,Fa,G)),H) = heap_Time_timeFrame(B,aa(product_prod(heap_ext(product_unit),nat),nat,product_snd(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,Fa),H)))),aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,aa(A,heap_Time_Heap(B),G,aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),A,product_fst(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,Fa),H))))),aa(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),product_fst(heap_ext(product_unit),nat),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),product_prod(heap_ext(product_unit),nat),product_snd(A,product_prod(heap_ext(product_unit),nat)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(A,product_prod(heap_ext(product_unit),nat)),the2(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,Fa),H)))))) ) ) ).

% execute_bind_success
tff(fact_6497_execute__assert_I1_J,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,H: heap_ext(product_unit)] :
      ( aa(A,$o,P,Xb)
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_assert(A,P,Xb)),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Xb),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H),one_one(nat)))) ) ) ).

% execute_assert(1)
tff(fact_6498_timeFrame__zero,axiom,
    ! [A: $tType,H: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_timeFrame(A,zero_zero(nat),H) = H ).

% timeFrame_zero
tff(fact_6499_assert__cong,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),P3: fun(A,$o),Fa: fun(A,heap_Time_Heap(B)),F3: fun(A,heap_Time_Heap(B)),Xb: A] :
      ( ( P = P3 )
     => ( ! [X2: A] :
            ( aa(A,$o,P3,X2)
           => ( aa(A,heap_Time_Heap(B),Fa,X2) = aa(A,heap_Time_Heap(B),F3,X2) ) )
       => ( heap_Time_bind(A,B,heap_Time_assert(A,P,Xb),Fa) = heap_Time_bind(A,B,heap_Time_assert(A,P3,Xb),F3) ) ) ) ).

% assert_cong
tff(fact_6500_effect__assertE,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: A,N: nat] :
      ( heap_Time_effect(A,heap_Time_assert(A,P,Xb),H,H2,R2,N)
     => ~ ( aa(A,$o,P,Xb)
         => ( ( R2 = Xb )
           => ( ( H2 = H )
             => ( N != one_one(nat) ) ) ) ) ) ).

% effect_assertE
tff(fact_6501_effect__assertI,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,H2: heap_ext(product_unit),H: heap_ext(product_unit),R2: A,N: nat] :
      ( aa(A,$o,P,Xb)
     => ( ( H2 = H )
       => ( ( R2 = Xb )
         => ( ( N = one_one(nat) )
           => heap_Time_effect(A,heap_Time_assert(A,P,Xb),H,H2,R2,N) ) ) ) ) ).

% effect_assertI
tff(fact_6502_execute__bind__case,axiom,
    ! [A: $tType,B: $tType,Fa: heap_Time_Heap(B),G: fun(B,heap_Time_Heap(A)),H: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,heap_Time_bind(B,A,Fa,G)),H) = case_option(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(B,product_prod(heap_ext(product_unit),nat)),none(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_case_prod(B,product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_akl(fun(B,heap_Time_Heap(A)),fun(B,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),G)),aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,Fa),H)) ).

% execute_bind_case
tff(fact_6503_Heap__Time__Monad_Obind__def,axiom,
    ! [A: $tType,B: $tType,Fa: heap_Time_Heap(B),G: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,Fa,G) = heap_Time_Heap2(A,aa(fun(B,heap_Time_Heap(A)),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_akm(heap_Time_Heap(B),fun(fun(B,heap_Time_Heap(A)),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),Fa),G)) ).

% Heap_Time_Monad.bind_def
tff(fact_6504_execute__bind_I1_J,axiom,
    ! [A: $tType,B: $tType,Fa: heap_Time_Heap(A),H: heap_ext(product_unit),Xb: A,H2: heap_ext(product_unit),N: nat,G: fun(A,heap_Time_Heap(B))] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,Fa),H) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Xb),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),N))) )
     => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,heap_Time_bind(A,B,Fa,G)),H) = heap_Time_timeFrame(B,N,aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,aa(A,heap_Time_Heap(B),G,Xb)),H2)) ) ) ).

% execute_bind(1)
tff(fact_6505_the__heap__def,axiom,
    ! [A: $tType,M: heap_Time_Heap(A),H: heap_ext(product_unit)] : time_the_heap(A,M,H) = case_option(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),undefined(heap_ext(product_unit)),product_case_prod(A,product_prod(heap_ext(product_unit),nat),heap_ext(product_unit),aTP_Lamp_ako(A,fun(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit)))),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,M),H)) ).

% the_heap_def
tff(fact_6506_sort__key__by__quicksort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Fa: fun(A,B),Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,B,Fa),Xs) = append(A,aa(list(A),list(A),linorder_sort_key(A,B,Fa),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_akp(fun(A,B),fun(list(A),fun(A,$o)),Fa),Xs),Xs)),append(A,filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_akq(fun(A,B),fun(list(A),fun(A,$o)),Fa),Xs),Xs),aa(list(A),list(A),linorder_sort_key(A,B,Fa),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_akr(fun(A,B),fun(list(A),fun(A,$o)),Fa),Xs),Xs)))) ) ).

% sort_key_by_quicksort
tff(fact_6507_sort__upt,axiom,
    ! [M: nat,N: nat] : aa(list(nat),list(nat),linorder_sort_key(nat,nat,aTP_Lamp_vj(nat,nat)),upt(M,N)) = upt(M,N) ).

% sort_upt
tff(fact_6508_sort__upto,axiom,
    ! [I: int,J: int] : aa(list(int),list(int),linorder_sort_key(int,int,aTP_Lamp_wv(int,int)),upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_6509_length__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Fa: fun(A,B),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),linorder_sort_key(A,B,Fa),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ).

% length_sort
tff(fact_6510_sort__key__const,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [C2: B,Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,B,aTP_Lamp_ahh(B,fun(A,B),C2)),Xs) = Xs ) ).

% sort_key_const
tff(fact_6511_sort__key__stable,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Fa: fun(A,B),K: B,Xs: list(A)] : filter2(A,aa(B,fun(A,$o),aTP_Lamp_aks(fun(A,B),fun(B,fun(A,$o)),Fa),K),aa(list(A),list(A),linorder_sort_key(A,B,Fa),Xs)) = filter2(A,aa(B,fun(A,$o),aTP_Lamp_aks(fun(A,B),fun(B,fun(A,$o)),Fa),K),Xs) ) ).

% sort_key_stable
tff(fact_6512_sort__by__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_akt(A,A)),Xs) = append(A,aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_akt(A,A)),filter2(A,aTP_Lamp_aku(list(A),fun(A,$o),Xs),Xs)),append(A,filter2(A,aTP_Lamp_akv(list(A),fun(A,$o),Xs),Xs),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_akt(A,A)),filter2(A,ord_less(A,aa(nat,A,nth(A,Xs),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Xs)))) ) ).

% sort_by_quicksort
tff(fact_6513_time__bind,axiom,
    ! [B: $tType,A: $tType,M: heap_Time_Heap(B),Fa: fun(B,heap_Time_Heap(A)),H: heap_ext(product_unit)] :
      time_time(A,heap_Time_bind(B,A,M,Fa),H) = $ite(
        ( ~ time_fails(B,M,H)
        & ~ time_fails(A,aa(B,heap_Time_Heap(A),Fa,time_the_res(B,M,H)),time_the_heap(B,M,H)) ),
        aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),time_time(B,M,H)),time_time(A,aa(B,heap_Time_Heap(A),Fa,time_the_res(B,M,H)),time_the_heap(B,M,H))),
        zero_zero(nat) ) ).

% time_bind
tff(fact_6514_ureturn__def,axiom,
    ! [A: $tType,Xb: A] : aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),Xb) = heap_Time_heap(A,aTP_Lamp_akw(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Xb)) ).

% ureturn_def
tff(fact_6515_fails__return,axiom,
    ! [A: $tType,Xb: A,H: heap_ext(product_unit)] : ~ time_fails(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),Xb),H) ).

% fails_return
tff(fact_6516_fails__assert_H,axiom,
    ! [P: $o,H: heap_ext(product_unit)] :
      ( time_fails(product_unit,refine_Imp_assert((P)),H)
    <=> ~ (P) ) ).

% fails_assert'
tff(fact_6517_fails__refines,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),C6: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( refine_Imp_refines(A,C2,C6)
     => ( time_fails(A,C2,H)
       => time_fails(A,C6,H) ) ) ).

% fails_refines
tff(fact_6518_bind__ureturn,axiom,
    ! [A: $tType,Fa: heap_Time_Heap(A)] : heap_Time_bind(A,A,Fa,heap_Time_ureturn(A)) = Fa ).

% bind_ureturn
tff(fact_6519_ureturn__bind,axiom,
    ! [A: $tType,B: $tType,Xb: B,Fa: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,aa(B,heap_Time_Heap(B),heap_Time_ureturn(B),Xb),Fa) = aa(B,heap_Time_Heap(A),Fa,Xb) ).

% ureturn_bind
tff(fact_6520_upd__ureturn,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xb: A,A3: array(A)] : heap_Time_bind(array(A),array(A),array_upd(A,I,Xb,A3),aTP_Lamp_akx(array(A),fun(array(A),heap_Time_Heap(array(A))),A3)) = array_upd(A,I,Xb,A3) ) ).

% upd_ureturn
tff(fact_6521_effect__ureturnE,axiom,
    ! [A: $tType,Xb: A,H: heap_ext(product_unit),H2: heap_ext(product_unit),R2: A,N: nat] :
      ( heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),Xb),H,H2,R2,N)
     => ~ ( ( R2 = Xb )
         => ( ( H2 = H )
           => ( N != zero_zero(nat) ) ) ) ) ).

% effect_ureturnE
tff(fact_6522_effect__ureturnI,axiom,
    ! [A: $tType,H: heap_ext(product_unit),H2: heap_ext(product_unit),Xb: A] :
      ( ( H = H2 )
     => heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),Xb),H,H2,Xb,zero_zero(nat)) ) ).

% effect_ureturnI
tff(fact_6523_time__refines,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),C6: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( refine_Imp_refines(A,C2,C6)
     => ( ~ time_fails(A,C6,H)
       => aa(nat,$o,ord_less_eq(nat,time_time(A,C2,H)),time_time(A,C6,H)) ) ) ).

% time_refines
tff(fact_6524_TBOUND__bind__weak,axiom,
    ! [A: $tType,B: $tType,M: heap_Time_Heap(A),T_1: nat,Fa: fun(A,heap_Time_Heap(B)),T_2: nat] :
      ( time_TBOUND(A,M,T_1)
     => ( ! [X2: A,H5: heap_ext(product_unit)] :
            ( ~ time_fails(A,M,H5)
           => time_TBOUND(B,aa(A,heap_Time_Heap(B),Fa,X2),T_2) )
       => time_TBOUND(B,heap_Time_bind(A,B,M,Fa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),T_1),T_2)) ) ) ).

% TBOUND_bind_weak
tff(fact_6525_time__array__upd,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xb: A,P2: array(A),H: heap_ext(product_unit)] :
          time_time(array(A),array_upd(A,I,Xb,P2),H) = $ite(time_fails(array(A),array_upd(A,I,Xb,P2),H),zero_zero(nat),one_one(nat)) ) ).

% time_array_upd
tff(fact_6526_time__array__nth,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P2: array(A),I: nat,H: heap_ext(product_unit)] :
          time_time(A,aa(nat,heap_Time_Heap(A),array_nth(A,P2),I),H) = $ite(time_fails(A,aa(nat,heap_Time_Heap(A),array_nth(A,P2),I),H),zero_zero(nat),one_one(nat)) ) ).

% time_array_nth
tff(fact_6527_fails__bind,axiom,
    ! [A: $tType,B: $tType,M: heap_Time_Heap(B),Fa: fun(B,heap_Time_Heap(A)),H: heap_ext(product_unit)] :
      ( time_fails(A,heap_Time_bind(B,A,M,Fa),H)
    <=> ( ~ time_fails(B,M,H)
       => time_fails(A,aa(B,heap_Time_Heap(A),Fa,time_the_res(B,M,H)),time_the_heap(B,M,H)) ) ) ).

% fails_bind
tff(fact_6528_TBOUND__bind,axiom,
    ! [A: $tType,B: $tType,M: heap_Time_Heap(A),T_1: nat,Fa: fun(A,heap_Time_Heap(B)),T_2: nat] :
      ( time_TBOUND(A,M,T_1)
     => ( ! [X2: A,H5: heap_ext(product_unit)] :
            ( ( X2 = time_the_res(A,M,H5) )
           => ( ~ time_fails(A,M,H5)
             => time_TBOUND(B,aa(A,heap_Time_Heap(B),Fa,X2),T_2) ) )
       => time_TBOUND(B,heap_Time_bind(A,B,M,Fa),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),T_1),T_2)) ) ) ).

% TBOUND_bind
tff(fact_6529_len_H__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semiring_1(A)
        & heap(B) )
     => ! [A3: array(B)] : array_len2(B,A,A3) = heap_Time_bind(nat,A,array_len(B,A3),aTP_Lamp_aky(nat,heap_Time_Heap(A))) ) ).

% len'_def
tff(fact_6530_signed_Ostable__sort__key__def,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [Sk: fun(fun(A,word(B)),fun(list(A),list(A)))] :
          ( stable_sort_key(A,word(B),Sk)
        <=> ! [F7: fun(A,word(B)),Xs3: list(A),K3: word(B)] : filter2(A,aa(word(B),fun(A,$o),aTP_Lamp_akz(fun(A,word(B)),fun(word(B),fun(A,$o)),F7),K3),aa(list(A),list(A),aa(fun(A,word(B)),fun(list(A),list(A)),Sk,F7),Xs3)) = filter2(A,aa(word(B),fun(A,$o),aTP_Lamp_akz(fun(A,word(B)),fun(word(B),fun(A,$o)),F7),K3),Xs3) ) ) ).

% signed.stable_sort_key_def
tff(fact_6531_time__array__map__entry,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Fa: fun(A,A),P2: array(A),H: heap_ext(product_unit)] :
          time_time(array(A),array_map_entry(A,I,Fa,P2),H) = $ite(time_fails(array(A),array_map_entry(A,I,Fa,P2),H),zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% time_array_map_entry
tff(fact_6532_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : foldr(list(A),nat,aTP_Lamp_ala(list(A),fun(nat,nat)),transpose(A,Xs),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_alb(list(A),$o),Xs)) ).

% transpose_max_length
tff(fact_6533_nth__transpose,axiom,
    ! [A: $tType,I: nat,Xs: list(list(A))] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)))
     => ( aa(nat,list(A),nth(list(A),transpose(A,Xs)),I) = map(list(A),A,aTP_Lamp_alc(nat,fun(list(A),A),I),filter2(list(A),aTP_Lamp_ald(nat,fun(list(A),$o),I),Xs)) ) ) ).

% nth_transpose
tff(fact_6534_length__transpose,axiom,
    ! [A: $tType,Xs: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),transpose(A,Xs)) = foldr(list(A),nat,aTP_Lamp_ala(list(A),fun(nat,nat)),Xs,zero_zero(nat)) ).

% length_transpose
tff(fact_6535_TBOUND__map__entry,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xb: fun(A,A),A3: array(A)] : time_TBOUND(array(A),array_map_entry(A,I,Xb,A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% TBOUND_map_entry
tff(fact_6536_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),N: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( N = zero_zero(nat) ) )
     => ( ! [I2: nat] :
            ( aa(nat,$o,ord_less(nat,I2),aa(list(list(A)),nat,size_size(list(list(A))),Xs))
           => ( aa(list(A),nat,size_size(list(A)),aa(nat,list(A),nth(list(A),Xs),I2)) = N ) )
       => ( transpose(A,Xs) = map(nat,list(A),aTP_Lamp_alf(list(list(A)),fun(nat,list(A)),Xs),upt(zero_zero(nat),N)) ) ) ) ).

% transpose_rectangle
tff(fact_6537_time__array__swap,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xb: A,P2: array(A),H: heap_ext(product_unit),Fa: fun(A,A)] :
          time_time(A,array_swap(A,I,Xb,P2),H) = $ite(time_fails(array(A),array_map_entry(A,I,Fa,P2),H),zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% time_array_swap
tff(fact_6538_length__product__lists,axiom,
    ! [A: $tType,Xss: list(list(A))] : aa(list(list(A)),nat,size_size(list(list(A))),product_lists(A,Xss)) = foldr(nat,nat,times_times(nat),map(list(A),nat,size_size(list(A)),Xss),one_one(nat)) ).

% length_product_lists
tff(fact_6539_in__set__product__lists__length,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] :
      ( member(list(A),Xs,set2(list(A),product_lists(A,Xss)))
     => ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(list(A)),nat,size_size(list(list(A))),Xss) ) ) ).

% in_set_product_lists_length
tff(fact_6540_TBOUND__swap,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xb: A,A3: array(A)] : time_TBOUND(A,array_swap(A,I,Xb,A3),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% TBOUND_swap
tff(fact_6541_length__subseqs,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),subseqs(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(list(A),nat,size_size(list(A)),Xs)) ).

% length_subseqs
tff(fact_6542_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),N: nat] :
      ( ! [X2: list(A)] :
          ( member(list(A),X2,set2(list(A),Xs))
         => ( aa(list(A),nat,size_size(list(A)),X2) = N ) )
     => ( aa(list(A),nat,size_size(list(A)),concat(A,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(list(list(A)),nat,size_size(list(list(A))),Xs)),N) ) ) ).

% length_mul_elem
tff(fact_6543_concat__map__singleton,axiom,
    ! [A: $tType,B: $tType,Fa: fun(B,A),Xs: list(B)] : concat(A,map(B,list(A),aTP_Lamp_alg(fun(B,A),fun(B,list(A)),Fa),Xs)) = map(B,A,Fa,Xs) ).

% concat_map_singleton
tff(fact_6544_subseqs_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      subseqs(A,aa(list(A),list(A),cons(A,Xb),Xs)) = $let(
        xss: list(list(A)),
        xss:= subseqs(A,Xs),
        append(list(A),map(list(A),list(A),cons(A,Xb),xss),xss) ) ).

% subseqs.simps(2)
tff(fact_6545_product__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] : product_lists(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs),Xss)) = concat(list(A),map(A,list(list(A)),aTP_Lamp_alh(list(list(A)),fun(A,list(list(A))),Xss),Xs)) ).

% product_lists.simps(2)
tff(fact_6546_concat__filter__neq__Nil,axiom,
    ! [A: $tType,Xs: list(list(A))] : concat(A,filter2(list(A),aTP_Lamp_alb(list(A),$o),Xs)) = concat(A,Xs) ).

% concat_filter_neq_Nil
tff(fact_6547_product__concat__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product(A,B,Xs,Ys) = concat(product_prod(A,B),map(A,list(product_prod(A,B)),aTP_Lamp_ali(list(B),fun(A,list(product_prod(A,B))),Ys),Xs)) ).

% product_concat_map
tff(fact_6548_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(A) )
       => ( ( append(A,Xs,Ys) = append(A,Ys,Xs) )
         => ? [N3: nat,Zs2: list(A)] :
              ( aa(nat,$o,ord_less(nat,one_one(nat)),N3)
              & ( concat(A,replicate(list(A),N3,Zs2)) = append(A,Xs,Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_6549_product__code,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product_product(A,B,set2(A,Xs),set2(B,Ys)) = set2(product_prod(A,B),concat(product_prod(A,B),map(A,list(product_prod(A,B)),aTP_Lamp_ali(list(B),fun(A,list(product_prod(A,B))),Ys),Xs))) ).

% product_code
tff(fact_6550_n__lists__Nil,axiom,
    ! [A: $tType,N: nat] :
      n_lists(A,N,nil(A)) = $ite(N = zero_zero(nat),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))),nil(list(A))) ).

% n_lists_Nil
tff(fact_6551_length__n__lists__elem,axiom,
    ! [A: $tType,Ys: list(A),N: nat,Xs: list(A)] :
      ( member(list(A),Ys,set2(list(A),n_lists(A,N,Xs)))
     => ( aa(list(A),nat,size_size(list(A)),Ys) = N ) ) ).

% length_n_lists_elem
tff(fact_6552_n__lists_Osimps_I1_J,axiom,
    ! [A: $tType,Xs: list(A)] : n_lists(A,zero_zero(nat),Xs) = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),nil(list(A))) ).

% n_lists.simps(1)
tff(fact_6553_length__n__lists,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),n_lists(A,N,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(list(A),nat,size_size(list(A)),Xs)),N) ).

% length_n_lists
tff(fact_6554_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : n_lists(A,aa(nat,nat,suc,N),Xs) = concat(list(A),map(list(A),list(list(A)),aTP_Lamp_alk(list(A),fun(list(A),list(list(A))),Xs),n_lists(A,N,Xs))) ).

% n_lists.simps(2)
tff(fact_6555_set__n__lists,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] : set2(list(A),n_lists(A,N,Xs)) = collect(list(A),aa(list(A),fun(list(A),$o),aTP_Lamp_all(nat,fun(list(A),fun(list(A),$o)),N),Xs)) ).

% set_n_lists
tff(fact_6556_sort__key__by__quicksort__code,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Fa: fun(A,B),Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,B,Fa),Xs) = aa(list(A),list(A),case_list(list(A),A,nil(A),aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_alq(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),Fa),Xs)),Xs) ) ).

% sort_key_by_quicksort_code
tff(fact_6557_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,Fa: fun(B,A),P: fun(B,$o),Xs: list(B)] : map(B,A,Fa,filter2(B,P,Xs)) = map_filter(B,A,aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_alr(fun(B,A),fun(fun(B,$o),fun(B,option(A))),Fa),P),Xs) ).

% map_filter_map_filter
tff(fact_6558_list_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType,H: fun(B,A),F1: B,F2: fun(C,fun(list(C),B)),List: list(C)] : aa(B,A,H,aa(list(C),B,case_list(B,C,F1,F2),List)) = aa(list(C),A,case_list(A,C,aa(B,A,H,F1),aa(fun(C,fun(list(C),B)),fun(C,fun(list(C),A)),aTP_Lamp_als(fun(B,A),fun(fun(C,fun(list(C),B)),fun(C,fun(list(C),A))),H),F2)),List) ).

% list.case_distrib
tff(fact_6559_map__filter__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,Fa: fun(B,option(A)),Xb: B,Xs: list(B)] : map_filter(B,A,Fa,aa(list(B),list(B),cons(B,Xb),Xs)) = case_option(list(A),A,map_filter(B,A,Fa,Xs),aa(list(B),fun(A,list(A)),aTP_Lamp_alt(fun(B,option(A)),fun(list(B),fun(A,list(A))),Fa),Xs),aa(B,option(A),Fa,Xb)) ).

% map_filter_simps(1)
tff(fact_6560_transpose_Oelims,axiom,
    ! [A: $tType,Xb: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,Xb) = Y )
     => ( ( ( Xb = nil(list(A)) )
         => ( Y != nil(list(A)) ) )
       => ( ! [Xss2: list(list(A))] :
              ( ( Xb = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2) )
             => ( Y != transpose(A,Xss2) ) )
         => ~ ! [X2: A,Xs2: list(A),Xss2: list(list(A))] :
                ( ( Xb = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X2),Xs2)),Xss2) )
               => ( Y != aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X2),concat(A,map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_alu(A,fun(list(A),list(A)))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs2),concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_alv(A,fun(list(A),list(list(A))))),Xss2))))) ) ) ) ) ) ).

% transpose.elims
tff(fact_6561_transpose_Osimps_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Xss: list(list(A))] : transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xb),Xs)),Xss)) = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xb),concat(A,map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_alu(A,fun(list(A),list(A)))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs),concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_alv(A,fun(list(A),list(list(A))))),Xss))))) ).

% transpose.simps(3)
tff(fact_6562_part__code_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Fa: fun(A,B),Pivot: B,Xb: A,Xs: list(A)] : linorder_part(A,B,Fa,Pivot,aa(list(A),list(A),cons(A,Xb),Xs)) = aa(product_prod(list(A),product_prod(list(A),list(A))),product_prod(list(A),product_prod(list(A),list(A))),product_case_prod(list(A),product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))),aa(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))),aa(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))))),aTP_Lamp_alx(fun(A,B),fun(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))))),Fa),Pivot),Xb)),linorder_part(A,B,Fa,Pivot,Xs)) ) ).

% part_code(2)
tff(fact_6563_part__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Fa: fun(A,B),Pivot: B,Xs: list(A)] : linorder_part(A,B,Fa,Pivot,Xs) = aa(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))),aa(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),product_Pair(list(A),product_prod(list(A),list(A))),filter2(A,aa(B,fun(A,$o),aTP_Lamp_aly(fun(A,B),fun(B,fun(A,$o)),Fa),Pivot),Xs)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),filter2(A,aa(B,fun(A,$o),aTP_Lamp_aks(fun(A,B),fun(B,fun(A,$o)),Fa),Pivot),Xs)),filter2(A,aa(B,fun(A,$o),aTP_Lamp_alz(fun(A,B),fun(B,fun(A,$o)),Fa),Pivot),Xs))) ) ).

% part_def
tff(fact_6564_Bleast__code,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] : bleast(A,set2(A,Xs),P) = aa(list(A),A,case_list(A,A,abort_Bleast(A,set2(A,Xs),P),aTP_Lamp_ama(A,fun(list(A),A))),filter2(A,P,aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_akt(A,A)),Xs))) ) ).

% Bleast_code
tff(fact_6565_transpose__aux__filter__tail,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_alv(A,fun(list(A),list(list(A))))),Xss)) = map(list(A),list(A),tl(A),filter2(list(A),aTP_Lamp_alb(list(A),$o),Xss)) ).

% transpose_aux_filter_tail
tff(fact_6566_length__tl,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs)) = aa(nat,nat,minus_minus(nat,aa(list(A),nat,size_size(list(A)),Xs)),one_one(nat)) ).

% length_tl
tff(fact_6567_tl__replicate,axiom,
    ! [A: $tType,N: nat,Xb: A] : aa(list(A),list(A),tl(A),replicate(A,N,Xb)) = replicate(A,aa(nat,nat,minus_minus(nat,N),one_one(nat)),Xb) ).

% tl_replicate
tff(fact_6568_list_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List != nil(A) )
    <=> aa(list(A),$o,case_list($o,A,$false,aTP_Lamp_amb(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(2)
tff(fact_6569_list_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,List: list(A)] :
      ( ( List = nil(A) )
    <=> aa(list(A),$o,case_list($o,A,$true,aTP_Lamp_amc(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(1)
tff(fact_6570_tl__def,axiom,
    ! [A: $tType,List: list(A)] : aa(list(A),list(A),tl(A),List) = aa(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_amd(A,fun(list(A),list(A)))),List) ).

% tl_def
tff(fact_6571_tl__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),tl(A),append(A,Xs,Ys)) = aa(list(A),list(A),case_list(list(A),A,aa(list(A),list(A),tl(A),Ys),aTP_Lamp_ame(list(A),fun(A,fun(list(A),list(A))),Ys)),Xs) ).

% tl_append
tff(fact_6572_Misc_Onth__tl,axiom,
    ! [A: $tType,Xs: list(A),N: nat] :
      ( ( Xs != nil(A) )
     => ( aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N)) ) ) ).

% Misc.nth_tl
tff(fact_6573_list__take__induct__tl2,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B),P: fun(B,fun(A,$o))] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(B),nat,size_size(list(B)),Ys) )
     => ( ! [N3: nat] :
            ( aa(nat,$o,ord_less(nat,N3),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,aa(B,fun(A,$o),P,aa(nat,B,nth(B,Ys),N3)),aa(nat,A,nth(A,Xs),N3)) )
       => ! [N4: nat] :
            ( aa(nat,$o,ord_less(nat,N4),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs)))
           => aa(A,$o,aa(B,fun(A,$o),P,aa(nat,B,nth(B,aa(list(B),list(B),tl(B),Ys)),N4)),aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N4)) ) ) ) ).

% list_take_induct_tl2
tff(fact_6574_Nitpick_Osize__list__simp_I2_J,axiom,
    ! [A: $tType,Xs: list(A)] :
      aa(list(A),nat,size_size(list(A)),Xs) = $ite(Xs = nil(A),zero_zero(nat),aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs)))) ).

% Nitpick.size_list_simp(2)
tff(fact_6575_List_Onth__tl,axiom,
    ! [A: $tType,N: nat,Xs: list(A)] :
      ( aa(nat,$o,ord_less(nat,N),aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),tl(A),Xs)))
     => ( aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,N)) ) ) ).

% List.nth_tl
tff(fact_6576_transpose_Opsimps_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Xss: list(list(A))] :
      ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xb),Xs)),Xss))
     => ( transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xb),Xs)),Xss)) = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,Xb),concat(A,map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_alu(A,fun(list(A),list(A)))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs),concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_alv(A,fun(list(A),list(list(A))))),Xss))))) ) ) ).

% transpose.psimps(3)
tff(fact_6577_transpose_Opelims,axiom,
    ! [A: $tType,Xb: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,Xb) = Y )
     => ( accp(list(list(A)),transpose_rel(A),Xb)
       => ( ( ( Xb = nil(list(A)) )
           => ( ( Y = nil(list(A)) )
             => ~ accp(list(list(A)),transpose_rel(A),nil(list(A))) ) )
         => ( ! [Xss2: list(list(A))] :
                ( ( Xb = aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2) )
               => ( ( Y = transpose(A,Xss2) )
                 => ~ accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2)) ) )
           => ~ ! [X2: A,Xs2: list(A),Xss2: list(list(A))] :
                  ( ( Xb = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X2),Xs2)),Xss2) )
                 => ( ( Y = aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X2),concat(A,map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_alu(A,fun(list(A),list(A)))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),cons(list(A),Xs2),concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_alv(A,fun(list(A),list(list(A))))),Xss2))))) )
                   => ~ accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X2),Xs2)),Xss2)) ) ) ) ) ) ) ).

% transpose.pelims
tff(fact_6578_transpose_Opinduct,axiom,
    ! [A: $tType,A0: list(list(A)),P: fun(list(list(A)),$o)] :
      ( accp(list(list(A)),transpose_rel(A),A0)
     => ( ( accp(list(list(A)),transpose_rel(A),nil(list(A)))
         => aa(list(list(A)),$o,P,nil(list(A))) )
       => ( ! [Xss2: list(list(A))] :
              ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2))
             => ( aa(list(list(A)),$o,P,Xss2)
               => aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),cons(list(A),nil(A)),Xss2)) ) )
         => ( ! [X2: A,Xs2: list(A),Xss2: list(list(A))] :
                ( accp(list(list(A)),transpose_rel(A),aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X2),Xs2)),Xss2))
               => ( aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),cons(list(A),Xs2),concat(list(A),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_alv(A,fun(list(A),list(list(A))))),Xss2))))
                 => aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),cons(list(A),aa(list(A),list(A),cons(A,X2),Xs2)),Xss2)) ) )
           => aa(list(list(A)),$o,P,A0) ) ) ) ) ).

% transpose.pinduct
tff(fact_6579_quicksort_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Xs: list(A)] : aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),cons(A,Xb),Xs)) = append(A,aa(list(A),list(A),linorder_quicksort(A),filter2(A,aTP_Lamp_amf(A,fun(A,$o),Xb),Xs)),append(A,aa(list(A),list(A),cons(A,Xb),nil(A)),aa(list(A),list(A),linorder_quicksort(A),filter2(A,ord_less_eq(A,Xb),Xs)))) ) ).

% quicksort.simps(2)
tff(fact_6580_quicksort_Oelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: list(A),Y: list(A)] :
          ( ( aa(list(A),list(A),linorder_quicksort(A),Xb) = Y )
         => ( ( ( Xb = nil(A) )
             => ( Y != nil(A) ) )
           => ~ ! [X2: A,Xs2: list(A)] :
                  ( ( Xb = aa(list(A),list(A),cons(A,X2),Xs2) )
                 => ( Y != append(A,aa(list(A),list(A),linorder_quicksort(A),filter2(A,aTP_Lamp_amf(A,fun(A,$o),X2),Xs2)),append(A,aa(list(A),list(A),cons(A,X2),nil(A)),aa(list(A),list(A),linorder_quicksort(A),filter2(A,ord_less_eq(A,X2),Xs2)))) ) ) ) ) ) ).

% quicksort.elims
tff(fact_6581_sort__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_akt(A,A)) = linorder_quicksort(A) ) ) ).

% sort_quicksort
tff(fact_6582_quicksort_Opelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: list(A),Y: list(A)] :
          ( ( aa(list(A),list(A),linorder_quicksort(A),Xb) = Y )
         => ( accp(list(A),linord6200660962353139674rt_rel(A),Xb)
           => ( ( ( Xb = nil(A) )
               => ( ( Y = nil(A) )
                 => ~ accp(list(A),linord6200660962353139674rt_rel(A),nil(A)) ) )
             => ~ ! [X2: A,Xs2: list(A)] :
                    ( ( Xb = aa(list(A),list(A),cons(A,X2),Xs2) )
                   => ( ( Y = append(A,aa(list(A),list(A),linorder_quicksort(A),filter2(A,aTP_Lamp_amf(A,fun(A,$o),X2),Xs2)),append(A,aa(list(A),list(A),cons(A,X2),nil(A)),aa(list(A),list(A),linorder_quicksort(A),filter2(A,ord_less_eq(A,X2),Xs2)))) )
                     => ~ accp(list(A),linord6200660962353139674rt_rel(A),aa(list(A),list(A),cons(A,X2),Xs2)) ) ) ) ) ) ) ).

% quicksort.pelims
tff(fact_6583_transpose__aux__filter__head,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(A,map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_alu(A,fun(list(A),list(A)))),Xss)) = map(list(A),A,hd(A),filter2(list(A),aTP_Lamp_alb(list(A),$o),Xss)) ).

% transpose_aux_filter_head
tff(fact_6584_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( aa(list(nat),nat,hd(nat),upt(I,J)) = I ) ) ).

% hd_upt
tff(fact_6585_hd__replicate,axiom,
    ! [A: $tType,N: nat,Xb: A] :
      ( ( N != zero_zero(nat) )
     => ( aa(list(A),A,hd(A),replicate(A,N,Xb)) = Xb ) ) ).

% hd_replicate
tff(fact_6586_hd__def,axiom,
    ! [A: $tType,List: list(A)] : aa(list(A),A,hd(A),List) = aa(list(A),A,case_list(A,A,undefined(A),aTP_Lamp_amg(A,fun(list(A),A))),List) ).

% hd_def
tff(fact_6587_hd__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( aa(list(A),A,hd(A),Xs) = aa(nat,A,nth(A,Xs),zero_zero(nat)) ) ) ).

% hd_conv_nth
tff(fact_6588_slice__head,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A)] :
      ( aa(nat,$o,ord_less(nat,From),To)
     => ( aa(nat,$o,ord_less_eq(nat,To),aa(list(A),nat,size_size(list(A)),Xs))
       => ( aa(list(A),A,hd(A),slice(A,From,To,Xs)) = aa(nat,A,nth(A,Xs),From) ) ) ) ).

% slice_head
tff(fact_6589_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType,Fa: fun(A,nat),Xs: list(A)] :
      size_list(A,Fa,Xs) = $ite(Xs = nil(A),zero_zero(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Fa,aa(list(A),A,hd(A),Xs))),size_list(A,Fa,aa(list(A),list(A),tl(A),Xs))))) ).

% Nitpick.size_list_simp(1)
tff(fact_6590_Cauchy__iff2,axiom,
    ! [X5: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X5)
    <=> ! [J3: nat] :
        ? [M11: nat] :
        ! [M6: nat] :
          ( aa(nat,$o,ord_less_eq(nat,M11),M6)
         => ! [N8: nat] :
              ( aa(nat,$o,ord_less_eq(nat,M11),N8)
             => aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,aa(nat,real,X5,M6)),aa(nat,real,X5,N8)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_6591_list_Osize__gen_I1_J,axiom,
    ! [A: $tType,Xb: fun(A,nat)] : size_list(A,Xb,nil(A)) = zero_zero(nat) ).

% list.size_gen(1)
tff(fact_6592_size__list__estimation,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: nat,Fa: fun(A,nat)] :
      ( member(A,Xb,set2(A,Xs))
     => ( aa(nat,$o,ord_less(nat,Y),aa(A,nat,Fa,Xb))
       => aa(nat,$o,ord_less(nat,Y),size_list(A,Fa,Xs)) ) ) ).

% size_list_estimation
tff(fact_6593_size__list__estimation_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: nat,Fa: fun(A,nat)] :
      ( member(A,Xb,set2(A,Xs))
     => ( aa(nat,$o,ord_less_eq(nat,Y),aa(A,nat,Fa,Xb))
       => aa(nat,$o,ord_less_eq(nat,Y),size_list(A,Fa,Xs)) ) ) ).

% size_list_estimation'
tff(fact_6594_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),Fa: fun(A,nat),G: fun(A,nat)] :
      ( ! [X2: A] :
          ( member(A,X2,set2(A,Xs))
         => aa(nat,$o,ord_less_eq(nat,aa(A,nat,Fa,X2)),aa(A,nat,G,X2)) )
     => aa(nat,$o,ord_less_eq(nat,size_list(A,Fa,Xs)),size_list(A,G,Xs)) ) ).

% size_list_pointwise
tff(fact_6595_CauchyD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),E3: real] :
          ( topolo3814608138187158403Cauchy(A,X5)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
           => ? [M10: nat] :
              ! [M2: nat] :
                ( aa(nat,$o,ord_less_eq(nat,M10),M2)
               => ! [N4: nat] :
                    ( aa(nat,$o,ord_less_eq(nat,M10),N4)
                   => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,M2)),aa(nat,A,X5,N4)))),E3) ) ) ) ) ) ).

% CauchyD
tff(fact_6596_CauchyI,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( ! [E2: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E2)
             => ? [M13: nat] :
                ! [M3: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,M13),M3)
                 => ! [N3: nat] :
                      ( aa(nat,$o,ord_less_eq(nat,M13),N3)
                     => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,M3)),aa(nat,A,X5,N3)))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% CauchyI
tff(fact_6597_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [E5: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E5)
             => ? [M11: nat] :
                ! [M6: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,M11),M6)
                 => ! [N8: nat] :
                      ( aa(nat,$o,ord_less_eq(nat,M11),N8)
                     => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,M6)),aa(nat,A,X5,N8)))),E5) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_6598_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,nat),X21: A,X222: list(A)] : size_list(A,Xb,aa(list(A),list(A),cons(A,X21),X222)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xb,X21)),size_list(A,Xb,X222))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_6599_VEBT_Osize_I3_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(vEBT_VEBT,size_size(vEBT_VEBT),X13)),aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size(3)
tff(fact_6600_VEBT_Osize__gen_I1_J,axiom,
    ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Node(X11,X12,X13,X14)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(vEBT_VEBT,vEBT_size_VEBT,X13)),aa(vEBT_VEBT,nat,vEBT_size_VEBT,X14))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size_gen(1)
tff(fact_6601_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),Fa: fun(A,B),B6: set(B)] :
      ( ! [X2: A] :
          ( aa(A,$o,P,X2)
         => member(B,aa(A,B,Fa,X2),B6) )
     => aa(set(B),$o,ord_less_eq(set(B),aa(set(A),set(B),image(A,B,Fa),collect(A,P))),B6) ) ).

% image_Collect_subsetI
tff(fact_6602_ssubst__Pair__rhs,axiom,
    ! [B: $tType,A: $tType,R2: A,S: B,R: set(product_prod(A,B)),S4: B] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R2),S),R)
     => ( ( S4 = S )
       => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R2),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_6603_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Leaf((X21),(X222))) = zero_zero(nat) ).

% VEBT.size_gen(2)
tff(fact_6604_smod__int__range,axiom,
    ! [B3: int,A3: int] :
      ( ( B3 != zero_zero(int) )
     => member(int,signed6721504322012087516modulo(int,A3,B3),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),abs_abs(int,B3))),one_one(int)),aa(int,int,minus_minus(int,abs_abs(int,B3)),one_one(int)))) ) ).

% smod_int_range
tff(fact_6605_Multiset_Ois__empty__def,axiom,
    ! [A: $tType,A4: multiset(A)] :
      ( is_empty(A,A4)
    <=> ( A4 = zero_zero(multiset(A)) ) ) ).

% Multiset.is_empty_def
tff(fact_6606_smod__int__0__mod,axiom,
    ! [Xb: int] : signed6721504322012087516modulo(int,zero_zero(int),Xb) = zero_zero(int) ).

% smod_int_0_mod
tff(fact_6607_smod__int__mod__0,axiom,
    ! [Xb: int] : signed6721504322012087516modulo(int,Xb,zero_zero(int)) = Xb ).

% smod_int_mod_0
tff(fact_6608_smod__int__numeral__numeral,axiom,
    ! [M: num,N: num] : signed6721504322012087516modulo(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),N)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),N)) ).

% smod_int_numeral_numeral
tff(fact_6609_smod__int__compares_I8_J,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,B3),zero_zero(int))
       => aa(int,$o,ord_less_eq(int,B3),signed6721504322012087516modulo(int,A3,B3)) ) ) ).

% smod_int_compares(8)
tff(fact_6610_smod__int__compares_I7_J,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,B3),zero_zero(int))
       => aa(int,$o,ord_less_eq(int,signed6721504322012087516modulo(int,A3,B3)),zero_zero(int)) ) ) ).

% smod_int_compares(7)
tff(fact_6611_smod__int__compares_I6_J,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less(int,B3),zero_zero(int))
       => aa(int,$o,ord_less_eq(int,zero_zero(int)),signed6721504322012087516modulo(int,A3,B3)) ) ) ).

% smod_int_compares(6)
tff(fact_6612_smod__int__compares_I4_J,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
       => aa(int,$o,ord_less_eq(int,signed6721504322012087516modulo(int,A3,B3)),zero_zero(int)) ) ) ).

% smod_int_compares(4)
tff(fact_6613_smod__int__compares_I2_J,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
       => aa(int,$o,ord_less_eq(int,zero_zero(int)),signed6721504322012087516modulo(int,A3,B3)) ) ) ).

% smod_int_compares(2)
tff(fact_6614_smod__int__compares_I1_J,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
       => aa(int,$o,ord_less(int,signed6721504322012087516modulo(int,A3,B3)),B3) ) ) ).

% smod_int_compares(1)
tff(fact_6615_smod__mod__positive,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),B3)
       => ( signed6721504322012087516modulo(int,A3,B3) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),B3) ) ) ) ).

% smod_mod_positive
tff(fact_6616_smod__int__compares_I5_J,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,zero_zero(int)),A3)
     => ( aa(int,$o,ord_less(int,B3),zero_zero(int))
       => aa(int,$o,ord_less(int,signed6721504322012087516modulo(int,A3,B3)),aa(int,int,uminus_uminus(int),B3)) ) ) ).

% smod_int_compares(5)
tff(fact_6617_smod__int__compares_I3_J,axiom,
    ! [A3: int,B3: int] :
      ( aa(int,$o,ord_less_eq(int,A3),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B3)
       => aa(int,$o,ord_less(int,aa(int,int,uminus_uminus(int),B3)),signed6721504322012087516modulo(int,A3,B3)) ) ) ).

% smod_int_compares(3)
tff(fact_6618_mod__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))))) ) ).

% mod_word_minus_1_minus_numeral
tff(fact_6619_valid__eq2,axiom,
    ! [Tb: vEBT_VEBT,D: nat] :
      ( vEBT_VEBT_valid(Tb,D)
     => vEBT_invar_vebt(Tb,D) ) ).

% valid_eq2
tff(fact_6620_valid__eq,axiom,
    ! [Tb: vEBT_VEBT,D: nat] :
      ( vEBT_VEBT_valid(Tb,D)
    <=> vEBT_invar_vebt(Tb,D) ) ).

% valid_eq
tff(fact_6621_valid__eq1,axiom,
    ! [Tb: vEBT_VEBT,D: nat] :
      ( vEBT_invar_vebt(Tb,D)
     => vEBT_VEBT_valid(Tb,D) ) ).

% valid_eq1
tff(fact_6622_drop__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N)
         => ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),N),W) = zero_zero(word(A)) ) ) ) ).

% drop_bit_word_beyond
tff(fact_6623_push__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N)
         => ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),N),W) = zero_zero(word(A)) ) ) ) ).

% push_bit_word_beyond
tff(fact_6624_uint__bintrunc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),Bin)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Bin)) ) ).

% uint_bintrunc
tff(fact_6625_signed__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ring_1(A)
        & type_len(B) )
     => ( ring_1_signed(B,A,one_one(word(B))) = $ite(type_len0_len_of(B,type2(B)) = one_one(nat),aa(A,A,uminus_uminus(A),one_one(A)),one_one(A)) ) ) ).

% signed_1
tff(fact_6626_word__exp__length__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A))) = zero_zero(word(A)) ) ) ).

% word_exp_length_eq_0
tff(fact_6627_less__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A3))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B3))) ) ) ).

% less_eq_word_numeral_numeral
tff(fact_6628_less__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A3))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B3))) ) ) ).

% less_word_numeral_numeral
tff(fact_6629_unat__bintrunc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(num,word(A),numeral_numeral(word(A)),Bin)) = nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Bin))) ) ).

% unat_bintrunc
tff(fact_6630_bit__numeral__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num,N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(num,word(A),numeral_numeral(word(A)),W)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W)),N) ) ) ) ).

% bit_numeral_word_iff
tff(fact_6631_ucast__bintr,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: num] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(num,word(B),numeral_numeral(word(B)),W)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B))),aa(num,int,numeral_numeral(int),W))) ) ).

% ucast_bintr
tff(fact_6632_unsigned__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [N: num] : aa(word(B),A,semiring_1_unsigned(B,A),aa(num,word(B),numeral_numeral(word(B)),N)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(B,type2(B))),aa(num,nat,numeral_numeral(nat),N))) ) ).

% unsigned_numeral
tff(fact_6633_unat__lt2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_lt2p
tff(fact_6634_uint__bounded,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_bounded
tff(fact_6635_uint__lt2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_lt2p
tff(fact_6636_exp__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) = zero_zero(word(A)) )
        <=> aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N) ) ) ).

% exp_eq_zero_iff
tff(fact_6637_of__nat__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) = zero_zero(word(A)) ) ) ).

% of_nat_2p
tff(fact_6638_signed__take__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: num] : aa(word(A),word(A),bit_ri4674362597316999326ke_bit(word(A),aa(nat,nat,suc,N)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% signed_take_bit_word_Suc_numeral
tff(fact_6639_signed__take__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,K: num] : aa(word(A),word(A),bit_ri4674362597316999326ke_bit(word(A),aa(num,nat,numeral_numeral(nat),N)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),N)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% signed_take_bit_word_numeral
tff(fact_6640_sint__sbintrunc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Bin)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),Bin)) ) ).

% sint_sbintrunc
tff(fact_6641_uint__bintrunc__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin))) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin))) ) ).

% uint_bintrunc_neg
tff(fact_6642_div__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A3))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B3)))) ) ).

% div_word_numeral_numeral
tff(fact_6643_mod__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A3))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B3)))) ) ).

% mod_word_numeral_numeral
tff(fact_6644_signed__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [N: num] : ring_1_signed(B,A,aa(num,word(B),numeral_numeral(word(B)),N)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),one_one(nat))),aa(num,int,numeral_numeral(int),N))) ) ).

% signed_numeral
tff(fact_6645_less__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)))
        <=> aa(int,$o,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ) ).

% less_eq_word_minus_numeral_minus_numeral
tff(fact_6646_less__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)))
        <=> aa(int,$o,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A3))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ) ).

% less_eq_word_numeral_minus_numeral
tff(fact_6647_less__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B3))) ) ) ).

% less_eq_word_minus_numeral_numeral
tff(fact_6648_less__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B3))) ) ) ).

% less_word_minus_numeral_numeral
tff(fact_6649_less__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)))
        <=> aa(int,$o,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A3))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ) ).

% less_word_numeral_minus_numeral
tff(fact_6650_less__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)))
        <=> aa(int,$o,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ) ).

% less_word_minus_numeral_minus_numeral
tff(fact_6651_unat__bintrunc__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin))) = nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin)))) ) ).

% unat_bintrunc_neg
tff(fact_6652_bit__neg__numeral__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num,N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W))),N)
        <=> ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),N) ) ) ) ).

% bit_neg_numeral_word_iff
tff(fact_6653_drop__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,suc,N)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,N)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% drop_bit_word_Suc_numeral
tff(fact_6654_drop__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(num,nat,numeral_numeral(nat),N)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(num,nat,numeral_numeral(nat),N)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% drop_bit_word_numeral
tff(fact_6655_unat__power__lower,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ) ) ) ).

% unat_power_lower
tff(fact_6656_unsigned__neg__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [N: num] : aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),uminus_uminus(word(B)),aa(num,word(B),numeral_numeral(word(B)),N))) = aa(nat,A,semiring_1_of_nat(A),nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(B,type2(B))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))))) ) ).

% unsigned_neg_numeral
tff(fact_6657_signed__take__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: num] : aa(word(A),word(A),bit_ri4674362597316999326ke_bit(word(A),aa(nat,nat,suc,N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,N)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_take_bit_word_Suc_minus_numeral
tff(fact_6658_signed__take__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,K: num] : aa(word(A),word(A),bit_ri4674362597316999326ke_bit(word(A),aa(num,nat,numeral_numeral(nat),N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(num,nat,numeral_numeral(nat),N)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_take_bit_word_minus_numeral
tff(fact_6659_sint__sbintrunc__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: num] : ring_1_signed(A,int,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),Bin))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Bin))) ) ).

% sint_sbintrunc_neg
tff(fact_6660_scast__sbintr,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: num] : ring_1_signed(B,word(A),aa(num,word(B),numeral_numeral(word(B)),W)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),aa(num,int,numeral_numeral(int),W))) ) ).

% scast_sbintr
tff(fact_6661_drop__bit__numeral__bit0__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,suc,zero_zero(nat))),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,zero_zero(nat))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),K)))) ) ).

% drop_bit_numeral_bit0_1
tff(fact_6662_div__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))))) ) ).

% div_word_minus_numeral_minus_numeral
tff(fact_6663_div__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A3))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))))) ) ).

% div_word_numeral_minus_numeral
tff(fact_6664_div__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B3)))) ) ).

% div_word_minus_numeral_numeral
tff(fact_6665_mod__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))))) ) ).

% mod_word_minus_numeral_minus_numeral
tff(fact_6666_mod__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A3))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))))) ) ).

% mod_word_numeral_minus_numeral
tff(fact_6667_mod__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B3)))) ) ).

% mod_word_minus_numeral_numeral
tff(fact_6668_word__less__sub__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),one_one(word(A))))
          <=> aa(word(A),$o,ord_less(word(A),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ) ) ).

% word_less_sub_le
tff(fact_6669_signed__neg__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [N: num] : ring_1_signed(B,A,aa(word(B),word(B),uminus_uminus(word(B)),aa(num,word(B),numeral_numeral(word(B)),N))) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N)))) ) ).

% signed_neg_numeral
tff(fact_6670_drop__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,suc,N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,N)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% drop_bit_word_Suc_minus_numeral
tff(fact_6671_drop__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(num,nat,numeral_numeral(nat),N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(num,nat,numeral_numeral(nat),N)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% drop_bit_word_minus_numeral
tff(fact_6672_less__word__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))
        <=> aa(int,$o,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A3))),aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))) ) ) ).

% less_word_numeral_minus_1
tff(fact_6673_less__word__minus__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3))),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))))
        <=> aa(int,$o,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))) ) ) ).

% less_word_minus_numeral_minus_1
tff(fact_6674_div__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B3)))) ) ).

% div_word_minus_1_numeral
tff(fact_6675_mod__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B3)))) ) ).

% mod_word_minus_1_numeral
tff(fact_6676_div__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(int))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))))) ) ).

% div_word_minus_1_minus_numeral
tff(fact_6677_num__of__sbintr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A3)) = aa(num,int,numeral_numeral(int),B3) )
         => ( aa(num,word(A),numeral_numeral(word(A)),A3) = aa(num,word(A),numeral_numeral(word(A)),B3) ) ) ) ).

% num_of_sbintr'
tff(fact_6678_signed__take__bit__decr__length__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( bit_ri3973907225187159222ations(A)
        & type_len(B) )
     => ! [K: A,L: A] :
          ( ( aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),K) = aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),L) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,type_len0_len_of(B,type2(B))),K) = aa(A,A,bit_se2584673776208193580ke_bit(A,type_len0_len_of(B,type2(B))),L) ) ) ) ).

% signed_take_bit_decr_length_iff
tff(fact_6679_bit__set__bit__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: nat,B3: $o,N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),W),M),(B3))),N)
        <=> $ite(
              M = N,
              ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
              & (B3) ),
              aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),N) ) ) ) ).

% bit_set_bit_word_iff
tff(fact_6680_nth__slice,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [N: nat,W: word(B),M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),slice2(B,A,N,W)),M)
        <=> ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N))
            & aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_slice
tff(fact_6681_neg__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xb)),N)
        <=> ( ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),N)
            & aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A))) ) ) ) ).

% neg_test_bit
tff(fact_6682_ucast__drop__bit__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [N: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),N),W)) = aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se4197421643247451524op_bit(word(B)),N),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),W)) ) ) ) ).

% ucast_drop_bit_eq
tff(fact_6683_num__abs__bintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: num] : aa(num,word(A),numeral_numeral(word(A)),Xb) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),Xb))) ) ).

% num_abs_bintr
tff(fact_6684_num__of__bintr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A3)) = aa(num,int,numeral_numeral(int),B3) )
         => ( aa(num,word(A),numeral_numeral(word(A)),A3) = aa(num,word(A),numeral_numeral(word(A)),B3) ) ) ) ).

% num_of_bintr'
tff(fact_6685_uint32_Osize__eq__length,axiom,
    aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2)))))) = type_len0_len_of(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))),type2(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))) ).

% uint32.size_eq_length
tff(fact_6686_scast__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( ring_1_signed(B,word(A),one_one(word(B))) = $ite(type_len0_len_of(B,type2(B)) = one_one(nat),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),one_one(word(A))) ) ) ).

% scast_1
tff(fact_6687_word__of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,N: nat] :
          ( aa(word(A),$o,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),M)),aa(nat,word(A),semiring_1_of_nat(word(A)),N))
        <=> aa(nat,$o,ord_less(nat,aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),M)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),N)) ) ) ).

% word_of_nat_less_iff
tff(fact_6688_word__of__nat__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,N: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),M)),aa(nat,word(A),semiring_1_of_nat(word(A)),N))
        <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),M)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,type_len0_len_of(A,type2(A))),N)) ) ) ).

% word_of_nat_less_eq_iff
tff(fact_6689_uint__word__arith__bintrs_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),one_one(word(A))) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),one_one(int)) ) ) ).

% uint_word_arith_bintrs(8)
tff(fact_6690_word__of__int__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,L: int] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(int,word(A),ring_1_of_int(word(A)),K)),aa(int,word(A),ring_1_of_int(word(A)),L))
        <=> aa(int,$o,ord_less_eq(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),L)) ) ) ).

% word_of_int_less_eq_iff
tff(fact_6691_word__of__int__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,L: int] :
          ( aa(word(A),$o,ord_less(word(A),aa(int,word(A),ring_1_of_int(word(A)),K)),aa(int,word(A),ring_1_of_int(word(A)),L))
        <=> aa(int,$o,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),L)) ) ) ).

% word_of_int_less_iff
tff(fact_6692_one__word_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),one_one(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),one_one(int)) ) ) ).

% one_word.rsp
tff(fact_6693_degenerate__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
         => ( ( Xb = zero_zero(word(A)) )
            | ( Xb = one_one(word(A)) ) ) ) ) ).

% degenerate_word
tff(fact_6694_size__0__same,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),V: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) = zero_zero(nat) )
         => ( W = V ) ) ) ).

% size_0_same
tff(fact_6695_zero__word_Orsp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),zero_zero(int)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),zero_zero(int)) ) ) ).

% zero_word.rsp
tff(fact_6696_bintr__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N)
         => ( aa(int,int,bit_se2584673776208193580ke_bit(int,N),aa(word(A),int,semiring_1_unsigned(A,int),W)) = aa(word(A),int,semiring_1_unsigned(A,int),W) ) ) ) ).

% bintr_uint
tff(fact_6697_less__ucast__ucast__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(B),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,ord_less(word(B),Xb),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
           => aa(word(A),$o,ord_less(word(A),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xb)),Y) ) ) ) ).

% less_ucast_ucast_less
tff(fact_6698_ucast__less__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,ord_less(word(A),Xb),Y) ) ) ) ).

% ucast_less_ucast
tff(fact_6699_ucast__up__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),$o,ord_less(word(A),Xb),Y)
           => aa(word(B),$o,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_up_mono
tff(fact_6700_up__ucast__inj,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xb: word(B),Y: word(B)] :
          ( ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xb) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y) )
         => ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( Xb = Y ) ) ) ) ).

% up_ucast_inj
tff(fact_6701_ucast__le__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,ord_less_eq(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,ord_less_eq(word(A),Xb),Y) ) ) ) ).

% ucast_le_ucast
tff(fact_6702_ucast__ucast__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [Xb: word(B),Y: word(C)] :
          ( ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xb) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(C),word(B),semiring_1_unsigned(C,word(B)),Y)) )
         => ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(C,type2(C)))
           => ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(C,type2(C))),type_len0_len_of(A,type2(A)))
             => ( Xb = aa(word(C),word(B),semiring_1_unsigned(C,word(B)),Y) ) ) ) ) ) ).

% ucast_ucast_eq
tff(fact_6703_up__ucast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb) = aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y) )
          <=> ( Xb = Y ) ) ) ) ).

% up_ucast_inj_eq
tff(fact_6704_ucast__up__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),Y)
           => aa(word(B),$o,ord_less_eq(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_up_mono_le
tff(fact_6705_eq__ucast__ucast__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(B),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( ( Xb = aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y) )
           => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xb) = Y ) ) ) ) ).

% eq_ucast_ucast_eq
tff(fact_6706_unat__ucast__up__simp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),nat,semiring_1_unsigned(B,nat),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb)) = aa(word(A),nat,semiring_1_unsigned(A,nat),Xb) ) ) ) ).

% unat_ucast_up_simp
tff(fact_6707_ucast__ucast__id,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb)) = Xb ) ) ) ).

% ucast_ucast_id
tff(fact_6708_ucast__less__ucast__weak,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y))
          <=> aa(word(A),$o,ord_less(word(A),Xb),Y) ) ) ) ).

% ucast_less_ucast_weak
tff(fact_6709_up__scast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( ( ring_1_signed(A,word(B),Xb) = ring_1_signed(A,word(B),Y) )
          <=> ( Xb = Y ) ) ) ) ).

% up_scast_inj_eq
tff(fact_6710_take__bit__word__beyond__length__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N)
         => ( aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),N),W) = W ) ) ) ).

% take_bit_word_beyond_length_eq
tff(fact_6711_wi__bintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: int] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N)
         => ( aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,N),W)) = aa(int,word(A),ring_1_of_int(word(A)),W) ) ) ) ).

% wi_bintr
tff(fact_6712_uint__word__arith__bintrs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),zero_zero(word(A))) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),zero_zero(int)) ) ) ).

% uint_word_arith_bintrs(7)
tff(fact_6713_signed__take__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [N: nat,W: word(B)] :
          ring_1_signed(B,A,aa(word(B),word(B),bit_se2584673776208193580ke_bit(word(B),N),W)) = $ite(aa(nat,$o,ord_less(nat,N),type_len0_len_of(B,type2(B))),aa(A,A,bit_se2584673776208193580ke_bit(A,N),ring_1_signed(B,A,W)),ring_1_signed(B,A,W)) ) ).

% signed_take_bit_eq
tff(fact_6714_mask__over__length,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N)
         => ( bit_se2239418461657761734s_mask(word(A),N) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% mask_over_length
tff(fact_6715_ucast__ucast__add,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xb: word(A),Y: word(B)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb)),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y)) ) ) ) ).

% ucast_ucast_add
tff(fact_6716_ucast__sub__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Y: word(A),Xb: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Y),Xb)
         => ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
           => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),aa(word(A),word(A),minus_minus(word(A),Xb),Y)) = aa(word(B),word(B),minus_minus(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ) ).

% ucast_sub_ucast
tff(fact_6717_test__bit__1_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),one_one(word(A))),N)
        <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),type_len0_len_of(A,type2(A)))
            & ( N = zero_zero(nat) ) ) ) ) ).

% test_bit_1'
tff(fact_6718_max__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),N)
        <=> aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A))) ) ) ).

% max_test_bit
tff(fact_6719_bit__uint__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),N) ) ) ) ).

% bit_uint_iff
tff(fact_6720_bit__ucast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [A3: word(B),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),A3)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),A3),N) ) ) ) ).

% bit_ucast_iff
tff(fact_6721_nth__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [W: word(B),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W)),N)
        <=> ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),N)
            & aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_ucast
tff(fact_6722_bin__nth__uint__imp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),N)
         => aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A))) ) ) ).

% bin_nth_uint_imp
tff(fact_6723_test__bit__bin,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),N)
        <=> ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),N) ) ) ) ).

% test_bit_bin
tff(fact_6724_bit__word__ucast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(B,type2(B)))
            & aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),N) ) ) ) ).

% bit_word_ucast_iff
tff(fact_6725_test__bit__conj__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),M: nat] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),M)
            & aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) )
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),M) ) ) ).

% test_bit_conj_lt
tff(fact_6726_bit__imp__le__length,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),N)
         => aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A))) ) ) ).

% bit_imp_le_length
tff(fact_6727_word__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( ( Xb = Y )
        <=> ! [N8: nat] :
              ( aa(nat,$o,ord_less(nat,N8),type_len0_len_of(A,type2(A)))
             => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),N8)
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),N8) ) ) ) ) ).

% word_eq_iff
tff(fact_6728_bit__word__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( ! [N3: nat] :
              ( aa(nat,$o,ord_less(nat,N3),type_len0_len_of(A,type2(A)))
             => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A3),N3)
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),B3),N3) ) )
         => ( A3 = B3 ) ) ) ).

% bit_word_eqI
tff(fact_6729_test__bit__wi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int,N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),ring_1_of_int(word(A)),Xb)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),Xb),N) ) ) ) ).

% test_bit_wi
tff(fact_6730_bit__word__of__int__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),ring_1_of_int(word(A)),K)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) ) ) ) ).

% bit_word_of_int_iff
tff(fact_6731_bit__set__bit__aux,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Fa: fun(nat,$o),N: nat,W: word(A),M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),code_T2661198915054445665ts_aux(A,Fa,N,W)),M)
        <=> ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
            & $ite(aa(nat,$o,ord_less(nat,M),N),aa(nat,$o,Fa,M),aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,minus_minus(nat,M),N))) ) ) ) ).

% bit_set_bit_aux
tff(fact_6732_ucast__mask__drop,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [N: nat,Xb: word(B)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N)
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),bit_se5824344872417868541ns_and(word(B)),Xb),bit_se2239418461657761734s_mask(word(B),N))) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xb) ) ) ) ).

% ucast_mask_drop
tff(fact_6733_bit__word__cat__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [V: word(B),W: word(C),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_cat(B,C,A,V,W)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & $ite(aa(nat,$o,ord_less(nat,N),type_len0_len_of(C,type2(C))),aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),W),N),aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),V),aa(nat,nat,minus_minus(nat,N),type_len0_len_of(C,type2(C))))) ) ) ) ).

% bit_word_cat_iff
tff(fact_6734_word__cat__inj,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [A3: word(A),B3: word(B),C2: word(A),D: word(B)] :
          ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),type_len0_len_of(C,type2(C)))
         => ( ( word_cat(A,B,C,A3,B3) = word_cat(A,B,C,C2,D) )
          <=> ( ( A3 = C2 )
              & ( B3 = D ) ) ) ) ) ).

% word_cat_inj
tff(fact_6735_two__power__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat] :
          ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
           => ( ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M) )
            <=> ( N = M ) ) ) ) ) ).

% two_power_eq
tff(fact_6736_up__scast__inj,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xb: word(B),Y: word(B)] :
          ( ( ring_1_signed(B,word(A),Xb) = ring_1_signed(B,word(A),Y) )
         => ( aa(nat,$o,ord_less_eq(nat,aa(word(B),nat,size_size(word(B)),Xb)),type_len0_len_of(A,type2(A)))
           => ( Xb = Y ) ) ) ) ).

% up_scast_inj
tff(fact_6737_word__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),nat,size_size(word(A)),W) = type_len0_len_of(A,type2(A)) ) ).

% word_size
tff(fact_6738_size__word_Orep__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(word(A),nat,size_size(word(A)),Xb) = type_len0_len_of(A,type2(A)) ) ).

% size_word.rep_eq
tff(fact_6739_word__sint__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,one_one(word(A))) = $ite(type_len0_len_of(A,type2(A)) = one_one(nat),aa(int,int,uminus_uminus(int),one_one(int)),one_one(int)) ) ) ).

% word_sint_1
tff(fact_6740_bin__nth__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N)
         => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),ring_1_signed(A,int,W)),N)
          <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),ring_1_signed(A,int,W)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) ) ) ) ).

% bin_nth_sint
tff(fact_6741_sint__sbintrunc_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bin: int] : ring_1_signed(A,int,aa(int,word(A),ring_1_of_int(word(A)),Bin)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),Bin) ) ).

% sint_sbintrunc'
tff(fact_6742_neg__mask__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N))),M)
        <=> ( aa(nat,$o,ord_less_eq(nat,N),M)
            & aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) ) ) ) ).

% neg_mask_test_bit
tff(fact_6743_word__of__int__bin__cat__eq__iff,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(C)
        & type_len(B) )
     => ! [B3: word(B),A3: word(A),D: word(B),C2: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),type_len0_len_of(C,type2(C)))
         => ( ( aa(int,word(C),ring_1_of_int(word(C)),aa(int,int,bit_concat_bit(type_len0_len_of(B,type2(B)),aa(word(B),int,semiring_1_unsigned(B,int),B3)),aa(word(A),int,semiring_1_unsigned(A,int),A3))) = aa(int,word(C),ring_1_of_int(word(C)),aa(int,int,bit_concat_bit(type_len0_len_of(B,type2(B)),aa(word(B),int,semiring_1_unsigned(B,int),D)),aa(word(A),int,semiring_1_unsigned(A,int),C2))) )
          <=> ( ( B3 = D )
              & ( A3 = C2 ) ) ) ) ) ).

% word_of_int_bin_cat_eq_iff
tff(fact_6744_mask__exceed,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),N))) = zero_zero(word(A)) ) ) ) ).

% mask_exceed
tff(fact_6745_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv2: $o,D: nat] :
      ( vEBT_VEBT_valid(vEBT_Leaf((Uu),(Uv2)),D)
    <=> ( D = one_one(nat) ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_6746_unsigned__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & unique1627219031080169319umeral(A) )
     => ! [W: word(B)] : aa(A,$o,ord_less(A,aa(word(B),A,semiring_1_unsigned(B,A),W)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),type_len0_len_of(B,type2(B)))) ) ).

% unsigned_less
tff(fact_6747_not__degenerate__imp__2__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,$o,ord_less(nat,one_one(nat)),type_len0_len_of(A,type2(A)))
       => ( aa(num,word(A),numeral_numeral(word(A)),bit0(one2)) != zero_zero(word(A)) ) ) ) ).

% not_degenerate_imp_2_neq_0
tff(fact_6748_word__of__nat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: nat,Y: nat] :
          ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Xb) = aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
             => ( Xb = Y ) ) ) ) ) ).

% word_of_nat_inj
tff(fact_6749_of__nat__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: nat,Y: nat] :
          ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Xb) = aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
            <=> ( Xb = Y ) ) ) ) ) ).

% of_nat_inj
tff(fact_6750_word__nat__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ~ ! [N3: nat] :
              ( ( Xb = aa(nat,word(A),semiring_1_of_nat(word(A)),N3) )
             => ~ aa(nat,$o,ord_less(nat,N3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% word_nat_cases
tff(fact_6751_word__nchotomy,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W3: word(A)] :
        ? [N3: nat] :
          ( ( W3 = aa(nat,word(A),semiring_1_of_nat(word(A)),N3) )
          & aa(nat,$o,ord_less(nat,N3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% word_nchotomy
tff(fact_6752_More__Word_Opower__not__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
         => ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) != zero_zero(word(A)) ) ) ) ).

% More_Word.power_not_zero
tff(fact_6753_word__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: nat,Y: nat] :
          ( aa(word(A),$o,ord_less(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Xb)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Y))
         => ( aa(nat,$o,ord_less(nat,Xb),type_len0_len_of(A,type2(A)))
           => ( aa(nat,$o,ord_less(nat,Y),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,ord_less(nat,Xb),Y) ) ) ) ) ).

% word_power_increasing
tff(fact_6754_power__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N)
         => ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N) = zero_zero(word(A)) ) ) ) ).

% power_overflow
tff(fact_6755_nth__w2p__same,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),N)
        <=> aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A))) ) ) ).

% nth_w2p_same
tff(fact_6756_nth__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),M)
        <=> ( ( M = N )
            & aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_w2p
tff(fact_6757_uint__idem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) = aa(word(A),int,semiring_1_unsigned(A,int),W) ) ).

% uint_idem
tff(fact_6758_word__of__int__2p__len,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) = zero_zero(word(A)) ) ) ).

% word_of_int_2p_len
tff(fact_6759_of__nat__neq__iff__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: nat,Y: nat] :
          ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) != aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) )
         => ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Xb) != aa(nat,word(A),semiring_1_of_nat(word(A)),Y) )
          <=> ( Xb != Y ) ) ) ) ).

% of_nat_neq_iff_word
tff(fact_6760_sint__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : ring_1_signed(A,int,W) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% sint_uint
tff(fact_6761_num__abs__sbintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: num] : aa(num,word(A),numeral_numeral(word(A)),Xb) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),Xb))) ) ).

% num_abs_sbintr
tff(fact_6762_ucast__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),Y)
         => ( aa(word(A),$o,ord_less(word(A),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(B,type2(B))))
           => aa(word(B),$o,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_mono
tff(fact_6763_ucast__ucast__len,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(B,type2(B))))
         => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb)) = Xb ) ) ) ).

% ucast_ucast_len
tff(fact_6764_horner__sum__uint__exp__Cons__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Ws: list(word(A))] : groups4207007520872428315er_sum(word(A),int,semiring_1_unsigned(A,int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))),aa(list(word(A)),list(word(A)),cons(word(A),W),Ws)) = aa(int,int,bit_concat_bit(type_len0_len_of(A,type2(A)),aa(word(A),int,semiring_1_unsigned(A,int),W)),groups4207007520872428315er_sum(word(A),int,semiring_1_unsigned(A,int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))),Ws)) ) ).

% horner_sum_uint_exp_Cons_eq
tff(fact_6765_sint__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,zero_zero(word(A))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),zero_zero(int)) ) ) ).

% sint_word_ariths(7)
tff(fact_6766_sint__word__ariths_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,one_one(word(A))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),one_one(int)) ) ) ).

% sint_word_ariths(8)
tff(fact_6767_sint__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : ring_1_signed(A,int,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A3),B3)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,aa(int,fun(int,int),plus_plus(int),ring_1_signed(A,int,A3)),ring_1_signed(A,int,B3))) ) ).

% sint_word_ariths(1)
tff(fact_6768_nth__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),ring_1_signed(A,int,W)),N)
        <=> $ite(aa(nat,$o,ord_less(nat,N),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),N),aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ) ).

% nth_sint
tff(fact_6769_bit__sint__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),ring_1_signed(A,int,W)),N)
        <=> ( ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N)
              & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) )
            | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),N) ) ) ) ).

% bit_sint_iff
tff(fact_6770_sint__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : ring_1_signed(A,int,aa(word(A),word(A),uminus_uminus(word(A)),A3)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),ring_1_signed(A,int,A3))) ) ).

% sint_word_ariths(4)
tff(fact_6771_drop__bit__eq__zero__iff__not__bit__last,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),W) = zero_zero(word(A)) )
        <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ).

% drop_bit_eq_zero_iff_not_bit_last
tff(fact_6772_sint__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : ring_1_signed(A,int,aa(word(A),word(A),minus_minus(word(A),A3),B3)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,minus_minus(int,ring_1_signed(A,int,A3)),ring_1_signed(A,int,B3))) ) ).

% sint_word_ariths(2)
tff(fact_6773_signed__scast__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & bit_ri3973907225187159222ations(A)
        & type_len(B) )
     => ! [W: word(C)] : ring_1_signed(B,A,ring_1_signed(C,word(B),W)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),ring_1_signed(C,A,W)) ) ).

% signed_scast_eq
tff(fact_6774_sint__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : ring_1_signed(A,int,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A3),B3)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,aa(int,fun(int,int),times_times(int),ring_1_signed(A,int,A3)),ring_1_signed(A,int,B3))) ) ).

% sint_word_ariths(3)
tff(fact_6775_less__Suc__unat__less__bound,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,N),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)))
         => aa(nat,$o,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% less_Suc_unat_less_bound
tff(fact_6776_uint__2__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))
       => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) = aa(num,int,numeral_numeral(int),bit0(one2)) ) ) ) ).

% uint_2_id
tff(fact_6777_lt2p__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N)
         => aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N)) ) ) ).

% lt2p_lem
tff(fact_6778_two__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N),M)
         => ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,ord_less_eq(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ).

% two_power_increasing
tff(fact_6779_power__le__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
         => ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
           => ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,ord_less_eq(nat,N),M) ) ) ) ) ).

% power_le_mono
tff(fact_6780_unat__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(num,nat,numeral_numeral(nat),B3)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_numeral
tff(fact_6781_of__nat__mono__maybe_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: nat,Y: nat] :
          ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,ord_less(nat,Y),Xb)
            <=> aa(word(A),$o,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),Xb)) ) ) ) ) ).

% of_nat_mono_maybe'
tff(fact_6782_of__nat__mono__maybe,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: nat,Y: nat] :
          ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,Y),Xb)
           => aa(word(A),$o,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),Xb)) ) ) ) ).

% of_nat_mono_maybe
tff(fact_6783_unat__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),Xb: word(A)] :
          ( aa(nat,$o,P,aa(word(A),nat,semiring_1_unsigned(A,nat),Xb))
        <=> ~ ? [N8: nat] :
                ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N8) = Xb )
                & aa(nat,$o,ord_less(nat,N8),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
                & ~ aa(nat,$o,P,N8) ) ) ) ).

% unat_split_asm
tff(fact_6784_of__nat__inverse,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [R2: nat,A3: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),R2) = A3 )
         => ( aa(nat,$o,ord_less(nat,R2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),nat,semiring_1_unsigned(A,nat),A3) = R2 ) ) ) ) ).

% of_nat_inverse
tff(fact_6785_unat__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),Xb: word(A)] :
          ( aa(nat,$o,P,aa(word(A),nat,semiring_1_unsigned(A,nat),Xb))
        <=> ! [N8: nat] :
              ( ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N8) = Xb )
                & aa(nat,$o,ord_less(nat,N8),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) )
             => aa(nat,$o,P,N8) ) ) ) ).

% unat_split
tff(fact_6786_unat__of__nat__len,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: nat] :
          ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),Xb)) = Xb ) ) ) ).

% unat_of_nat_len
tff(fact_6787_unat__eq__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),Xb) = N )
          <=> ( Xb = aa(nat,word(A),semiring_1_of_nat(word(A)),N) ) ) ) ) ).

% unat_eq_of_nat
tff(fact_6788_UNIV__word__eq__word__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( top_top(set(word(A))) = aa(set(nat),set(word(A)),image(nat,word(A),semiring_1_of_nat(word(A))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% UNIV_word_eq_word_of_nat
tff(fact_6789_x__less__2__0__1_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( type_len0_len_of(A,type2(A)) != one_one(nat) )
         => ( aa(word(A),$o,ord_less(word(A),Xb),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)))
           => ( ( Xb = zero_zero(word(A)) )
              | ( Xb = one_one(word(A)) ) ) ) ) ) ).

% x_less_2_0_1'
tff(fact_6790_test__bit__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N))),M)
        <=> ( ( M = N )
            & aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) ) ) ) ).

% test_bit_2p
tff(fact_6791_Word_Oof__nat__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),M) = zero_zero(word(A)) )
        <=> ? [Q4: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% Word.of_nat_0
tff(fact_6792_word__1__le__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,ord_less_eq(word(A),one_one(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ) ).

% word_1_le_power
tff(fact_6793_uint__sub__lt2p,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(B)] : aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(B),int,semiring_1_unsigned(B,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_sub_lt2p
tff(fact_6794_ucast__of__nat__small,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xb: nat] :
          ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(word(A),word(B),semiring_1_unsigned(A,word(B)),aa(nat,word(A),semiring_1_of_nat(word(A)),Xb)) = aa(nat,word(B),semiring_1_of_nat(word(B)),Xb) ) ) ) ).

% ucast_of_nat_small
tff(fact_6795_uint__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),B3)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_numeral
tff(fact_6796_p2__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))
        <=> aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A))) ) ) ).

% p2_gt_0
tff(fact_6797_word__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N) = zero_zero(word(A)) )
        <=> aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))),N) ) ) ).

% word_of_nat_eq_0_iff
tff(fact_6798_word__of__int__minus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int] : aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),I)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),I)) ) ).

% word_of_int_minus
tff(fact_6799_bit__last__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
        <=> aa(int,$o,ord_less(int,ring_1_signed(A,int,W)),zero_zero(int)) ) ) ).

% bit_last_iff
tff(fact_6800_unat__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(B)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xb)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(word(B),nat,semiring_1_unsigned(B,nat),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_ucast
tff(fact_6801_mask__lt__2pn,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,ord_less(word(A),bit_se2239418461657761734s_mask(word(A),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ) ).

% mask_lt_2pn
tff(fact_6802_uint__word__of__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int] : aa(word(A),int,semiring_1_unsigned(A,int),aa(int,word(A),ring_1_of_int(word(A)),K)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_of_int
tff(fact_6803_unat__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(nat,word(A),semiring_1_of_nat(word(A)),Xb)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_of_nat
tff(fact_6804_ucast__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => aa(word(B),$o,ord_less(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb)),aa(nat,word(B),aa(word(B),fun(nat,word(B)),power_power(word(B)),aa(num,word(B),numeral_numeral(word(B)),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% ucast_less
tff(fact_6805_signed__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [N: int] : ring_1_signed(B,A,aa(int,word(B),ring_1_of_int(word(B)),N)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),N)) ) ).

% signed_of_int
tff(fact_6806_word__of__int__eq__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int] :
          ( ( aa(int,word(A),ring_1_of_int(word(A)),K) = zero_zero(word(A)) )
        <=> aa(int,$o,dvd_dvd(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),K) ) ) ).

% word_of_int_eq_0_iff
tff(fact_6807_of__nat__n__less__equal__power__2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ) ).

% of_nat_n_less_equal_power_2
tff(fact_6808_signed__ucast__eq,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & bit_ri3973907225187159222ations(A)
        & type_len(B) )
     => ! [W: word(C)] : ring_1_signed(B,A,aa(word(C),word(B),semiring_1_unsigned(C,word(B)),W)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),aa(word(C),A,semiring_1_unsigned(C,A),W)) ) ).

% signed_ucast_eq
tff(fact_6809_complement__nth__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N5: nat,N: nat] :
          ( aa(nat,$o,ord_less(nat,N5),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))),N5)
          <=> ( N5 != N ) ) ) ) ).

% complement_nth_w2p
tff(fact_6810_upper__trivial,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( Xb != aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(word(A))) )
         => aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(word(A)))) ) ) ).

% upper_trivial
tff(fact_6811_range__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(set(word(A)),set(int),image(word(A),int,semiring_1_unsigned(A,int)),top_top(set(word(A)))) = set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% range_uint
tff(fact_6812_bit__word__scast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),ring_1_signed(B,word(A),W)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),N)
              | ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(B,type2(B))),N)
                & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ) ) ).

% bit_word_scast_iff
tff(fact_6813_minus__one__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) = aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(word(A))) ) ) ).

% minus_one_word
tff(fact_6814_UNIV__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( top_top(set(word(A))) = aa(set(int),set(word(A)),image(int,word(A),ring_1_of_int(word(A))),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% UNIV_eq
tff(fact_6815_word__power__less__diff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,Q2: word(A),M: nat] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),Q2)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
         => ( aa(word(A),$o,ord_less(word(A),Q2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),N)))
           => aa(word(A),$o,ord_less(word(A),Q2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,M),N))) ) ) ) ).

% word_power_less_diff
tff(fact_6816_ucast__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),Y)
         => ( aa(word(A),$o,ord_less(word(A),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),type_len0_len_of(B,type2(B))))
           => aa(word(B),$o,ord_less_eq(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ).

% ucast_mono_le
tff(fact_6817_unat__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(7)
tff(fact_6818_take__bit__word__eq__self__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( ( aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),N),W) = W )
        <=> ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N)
            | aa(word(A),$o,ord_less(word(A),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ) ) ).

% take_bit_word_eq_self_iff
tff(fact_6819_signed__push__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [N: nat,W: word(B)] : ring_1_signed(B,A,aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se4730199178511100633sh_bit(word(B)),N),W)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),N),ring_1_signed(B,A,W))) ) ).

% signed_push_bit_eq
tff(fact_6820_ucast__range__less,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( aa(nat,$o,ord_less(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
       => ( aa(set(word(A)),set(word(B)),image(word(A),word(B),semiring_1_unsigned(A,word(B))),top_top(set(word(A)))) = collect(word(B),aTP_Lamp_amh(word(B),$o)) ) ) ) ).

% ucast_range_less
tff(fact_6821_unat__add__lem_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_add_lem'
tff(fact_6822_unat__add__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
        <=> ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_add_lem
tff(fact_6823_Word_Oof__nat__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
         => ( aa(nat,$o,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,word(A),semiring_1_of_nat(word(A)),K) != zero_zero(word(A)) ) ) ) ) ).

% Word.of_nat_neq_0
tff(fact_6824_More__Word_Oof__nat__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N) = zero_zero(word(A)) )
         => ( aa(nat,$o,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( N = zero_zero(nat) ) ) ) ) ).

% More_Word.of_nat_0
tff(fact_6825_of__nat__mono__maybe__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: nat,Y: nat] :
          ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,ord_less_eq(nat,Y),Xb)
            <=> aa(word(A),$o,ord_less_eq(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),Y)),aa(nat,word(A),semiring_1_of_nat(word(A)),Xb)) ) ) ) ) ).

% of_nat_mono_maybe_le
tff(fact_6826_unat__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),zero_zero(word(A))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% unat_word_ariths(4)
tff(fact_6827_bool__mask_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),one_one(word(A))))
          <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),one_one(word(A))) = one_one(word(A)) ) ) ) ) ).

% bool_mask'
tff(fact_6828_unat__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(1)
tff(fact_6829_uint__range_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),Xb))
          & aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% uint_range'
tff(fact_6830_sint__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(int,$o,ord_less(int,ring_1_signed(A,int,Xb)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ).

% sint_lt
tff(fact_6831_word__int__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ~ ! [N3: int] :
              ( ( Xb = aa(int,word(A),ring_1_of_int(word(A)),N3) )
             => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),N3)
               => ~ aa(int,$o,ord_less(int,N3),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ) ).

% word_int_cases
tff(fact_6832_word__of__int__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int,Y: int] :
          ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Xb)
            & aa(int,$o,ord_less(int,Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) )
         => ( ( aa(int,$o,ord_less_eq(int,zero_zero(int)),Y)
              & aa(int,$o,ord_less(int,Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) )
           => ( ( aa(int,word(A),ring_1_of_int(word(A)),Xb) = aa(int,word(A),ring_1_of_int(word(A)),Y) )
            <=> ( Xb = Y ) ) ) ) ) ).

% word_of_int_inj
tff(fact_6833_ucast__mono__le_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Y: word(A),Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(B,type2(B))))
         => ( aa(nat,$o,ord_less(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),$o,ord_less_eq(word(A),Xb),Y)
             => aa(word(B),$o,ord_less_eq(word(B),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Y)) ) ) ) ) ).

% ucast_mono_le'
tff(fact_6834_of__nat__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N) = W )
        <=> ? [Q4: nat] : N = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),W)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% of_nat_eq
tff(fact_6835_unat__mult__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
        <=> ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xb),Y)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)) ) ) ) ).

% unat_mult_lem
tff(fact_6836_unat__ucast__no__overflow__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [B3: word(A),Fa: word(B)] :
          ( aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),B3)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(B,type2(B))))
         => ( aa(nat,$o,ord_less(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),$o,ord_less(word(A),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Fa)),B3)
            <=> aa(nat,$o,ord_less(nat,aa(word(B),nat,semiring_1_unsigned(B,nat),Fa)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3)) ) ) ) ) ).

% unat_ucast_no_overflow_le
tff(fact_6837_uint__m2p__not__non__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : ~ aa(int,$o,ord_less_eq(int,zero_zero(int)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% uint_m2p_not_non_neg
tff(fact_6838_uint__m2p__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),zero_zero(int)) ) ).

% uint_m2p_neg
tff(fact_6839_unat__ucast__less__no__overflow__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,Fa: word(A)] :
          ( aa(nat,$o,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Fa)),N)
          <=> aa(word(A),$o,ord_less(word(A),Fa),aa(nat,word(A),semiring_1_of_nat(word(A)),N)) ) ) ) ).

% unat_ucast_less_no_overflow_simp
tff(fact_6840_unat__ucast__less__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,Fa: word(A)] :
          ( aa(nat,$o,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Fa)),N)
           => aa(word(A),$o,ord_less(word(A),Fa),aa(nat,word(A),semiring_1_of_nat(word(A)),N)) ) ) ) ).

% unat_ucast_less_no_overflow
tff(fact_6841_uint__power__lower,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),int,semiring_1_unsigned(A,int),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),N) ) ) ) ).

% uint_power_lower
tff(fact_6842_less__2p__is__upper__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),N: nat] :
          ( aa(word(A),$o,ord_less(word(A),P2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))
        <=> ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & ! [N12: nat] :
                ( aa(nat,$o,ord_less_eq(nat,N),N12)
               => ( aa(nat,$o,ord_less(nat,N12),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P2),N12) ) ) ) ) ) ).

% less_2p_is_upper_bits_unset
tff(fact_6843_upper__bits__unset__is__l2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,P2: word(A)] :
          ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
         => ( ! [N12: nat] :
                ( aa(nat,$o,ord_less_eq(nat,N),N12)
               => ( aa(nat,$o,ord_less(nat,N12),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P2),N12) ) )
          <=> aa(word(A),$o,ord_less(word(A),P2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ) ) ).

% upper_bits_unset_is_l2p
tff(fact_6844_nth__bounded,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),N)
         => ( aa(word(A),$o,ord_less(word(A),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
           => ( aa(nat,$o,ord_less_eq(nat,M),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,ord_less(nat,N),M) ) ) ) ) ).

% nth_bounded
tff(fact_6845_uint__add__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_add_lem
tff(fact_6846_uint__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),zero_zero(word(A))) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),zero_zero(int)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% uint_word_ariths(7)
tff(fact_6847_uint__word__ariths_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),int,semiring_1_unsigned(A,int),one_one(word(A))) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),one_one(int)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% uint_word_ariths(8)
tff(fact_6848_wi__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: int,M: int] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(int,word(A),ring_1_of_int(word(A)),N)),aa(int,word(A),ring_1_of_int(word(A)),M))
        <=> aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% wi_le
tff(fact_6849_uint__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A3),B3)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(1)
tff(fact_6850_word__2p__mult__inc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)))
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,N)),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,ord_less(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ).

% word_2p_mult_inc
tff(fact_6851_wi__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: int,M: int] :
          ( aa(word(A),$o,ord_less(word(A),aa(int,word(A),ring_1_of_int(word(A)),N)),aa(int,word(A),ring_1_of_int(word(A)),M))
        <=> aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ) ).

% wi_less
tff(fact_6852_power__2__ge__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))
        <=> aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A))) ) ) ).

% power_2_ge_iff
tff(fact_6853_word__power__less__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat] :
          ( aa(nat,$o,ord_less(nat,Sz),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)) ) ) ).

% word_power_less_1
tff(fact_6854_unat__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(2)
tff(fact_6855_uint__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),uminus_uminus(word(A)),A3)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(word(A),int,semiring_1_unsigned(A,int),A3))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(4)
tff(fact_6856_unat__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A3),B3)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(6)
tff(fact_6857_le__mask__iff__lt__2n,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
        <=> ( aa(word(A),$o,ord_less_eq(word(A),W),bit_se2239418461657761734s_mask(word(A),N))
          <=> aa(word(A),$o,ord_less(word(A),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ) ) ).

% le_mask_iff_lt_2n
tff(fact_6858_and__mask__less_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ) ).

% and_mask_less'
tff(fact_6859_eq__mask__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),N: nat] :
          ( ( W = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),N)) )
         => ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,ord_less(word(A),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) ) ) ) ).

% eq_mask_less
tff(fact_6860_sint__1__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
           => ( ( A3 = zero_zero(word(A)) )
             => ( ring_1_signed(A,int,A3) != zero_zero(int) ) ) )
         => ( ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
             => ( ( A3 = one_one(word(A)) )
               => ( ring_1_signed(A,int,one_one(word(A))) != aa(int,int,uminus_uminus(int),one_one(int)) ) ) )
           => ~ ( aa(nat,$o,ord_less(nat,one_one(nat)),type_len0_len_of(A,type2(A)))
               => ( ring_1_signed(A,int,one_one(word(A))) != one_one(int) ) ) ) ) ) ).

% sint_1_cases
tff(fact_6861_uint__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),A3),B3)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(2)
tff(fact_6862_uint__mult__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))
        <=> ( aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xb),Y)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(A),int,semiring_1_unsigned(A,int),Y)) ) ) ) ).

% uint_mult_lem
tff(fact_6863_uint__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A3),B3)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(3)
tff(fact_6864_signed__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [N: nat] : ring_1_signed(B,A,aa(nat,word(B),semiring_1_of_nat(word(B)),N)) = aa(int,A,ring_1_of_int(A),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),aa(nat,int,semiring_1_of_nat(int),N))) ) ).

% signed_of_nat
tff(fact_6865_word__power__mod__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ) ).

% word_power_mod_div
tff(fact_6866_scast__1_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ( ring_1_signed(B,word(A),one_one(word(B))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat)))),one_one(int))) ) ) ).

% scast_1'
tff(fact_6867_msb1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xb),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xb),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xb)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
          <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ).

% msb1
tff(fact_6868_unat__plus__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A3),B3)) = $ite(aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3)),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A3)),aa(word(A),nat,semiring_1_unsigned(A,nat),B3))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% unat_plus_if'
tff(fact_6869_unat__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),one_one(word(A))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% unat_word_ariths(5)
tff(fact_6870_unat__sub__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),Xb),Y)) = $ite(aa(nat,$o,ord_less_eq(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))) ) ).

% unat_sub_if'
tff(fact_6871_sint__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,ord_less(int,ring_1_signed(A,int,W)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% sint_less
tff(fact_6872_word__add__le__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,ord_less_eq(word(A),I),J) ) ) ) ) ).

% word_add_le_dest
tff(fact_6873_no__olen__add__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y))
        <=> aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% no_olen_add_nat
tff(fact_6874_word__add__le__mono2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),J)
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),I)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),J)) ) ) ) ).

% word_add_le_mono2
tff(fact_6875_word__add__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),J)
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K)) ) ) ) ).

% word_add_le_mono1
tff(fact_6876_word__add__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
            <=> aa(word(A),$o,ord_less_eq(word(A),I),J) ) ) ) ) ).

% word_add_le_iff
tff(fact_6877_word__add__less__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,ord_less(word(A),I),J) ) ) ) ) ).

% word_add_less_dest
tff(fact_6878_word__add__less__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),I),J)
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K)) ) ) ) ).

% word_add_less_mono1
tff(fact_6879_word__add__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
            <=> aa(word(A),$o,ord_less(word(A),I),J) ) ) ) ) ).

% word_add_less_iff
tff(fact_6880_unat__minus__one__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(nat)) ) ) ).

% unat_minus_one_word
tff(fact_6881_sint__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))),ring_1_signed(A,int,Xb)) ) ).

% sint_ge
tff(fact_6882_unat__less__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat,K: word(A)] :
          ( aa(nat,$o,ord_less(nat,Sz),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,ord_less(word(A),K),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz))
           => aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),K)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Sz)) ) ) ) ).

% unat_less_power
tff(fact_6883_word__of__int__inverse,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [R2: int,A3: word(A)] :
          ( ( aa(int,word(A),ring_1_of_int(word(A)),R2) = A3 )
         => ( aa(int,$o,ord_less_eq(int,zero_zero(int)),R2)
           => ( aa(int,$o,ord_less(int,R2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(word(A),int,semiring_1_unsigned(A,int),A3) = R2 ) ) ) ) ) ).

% word_of_int_inverse
tff(fact_6884_uint__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(int,$o),Xb: word(A)] :
          ( aa(int,$o,P,aa(word(A),int,semiring_1_unsigned(A,int),Xb))
        <=> ~ ? [I4: int] :
                ( ( aa(int,word(A),ring_1_of_int(word(A)),I4) = Xb )
                & aa(int,$o,ord_less_eq(int,zero_zero(int)),I4)
                & aa(int,$o,ord_less(int,I4),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))
                & ~ aa(int,$o,P,I4) ) ) ) ).

% uint_split_asm
tff(fact_6885_uint__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(int,$o),Xb: word(A)] :
          ( aa(int,$o,P,aa(word(A),int,semiring_1_unsigned(A,int),Xb))
        <=> ! [I4: int] :
              ( ( ( aa(int,word(A),ring_1_of_int(word(A)),I4) = Xb )
                & aa(int,$o,ord_less_eq(int,zero_zero(int)),I4)
                & aa(int,$o,ord_less(int,I4),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) )
             => aa(int,$o,P,I4) ) ) ) ).

% uint_split
tff(fact_6886_word__mult__less__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,ord_less(word(A),I),J) ) ) ) ) ).

% word_mult_less_dest
tff(fact_6887_div__lt_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xb: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xb))
         => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% div_lt'
tff(fact_6888_div__lt_H_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xb))
         => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% div_lt''
tff(fact_6889_double__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A3) = zero_zero(word(A)) )
        <=> ( ( A3 = zero_zero(word(A)) )
            | ( A3 = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ) ) ).

% double_eq_zero_iff
tff(fact_6890_word__le__exists_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),Y)
         => ? [Z2: word(A)] :
              ( ( Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Z2) )
              & aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(A),int,semiring_1_unsigned(A,int),Z2))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ) ).

% word_le_exists'
tff(fact_6891_no__olen__add_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Xb))
        <=> aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Y)),aa(word(A),int,semiring_1_unsigned(A,int),Xb))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% no_olen_add'
tff(fact_6892_no__olen__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y))
        <=> aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),Xb)),aa(word(A),int,semiring_1_unsigned(A,int),Y))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% no_olen_add
tff(fact_6893_More__Word_Oof__nat__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: nat,Xb: nat] :
          ( aa(nat,$o,ord_less(nat,P2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xb))
         => ( aa(nat,$o,ord_less(nat,Xb),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,ord_less(word(A),aa(nat,word(A),semiring_1_of_nat(word(A)),P2)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Xb)) ) ) ) ).

% More_Word.of_nat_power
tff(fact_6894_uint__plus__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A3),B3)) = $ite(aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3)),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% uint_plus_if'
tff(fact_6895_word__less__power__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),M: nat,K: nat] :
          ( aa(word(A),$o,ord_less(word(A),N),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
         => ( aa(nat,$o,ord_less_eq(nat,K),M)
           => ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),K)),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ) ).

% word_less_power_trans
tff(fact_6896_word__less__power__trans2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),M: nat,K: nat] :
          ( aa(word(A),$o,ord_less(word(A),N),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
         => ( aa(nat,$o,ord_less_eq(nat,K),M)
           => ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),N),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),K))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ) ).

% word_less_power_trans2
tff(fact_6897_uint__sub__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),A3),B3)) = $ite(aa(int,$o,ord_less_eq(int,aa(word(A),int,semiring_1_unsigned(A,int),B3)),aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),aa(word(A),int,semiring_1_unsigned(A,int),B3))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% uint_sub_if'
tff(fact_6898_word__less__two__pow__divI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: nat,M: nat] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M)))
         => ( aa(nat,$o,ord_less_eq(nat,M),N)
           => ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))) ) ) ) ) ).

% word_less_two_pow_divI
tff(fact_6899_uint__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_neg_numeral
tff(fact_6900_word__power__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),N: nat] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),N)))
         => ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
           => ( ( Xb != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) != zero_zero(word(A)) ) ) ) ) ) ).

% word_power_nonzero
tff(fact_6901_div__lt__uint_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xb: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xb))
         => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),I)),aa(word(A),int,semiring_1_unsigned(A,int),Xb))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% div_lt_uint'
tff(fact_6902_mult__pow2__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,N: nat,Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,ord_less_eq(word(A),Xb),bit_se2239418461657761734s_mask(word(A),M))
           => ( aa(word(A),$o,ord_less_eq(word(A),Y),bit_se2239418461657761734s_mask(word(A),M))
             => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) )
               => ( Xb = Y ) ) ) ) ) ) ).

% mult_pow2_inj
tff(fact_6903_div__lt__uint_H_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),K),Xb))
         => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),I)),aa(word(A),int,semiring_1_unsigned(A,int),Xb))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% div_lt_uint''
tff(fact_6904_push__bit__word__eq__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: nat,N: nat] :
          ( aa(word(A),$o,ord_less(word(A),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N)),type_len0_len_of(A,type2(A)))
           => ( ( W != zero_zero(word(A)) )
             => ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),N),W) != zero_zero(word(A)) ) ) ) ) ) ).

% push_bit_word_eq_nonzero
tff(fact_6905_uint__and__mask__or__full,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: word(A),Mask1: word(A),Mask2: int] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),N),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))
         => ( ( Mask1 = bit_se2239418461657761734s_mask(word(A),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) )
           => ( ( Mask2 = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),one_one(int)) )
             => ( aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),N),Mask1))),Mask2) = aa(word(A),int,semiring_1_unsigned(A,int),N) ) ) ) ) ) ).

% uint_and_mask_or_full
tff(fact_6906_sint__greater__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))),ring_1_signed(A,int,W)) ) ).

% sint_greater_eq
tff(fact_6907_int__eq__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: nat] :
          ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
         => ( ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),Xb)) = aa(nat,int,semiring_1_of_nat(int),Xb) ) ) ) ).

% int_eq_sint
tff(fact_6908_word__mult__less__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),I: word(A),J: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),K)
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))
              <=> aa(word(A),$o,ord_less(word(A),I),J) ) ) ) ) ) ).

% word_mult_less_cancel
tff(fact_6909_word__mult__less__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),I),J)
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),K)
           => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K)) ) ) ) ) ).

% word_mult_less_mono1
tff(fact_6910_smod__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(int,$o,ord_less(int,signed6721504322012087516modulo(int,ring_1_signed(A,int,A3),ring_1_signed(A,int,B3))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% smod_word_max
tff(fact_6911_le2p__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),N: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),P2),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),one_one(word(A))))
         => ! [N6: nat] :
              ( aa(nat,$o,ord_less_eq(nat,N),N6)
             => ( aa(nat,$o,ord_less(nat,N6),type_len0_len_of(A,type2(A)))
               => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P2),N6) ) ) ) ) ).

% le2p_bits_unset
tff(fact_6912_le__2p__upper__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),N: nat] :
          ( aa(word(A),$o,ord_less_eq(word(A),P2),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),one_one(word(A))))
         => ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
           => ! [N6: nat] :
                ( aa(nat,$o,ord_less_eq(nat,N),N6)
               => ( aa(nat,$o,ord_less(nat,N6),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P2),N6) ) ) ) ) ) ).

% le_2p_upper_bits
tff(fact_6913_word__add__offset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),N: nat,Xb: word(A),M: nat,Sz: nat] :
          ( aa(word(A),$o,ord_less(word(A),Y),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))
         => ( aa(word(A),$o,ord_less(word(A),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
           => ( aa(nat,$o,ord_less(nat,Sz),type_len0_len_of(A,type2(A)))
             => ( aa(word(A),$o,ord_less(word(A),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),N)))
               => ( ( Sz = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) )
                 => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))),Y)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)) ) ) ) ) ) ) ).

% word_add_offset_less
tff(fact_6914_bit__horner__sum__uint__exp__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ws: list(word(A)),N: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),groups4207007520872428315er_sum(word(A),int,semiring_1_unsigned(A,int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))),Ws)),N)
        <=> ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),type_len0_len_of(A,type2(A)))),aa(list(word(A)),nat,size_size(list(word(A))),Ws))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),nth(word(A),Ws),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),N),type_len0_len_of(A,type2(A))))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),N),type_len0_len_of(A,type2(A)))) ) ) ) ).

% bit_horner_sum_uint_exp_iff
tff(fact_6915_div__power__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: nat,Y: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Xb),Y)
         => ( aa(nat,$o,ord_less(nat,Y),type_len0_len_of(A,type2(A)))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Y)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Xb)) = aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,Y),Xb))),one_one(word(A))) ) ) ) ) ).

% div_power_helper
tff(fact_6916_even__mult__exp__div__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),M: nat,N: nat] :
          ( aa(word(A),$o,dvd_dvd(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A3),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)))
        <=> ~ ( aa(nat,$o,ord_less_eq(nat,M),N)
              & aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
              & ~ aa(word(A),$o,dvd_dvd(word(A),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A3),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M)))) ) ) ) ).

% even_mult_exp_div_word_iff
tff(fact_6917_Suc__2p__unat__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: nat] :
          ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
         => ( aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),K)),aa(word(A),nat,semiring_1_unsigned(A,nat),bit_se2239418461657761734s_mask(word(A),N)))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ) ) ).

% Suc_2p_unat_mask
tff(fact_6918_sint__of__nat__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: nat,A3: nat] :
          ( aa(nat,$o,ord_less(nat,B3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
         => ( aa(nat,$o,ord_less_eq(nat,A3),B3)
           => aa(int,$o,ord_less_eq(int,ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),A3))),ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),B3))) ) ) ) ).

% sint_of_nat_le
tff(fact_6919_sint__of__nat__ge__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: nat] :
          ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
         => aa(int,$o,ord_less_eq(int,zero_zero(int)),ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),Xb))) ) ) ).

% sint_of_nat_ge_zero
tff(fact_6920_sint__int__max__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) = aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% sint_int_max_plus_1
tff(fact_6921_sint__of__int__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int] :
          ( aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))),Xb)
         => ( aa(int,$o,ord_less(int,Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
           => ( ring_1_signed(A,int,aa(int,word(A),ring_1_of_int(word(A)),Xb)) = Xb ) ) ) ) ).

% sint_of_int_eq
tff(fact_6922_sint__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B3: num] : ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(num,int,numeral_numeral(int),B3)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ).

% sint_numeral
tff(fact_6923_word__mult__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less_eq(word(A),I),J)
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),K)
           => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K)) ) ) ) ) ).

% word_mult_le_mono1
tff(fact_6924_word__mult__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),I: word(A),J: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),K)
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),I)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),J)),aa(word(A),nat,semiring_1_unsigned(A,nat),K))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
             => ( aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),I),K)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),J),K))
              <=> aa(word(A),$o,ord_less_eq(word(A),I),J) ) ) ) ) ) ).

% word_mult_le_iff
tff(fact_6925_smod__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : aa(int,$o,ord_less_eq(int,aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))),signed6721504322012087516modulo(int,ring_1_signed(A,int,A3),ring_1_signed(A,int,B3))) ) ).

% smod_word_min
tff(fact_6926_int__word__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int] : ring_1_signed(A,int,aa(int,word(A),ring_1_of_int(word(A)),Xb)) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)))) ) ).

% int_word_sint
tff(fact_6927_Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Y)
         => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),aa(word(A),nat,semiring_1_unsigned(A,nat),Y))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))
           => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),Xb),Y)),Y) = Xb ) ) ) ) ).

% Word.word_div_mult
tff(fact_6928_of__nat__less__two__pow__div__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat] :
          ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
         => ( collect(word(A),aa(nat,fun(word(A),$o),aTP_Lamp_ami(nat,fun(nat,fun(word(A),$o)),N),M)) = aa(set(nat),set(word(A)),image(nat,word(A),semiring_1_of_nat(word(A))),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_amj(nat,fun(nat,fun(nat,$o)),N),M))) ) ) ) ).

% of_nat_less_two_pow_div_set
tff(fact_6929_sint__int__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ring_1_signed(A,int,aa(word(A),word(A),uminus_uminus(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))) = aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% sint_int_min
tff(fact_6930_word__bit__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),A3: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [A2: word(A)] :
                ( aa(word(A),$o,P,A2)
               => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),A2)
                 => ( aa(word(A),$o,ord_less(word(A),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
                   => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A2)) ) ) )
           => ( ! [A2: word(A)] :
                  ( aa(word(A),$o,P,A2)
                 => ( aa(word(A),$o,ord_less(word(A),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
                   => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A2))) ) )
             => aa(word(A),$o,P,A3) ) ) ) ) ).

% word_bit_induct
tff(fact_6931_word__less__power__trans__ofnat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,M: nat,K: nat] :
          ( aa(nat,$o,ord_less(nat,N),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
         => ( aa(nat,$o,ord_less_eq(nat,K),M)
           => ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
             => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),N)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),K))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M)) ) ) ) ) ).

% word_less_power_trans_ofnat
tff(fact_6932_unat__mult__power__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,Sz: nat] :
          ( aa(nat,$o,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),Sz)))
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)),aa(nat,word(A),semiring_1_of_nat(word(A)),K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Sz)),K) ) ) ) ).

% unat_mult_power_lem
tff(fact_6933_bit__word__half__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: $o] :
          ( aa(word(A),$o,ord_less(word(A),A3),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa($o,word(A),zero_neq_one_of_bool(word(A)),(B3))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A3),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))))),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) = A3 ) ) ) ).

% bit_word_half_eq
tff(fact_6934_word__of__int__via__signed,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Mask: int,Shift: int,Index: nat,Overflow: int,Least: int,I: int,Arbitrary1: fun(int,word(A)),Arbitrary2: fun(int,word(A))] :
          ( ( Mask = bit_se2239418461657761734s_mask(int,type_len0_len_of(A,type2(A))) )
         => ( ( Shift = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),type_len0_len_of(A,type2(A))),one_one(int)) )
           => ( ( Index = aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)) )
             => ( ( Overflow = aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),one_one(int)) )
               => ( ( Least = aa(int,int,uminus_uminus(int),Overflow) )
                 => ( aa(int,word(A),ring_1_of_int(word(A)),I) = $let(
                        i: int,
                        i:= aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),I),Mask),
                        $ite(
                          aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),i),Index),
                          $ite(
                            ( aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,i),Shift)),Least)
                            | aa(int,$o,ord_less_eq(int,Overflow),aa(int,int,minus_minus(int,i),Shift)) ),
                            aa(int,word(A),Arbitrary1,i),
                            aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,i),Shift)) ),
                          $ite(
                            ( aa(int,$o,ord_less(int,i),Least)
                            | aa(int,$o,ord_less_eq(int,Overflow),i) ),
                            aa(int,word(A),Arbitrary2,i),
                            aa(int,word(A),ring_1_of_int(word(A)),i) ) ) ) ) ) ) ) ) ) ) ).

% word_of_int_via_signed
tff(fact_6935_Suc__div__unat__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Sz: nat,Us: nat] :
          ( aa(nat,$o,ord_less(nat,Sz),type_len0_len_of(A,type2(A)))
         => ( aa(nat,$o,ord_less_eq(nat,Us),Sz)
           => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Sz),Us)) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Sz)),one_one(word(A)))),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Us)))) ) ) ) ) ).

% Suc_div_unat_helper
tff(fact_6936_alignUp__div__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,N: nat,Xb: word(A),A3: word(A)] :
          ( aa(nat,$o,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),N)))
         => ( ( Xb = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)),aa(nat,word(A),semiring_1_of_nat(word(A)),K)) )
           => ( aa(word(A),$o,ord_less_eq(word(A),A3),Xb)
             => ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
               => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),A3),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N)) != zero_zero(word(A)) )
                 => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A3),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),N))),aa(nat,word(A),semiring_1_of_nat(word(A)),K)) ) ) ) ) ) ) ).

% alignUp_div_helper
tff(fact_6937_less__eq__decr__length__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
        <=> aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A))) ) ) ).

% less_eq_decr_length_iff
tff(fact_6938_decr__length__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          ( aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),N)
        <=> aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N) ) ) ).

% decr_length_less_iff
tff(fact_6939_len__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),type_len0_len_of(A,type2(A))) ) ).

% len_gt_0
tff(fact_6940_length__not__greater__eq__2__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ~ aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))
      <=> ( type_len0_len_of(A,type2(A)) = one_one(nat) ) ) ) ).

% length_not_greater_eq_2_iff
tff(fact_6941_len__num0,axiom,
    ! [Uu: itself(numeral_num0)] : type_len0_len_of(numeral_num0,Uu) = zero_zero(nat) ).

% len_num0
tff(fact_6942_len__num1,axiom,
    ! [Uu: itself(numeral_num1)] : type_len0_len_of(numeral_num1,Uu) = one_one(nat) ).

% len_num1
tff(fact_6943_len__of__finite__1__def,axiom,
    ! [Xb: itself(finite_1)] : type_len0_len_of(finite_1,Xb) = one_one(nat) ).

% len_of_finite_1_def
tff(fact_6944_len__of__finite__3__def,axiom,
    ! [Xb: itself(finite_3)] : type_len0_len_of(finite_3,Xb) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% len_of_finite_3_def
tff(fact_6945_len__of__finite__2__def,axiom,
    ! [Xb: itself(finite_2)] : type_len0_len_of(finite_2,Xb) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% len_of_finite_2_def
tff(fact_6946_len__not__eq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( type_len0_len_of(A,type2(A)) != zero_zero(nat) ) ) ).

% len_not_eq_0
tff(fact_6947_len__bit0,axiom,
    ! [A: $tType] :
      ( type_len0(A)
     => ! [Uu: itself(numeral_bit0(A))] : type_len0_len_of(numeral_bit0(A),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))) ) ).

% len_bit0
tff(fact_6948_two__less__eq__exp__length,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & linordered_idom(A) )
     => aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),type_len0_len_of(B,type2(B)))) ) ).

% two_less_eq_exp_length
tff(fact_6949_len__bit1,axiom,
    ! [A: $tType] :
      ( type_len0(A)
     => ! [Uu: itself(numeral_bit1(A))] : type_len0_len_of(numeral_bit1(A),Uu) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))),one_one(nat)) ) ).

% len_bit1
tff(fact_6950_divmod__via__sdivmod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A)] :
          ( ( Y != zero_zero(word(A)) )
         => ( aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xb),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Xb),Y)) = $ite(
                aa(word(A),$o,ord_less_eq(word(A),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),one_one(word(A)))),Y),
                $ite(aa(word(A),$o,ord_less(word(A),Xb),Y),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),zero_zero(word(A))),Xb),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),one_one(word(A))),aa(word(A),word(A),minus_minus(word(A),Xb),Y))),
                $let(
                  q: word(A),
                  q:= aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),one_one(nat)),signed7115095781618012415divide(word(A),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),one_one(nat)),Xb),Y)),
                  $let(
                    r: word(A),
                    r:= aa(word(A),word(A),minus_minus(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),q),Y)),
                    $ite(aa(word(A),$o,ord_less_eq(word(A),Y),r),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),q),one_one(word(A)))),aa(word(A),word(A),minus_minus(word(A),r),Y)),aa(word(A),product_prod(word(A),word(A)),aa(word(A),fun(word(A),product_prod(word(A),word(A))),product_Pair(word(A),word(A)),q),r)) ) ) ) ) ) ) ).

% divmod_via_sdivmod
tff(fact_6951_signed__drop__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,K: num] : signed_drop_bit(A,aa(num,nat,numeral_numeral(nat),N),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(num,nat,numeral_numeral(nat),N)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_drop_bit_word_minus_numeral
tff(fact_6952_signed__drop__bit__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed_drop_bit(A,zero_zero(nat),W) = W ) ).

% signed_drop_bit_0
tff(fact_6953_signed__drop__bit__of__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : signed_drop_bit(A,N,zero_zero(word(A))) = zero_zero(word(A)) ) ).

% signed_drop_bit_of_0
tff(fact_6954_word__sdiv__div0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : signed7115095781618012415divide(word(A),A3,zero_zero(word(A))) = zero_zero(word(A)) ) ).

% word_sdiv_div0
tff(fact_6955_signed__drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          signed_drop_bit(A,N,one_one(word(A))) = aa($o,word(A),zero_neq_one_of_bool(word(A)),
            ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
            | ( N = zero_zero(nat) ) )) ) ).

% signed_drop_bit_of_1
tff(fact_6956_sdiv__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),A3),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A3)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B3)))) ) ).

% sdiv_word_numeral_numeral
tff(fact_6957_signed__drop__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: num] : signed_drop_bit(A,aa(nat,nat,suc,N),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,N)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),K)))) ) ).

% signed_drop_bit_word_Suc_numeral
tff(fact_6958_signed__drop__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,K: num] : signed_drop_bit(A,aa(num,nat,numeral_numeral(nat),N),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(num,nat,numeral_numeral(nat),N)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),K)))) ) ).

% signed_drop_bit_word_numeral
tff(fact_6959_sdiv__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : signed7115095781618012415divide(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B3)))) ) ).

% sdiv_word_minus_numeral_numeral
tff(fact_6960_sdiv__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),A3),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A3)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))))) ) ).

% sdiv_word_numeral_minus_numeral
tff(fact_6961_sdiv__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : signed7115095781618012415divide(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))))) ) ).

% sdiv_word_minus_numeral_minus_numeral
tff(fact_6962_signed__drop__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: num] : signed_drop_bit(A,aa(nat,nat,suc,N),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,N)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))))) ) ).

% signed_drop_bit_word_Suc_minus_numeral
tff(fact_6963_word__sdiv__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed7115095781618012415divide(word(A),zero_zero(word(A)),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,zero_zero(word(A))))) ) ) ).

% word_sdiv_numerals(5)
tff(fact_6964_word__sdiv__numerals__lhs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed7115095781618012415divide(word(A),zero_zero(word(A)),W) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,W))) ) ).

% word_sdiv_numerals_lhs(2)
tff(fact_6965_word__sdiv__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] : signed7115095781618012415divide(word(A),zero_zero(word(A)),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_sdiv_numerals(2)
tff(fact_6966_word__sdiv__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),Xb),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Xb)),ring_1_signed(A,int,zero_zero(word(A))))) ) ).

% word_sdiv_numerals(4)
tff(fact_6967_word__sdiv__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed7115095781618012415divide(word(A),zero_zero(word(A)),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,one_one(word(A))))) ) ) ).

% word_sdiv_numerals(8)
tff(fact_6968_word__sdiv__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed7115095781618012415divide(word(A),one_one(word(A)),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed7115095781618012415divide(int,ring_1_signed(A,int,one_one(word(A))),ring_1_signed(A,int,zero_zero(word(A))))) ) ) ).

% word_sdiv_numerals(6)
tff(fact_6969_signed__div__arith,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : ring_1_signed(A,int,signed7115095781618012415divide(word(A),A3,B3)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),signed7115095781618012415divide(int,ring_1_signed(A,int,A3),ring_1_signed(A,int,B3))) ) ).

% signed_div_arith
tff(fact_6970_bit__signed__drop__bit__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),signed_drop_bit(A,M,W)),N)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),
              $ite(
                ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M)),N)
                & aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A))) ),
                aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),
                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) )) ) ) ).

% bit_signed_drop_bit_iff
tff(fact_6971_signed__drop__bit__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,W: word(A)] :
          ( aa(nat,$o,ord_less_eq(nat,type_len0_len_of(A,type2(A))),N)
         => ( signed_drop_bit(A,N,W) = $ite(aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),zero_zero(word(A))) ) ) ) ).

% signed_drop_bit_beyond
tff(fact_6972_word__int__split__asm,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P: fun(A,$o),Fa: fun(int,A),Xb: word(B)] :
          ( aa(A,$o,P,word_int_case(A,B,Fa,Xb))
        <=> ~ ? [N8: int] :
                ( ( Xb = aa(int,word(B),ring_1_of_int(word(B)),N8) )
                & aa(int,$o,ord_less_eq(int,zero_zero(int)),N8)
                & aa(int,$o,ord_less(int,N8),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(B,type2(B))))
                & ~ aa(A,$o,P,aa(int,A,Fa,N8)) ) ) ) ).

% word_int_split_asm
tff(fact_6973_word__int__split,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P: fun(A,$o),Fa: fun(int,A),Xb: word(B)] :
          ( aa(A,$o,P,word_int_case(A,B,Fa,Xb))
        <=> ! [I4: int] :
              ( ( ( Xb = aa(int,word(B),ring_1_of_int(word(B)),I4) )
                & aa(int,$o,ord_less_eq(int,zero_zero(int)),I4)
                & aa(int,$o,ord_less(int,I4),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(B,type2(B)))) )
             => aa(A,$o,P,aa(int,A,Fa,I4)) ) ) ) ).

% word_int_split
tff(fact_6974_word__int__case__wi,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [Fa: fun(int,A),I: int] : word_int_case(A,B,Fa,aa(int,word(B),ring_1_of_int(word(B)),I)) = aa(int,A,Fa,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),I),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(B,type2(B))))) ) ).

% word_int_case_wi
tff(fact_6975_smod__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : signed6721504322012087516modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B3)))) ) ).

% smod_word_minus_numeral_numeral
tff(fact_6976_smod__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),A3),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A3)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))))) ) ).

% smod_word_numeral_minus_numeral
tff(fact_6977_smod__word__mod__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : signed6721504322012087516modulo(word(A),Xb,zero_zero(word(A))) = Xb ) ).

% smod_word_mod_0
tff(fact_6978_smod__word__0__mod,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : signed6721504322012087516modulo(word(A),zero_zero(word(A)),Xb) = zero_zero(word(A)) ) ).

% smod_word_0_mod
tff(fact_6979_smod__word__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),W,zero_zero(word(A))) = W ) ).

% smod_word_zero
tff(fact_6980_smod__word__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),W,one_one(word(A))) = zero_zero(word(A)) ) ).

% smod_word_one
tff(fact_6981_smod__word__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),W,aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = zero_zero(word(A)) ) ).

% smod_word_minus_one
tff(fact_6982_smod__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),A3),aa(num,word(A),numeral_numeral(word(A)),B3)) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A3)),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B3)))) ) ).

% smod_word_numeral_numeral
tff(fact_6983_smod__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] : signed6721504322012087516modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3))))) ) ).

% smod_word_minus_numeral_minus_numeral
tff(fact_6984_word__smod__numerals__lhs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),zero_zero(word(A)),W) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,W))) ) ).

% word_smod_numerals_lhs(2)
tff(fact_6985_word__smod__numerals_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed6721504322012087516modulo(word(A),zero_zero(word(A)),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,zero_zero(word(A))))) ) ) ).

% word_smod_numerals(5)
tff(fact_6986_word__smod__numerals_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),Xb),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Xb)),ring_1_signed(A,int,zero_zero(word(A))))) ) ).

% word_smod_numerals(4)
tff(fact_6987_word__smod__numerals_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] : signed6721504322012087516modulo(word(A),zero_zero(word(A)),aa(num,word(A),numeral_numeral(word(A)),Y)) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_smod_numerals(2)
tff(fact_6988_word__smod__numerals_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed6721504322012087516modulo(word(A),one_one(word(A)),zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,ring_1_signed(A,int,one_one(word(A))),ring_1_signed(A,int,zero_zero(word(A))))) ) ) ).

% word_smod_numerals(6)
tff(fact_6989_word__smod__numerals_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed6721504322012087516modulo(word(A),zero_zero(word(A)),one_one(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),signed6721504322012087516modulo(int,ring_1_signed(A,int,zero_zero(word(A))),ring_1_signed(A,int,one_one(word(A))))) ) ) ).

% word_smod_numerals(8)
tff(fact_6990_signed__mod__arith,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] : ring_1_signed(A,int,signed6721504322012087516modulo(word(A),A3,B3)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),signed6721504322012087516modulo(int,ring_1_signed(A,int,A3),ring_1_signed(A,int,B3))) ) ).

% signed_mod_arith
tff(fact_6991_uint__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_pred(A,A3)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),one_one(int))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(6)
tff(fact_6992_slice1__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [N: nat,W: word(B)] :
          slice1(B,A,N,W) = $ite(aa(nat,$o,ord_less(nat,N),type_len0_len_of(B,type2(B))),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se4197421643247451524op_bit(word(B)),aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),N)),W)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),aa(nat,nat,minus_minus(nat,N),type_len0_len_of(B,type2(B)))),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W))) ) ).

% slice1_def
tff(fact_6993_slice1__0,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [N: nat] : slice1(B,A,N,zero_zero(word(B))) = zero_zero(word(A)) ) ).

% slice1_0
tff(fact_6994_word__m1__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,ord_less_eq(word(A),Y),word_pred(A,zero_zero(word(A)))) ) ).

% word_m1_ge
tff(fact_6995_word__not__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : ~ aa(word(A),$o,ord_less(word(A),word_pred(A,zero_zero(word(A)))),Y) ) ).

% word_not_simps(2)
tff(fact_6996_ucast__slice1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W) = slice1(B,A,aa(word(B),nat,size_size(word(B)),W),W) ) ).

% ucast_slice1
tff(fact_6997_wi__hom__pred,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: int] : word_pred(A,aa(int,word(A),ring_1_of_int(word(A)),A3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,A3),one_one(int))) ) ).

% wi_hom_pred
tff(fact_6998_word__pred__0__n1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_pred(A,zero_zero(word(A))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),one_one(int))) ) ) ).

% word_pred_0_n1
tff(fact_6999_word__pred__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : word_pred(A,A3) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),one_one(int))) ) ).

% word_pred_alt
tff(fact_7000_uint__word__arith__bintrs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_pred(A,A3)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A3)),one_one(int))) ) ).

% uint_word_arith_bintrs(6)
tff(fact_7001_sint__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : ring_1_signed(A,int,word_pred(A,A3)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,minus_minus(int,ring_1_signed(A,int,A3)),one_one(int))) ) ).

% sint_word_ariths(6)
tff(fact_7002_uint__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_succ(A,A3)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),one_one(int))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% uint_word_ariths(5)
tff(fact_7003_bit__word__roti__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,W: word(A),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_roti(A,K,W)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),nat2(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),N)),K)),aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))) ) ) ) ).

% bit_word_roti_iff
tff(fact_7004_word__roti__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : word_roti(A,zero_zero(int),W) = W ) ).

% word_roti_0
tff(fact_7005_word__roti__0_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: int] : word_roti(A,N,zero_zero(word(A))) = zero_zero(word(A)) ) ).

% word_roti_0'
tff(fact_7006_wi__hom__succ,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: int] : word_succ(A,aa(int,word(A),ring_1_of_int(word(A)),A3)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),A3),one_one(int))) ) ).

% wi_hom_succ
tff(fact_7007_word__roti__conv__mod_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: int,W: word(A)] : word_roti(A,N,W) = word_roti(A,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),N),aa(nat,int,semiring_1_of_nat(int),aa(word(A),nat,size_size(word(A)),W))),W) ) ).

% word_roti_conv_mod'
tff(fact_7008_word__succ__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : word_succ(A,A3) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),one_one(int))) ) ).

% word_succ_alt
tff(fact_7009_word__sp__01,axiom,
    ! [C: $tType,A: $tType,B: $tType,D2: $tType] :
      ( ( type_len(D2)
        & type_len(B)
        & type_len(A)
        & type_len(C) )
     => ( ( word_succ(A,aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = zero_zero(word(A)) )
        & ( word_succ(B,zero_zero(word(B))) = one_one(word(B)) )
        & ( word_pred(C,zero_zero(word(C))) = aa(word(C),word(C),uminus_uminus(word(C)),one_one(word(C))) )
        & ( word_pred(D2,one_one(word(D2))) = zero_zero(word(D2)) ) ) ) ).

% word_sp_01
tff(fact_7010_uint__word__arith__bintrs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_succ(A,A3)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A3)),one_one(int))) ) ).

% uint_word_arith_bintrs(5)
tff(fact_7011_sint__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : ring_1_signed(A,int,word_succ(A,A3)) = aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,aa(int,fun(int,int),plus_plus(int),ring_1_signed(A,int,A3)),one_one(int))) ) ).

% sint_word_ariths(5)
tff(fact_7012_unat__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),word_succ(A,A3)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),A3))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ).

% unat_word_ariths(3)
tff(fact_7013_sless__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sle(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)))
        <=> aa(int,$o,ord_less_eq(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ) ).

% sless_eq_word_minus_numeral_minus_numeral
tff(fact_7014_sless__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),A3),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)))
        <=> aa(int,$o,ord_less_eq(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A3))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ) ).

% sless_eq_word_numeral_minus_numeral
tff(fact_7015_word__sle__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),A3),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less_eq(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),A3))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),B3))) ) ) ).

% word_sle_no
tff(fact_7016_extra__sle__sless__unfolds_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num] :
          ( word_sle(A,one_one(word(A)),aa(num,word(A),numeral_numeral(word(A)),N))
        <=> aa(int,$o,ord_less_eq(int,ring_1_signed(A,int,one_one(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),N))) ) ) ).

% extra_sle_sless_unfolds(4)
tff(fact_7017_extra__sle__sless__unfolds_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),N),one_one(word(A)))
        <=> aa(int,$o,ord_less_eq(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),N))),ring_1_signed(A,int,one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(6)
tff(fact_7018_extra__sle__sless__unfolds_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num] :
          ( word_sle(A,zero_zero(word(A)),aa(num,word(A),numeral_numeral(word(A)),N))
        <=> aa(int,$o,ord_less_eq(int,ring_1_signed(A,int,zero_zero(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),N))) ) ) ).

% extra_sle_sless_unfolds(2)
tff(fact_7019_extra__sle__sless__unfolds_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),N),zero_zero(word(A)))
        <=> aa(int,$o,ord_less_eq(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),N))),ring_1_signed(A,int,zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(5)
tff(fact_7020_extra__sle__sless__unfolds_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_sle(A,zero_zero(word(A)),one_one(word(A)))
      <=> aa(int,$o,ord_less_eq(int,ring_1_signed(A,int,zero_zero(word(A)))),ring_1_signed(A,int,one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(1)
tff(fact_7021_extra__sle__sless__unfolds_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_sle(A,one_one(word(A)),zero_zero(word(A)))
      <=> aa(int,$o,ord_less_eq(int,ring_1_signed(A,int,one_one(word(A)))),ring_1_signed(A,int,zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(3)
tff(fact_7022_sless__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),A3),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less_eq(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A3))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B3))) ) ) ).

% sless_eq_word_numeral_numeral
tff(fact_7023_sless__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sle(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less_eq(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B3))) ) ) ).

% sless_eq_word_minus_numeral_numeral
tff(fact_7024_word__sle__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( word_sle(A,A3,B3)
        <=> aa(int,$o,ord_less_eq(int,ring_1_signed(A,int,A3)),ring_1_signed(A,int,B3)) ) ) ).

% word_sle_eq
tff(fact_7025_signed_Olift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Fa: fun(nat,word(A)),N: nat,N5: nat] :
          ( ! [N3: nat] : word_sle(A,aa(nat,word(A),Fa,N3),aa(nat,word(A),Fa,aa(nat,nat,suc,N3)))
         => ( aa(nat,$o,ord_less_eq(nat,N),N5)
           => word_sle(A,aa(nat,word(A),Fa,N),aa(nat,word(A),Fa,N5)) ) ) ) ).

% signed.lift_Suc_mono_le
tff(fact_7026_signed_Olift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Fa: fun(nat,word(A)),N: nat,N5: nat] :
          ( ! [N3: nat] : word_sle(A,aa(nat,word(A),Fa,aa(nat,nat,suc,N3)),aa(nat,word(A),Fa,N3))
         => ( aa(nat,$o,ord_less_eq(nat,N),N5)
           => word_sle(A,aa(nat,word(A),Fa,N5),aa(nat,word(A),Fa,N)) ) ) ) ).

% signed.lift_Suc_antimono_le
tff(fact_7027_word__0__sle__from__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
         => word_sle(A,zero_zero(word(A)),Xb) ) ) ).

% word_0_sle_from_less
tff(fact_7028_sless__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sless(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B3))) ) ) ).

% sless_word_minus_numeral_numeral
tff(fact_7029_sless__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),A3),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)))
        <=> aa(int,$o,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A3))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ) ).

% sless_word_numeral_minus_numeral
tff(fact_7030_extra__sle__sless__unfolds_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num] :
          ( word_sless(A,one_one(word(A)),aa(num,word(A),numeral_numeral(word(A)),N))
        <=> aa(int,$o,ord_less(int,ring_1_signed(A,int,one_one(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),N))) ) ) ).

% extra_sle_sless_unfolds(10)
tff(fact_7031_extra__sle__sless__unfolds_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),N),one_one(word(A)))
        <=> aa(int,$o,ord_less(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),N))),ring_1_signed(A,int,one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(12)
tff(fact_7032_extra__sle__sless__unfolds_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),N),zero_zero(word(A)))
        <=> aa(int,$o,ord_less(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),N))),ring_1_signed(A,int,zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(11)
tff(fact_7033_extra__sle__sless__unfolds_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num] :
          ( word_sless(A,zero_zero(word(A)),aa(num,word(A),numeral_numeral(word(A)),N))
        <=> aa(int,$o,ord_less(int,ring_1_signed(A,int,zero_zero(word(A)))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),N))) ) ) ).

% extra_sle_sless_unfolds(8)
tff(fact_7034_extra__sle__sless__unfolds_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_sless(A,zero_zero(word(A)),one_one(word(A)))
      <=> aa(int,$o,ord_less(int,ring_1_signed(A,int,zero_zero(word(A)))),ring_1_signed(A,int,one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(7)
tff(fact_7035_extra__sle__sless__unfolds_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( word_sless(A,one_one(word(A)),zero_zero(word(A)))
      <=> aa(int,$o,ord_less(int,ring_1_signed(A,int,one_one(word(A)))),ring_1_signed(A,int,zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(9)
tff(fact_7036_sless__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),A3),aa(num,word(A),numeral_numeral(word(A)),B3))
        <=> aa(int,$o,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),A3))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(num,int,numeral_numeral(int),B3))) ) ) ).

% sless_word_numeral_numeral
tff(fact_7037_sless__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: num,B3: num] :
          ( word_sless(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A3)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B3)))
        <=> aa(int,$o,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A3)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B3)))) ) ) ).

% sless_word_minus_numeral_minus_numeral
tff(fact_7038_word__sless__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A),B3: word(A)] :
          ( word_sless(A,A3,B3)
        <=> aa(int,$o,ord_less(int,ring_1_signed(A,int,A3)),ring_1_signed(A,int,B3)) ) ) ).

% word_sless_alt
tff(fact_7039_signed_Olift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Fa: fun(nat,word(A)),N: nat,N5: nat] :
          ( ! [N3: nat] : word_sless(A,aa(nat,word(A),Fa,N3),aa(nat,word(A),Fa,aa(nat,nat,suc,N3)))
         => ( aa(nat,$o,ord_less(nat,N),N5)
           => word_sless(A,aa(nat,word(A),Fa,N),aa(nat,word(A),Fa,N5)) ) ) ) ).

% signed.lift_Suc_mono_less
tff(fact_7040_signed_Olift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Fa: fun(nat,word(A)),N: nat,M: nat] :
          ( ! [N3: nat] : word_sless(A,aa(nat,word(A),Fa,N3),aa(nat,word(A),Fa,aa(nat,nat,suc,N3)))
         => ( word_sless(A,aa(nat,word(A),Fa,N),aa(nat,word(A),Fa,M))
          <=> aa(nat,$o,ord_less(nat,N),M) ) ) ) ).

% signed.lift_Suc_mono_less_iff
tff(fact_7041_word__sless__sint__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( word_sless(A,Xb,Y)
         => aa(int,$o,ord_less_eq(int,ring_1_signed(A,int,Xb)),aa(int,int,minus_minus(int,ring_1_signed(A,int,Y)),one_one(int))) ) ) ).

% word_sless_sint_le
tff(fact_7042_take__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,K: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,ord_min(nat,type_len0_len_of(A,type2(A)),aa(num,nat,numeral_numeral(nat),N))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))) ) ).

% take_bit_word_minus_numeral
tff(fact_7043_take__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(nat,nat,suc,N)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),K))) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,ord_min(nat,type_len0_len_of(A,type2(A)),aa(nat,nat,suc,N))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))) ) ).

% take_bit_word_Suc_minus_numeral
tff(fact_7044_min__arg__le_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [M: A,N: A] :
          ( aa(A,$o,ord_less_eq(A,M),ord_min(A,M,N))
        <=> ( ord_min(A,M,N) = M ) ) ) ).

% min_arg_le(2)
tff(fact_7045_min__arg__le_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [N: A,M: A] :
          ( aa(A,$o,ord_less_eq(A,N),ord_min(A,M,N))
        <=> ( ord_min(A,M,N) = N ) ) ) ).

% min_arg_le(1)
tff(fact_7046_min__eq__arg_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N: A] :
          ( ( ord_min(A,M,N) = N )
        <=> aa(A,$o,ord_less_eq(A,N),M) ) ) ).

% min_eq_arg(2)
tff(fact_7047_min__eq__arg_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N: A] :
          ( ( ord_min(A,M,N) = M )
        <=> aa(A,$o,ord_less_eq(A,M),N) ) ) ).

% min_eq_arg(1)
tff(fact_7048_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),ord_min(A,B3,C2))
        <=> ( aa(A,$o,ord_less_eq(A,A3),B3)
            & aa(A,$o,ord_less_eq(A,A3),C2) ) ) ) ).

% min.bounded_iff
tff(fact_7049_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
         => ( ord_min(A,A3,B3) = B3 ) ) ) ).

% min.absorb2
tff(fact_7050_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( ord_min(A,A3,B3) = A3 ) ) ) ).

% min.absorb1
tff(fact_7051_min__simps_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( ord_min(A,A3,B3) = B3 ) ) ) ).

% min_simps(2)
tff(fact_7052_min__simps_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( ord_min(A,A3,B3) = A3 ) ) ) ).

% min_simps(1)
tff(fact_7053_min__less__self__conv_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,ord_min(A,A3,B3)),B3)
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% min_less_self_conv(2)
tff(fact_7054_min__less__self__conv_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,ord_min(A,A3,B3)),A3)
        <=> aa(A,$o,ord_less(A,B3),A3) ) ) ).

% min_less_self_conv(1)
tff(fact_7055_min__arg__not__ge_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N: A] :
          ( ~ aa(A,$o,ord_less(A,ord_min(A,M,N)),N)
        <=> ( ord_min(A,M,N) = N ) ) ) ).

% min_arg_not_ge(2)
tff(fact_7056_min__arg__not__ge_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,N: A] :
          ( ~ aa(A,$o,ord_less(A,ord_min(A,M,N)),M)
        <=> ( ord_min(A,M,N) = M ) ) ) ).

% min_arg_not_ge(1)
tff(fact_7057_min__less__iff__conj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Z),ord_min(A,Xb,Y))
        <=> ( aa(A,$o,ord_less(A,Z),Xb)
            & aa(A,$o,ord_less(A,Z),Y) ) ) ) ).

% min_less_iff_conj
tff(fact_7058_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),A3)
         => ( ord_min(A,A3,B3) = B3 ) ) ) ).

% min.absorb4
tff(fact_7059_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( ord_min(A,A3,B3) = A3 ) ) ) ).

% min.absorb3
tff(fact_7060_min__Suc__Suc,axiom,
    ! [M: nat,N: nat] : ord_min(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N)) = aa(nat,nat,suc,ord_min(nat,M,N)) ).

% min_Suc_Suc
tff(fact_7061_min__0L,axiom,
    ! [N: nat] : ord_min(nat,zero_zero(nat),N) = zero_zero(nat) ).

% min_0L
tff(fact_7062_min__0R,axiom,
    ! [N: nat] : ord_min(nat,N,zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_7063_take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat,A3: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,N),A3)) = aa(A,A,bit_se2584673776208193580ke_bit(A,ord_min(nat,M,N)),A3) ) ).

% take_bit_take_bit
tff(fact_7064_signed__take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat,A3: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,N),A3)) = aa(A,A,bit_ri4674362597316999326ke_bit(A,ord_min(nat,M,N)),A3) ) ).

% signed_take_bit_signed_take_bit
tff(fact_7065_min__0__1_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : ord_min(A,aa(num,A,numeral_numeral(A),Xb),zero_zero(A)) = zero_zero(A) ) ).

% min_0_1(4)
tff(fact_7066_min__0__1_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : ord_min(A,zero_zero(A),aa(num,A,numeral_numeral(A),Xb)) = zero_zero(A) ) ).

% min_0_1(3)
tff(fact_7067_min__number__of_I1_J,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ord_min(A,aa(num,A,numeral_numeral(A),U),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)),aa(num,A,numeral_numeral(A),U),aa(num,A,numeral_numeral(A),V)) ) ).

% min_number_of(1)
tff(fact_7068_min__0__1_I1_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( ord_min(A,zero_zero(A),one_one(A)) = zero_zero(A) ) ) ).

% min_0_1(1)
tff(fact_7069_min__0__1_I2_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ( ord_min(A,one_one(A),zero_zero(A)) = zero_zero(A) ) ) ).

% min_0_1(2)
tff(fact_7070_min__0__1_I6_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : ord_min(A,aa(num,A,numeral_numeral(A),Xb),one_one(A)) = one_one(A) ) ).

% min_0_1(6)
tff(fact_7071_min__0__1_I5_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: num] : ord_min(A,one_one(A),aa(num,A,numeral_numeral(A),Xb)) = one_one(A) ) ).

% min_0_1(5)
tff(fact_7072_min__Suc__gt_I1_J,axiom,
    ! [A3: nat,B3: nat] :
      ( aa(nat,$o,ord_less(nat,A3),B3)
     => ( ord_min(nat,aa(nat,nat,suc,A3),B3) = aa(nat,nat,suc,A3) ) ) ).

% min_Suc_gt(1)
tff(fact_7073_min__Suc__gt_I2_J,axiom,
    ! [A3: nat,B3: nat] :
      ( aa(nat,$o,ord_less(nat,A3),B3)
     => ( ord_min(nat,B3,aa(nat,nat,suc,A3)) = aa(nat,nat,suc,A3) ) ) ).

% min_Suc_gt(2)
tff(fact_7074_take__bit__of__mask,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),bit_se2239418461657761734s_mask(A,N)) = bit_se2239418461657761734s_mask(A,ord_min(nat,M,N)) ) ).

% take_bit_of_mask
tff(fact_7075_min__number__of_I2_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ord_min(A,aa(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(num,A,numeral_numeral(A),U),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ).

% min_number_of(2)
tff(fact_7076_min__number__of_I3_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ord_min(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) = $ite(aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(num,A,numeral_numeral(A),V)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(num,A,numeral_numeral(A),V)) ) ).

% min_number_of(3)
tff(fact_7077_min__number__of_I4_J,axiom,
    ! [A: $tType] :
      ( ( uminus(A)
        & numeral(A)
        & ord(A) )
     => ! [U: num,V: num] :
          ord_min(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) = $ite(aa(A,$o,ord_less_eq(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),U)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),V))) ) ).

% min_number_of(4)
tff(fact_7078_min__Suc__numeral,axiom,
    ! [N: nat,K: num] : ord_min(nat,aa(nat,nat,suc,N),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,ord_min(nat,N,pred_numeral(K))) ).

% min_Suc_numeral
tff(fact_7079_min__numeral__Suc,axiom,
    ! [K: num,N: nat] : ord_min(nat,aa(num,nat,numeral_numeral(nat),K),aa(nat,nat,suc,N)) = aa(nat,nat,suc,ord_min(nat,pred_numeral(K),N)) ).

% min_numeral_Suc
tff(fact_7080_take__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat,K: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(nat,nat,suc,N)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,ord_min(nat,type_len0_len_of(A,type2(A)),aa(nat,nat,suc,N))),aa(num,int,numeral_numeral(int),K))) ) ).

% take_bit_word_Suc_numeral
tff(fact_7081_take__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: num,K: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),N)),aa(num,word(A),numeral_numeral(word(A)),K)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,ord_min(nat,type_len0_len_of(A,type2(A)),aa(num,nat,numeral_numeral(nat),N))),aa(num,int,numeral_numeral(int),K))) ) ).

% take_bit_word_numeral
tff(fact_7082_concat__bit__assoc__sym,axiom,
    ! [M: nat,N: nat,K: int,L: int,R2: int] : aa(int,int,bit_concat_bit(M,aa(int,int,bit_concat_bit(N,K),L)),R2) = aa(int,int,bit_concat_bit(ord_min(nat,M,N),K),aa(int,int,bit_concat_bit(aa(nat,nat,minus_minus(nat,M),N),L),R2)) ).

% concat_bit_assoc_sym
tff(fact_7083_min__diff,axiom,
    ! [M: nat,I: nat,N: nat] : ord_min(nat,aa(nat,nat,minus_minus(nat,M),I),aa(nat,nat,minus_minus(nat,N),I)) = aa(nat,nat,minus_minus(nat,ord_min(nat,M,N)),I) ).

% min_diff
tff(fact_7084_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,ord_less_eq(A,Y),Xb)
         => ( ord_min(A,Xb,Y) = Y ) ) ) ).

% min_absorb2
tff(fact_7085_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less_eq(A,Xb),Y)
         => ( ord_min(A,Xb,Y) = Xb ) ) ) ).

% min_absorb1
tff(fact_7086_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A3: A,B3: A] :
          ord_min(A,A3,B3) = $ite(aa(A,$o,ord_less_eq(A,A3),B3),A3,B3) ) ).

% min_def
tff(fact_7087_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less_eq(A,ord_min(A,Xb,Y)),Z)
        <=> ( aa(A,$o,ord_less_eq(A,Xb),Z)
            | aa(A,$o,ord_less_eq(A,Y),Z) ) ) ) ).

% min_le_iff_disj
tff(fact_7088_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),C2)
         => aa(A,$o,ord_less_eq(A,ord_min(A,A3,B3)),C2) ) ) ).

% min.coboundedI2
tff(fact_7089_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),C2)
         => aa(A,$o,ord_less_eq(A,ord_min(A,A3,B3)),C2) ) ) ).

% min.coboundedI1
tff(fact_7090_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,A3: A] :
          ( aa(A,$o,ord_less_eq(A,B3),A3)
        <=> ( ord_min(A,A3,B3) = B3 ) ) ) ).

% min.absorb_iff2
tff(fact_7091_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
        <=> ( ord_min(A,A3,B3) = A3 ) ) ) ).

% min.absorb_iff1
tff(fact_7092_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,ord_min(A,A3,B3)),B3) ) ).

% min.cobounded2
tff(fact_7093_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] : aa(A,$o,ord_less_eq(A,ord_min(A,A3,B3)),A3) ) ).

% min.cobounded1
tff(fact_7094_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
        <=> ( A3 = ord_min(A,A3,B3) ) ) ) ).

% min.order_iff
tff(fact_7095_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( aa(A,$o,ord_less_eq(A,A3),C2)
           => aa(A,$o,ord_less_eq(A,A3),ord_min(A,B3,C2)) ) ) ) ).

% min.boundedI
tff(fact_7096_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less_eq(A,A3),ord_min(A,B3,C2))
         => ~ ( aa(A,$o,ord_less_eq(A,A3),B3)
             => ~ aa(A,$o,ord_less_eq(A,A3),C2) ) ) ) ).

% min.boundedE
tff(fact_7097_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = ord_min(A,A3,B3) )
         => aa(A,$o,ord_less_eq(A,A3),B3) ) ) ).

% min.orderI
tff(fact_7098_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less_eq(A,A3),B3)
         => ( A3 = ord_min(A,A3,B3) ) ) ) ).

% min.orderE
tff(fact_7099_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B3: A,D: A] :
          ( aa(A,$o,ord_less_eq(A,A3),C2)
         => ( aa(A,$o,ord_less_eq(A,B3),D)
           => aa(A,$o,ord_less_eq(A,ord_min(A,A3,B3)),ord_min(A,C2,D)) ) ) ) ).

% min.mono
tff(fact_7100_min__less__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,ord_less(A,ord_min(A,Xb,Y)),Z)
        <=> ( aa(A,$o,ord_less(A,Xb),Z)
            | aa(A,$o,ord_less(A,Y),Z) ) ) ) ).

% min_less_iff_disj
tff(fact_7101_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A] :
          ( aa(A,$o,ord_less(A,A3),ord_min(A,B3,C2))
         => ~ ( aa(A,$o,ord_less(A,A3),B3)
             => ~ aa(A,$o,ord_less(A,A3),C2) ) ) ) ).

% min.strict_boundedE
tff(fact_7102_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
        <=> ( ( A3 = ord_min(A,A3,B3) )
            & ( A3 != B3 ) ) ) ) ).

% min.strict_order_iff
tff(fact_7103_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,C2: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),C2)
         => aa(A,$o,ord_less(A,ord_min(A,A3,B3)),C2) ) ) ).

% min.strict_coboundedI1
tff(fact_7104_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B3: A,C2: A,A3: A] :
          ( aa(A,$o,ord_less(A,B3),C2)
         => aa(A,$o,ord_less(A,ord_min(A,A3,B3)),C2) ) ) ).

% min.strict_coboundedI2
tff(fact_7105_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X3: A,Xa: A] :
          ord_min(A,X3,Xa) = $ite(aa(A,$o,ord_less_eq(A,X3),Xa),X3,Xa) ) ).

% min_def_raw
tff(fact_7106_of__nat__min,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Xb: nat,Y: nat] : aa(nat,A,semiring_1_of_nat(A),ord_min(nat,Xb,Y)) = ord_min(A,aa(nat,A,semiring_1_of_nat(A),Xb),aa(nat,A,semiring_1_of_nat(A),Y)) ) ).

% of_nat_min
tff(fact_7107_nat__mult__min__right,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),ord_min(nat,N,Q2)) = ord_min(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),N),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)) ).

% nat_mult_min_right
tff(fact_7108_nat__mult__min__left,axiom,
    ! [M: nat,N: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),ord_min(nat,M,N)),Q2) = ord_min(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),N),Q2)) ).

% nat_mult_min_left
tff(fact_7109_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,M),aa(int,int,bit_concat_bit(N,K),L)) = aa(int,int,bit_concat_bit(ord_min(nat,M,N),K),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,minus_minus(nat,M),N)),L)) ).

% take_bit_concat_bit_eq
tff(fact_7110_max__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,Xb: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P2),aa(A,A,ord_max(A,Xb),Y)) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P2),aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),times_times(A),P2),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)),ord_min(A,aa(A,A,aa(A,fun(A,A),times_times(A),P2),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y))) ) ).

% max_mult_distrib_left
tff(fact_7111_min__mult__distrib__left,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P2: A,Xb: A,Y: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),P2),ord_min(A,Xb,Y)) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P2),ord_min(A,aa(A,A,aa(A,fun(A,A),times_times(A),P2),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y)),aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),times_times(A),P2),Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),P2),Y))) ) ).

% min_mult_distrib_left
tff(fact_7112_max__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A,P2: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,ord_max(A,Xb),Y)),P2) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P2),aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)),ord_min(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2))) ) ).

% max_mult_distrib_right
tff(fact_7113_min__mult__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A,P2: A] :
          aa(A,A,aa(A,fun(A,A),times_times(A),ord_min(A,Xb,Y)),P2) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P2),ord_min(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P2),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2)),aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),P2)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),P2))) ) ).

% min_mult_distrib_right
tff(fact_7114_max__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,P2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,ord_max(A,Xb),Y)),P2) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P2),aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2)),ord_min(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),P2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2))) ) ).

% max_divide_distrib_right
tff(fact_7115_min__divide__distrib__right,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,P2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),ord_min(A,Xb,Y)),P2) = $ite(aa(A,$o,ord_less_eq(A,zero_zero(A)),P2),ord_min(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),P2),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2)),aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),P2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),Y),P2))) ) ).

% min_divide_distrib_right
tff(fact_7116_min__Suc2,axiom,
    ! [M: nat,N: nat] : ord_min(nat,M,aa(nat,nat,suc,N)) = case_nat(nat,zero_zero(nat),aTP_Lamp_amk(nat,fun(nat,nat),N),M) ).

% min_Suc2
tff(fact_7117_min__Suc1,axiom,
    ! [N: nat,M: nat] : ord_min(nat,aa(nat,nat,suc,N),M) = case_nat(nat,zero_zero(nat),aTP_Lamp_aml(nat,fun(nat,nat),N),M) ).

% min_Suc1
tff(fact_7118_mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,M: nat,N: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),ord_min(nat,M,N))) ) ).

% mod_exp_eq
tff(fact_7119_mod__mod__power,axiom,
    ! [K: nat,M: nat,N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),ord_min(nat,M,N))) ).

% mod_mod_power
tff(fact_7120_Word_Obit__mask__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_se2239418461657761734s_mask(word(A),M)),N)
        <=> aa(nat,$o,ord_less(nat,N),ord_min(nat,type_len0_len_of(A,type2(A)),M)) ) ) ).

% Word.bit_mask_iff
tff(fact_7121_mask__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [N: nat,M: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)),one_one(A))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)) = aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),ord_min(nat,M,N))),one_one(A)) ) ).

% mask_mod_exp
tff(fact_7122_bit__slice__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [M: nat,W: word(B),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),slice2(B,A,M,W)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),ord_min(nat,type_len0_len_of(A,type2(A)),aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),M)))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),type_len0_len_of(B,type2(B)))),aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),M))) ) ) ) ).

% bit_slice_iff
tff(fact_7123_bit__slice1__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [M: nat,W: word(B),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),slice1(B,A,M,W)),N)
        <=> ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,M),type_len0_len_of(B,type2(B)))),N)
            & aa(nat,$o,ord_less(nat,N),ord_min(nat,type_len0_len_of(A,type2(A)),M))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),M))),aa(nat,nat,minus_minus(nat,M),type_len0_len_of(B,type2(B))))) ) ) ) ).

% bit_slice1_iff
tff(fact_7124_unat__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),bit_se2239418461657761734s_mask(word(A),N)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),ord_min(nat,N,type_len0_len_of(A,type2(A))))),one_one(nat)) ) ).

% unat_mask
tff(fact_7125_bit__horner__sum__bit__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bs: list($o),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),groups4207007520872428315er_sum($o,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)),Bs)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),ord_min(nat,type_len0_len_of(A,type2(A)),aa(list($o),nat,size_size(list($o)),Bs)))
            & aa(nat,$o,nth($o,Bs),N) ) ) ) ).

% bit_horner_sum_bit_word_iff
tff(fact_7126_min__list_Oelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: list(A),Y: A] :
          ( ( min_list(A,Xb) = Y )
         => ( ! [X2: A,Xs2: list(A)] :
                ( ( Xb = aa(list(A),list(A),cons(A,X2),Xs2) )
               => ( Y != aa(list(A),A,case_list(A,A,X2,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_amm(A,fun(list(A),fun(A,fun(list(A),A))),X2),Xs2)),Xs2) ) )
           => ~ ( ( Xb = nil(A) )
               => ( Y != undefined(A) ) ) ) ) ) ).

% min_list.elims
tff(fact_7127_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] :
          aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( aa(nat,$o,ord_less_eq(nat,M),N)
                & bit_se6407376104438227557le_bit(A,type2(A),N) ))),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,nat,minus_minus(nat,N),M))) ) ).

% drop_bit_exp_eq
tff(fact_7128_min__enat__simps_I3_J,axiom,
    ! [Q2: extended_enat] : ord_min(extended_enat,zero_zero(extended_enat),Q2) = zero_zero(extended_enat) ).

% min_enat_simps(3)
tff(fact_7129_min__enat__simps_I2_J,axiom,
    ! [Q2: extended_enat] : ord_min(extended_enat,Q2,zero_zero(extended_enat)) = zero_zero(extended_enat) ).

% min_enat_simps(2)
tff(fact_7130_possible__bit__min,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,ord_min(nat,I,J))
        <=> ( bit_se6407376104438227557le_bit(A,Tyrep,I)
            | bit_se6407376104438227557le_bit(A,Tyrep,J) ) ) ) ).

% possible_bit_min
tff(fact_7131_possible__bit__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat] :
          ( bit_se6407376104438227557le_bit(word(A),type2(word(A)),M)
        <=> aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) ) ) ).

% possible_bit_word
tff(fact_7132_bit__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),one_one(A))),N)
        <=> bit_se6407376104438227557le_bit(A,type2(A),N) ) ) ).

% bit_minus_1_iff
tff(fact_7133_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2)))),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & aa(nat,$o,ord_less(nat,zero_zero(nat)),N) ) ) ) ).

% bit_minus_2_iff
tff(fact_7134_bit__imp__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
         => bit_se6407376104438227557le_bit(A,type2(A),N) ) ) ).

% bit_imp_possible_bit
tff(fact_7135_impossible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat,A3: A] :
          ( ~ bit_se6407376104438227557le_bit(A,type2(A),N)
         => ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N) ) ) ).

% impossible_bit
tff(fact_7136_bit__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B3: A] :
          ( ( A3 = B3 )
        <=> ! [N8: nat] :
              ( bit_se6407376104438227557le_bit(A,type2(A),N8)
             => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N8)
              <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B3),N8) ) ) ) ) ).

% bit_eq_iff
tff(fact_7137_bit__eqI,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,B3: A] :
          ( ! [N3: nat] :
              ( bit_se6407376104438227557le_bit(A,type2(A),N3)
             => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N3)
              <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B3),N3) ) )
         => ( A3 = B3 ) ) ) ).

% bit_eqI
tff(fact_7138_possible__bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Ty: itself(A)] : bit_se6407376104438227557le_bit(A,Ty,zero_zero(nat)) ) ).

% possible_bit_0
tff(fact_7139_possible__bit__less__imp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),I: nat,J: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,I)
         => ( aa(nat,$o,ord_less_eq(nat,J),I)
           => bit_se6407376104438227557le_bit(A,Tyrep,J) ) ) ) ).

% possible_bit_less_imp
tff(fact_7140_bit__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A3)),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N) ) ) ) ).

% bit_not_iff
tff(fact_7141_semiring__bit__operations__class_Obit__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se5668285175392031749et_bit(A),M),A3)),N)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N)
            | ( ( M = N )
              & bit_se6407376104438227557le_bit(A,type2(A),N) ) ) ) ) ).

% semiring_bit_operations_class.bit_set_bit_iff
tff(fact_7142_bit__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),bit_se8732182000553998342ip_bit(A,M,A3)),N)
        <=> ( ( ( M = N )
            <=> ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),N) )
            & bit_se6407376104438227557le_bit(A,type2(A),N) ) ) ) ).

% bit_flip_bit_iff
tff(fact_7143_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),bit_se2239418461657761734s_mask(A,M)),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & aa(nat,$o,ord_less(nat,N),M) ) ) ) ).

% bit_mask_iff
tff(fact_7144_bit__of__int__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [K: int,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(int,A,ring_1_of_int(A),K)),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N) ) ) ) ).

% bit_of_int_iff
tff(fact_7145_bit__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,bit_ri4674362597316999326ke_bit(A,M),A3)),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),ord_min(nat,M,N)) ) ) ) ).

% bit_signed_take_bit_iff
tff(fact_7146_bit__of__nat__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(nat,A,semiring_1_of_nat(A),M)),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),M),N) ) ) ) ).

% bit_of_nat_iff
tff(fact_7147_possible__bit__def,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),N: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,N)
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) != zero_zero(A) ) ) ) ).

% possible_bit_def
tff(fact_7148_bit__minus__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),A3)),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,minus_minus(A,A3),one_one(A))),N) ) ) ) ).

% bit_minus_iff
tff(fact_7149_bit__twiddle__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A)] :
          aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Y),
            aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xb),Y)),
              $ite(aa(word(A),$o,ord_less(word(A),Xb),Y),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),zero_zero(word(A))))) = ord_min(word(A),Xb,Y) ) ).

% bit_twiddle_min
tff(fact_7150_bit__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),A3)),N)
        <=> ( aa(nat,$o,ord_less_eq(nat,M),N)
            & bit_se6407376104438227557le_bit(A,type2(A),N)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),aa(nat,nat,minus_minus(nat,N),M)) ) ) ) ).

% bit_push_bit_iff
tff(fact_7151_fold__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) = zero_zero(A) )
        <=> ~ bit_se6407376104438227557le_bit(A,type2(A),N) ) ) ).

% fold_possible_bit
tff(fact_7152_bit__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & ( M = N ) ) ) ) ).

% bit_exp_iff
tff(fact_7153_bit__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),one_one(nat))
            & ( N = one_one(nat) ) ) ) ) ).

% bit_2_iff
tff(fact_7154_min__list_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Xs: list(A)] : min_list(A,aa(list(A),list(A),cons(A,Xb),Xs)) = aa(list(A),A,case_list(A,A,Xb,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_amm(A,fun(list(A),fun(A,fun(list(A),A))),Xb),Xs)),Xs) ) ).

% min_list.simps
tff(fact_7155_bit__not__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & ( N != M ) ) ) ) ).

% bit_not_exp_iff
tff(fact_7156_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & aa(nat,$o,ord_less_eq(nat,M),N) ) ) ) ).

% bit_minus_exp_iff
tff(fact_7157_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & aa(nat,$o,ord_less(nat,N),M) ) ) ) ).

% bit_mask_sub_iff
tff(fact_7158_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A3: A,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A3)),N)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),aa(nat,nat,minus_minus(nat,N),one_one(nat)))
            & ( N != zero_zero(nat) )
            & bit_se6407376104438227557le_bit(A,type2(A),N) ) ) ) ).

% bit_double_iff
tff(fact_7159_bit__signed__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [W: word(B),N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),ring_1_signed(B,A,W)),N)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),N)
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),ord_min(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),aa(nat,nat,suc,zero_zero(nat))),N)) ) ) ) ).

% bit_signed_iff
tff(fact_7160_min__list_Opelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: list(A),Y: A] :
          ( ( min_list(A,Xb) = Y )
         => ( accp(list(A),min_list_rel(A),Xb)
           => ( ! [X2: A,Xs2: list(A)] :
                  ( ( Xb = aa(list(A),list(A),cons(A,X2),Xs2) )
                 => ( ( Y = aa(list(A),A,case_list(A,A,X2,aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_amm(A,fun(list(A),fun(A,fun(list(A),A))),X2),Xs2)),Xs2) )
                   => ~ accp(list(A),min_list_rel(A),aa(list(A),list(A),cons(A,X2),Xs2)) ) )
             => ~ ( ( Xb = nil(A) )
                 => ( ( Y = undefined(A) )
                   => ~ accp(list(A),min_list_rel(A),nil(A)) ) ) ) ) ) ) ).

% min_list.pelims
tff(fact_7161_word__set__bits__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o)] : bit_bi4170147762399595738t_bits(word(A),P) = groups4207007520872428315er_sum($o,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)),map(nat,$o,P,upt(zero_zero(nat),type_len0_len_of(A,type2(A))))) ) ).

% word_set_bits_def
tff(fact_7162_set__bits__False__eq,axiom,
    ! [A: $tType] :
      ( bit_bi6583157726757044596ension(A)
     => ( bit_bi4170147762399595738t_bits(A,aTP_Lamp_adh(nat,$o)) = zero_zero(A) ) ) ).

% set_bits_False_eq
tff(fact_7163_set__bits__K__False,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( bit_bi4170147762399595738t_bits(word(A),aTP_Lamp_adh(nat,$o)) = zero_zero(word(A)) ) ) ).

% set_bits_K_False
tff(fact_7164_bit__set__bits__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_bi4170147762399595738t_bits(word(A),P)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,P,N) ) ) ) ).

% bit_set_bits_word_iff
tff(fact_7165_word__of__int__conv__set__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int] : aa(int,word(A),ring_1_of_int(word(A)),I) = bit_bi4170147762399595738t_bits(word(A),aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),I)) ) ).

% word_of_int_conv_set_bits
tff(fact_7166_set__bits__conv__set__bits__aux,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Fa: fun(nat,$o)] : bit_bi4170147762399595738t_bits(word(A),Fa) = code_T2661198915054445665ts_aux(A,Fa,type_len0_len_of(A,type2(A)),zero_zero(word(A))) ) ).

% set_bits_conv_set_bits_aux
tff(fact_7167_word__test__bit__set__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Fa: fun(nat,$o),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_bi4170147762399595738t_bits(word(A),Fa)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,Fa,N) ) ) ) ).

% word_test_bit_set_bits
tff(fact_7168_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xb,Xaa)
     => ( ( ? [Uu2: $o,Uv: $o] : Xb = vEBT_Leaf((Uu2),(Uv))
         => ( Xaa = one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList2,Summary) )
             => ( ( Deg = Xaa )
                & $let(
                    n2: nat,
                    n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                    $let(
                      m: nat,
                      m:= aa(nat,nat,minus_minus(nat,Deg),n2),
                      ( ! [X: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList2))
                         => vEBT_VEBT_valid(X,n2) )
                      & vEBT_VEBT_valid(Summary,m)
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                      & case_option($o,product_prod(nat,nat),
                          ( ~ ? [X6: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X6)
                          & ! [X: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList2))
                             => ~ ? [X6: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X6) ) ),
                          product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_amn(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList2),Summary),n2),m)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_7169_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xb,Xaa)
     => ( ( ? [Uu2: $o,Uv: $o] : Xb = vEBT_Leaf((Uu2),(Uv))
         => ( Xaa != one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList2,Summary) )
             => ~ ( ( Deg = Xaa )
                  & $let(
                      n2: nat,
                      n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                      $let(
                        m: nat,
                        m:= aa(nat,nat,minus_minus(nat,Deg),n2),
                        ( ! [X: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList2))
                           => vEBT_VEBT_valid(X,n2) )
                        & vEBT_VEBT_valid(Summary,m)
                        & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                        & case_option($o,product_prod(nat,nat),
                            ( ~ ? [X6: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X6)
                            & ! [X: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList2))
                               => ~ ? [X6: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X6) ) ),
                            product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_amn(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList2),Summary),n2),m)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_7170_int__set__bits__K__False,axiom,
    bit_bi4170147762399595738t_bits(int,aTP_Lamp_adh(nat,$o)) = zero_zero(int) ).

% int_set_bits_K_False
tff(fact_7171_finite__Collect__bounded__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),Q: fun(B,fun(A,$o))] :
      ( finite_finite2(A,collect(A,P))
     => ( finite_finite2(B,collect(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_amo(fun(A,$o),fun(fun(B,fun(A,$o)),fun(B,$o)),P),Q)))
      <=> ! [Y3: A] :
            ( aa(A,$o,P,Y3)
           => finite_finite2(B,collect(B,aa(A,fun(B,$o),aTP_Lamp_aiv(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Q),Y3))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_7172_eq__or__mem__image__simp,axiom,
    ! [A: $tType,B: $tType,Fa: fun(B,A),A3: B,B6: set(B)] : collect(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_amp(fun(B,A),fun(B,fun(set(B),fun(A,$o))),Fa),A3),B6)) = aa(set(A),set(A),insert(A,aa(B,A,Fa,A3)),collect(A,aa(set(B),fun(A,$o),aTP_Lamp_amq(fun(B,A),fun(set(B),fun(A,$o)),Fa),B6))) ).

% eq_or_mem_image_simp
tff(fact_7173_int__set__bits__K__True,axiom,
    bit_bi4170147762399595738t_bits(int,aTP_Lamp_adg(nat,$o)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% int_set_bits_K_True
tff(fact_7174_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,Fa: fun(B,A),P: fun(B,$o)] : collect(A,aa(fun(B,$o),fun(A,$o),aTP_Lamp_amr(fun(B,A),fun(fun(B,$o),fun(A,$o)),Fa),P)) = aa(set(B),set(A),image(B,A,Fa),collect(B,P)) ).

% setcompr_eq_image
tff(fact_7175_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,Fa: fun(B,A),A4: set(B)] : collect(A,aa(set(B),fun(A,$o),aTP_Lamp_amq(fun(B,A),fun(set(B),fun(A,$o)),Fa),A4)) = aa(set(B),set(A),image(B,A,Fa),A4) ).

% Setcompr_eq_image
tff(fact_7176_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,Fa: fun(B,A)] : collect(A,aTP_Lamp_ams(fun(B,A),fun(A,$o),Fa)) = aa(set(B),set(A),image(B,A,Fa),top_top(set(B))) ).

% full_SetCompr_eq
tff(fact_7177_fs__contract,axiom,
    ! [A: $tType,B: $tType,C: $tType,Fa: fun(A,fun(B,C)),S3: set(C)] : aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),collect(product_prod(A,B),aa(set(C),fun(product_prod(A,B),$o),aTP_Lamp_amt(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),$o)),Fa),S3))) = collect(A,aa(set(C),fun(A,$o),aTP_Lamp_amu(fun(A,fun(B,C)),fun(set(C),fun(A,$o)),Fa),S3)) ).

% fs_contract
tff(fact_7178_set__Cons__def,axiom,
    ! [A: $tType,A4: set(A),XS: set(list(A))] : set_Cons(A,A4,XS) = collect(list(A),aa(set(list(A)),fun(list(A),$o),aTP_Lamp_amv(set(A),fun(set(list(A)),fun(list(A),$o)),A4),XS)) ).

% set_Cons_def
tff(fact_7179_some__theI,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] :
      ( ? [A6: A,X_13: B] : aa(B,$o,aa(A,fun(B,$o),P,A6),X_13)
     => ( ! [B1: B,B22: B] :
            ( ? [A2: A] : aa(B,$o,aa(A,fun(B,$o),P,A2),B1)
           => ( ? [A2: A] : aa(B,$o,aa(A,fun(B,$o),P,A2),B22)
             => ( B1 = B22 ) ) )
       => aa(B,$o,aa(A,fun(B,$o),P,fChoice(A,aTP_Lamp_amw(fun(A,fun(B,$o)),fun(A,$o),P))),the(B,aTP_Lamp_amx(fun(A,fun(B,$o)),fun(B,$o),P))) ) ) ).

% some_theI
tff(fact_7180_ran__def,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A))] : ran(B,A,M) = collect(A,aTP_Lamp_amy(fun(B,option(A)),fun(A,$o),M)) ).

% ran_def
tff(fact_7181_finite__image__set,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),Fa: fun(A,B)] :
      ( finite_finite2(A,collect(A,P))
     => finite_finite2(B,collect(B,aa(fun(A,B),fun(B,$o),aTP_Lamp_amz(fun(A,$o),fun(fun(A,B),fun(B,$o)),P),Fa))) ) ).

% finite_image_set
tff(fact_7182_finite__image__set2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,$o),Q: fun(B,$o),Fa: fun(A,fun(B,C))] :
      ( finite_finite2(A,collect(A,P))
     => ( finite_finite2(B,collect(B,Q))
       => finite_finite2(C,collect(C,aa(fun(A,fun(B,C)),fun(C,$o),aa(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o)),aTP_Lamp_ana(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),P),Q),Fa))) ) ) ).

% finite_image_set2
tff(fact_7183_admissible__ball,axiom,
    ! [A: $tType,B: $tType,A4: set(A),Lub: fun(set(B),B),Ord: fun(B,fun(B,$o)),P: fun(B,fun(A,$o))] :
      ( ! [Y2: A] :
          ( member(A,Y2,A4)
         => comple1908693960933563346ssible(B,Lub,Ord,aa(A,fun(B,$o),aTP_Lamp_aiv(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),Y2)) )
     => comple1908693960933563346ssible(B,Lub,Ord,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_anb(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A4),P)) ) ).

% admissible_ball
tff(fact_7184_set__map__filter,axiom,
    ! [A: $tType,B: $tType,G: fun(B,option(A)),Xs: list(B)] : set2(A,map_filter(B,A,G,Xs)) = collect(A,aa(list(B),fun(A,$o),aTP_Lamp_anc(fun(B,option(A)),fun(list(B),fun(A,$o)),G),Xs)) ).

% set_map_filter
tff(fact_7185_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : set2(A,Xs) = collect(A,aTP_Lamp_and(list(A),fun(A,$o),Xs)) ).

% set_conv_nth
tff(fact_7186_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary2: vEBT_VEBT,Deg4: nat] :
      ( vEBT_VEBT_valid(vEBT_Node(Mima2,Deg2,TreeList,Summary2),Deg4)
    <=> ( ( Deg2 = Deg4 )
        & $let(
            n2: nat,
            n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))),
            $let(
              m: nat,
              m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
              ( ! [X: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList))
                 => vEBT_VEBT_valid(X,n2) )
              & vEBT_VEBT_valid(Summary2,m)
              & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
              & case_option($o,product_prod(nat,nat),
                  ( ~ ? [X6: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X6)
                  & ! [X: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList))
                     => ~ ? [X6: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X6) ) ),
                  product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_amn(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList),Summary2),n2),m)),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_7187_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_valid(Xb,Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv: $o] : Xb = vEBT_Leaf((Uu2),(Uv))
         => ( (Y)
          <=> ( Xaa != one_one(nat) ) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg,TreeList2,Summary) )
             => ( (Y)
              <=> ~ ( ( Deg = Xaa )
                    & $let(
                        n2: nat,
                        n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        $let(
                          m: nat,
                          m:= aa(nat,nat,minus_minus(nat,Deg),n2),
                          ( ! [X: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList2))
                             => vEBT_VEBT_valid(X,n2) )
                          & vEBT_VEBT_valid(Summary,m)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X6: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X6)
                              & ! [X: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList2))
                                 => ~ ? [X6: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X6) ) ),
                              product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_amn(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList2),Summary),n2),m)),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_7188_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_valid(Xb,Xaa)
      <=> (Y) )
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv)) )
             => ( ( (Y)
                <=> ( Xaa = one_one(nat) ) )
               => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv))),Xaa)) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList2,Summary) )
               => ( ( (Y)
                  <=> ( ( Deg = Xaa )
                      & $let(
                          n2: nat,
                          n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $let(
                            m: nat,
                            m:= aa(nat,nat,minus_minus(nat,Deg),n2),
                            ( ! [X: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList2))
                               => vEBT_VEBT_valid(X,n2) )
                            & vEBT_VEBT_valid(Summary,m)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X6: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X6)
                                & ! [X: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList2))
                                   => ~ ? [X6: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X6) ) ),
                                product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_amn(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList2),Summary),n2),m)),Mima) ) ) ) ) )
                 => ~ accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg,TreeList2,Summary)),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_7189_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv))),Xaa))
               => ( Xaa != one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList2,Summary) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg,TreeList2,Summary)),Xaa))
                 => ~ ( ( Deg = Xaa )
                      & $let(
                          n2: nat,
                          n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $let(
                            m: nat,
                            m:= aa(nat,nat,minus_minus(nat,Deg),n2),
                            ( ! [X: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList2))
                               => vEBT_VEBT_valid(X,n2) )
                            & vEBT_VEBT_valid(Summary,m)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X6: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X6)
                                & ! [X: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList2))
                                   => ~ ? [X6: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X6) ) ),
                                product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_amn(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList2),Summary),n2),m)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_7190_fi__match__entails,axiom,
    ! [M: list(product_prod(assn,assn))] :
      ( ! [X2: product_prod(assn,assn)] :
          ( member(product_prod(assn,assn),X2,set2(product_prod(assn,assn),M))
         => aa(product_prod(assn,assn),$o,product_case_prod(assn,assn,$o,entails),X2) )
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,foldr(assn,assn,times_times(assn),map(product_prod(assn,assn),assn,product_fst(assn,assn),M),one_one(assn))),foldr(assn,assn,times_times(assn),map(product_prod(assn,assn),assn,product_snd(assn,assn),M),one_one(assn))) ) ).

% fi_match_entails
tff(fact_7191_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xb,Xaa)
     => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Uu2: $o,Uv: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv)) )
             => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((Uu2),(Uv))),Xaa))
               => ( Xaa = one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg: nat,TreeList2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg,TreeList2,Summary) )
               => ( accp(product_prod(vEBT_VEBT,nat),vEBT_VEBT_valid_rel,aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Mima,Deg,TreeList2,Summary)),Xaa))
                 => ( ( Deg = Xaa )
                    & $let(
                        n2: nat,
                        n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        $let(
                          m: nat,
                          m:= aa(nat,nat,minus_minus(nat,Deg),n2),
                          ( ! [X: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList2))
                             => vEBT_VEBT_valid(X,n2) )
                          & vEBT_VEBT_valid(Summary,m)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X6: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X6)
                              & ! [X: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X,set2(vEBT_VEBT,TreeList2))
                                 => ~ ? [X6: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X6) ) ),
                              product_case_prod(nat,nat,$o,aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_amn(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeList2),Summary),n2),m)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_7192_set__bits__int__def,axiom,
    ! [Fa: fun(nat,$o)] :
      bit_bi4170147762399595738t_bits(int,Fa) = $ite(
        ? [N8: nat] :
        ! [M6: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N8),M6)
         => ( aa(nat,$o,Fa,M6) = aa(nat,$o,Fa,N8) ) ),
        $let(
          n2: nat,
          n2:= ord_Least(nat,aTP_Lamp_ane(fun(nat,$o),fun(nat,$o),Fa)),
          aa(int,int,bit_ri4674362597316999326ke_bit(int,n2),groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),map(nat,$o,Fa,upt(zero_zero(nat),aa(nat,nat,suc,n2))))) ),
        zero_zero(int) ) ).

% set_bits_int_def
tff(fact_7193_set__bits__int__unfold_H,axiom,
    ! [Fa: fun(nat,$o)] :
      bit_bi4170147762399595738t_bits(int,Fa) = $ite(
        ? [N8: nat] :
        ! [N12: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N8),N12)
         => ~ aa(nat,$o,Fa,N12) ),
        groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),map(nat,$o,Fa,upt(zero_zero(nat),ord_Least(nat,aTP_Lamp_anf(fun(nat,$o),fun(nat,$o),Fa))))),
        $ite(
          ? [N8: nat] :
          ! [N12: nat] :
            ( aa(nat,$o,ord_less_eq(nat,N8),N12)
           => aa(nat,$o,Fa,N12) ),
          $let(
            n2: nat,
            n2:= ord_Least(nat,aTP_Lamp_ang(fun(nat,$o),fun(nat,$o),Fa)),
            aa(int,int,bit_ri4674362597316999326ke_bit(int,n2),groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),append($o,map(nat,$o,Fa,upt(zero_zero(nat),n2)),aa(list($o),list($o),cons($o,$true),nil($o))))) ),
          zero_zero(int) ) ) ).

% set_bits_int_unfold'
tff(fact_7194_Least__eq__0,axiom,
    ! [P: fun(nat,$o)] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ord_Least(nat,P) = zero_zero(nat) ) ) ).

% Least_eq_0
tff(fact_7195_Least__Suc,axiom,
    ! [P: fun(nat,$o),N: nat] :
      ( aa(nat,$o,P,N)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_anh(fun(nat,$o),fun(nat,$o),P))) ) ) ) ).

% Least_Suc
tff(fact_7196_Least__Suc2,axiom,
    ! [P: fun(nat,$o),N: nat,Q: fun(nat,$o),M: nat] :
      ( aa(nat,$o,P,N)
     => ( aa(nat,$o,Q,M)
       => ( ~ aa(nat,$o,P,zero_zero(nat))
         => ( ! [K2: nat] :
                ( aa(nat,$o,P,aa(nat,nat,suc,K2))
              <=> aa(nat,$o,Q,K2) )
           => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_7197_LeastI,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),K: A] :
          ( aa(A,$o,P,K)
         => aa(A,$o,P,ord_Least(A,P)) ) ) ).

% LeastI
tff(fact_7198_Least__le,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),K: A] :
          ( aa(A,$o,P,K)
         => aa(A,$o,ord_less_eq(A,ord_Least(A,P)),K) ) ) ).

% Least_le
tff(fact_7199_not__less__Least,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [K: A,P: fun(A,$o)] :
          ( aa(A,$o,ord_less(A,K),ord_Least(A,P))
         => ~ aa(A,$o,P,K) ) ) ).

% not_less_Least
tff(fact_7200_LeastI2__wellorder__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( ? [X_13: A] : aa(A,$o,P,X_13)
         => ( ! [A2: A] :
                ( aa(A,$o,P,A2)
               => ( ! [B5: A] :
                      ( aa(A,$o,P,B5)
                     => aa(A,$o,ord_less_eq(A,A2),B5) )
                 => aa(A,$o,Q,A2) ) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2_wellorder_ex
tff(fact_7201_LeastI2__wellorder,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A3: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,A3)
         => ( ! [A2: A] :
                ( aa(A,$o,P,A2)
               => ( ! [B5: A] :
                      ( aa(A,$o,P,B5)
                     => aa(A,$o,ord_less_eq(A,A2),B5) )
                 => aa(A,$o,Q,A2) ) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2_wellorder
tff(fact_7202_Least__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y2: A] :
                ( aa(A,$o,P,Y2)
               => aa(A,$o,ord_less_eq(A,Xb),Y2) )
           => ( ord_Least(A,P) = Xb ) ) ) ) ).

% Least_equality
tff(fact_7203_LeastI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y2: A] :
                ( aa(A,$o,P,Y2)
               => aa(A,$o,ord_less_eq(A,Xb),Y2) )
           => ( ! [X2: A] :
                  ( aa(A,$o,P,X2)
                 => ( ! [Y4: A] :
                        ( aa(A,$o,P,Y4)
                       => aa(A,$o,ord_less_eq(A,X2),Y4) )
                   => aa(A,$o,Q,X2) ) )
             => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ) ).

% LeastI2_order
tff(fact_7204_Least1__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Z: A] :
          ( ? [X3: A] :
              ( aa(A,$o,P,X3)
              & ! [Y2: A] :
                  ( aa(A,$o,P,Y2)
                 => aa(A,$o,ord_less_eq(A,X3),Y2) )
              & ! [Y2: A] :
                  ( ( aa(A,$o,P,Y2)
                    & ! [Ya2: A] :
                        ( aa(A,$o,P,Ya2)
                       => aa(A,$o,ord_less_eq(A,Y2),Ya2) ) )
                 => ( Y2 = X3 ) ) )
         => ( aa(A,$o,P,Z)
           => aa(A,$o,ord_less_eq(A,ord_Least(A,P)),Z) ) ) ) ).

% Least1_le
tff(fact_7205_Least1I,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] :
          ( ? [X3: A] :
              ( aa(A,$o,P,X3)
              & ! [Y2: A] :
                  ( aa(A,$o,P,Y2)
                 => aa(A,$o,ord_less_eq(A,X3),Y2) )
              & ! [Y2: A] :
                  ( ( aa(A,$o,P,Y2)
                    & ! [Ya2: A] :
                        ( aa(A,$o,P,Ya2)
                       => aa(A,$o,ord_less_eq(A,Y2),Ya2) ) )
                 => ( Y2 = X3 ) ) )
         => aa(A,$o,P,ord_Least(A,P)) ) ) ).

% Least1I
tff(fact_7206_Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S3: set(A),P: fun(A,$o)] : bleast(A,S3,P) = ord_Least(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ani(set(A),fun(fun(A,$o),fun(A,$o)),S3),P)) ) ).

% Bleast_def
tff(fact_7207_abort__Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S3: set(A),P: fun(A,$o)] : abort_Bleast(A,S3,P) = ord_Least(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ani(set(A),fun(fun(A,$o),fun(A,$o)),S3),P)) ) ).

% abort_Bleast_def
tff(fact_7208_Least__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [P: fun(A,$o)] : ord_Least(A,P) = the(A,aTP_Lamp_anj(fun(A,$o),fun(A,$o),P)) ) ).

% Least_def
tff(fact_7209_FI__RESULT__def,axiom,
    ! [M8: list(product_prod(assn,assn)),UP: assn,UQ: assn,F5: assn] :
      ( fI_RESULT(M8,UP,UQ,F5)
    <=> ( ! [X: product_prod(assn,assn)] :
            ( member(product_prod(assn,assn),X,set2(product_prod(assn,assn),M8))
           => aa(product_prod(assn,assn),$o,product_case_prod(assn,assn,$o,entails),X) )
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),foldr(assn,assn,times_times(assn),map(product_prod(assn,assn),assn,product_fst(assn,assn),M8),one_one(assn))),UP)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),foldr(assn,assn,times_times(assn),map(product_prod(assn,assn),assn,product_snd(assn,assn),M8),one_one(assn))),UQ)),F5)) ) ) ).

% FI_RESULT_def
tff(fact_7210_FI__def,axiom,
    ! [M: list(product_prod(assn,assn)),P2: assn,Q2: assn,Up: assn,Uq: assn,Fa: assn] :
      ( fi(M,P2,Q2,Up,Uq,Fa)
    <=> ( ! [X: product_prod(assn,assn)] :
            ( member(product_prod(assn,assn),X,set2(product_prod(assn,assn),M))
           => aa(product_prod(assn,assn),$o,product_case_prod(assn,assn,$o,entails),X) )
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),foldr(assn,assn,times_times(assn),map(product_prod(assn,assn),assn,product_fst(assn,assn),M),one_one(assn))),P2)),Up)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),foldr(assn,assn,times_times(assn),map(product_prod(assn,assn),assn,product_snd(assn,assn),M),one_one(assn))),Q2)),Uq)),Fa)) ) ) ).

% FI_def
tff(fact_7211_lexn__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),N: nat] : lexn(A,R2,N) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),$o,aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_ank(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),R2),N))) ).

% lexn_conv
tff(fact_7212_bin__last__set__bits,axiom,
    ! [Fa: fun(nat,$o)] :
      ( bit_wf_set_bits_int(Fa)
     => ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),bit_bi4170147762399595738t_bits(int,Fa))
      <=> aa(nat,$o,Fa,zero_zero(nat)) ) ) ).

% bin_last_set_bits
tff(fact_7213_wf__set__bits__int__Suc,axiom,
    ! [Fa: fun(nat,$o)] :
      ( bit_wf_set_bits_int(aTP_Lamp_anh(fun(nat,$o),fun(nat,$o),Fa))
    <=> bit_wf_set_bits_int(Fa) ) ).

% wf_set_bits_int_Suc
tff(fact_7214_wf__set__bits__int__simps,axiom,
    ! [Fa: fun(nat,$o)] :
      ( bit_wf_set_bits_int(Fa)
    <=> ? [N8: nat] :
          ( ! [N12: nat] :
              ( aa(nat,$o,ord_less_eq(nat,N8),N12)
             => ~ aa(nat,$o,Fa,N12) )
          | ! [N12: nat] :
              ( aa(nat,$o,ord_less_eq(nat,N8),N12)
             => aa(nat,$o,Fa,N12) ) ) ) ).

% wf_set_bits_int_simps
tff(fact_7215_zeros,axiom,
    ! [N: nat,Fa: fun(nat,$o)] :
      ( ! [N7: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N),N7)
         => ~ aa(nat,$o,Fa,N7) )
     => bit_wf_set_bits_int(Fa) ) ).

% zeros
tff(fact_7216_wf__set__bits__int_Osimps,axiom,
    ! [Fa: fun(nat,$o)] :
      ( bit_wf_set_bits_int(Fa)
    <=> ( ? [N8: nat] :
          ! [N12: nat] :
            ( aa(nat,$o,ord_less_eq(nat,N8),N12)
           => ~ aa(nat,$o,Fa,N12) )
        | ? [N8: nat] :
          ! [N12: nat] :
            ( aa(nat,$o,ord_less_eq(nat,N8),N12)
           => aa(nat,$o,Fa,N12) ) ) ) ).

% wf_set_bits_int.simps
tff(fact_7217_wf__set__bits__int_Ocases,axiom,
    ! [Fa: fun(nat,$o)] :
      ( bit_wf_set_bits_int(Fa)
     => ( ! [N3: nat] :
            ~ ! [N6: nat] :
                ( aa(nat,$o,ord_less_eq(nat,N3),N6)
               => ~ aa(nat,$o,Fa,N6) )
       => ~ ! [N3: nat] :
              ~ ! [N6: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,N3),N6)
                 => aa(nat,$o,Fa,N6) ) ) ) ).

% wf_set_bits_int.cases
tff(fact_7218_ones,axiom,
    ! [N: nat,Fa: fun(nat,$o)] :
      ( ! [N7: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N),N7)
         => aa(nat,$o,Fa,N7) )
     => bit_wf_set_bits_int(Fa) ) ).

% ones
tff(fact_7219_wf__set__bits__int__const,axiom,
    ! [B3: $o] : bit_wf_set_bits_int(aTP_Lamp_anl($o,fun(nat,$o),(B3))) ).

% wf_set_bits_int_const
tff(fact_7220_lexn_Osimps_I1_J,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lexn(A,R2,zero_zero(nat)) = bot_bot(set(product_prod(list(A),list(A)))) ).

% lexn.simps(1)
tff(fact_7221_lexn__length,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),N: nat] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lexn(A,R2,N))
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = N )
        & ( aa(list(A),nat,size_size(list(A)),Ys) = N ) ) ) ).

% lexn_length
tff(fact_7222_pairself__image__eq,axiom,
    ! [A: $tType,B: $tType,Fa: fun(B,A),P: fun(B,fun(B,$o))] : aa(set(product_prod(B,B)),set(product_prod(A,A)),image(product_prod(B,B),product_prod(A,A),pairself(B,A,Fa)),collect(product_prod(B,B),product_case_prod(B,B,$o,P))) = collect(product_prod(A,A),aa(fun(B,fun(B,$o)),fun(product_prod(A,A),$o),aTP_Lamp_anm(fun(B,A),fun(fun(B,fun(B,$o)),fun(product_prod(A,A),$o)),Fa),P)) ).

% pairself_image_eq
tff(fact_7223_lex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lex(A,R2) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),$o,aTP_Lamp_ann(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lex_conv
tff(fact_7224_Cons__in__lex,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A,Ys: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,Xb),Xs)),aa(list(A),list(A),cons(A,Y),Ys)),lex(A,R2))
    <=> ( ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),R2)
          & ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) ) )
        | ( ( Xb = Y )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lex(A,R2)) ) ) ) ).

% Cons_in_lex
tff(fact_7225_pairself_Osimps,axiom,
    ! [A: $tType,B: $tType,Fa: fun(B,A),A3: B,B3: B] : aa(product_prod(B,B),product_prod(A,A),pairself(B,A,Fa),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A3),B3)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Fa,A3)),aa(B,A,Fa,B3)) ).

% pairself.simps
tff(fact_7226_pairself_Oelims,axiom,
    ! [A: $tType,B: $tType,Xb: fun(B,A),Xaa: product_prod(B,B),Y: product_prod(A,A)] :
      ( ( aa(product_prod(B,B),product_prod(A,A),pairself(B,A,Xb),Xaa) = Y )
     => ~ ! [A2: B,B2: B] :
            ( ( Xaa = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),B2) )
           => ( Y != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Xb,A2)),aa(B,A,Xb,B2)) ) ) ) ).

% pairself.elims
tff(fact_7227_lex__append__leftD,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X2: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2),R2)
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Ys)),append(A,Xs,Zs)),lex(A,R2))
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lex(A,R2)) ) ) ).

% lex_append_leftD
tff(fact_7228_lex__append__left__iff,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ! [X2: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2),R2)
     => ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Ys)),append(A,Xs,Zs)),lex(A,R2))
      <=> member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ys),Zs),lex(A,R2)) ) ) ).

% lex_append_left_iff
tff(fact_7229_lex__append__rightI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R2: set(product_prod(A,A)),Vs: list(A),Us: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Ys),lex(A,R2))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Us) )
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Xs,Us)),append(A,Ys,Vs)),lex(A,R2)) ) ) ).

% lex_append_rightI
tff(fact_7230_int__set__bits__unfold__BIT,axiom,
    ! [Fa: fun(nat,$o)] :
      ( bit_wf_set_bits_int(Fa)
     => ( bit_bi4170147762399595738t_bits(int,Fa) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,Fa,zero_zero(nat)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),bit_bi4170147762399595738t_bits(int,comp(nat,$o,nat,Fa,suc)))) ) ) ).

% int_set_bits_unfold_BIT
tff(fact_7231_less__eq__char__simp,axiom,
    ! [B0: $o,B12: $o,B23: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C62: $o,C7: $o] :
      ( aa(char,$o,ord_less_eq(char,aa($o,char,char2((B0),(B12),(B23),(B32),(B42),(B52),(B62)),(B72))),aa($o,char,char2((C0),(C1),(C22),(C32),(C42),(C52),(C62)),(C7)))
    <=> aa(nat,$o,ord_less_eq(nat,foldr($o,nat,aTP_Lamp_ano($o,fun(nat,nat)),aa(list($o),list($o),cons($o,(B0)),aa(list($o),list($o),cons($o,(B12)),aa(list($o),list($o),cons($o,(B23)),aa(list($o),list($o),cons($o,(B32)),aa(list($o),list($o),cons($o,(B42)),aa(list($o),list($o),cons($o,(B52)),aa(list($o),list($o),cons($o,(B62)),aa(list($o),list($o),cons($o,(B72)),nil($o))))))))),zero_zero(nat))),foldr($o,nat,aTP_Lamp_ano($o,fun(nat,nat)),aa(list($o),list($o),cons($o,(C0)),aa(list($o),list($o),cons($o,(C1)),aa(list($o),list($o),cons($o,(C22)),aa(list($o),list($o),cons($o,(C32)),aa(list($o),list($o),cons($o,(C42)),aa(list($o),list($o),cons($o,(C52)),aa(list($o),list($o),cons($o,(C62)),aa(list($o),list($o),cons($o,(C7)),nil($o))))))))),zero_zero(nat))) ) ).

% less_eq_char_simp
tff(fact_7232_length__filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),Fa: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),filter2(A,P,map(B,A,Fa,Xs))) = aa(list(B),nat,size_size(list(B)),filter2(B,comp(A,$o,B,P,Fa),Xs)) ).

% length_filter_map
tff(fact_7233_bin__rest__set__bits,axiom,
    ! [Fa: fun(nat,$o)] :
      ( bit_wf_set_bits_int(Fa)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_bi4170147762399595738t_bits(int,Fa)),aa(num,int,numeral_numeral(int),bit0(one2))) = bit_bi4170147762399595738t_bits(int,comp(nat,$o,nat,Fa,suc)) ) ) ).

% bin_rest_set_bits
tff(fact_7234_less__char__simp,axiom,
    ! [B0: $o,B12: $o,B23: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o,C0: $o,C1: $o,C22: $o,C32: $o,C42: $o,C52: $o,C62: $o,C7: $o] :
      ( aa(char,$o,ord_less(char,aa($o,char,char2((B0),(B12),(B23),(B32),(B42),(B52),(B62)),(B72))),aa($o,char,char2((C0),(C1),(C22),(C32),(C42),(C52),(C62)),(C7)))
    <=> aa(nat,$o,ord_less(nat,foldr($o,nat,aTP_Lamp_ano($o,fun(nat,nat)),aa(list($o),list($o),cons($o,(B0)),aa(list($o),list($o),cons($o,(B12)),aa(list($o),list($o),cons($o,(B23)),aa(list($o),list($o),cons($o,(B32)),aa(list($o),list($o),cons($o,(B42)),aa(list($o),list($o),cons($o,(B52)),aa(list($o),list($o),cons($o,(B62)),aa(list($o),list($o),cons($o,(B72)),nil($o))))))))),zero_zero(nat))),foldr($o,nat,aTP_Lamp_ano($o,fun(nat,nat)),aa(list($o),list($o),cons($o,(C0)),aa(list($o),list($o),cons($o,(C1)),aa(list($o),list($o),cons($o,(C22)),aa(list($o),list($o),cons($o,(C32)),aa(list($o),list($o),cons($o,(C42)),aa(list($o),list($o),cons($o,(C52)),aa(list($o),list($o),cons($o,(C62)),aa(list($o),list($o),cons($o,(C7)),nil($o))))))))),zero_zero(nat))) ) ).

% less_char_simp
tff(fact_7235_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat,A3: A] : aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),N),A3)) = comp(nat,$o,nat,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),aa(nat,fun(nat,nat),plus_plus(nat),N)) ) ).

% bit_drop_bit_eq
tff(fact_7236_sum__comp__morphism,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( comm_monoid_add(B)
        & comm_monoid_add(A) )
     => ! [H: fun(B,A),G: fun(C,B),A4: set(C)] :
          ( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
         => ( ! [X2: B,Y2: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),Y2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X2)),aa(B,A,H,Y2))
           => ( aa(set(C),A,groups7311177749621191930dd_sum(C,A,comp(B,A,C,H,G)),A4) = aa(B,A,H,aa(set(C),B,groups7311177749621191930dd_sum(C,B,G),A4)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_7237_summable__inverse__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(nat,A),C2: A] :
          ( summable(A,comp(A,A,nat,inverse_inverse(A),Fa))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_anp(fun(nat,A),fun(A,fun(nat,A)),Fa),C2)) ) ) ).

% summable_inverse_divide
tff(fact_7238_sum_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [A4: set(A),H: fun(A,B),G: fun(B,C)] :
          ( finite_finite2(A,A4)
         => ( ! [X2: A,Y2: A] :
                ( member(A,X2,A4)
               => ( member(A,Y2,A4)
                 => ( ( X2 != Y2 )
                   => ( ( aa(A,B,H,X2) = aa(A,B,H,Y2) )
                     => ( aa(B,C,G,aa(A,B,H,X2)) = zero_zero(C) ) ) ) ) )
           => ( aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),aa(set(A),set(B),image(A,B,H),A4)) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,comp(B,C,A,G,H)),A4) ) ) ) ) ).

% sum.reindex_nontrivial
tff(fact_7239_prod_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A4: set(A),H: fun(A,B),G: fun(B,C)] :
          ( finite_finite2(A,A4)
         => ( ! [X2: A,Y2: A] :
                ( member(A,X2,A4)
               => ( member(A,Y2,A4)
                 => ( ( X2 != Y2 )
                   => ( ( aa(A,B,H,X2) = aa(A,B,H,Y2) )
                     => ( aa(B,C,G,aa(A,B,H,X2)) = one_one(C) ) ) ) ) )
           => ( groups7121269368397514597t_prod(B,C,G,aa(set(A),set(B),image(A,B,H),A4)) = groups7121269368397514597t_prod(A,C,comp(B,C,A,G,H),A4) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_7240_sum__image__le,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),G: fun(C,B),Fa: fun(A,C)] :
          ( finite_finite2(A,I5)
         => ( ! [I2: A] :
                ( member(A,I2,I5)
               => aa(B,$o,ord_less_eq(B,zero_zero(B)),aa(C,B,G,aa(A,C,Fa,I2))) )
           => aa(B,$o,ord_less_eq(B,aa(set(C),B,groups7311177749621191930dd_sum(C,B,G),aa(set(A),set(C),image(A,C,Fa),I5))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,comp(C,B,A,G,Fa)),I5)) ) ) ) ).

% sum_image_le
tff(fact_7241_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,suc)),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_7242_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,zero_zero(nat))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,suc)),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_7243_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,suc),set_or7035219750837199246ssThan(nat,zero_zero(nat),N))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_7244_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,N))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,suc),set_or1337092689740270186AtMost(nat,zero_zero(nat),N))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_7245_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,N),M))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_7246_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N)) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),M)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,N),M))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_7247_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aTP_Lamp_ajh(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,N)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_7248_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aTP_Lamp_ajh(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_7249_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aTP_Lamp_ajh(nat,nat)),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,N)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_7250_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,N: nat] : groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aTP_Lamp_ajh(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,N))) = groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_7251_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),M))),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_7252_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),M)),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,minus_minus(nat,N),M))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_7253_integer__of__char__code,axiom,
    ! [B0: $o,B12: $o,B23: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : integer_of_char(aa($o,char,char2((B0),(B12),(B23),(B32),(B42),(B52),(B62)),(B72))) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B72))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B62)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B52)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B42)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B32)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B23)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B12)))),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),aa($o,code_integer,zero_neq_one_of_bool(code_integer),(B0))) ).

% integer_of_char_code
tff(fact_7254_of__char__Char,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [B0: $o,B12: $o,B23: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] : aa(char,A,comm_s6883823935334413003f_char(A),aa($o,char,char2((B0),(B12),(B23),(B32),(B42),(B52),(B62)),(B72))) = groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),aa(list($o),list($o),cons($o,(B0)),aa(list($o),list($o),cons($o,(B12)),aa(list($o),list($o),cons($o,(B23)),aa(list($o),list($o),cons($o,(B32)),aa(list($o),list($o),cons($o,(B42)),aa(list($o),list($o),cons($o,(B52)),aa(list($o),list($o),cons($o,(B62)),aa(list($o),list($o),cons($o,(B72)),nil($o)))))))))) ) ).

% of_char_Char
tff(fact_7255_of__char__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [C2: char] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(char,A,comm_s6883823935334413003f_char(A),C2)),aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ).

% of_char_mod_256
tff(fact_7256_less__char__def,axiom,
    ! [C1: char,C22: char] :
      ( aa(char,$o,ord_less(char,C1),C22)
    <=> aa(nat,$o,ord_less(nat,aa(char,nat,comm_s6883823935334413003f_char(nat),C1)),aa(char,nat,comm_s6883823935334413003f_char(nat),C22)) ) ).

% less_char_def
tff(fact_7257_snd__diag__fst,axiom,
    ! [B: $tType,A: $tType] : comp(product_prod(A,A),A,product_prod(A,B),product_snd(A,A),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_anq(A,product_prod(A,A)),product_fst(A,B))) = product_fst(A,B) ).

% snd_diag_fst
tff(fact_7258_fst__diag__snd,axiom,
    ! [B: $tType,A: $tType] : comp(product_prod(B,B),B,product_prod(A,B),product_fst(B,B),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_anr(B,product_prod(B,B)),product_snd(A,B))) = product_snd(A,B) ).

% fst_diag_snd
tff(fact_7259_fst__diag__fst,axiom,
    ! [B: $tType,A: $tType] : comp(product_prod(A,A),A,product_prod(A,B),product_fst(A,A),comp(A,product_prod(A,A),product_prod(A,B),aTP_Lamp_anq(A,product_prod(A,A)),product_fst(A,B))) = product_fst(A,B) ).

% fst_diag_fst
tff(fact_7260_snd__diag__snd,axiom,
    ! [B: $tType,A: $tType] : comp(product_prod(B,B),B,product_prod(A,B),product_snd(B,B),comp(B,product_prod(B,B),product_prod(A,B),aTP_Lamp_anr(B,product_prod(B,B)),product_snd(A,B))) = product_snd(A,B) ).

% snd_diag_snd
tff(fact_7261_nat__of__char__less__256,axiom,
    ! [C2: char] : aa(nat,$o,ord_less(nat,aa(char,nat,comm_s6883823935334413003f_char(nat),C2)),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ).

% nat_of_char_less_256
tff(fact_7262_case__prod__comp,axiom,
    ! [D2: $tType,A: $tType,C: $tType,B: $tType,Fa: fun(D2,fun(C,A)),G: fun(B,D2),Xb: product_prod(B,C)] : aa(product_prod(B,C),A,product_case_prod(B,C,A,comp(D2,fun(C,A),B,Fa,G)),Xb) = aa(C,A,aa(D2,fun(C,A),Fa,aa(B,D2,G,aa(product_prod(B,C),B,product_fst(B,C),Xb))),aa(product_prod(B,C),C,product_snd(B,C),Xb)) ).

% case_prod_comp
tff(fact_7263_less__eq__char__def,axiom,
    ! [C1: char,C22: char] :
      ( aa(char,$o,ord_less_eq(char,C1),C22)
    <=> aa(nat,$o,ord_less_eq(nat,aa(char,nat,comm_s6883823935334413003f_char(nat),C1)),aa(char,nat,comm_s6883823935334413003f_char(nat),C22)) ) ).

% less_eq_char_def
tff(fact_7264_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A4: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A4) = finite_fold(B,A,comp(A,fun(A,A),B,plus_plus(A),G),zero_zero(A),A4) ) ).

% sum.eq_fold
tff(fact_7265_range__nat__of__char,axiom,
    aa(set(char),set(nat),image(char,nat,comm_s6883823935334413003f_char(nat)),top_top(set(char))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ).

% range_nat_of_char
tff(fact_7266_prod_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A4: set(B)] : groups7121269368397514597t_prod(B,A,G,A4) = finite_fold(B,A,comp(A,fun(A,A),B,times_times(A),G),one_one(A),A4) ) ).

% prod.eq_fold
tff(fact_7267_map__filter__def,axiom,
    ! [A: $tType,B: $tType,Fa: fun(B,option(A)),Xs: list(B)] : map_filter(B,A,Fa,Xs) = map(B,A,comp(option(A),A,B,the2(A),Fa),filter2(B,aTP_Lamp_ans(fun(B,option(A)),fun(B,$o),Fa),Xs)) ).

% map_filter_def
tff(fact_7268_finite__range__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType,Fa: fun(B,product_prod(A,C))] :
      ( finite_finite2(A,aa(set(B),set(A),image(B,A,comp(product_prod(A,C),A,B,product_fst(A,C),Fa)),top_top(set(B))))
     => ( finite_finite2(C,aa(set(B),set(C),image(B,C,comp(product_prod(A,C),C,B,product_snd(A,C),Fa)),top_top(set(B))))
       => finite_finite2(product_prod(A,C),aa(set(B),set(product_prod(A,C)),image(B,product_prod(A,C),Fa),top_top(set(B)))) ) ) ).

% finite_range_prod
tff(fact_7269_execute__return,axiom,
    ! [A: $tType,Xb: A] : heap_Time_execute(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),Xb)) = comp(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ajv(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Xb)) ).

% execute_return
tff(fact_7270_execute__ureturn,axiom,
    ! [A: $tType,Xb: A] : heap_Time_execute(A,aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),Xb)) = comp(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_ext(product_unit),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_akw(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Xb)) ).

% execute_ureturn
tff(fact_7271_char_Osize_I2_J,axiom,
    ! [X1: $o,X22: $o,X33: $o,X42: $o,X52: $o,X62: $o,X7: $o,X8: $o] : aa(char,nat,size_size(char),aa($o,char,char2((X1),(X22),(X33),(X42),(X52),(X62),(X7)),(X8))) = zero_zero(nat) ).

% char.size(2)
tff(fact_7272_fst__snd__flip,axiom,
    ! [B: $tType,A: $tType,Xy: product_prod(A,B)] : aa(product_prod(A,B),A,product_fst(A,B),Xy) = aa(product_prod(A,B),A,comp(product_prod(B,A),A,product_prod(A,B),product_snd(B,A),product_case_prod(A,B,product_prod(B,A),aTP_Lamp_tq(A,fun(B,product_prod(B,A))))),Xy) ).

% fst_snd_flip
tff(fact_7273_snd__fst__flip,axiom,
    ! [A: $tType,B: $tType,Xy: product_prod(B,A)] : aa(product_prod(B,A),A,product_snd(B,A),Xy) = aa(product_prod(B,A),A,comp(product_prod(A,B),A,product_prod(B,A),product_fst(A,B),product_case_prod(B,A,product_prod(A,B),aTP_Lamp_gp(B,fun(A,product_prod(A,B))))),Xy) ).

% snd_fst_flip
tff(fact_7274_Ball__fold,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,$o)] :
      ( finite_finite2(A,A4)
     => ( ! [X: A] :
            ( member(A,X,A4)
           => aa(A,$o,P,X) )
      <=> finite_fold(A,$o,aTP_Lamp_ant(fun(A,$o),fun(A,fun($o,$o)),P),$true,A4) ) ) ).

% Ball_fold
tff(fact_7275_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,A4: set(A),Fa: fun(A,B)] :
      ( finite_finite2(A,A4)
     => ( aa(set(A),set(B),image(A,B,Fa),A4) = finite_fold(A,set(B),aTP_Lamp_anu(fun(A,B),fun(A,fun(set(B),set(B))),Fa),bot_bot(set(B)),A4) ) ) ).

% image_fold_insert
tff(fact_7276_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A4: set(A)] : lattic643756798349783984er_Max(A,A4) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_anv(A,fun(option(A),option(A))),none(A),A4)) ) ).

% Max.eq_fold'
tff(fact_7277_fstI,axiom,
    ! [B: $tType,A: $tType,Xb: product_prod(A,B),Y: A,Z: B] :
      ( ( Xb = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z) )
     => ( aa(product_prod(A,B),A,product_fst(A,B),Xb) = Y ) ) ).

% fstI
tff(fact_7278_sndI,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(A,B),Y: A,Z: B] :
      ( ( Xb = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y),Z) )
     => ( aa(product_prod(A,B),B,product_snd(A,B),Xb) = Z ) ) ).

% sndI
tff(fact_7279_char__of__def,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: A] : aa(A,char,unique5772411509450598832har_of(A),N) = aa($o,char,char2(~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),N),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),N),one_one(nat)),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),N),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),N),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),N),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2))))),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2))))) ) ).

% char_of_def
tff(fact_7280_char_Osize__gen,axiom,
    ! [X1: $o,X22: $o,X33: $o,X42: $o,X52: $o,X62: $o,X7: $o,X8: $o] : size_char(aa($o,char,char2((X1),(X22),(X33),(X42),(X52),(X62),(X7)),(X8))) = zero_zero(nat) ).

% char.size_gen
tff(fact_7281_char__of__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: A] : aa(A,char,unique5772411509450598832har_of(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),N),aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) = aa(A,char,unique5772411509450598832har_of(A),N) ) ).

% char_of_mod_256
tff(fact_7282_char__of__quasi__inj,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: A,N: A] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),M) = aa(A,char,unique5772411509450598832har_of(A),N) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),M),aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),N),aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ) ) ) ).

% char_of_quasi_inj
tff(fact_7283_of__char__of,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [A3: A] : aa(char,A,comm_s6883823935334413003f_char(A),aa(A,char,unique5772411509450598832har_of(A),A3)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A3),aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ) ).

% of_char_of
tff(fact_7284_char__of__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: nat,M: A] :
          ( aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),N)
         => ( aa(A,char,unique5772411509450598832har_of(A),aa(A,A,bit_se2584673776208193580ke_bit(A,N),M)) = aa(A,char,unique5772411509450598832har_of(A),M) ) ) ) ).

% char_of_take_bit_eq
tff(fact_7285_char__of__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [N: A,C2: char] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),N) = C2 )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),N) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ) ) ).

% char_of_eq_iff
tff(fact_7286_UNIV__char__of__nat,axiom,
    top_top(set(char)) = aa(set(nat),set(char),image(nat,char,unique5772411509450598832har_of(nat)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).

% UNIV_char_of_nat
tff(fact_7287_Code__Target__Int_Onegative__def,axiom,
    code_Target_negative = comp(int,int,num,uminus_uminus(int),numeral_numeral(int)) ).

% Code_Target_Int.negative_def
tff(fact_7288_String_Ochar__of__ascii__of,axiom,
    ! [C2: char] : aa(char,nat,comm_s6883823935334413003f_char(nat),ascii_of(C2)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2)))),aa(char,nat,comm_s6883823935334413003f_char(nat),C2)) ).

% String.char_of_ascii_of
tff(fact_7289_bit__word__rotl__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_rotl(A,M,W)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),type_len0_len_of(A,type2(A)))))),type_len0_len_of(A,type2(A)))) ) ) ) ).

% bit_word_rotl_iff
tff(fact_7290_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N8: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N8)
           => ( aa(nat,A,semiring_1_of_nat(A),N8) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_7291_CHAR__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) ) ) ).

% CHAR_eq_0
tff(fact_7292_of__nat__CHAR,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,A,semiring_1_of_nat(A),semiri4206861660011772517g_char(A,type2(A))) = zero_zero(A) ) ) ).

% of_nat_CHAR
tff(fact_7293_CHAR__eqI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
         => ( ! [X2: nat] :
                ( ( aa(nat,A,semiring_1_of_nat(A),X2) = zero_zero(A) )
               => aa(nat,$o,dvd_dvd(nat,C2),X2) )
           => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ).

% CHAR_eqI
tff(fact_7294_of__nat__eq__0__iff__char__dvd,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [N: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),N) = zero_zero(A) )
        <=> aa(nat,$o,dvd_dvd(nat,semiri4206861660011772517g_char(A,type2(A))),N) ) ) ).

% of_nat_eq_0_iff_char_dvd
tff(fact_7295_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A)))
      <=> ? [N8: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N8)
            & ( aa(nat,A,semiring_1_of_nat(A),N8) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_7296_CHAR__eq__posI,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [C2: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
         => ( ( aa(nat,A,semiring_1_of_nat(A),C2) = zero_zero(A) )
           => ( ! [X2: nat] :
                  ( aa(nat,$o,ord_less(nat,zero_zero(nat)),X2)
                 => ( aa(nat,$o,ord_less(nat,X2),C2)
                   => ( aa(nat,A,semiring_1_of_nat(A),X2) != zero_zero(A) ) ) )
             => ( semiri4206861660011772517g_char(A,type2(A)) = C2 ) ) ) ) ) ).

% CHAR_eq_posI
tff(fact_7297_word__roti__eq__word__rotr__word__rotl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: int,W: word(A)] :
          word_roti(A,I,W) = $ite(aa(int,$o,ord_less_eq(int,zero_zero(int)),I),word_rotr(A,nat2(I),W),word_rotl(A,nat2(aa(int,int,uminus_uminus(int),I)),W)) ) ).

% word_roti_eq_word_rotr_word_rotl
tff(fact_7298_bit__sshiftr__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: nat,N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_Sh8784991116023147202shiftr(A,W,M)),N)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),
              $ite(
                ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M)),N)
                & aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A))) ),
                aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat)),
                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),N) )) ) ) ).

% bit_sshiftr_iff
tff(fact_7299_sshiftr__of__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : bit_Sh8784991116023147202shiftr(A,W,zero_zero(nat)) = W ) ).

% sshiftr_of_0
tff(fact_7300_sshiftr__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : bit_Sh8784991116023147202shiftr(A,zero_zero(word(A)),N) = zero_zero(word(A)) ) ).

% sshiftr_0
tff(fact_7301_sshiftr__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,N: num] : bit_Sh8784991116023147202shiftr(A,aa(num,word(A),numeral_numeral(word(A)),M),aa(num,nat,numeral_numeral(nat),N)) = signed_drop_bit(A,aa(num,nat,numeral_numeral(nat),N),aa(num,word(A),numeral_numeral(word(A)),M)) ) ).

% sshiftr_numeral_numeral
tff(fact_7302_sshiftr__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,N: num] : bit_Sh8784991116023147202shiftr(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M)),aa(num,nat,numeral_numeral(nat),N)) = signed_drop_bit(A,aa(num,nat,numeral_numeral(nat),N),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M))) ) ).

% sshiftr_minus_numeral_numeral
tff(fact_7303_sshiftr__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] :
          bit_Sh8784991116023147202shiftr(A,one_one(word(A)),N) = aa($o,word(A),zero_neq_one_of_bool(word(A)),
            ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
            | ( N = zero_zero(nat) ) )) ) ).

% sshiftr_1
tff(fact_7304_word__rotx__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [I: nat] :
          ( ( word_rotr(A,I,zero_zero(word(A))) = zero_zero(word(A)) )
          & ( word_rotl(B,I,zero_zero(word(B))) = zero_zero(word(B)) ) ) ) ).

% word_rotx_0
tff(fact_7305_bit__word__rotr__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_rotr(A,M,W)),N)
        <=> ( aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),M)),type_len0_len_of(A,type2(A)))) ) ) ) ).

% bit_word_rotr_iff
tff(fact_7306_shiftl__Suc__0,axiom,
    ! [N: nat] : bit_Sh4282982442137083160shiftl(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N) ).

% shiftl_Suc_0
tff(fact_7307_shiftl__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [N: num] : bit_Sh4282982442137083160shiftl(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,nat,numeral_numeral(nat),N)) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),N))) ) ).

% shiftl_minus_1_numeral
tff(fact_7308_shiftl__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_Sh4282982442137083160shiftl(A,zero_zero(A),N) = zero_zero(A) ) ).

% shiftl_0
tff(fact_7309_shiftl__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_Sh4282982442137083160shiftl(A,A3,zero_zero(nat)) = A3 ) ).

% shiftl_of_0
tff(fact_7310_shiftl__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N: nat] : bit_Sh4282982442137083160shiftl(A,aa(num,A,numeral_numeral(A),M),aa(nat,nat,suc,N)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftl_numeral_Suc
tff(fact_7311_shiftl__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N: num] : bit_Sh4282982442137083160shiftl(A,aa(num,A,numeral_numeral(A),M),aa(num,nat,numeral_numeral(nat),N)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(num,nat,numeral_numeral(nat),N)),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftl_numeral_numeral
tff(fact_7312_shiftl__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,N: nat] : bit_Sh4282982442137083160shiftl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(nat,nat,suc,N)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% shiftl_minus_numeral_Suc
tff(fact_7313_shiftl__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,N: num] : bit_Sh4282982442137083160shiftl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(num,nat,numeral_numeral(nat),N)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(num,nat,numeral_numeral(nat),N)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% shiftl_minus_numeral_numeral
tff(fact_7314_shiftl__of__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,N: nat] : bit_Sh4282982442137083160shiftl(A,A3,aa(nat,nat,suc,N)) = bit_Sh4282982442137083160shiftl(A,aa(A,A,aa(A,fun(A,A),times_times(A),A3),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ).

% shiftl_of_Suc
tff(fact_7315_shiftl__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_Sh4282982442137083160shiftl(A,one_one(A),N) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N) ) ).

% shiftl_1
tff(fact_7316_one__bit__shiftl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N: nat] : aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),zero_zero(word(A))),N),$true) = bit_Sh4282982442137083160shiftl(word(A),one_one(word(A)),N) ) ).

% one_bit_shiftl
tff(fact_7317_shiftl__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Xb: A,N: nat] : bit_Sh4282982442137083160shiftl(A,Xb,N) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% shiftl_eq_mult
tff(fact_7318_bit__shiftl__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A,M: nat,N: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),bit_Sh4282982442137083160shiftl(A,A3,M)),N)
        <=> ( aa(nat,$o,ord_less_eq(nat,M),N)
            & bit_se6407376104438227557le_bit(A,type2(A),N)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A3),aa(nat,nat,minus_minus(nat,N),M)) ) ) ) ).

% bit_shiftl_iff
tff(fact_7319_shiftr__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,N: num] : bit_Sh4282982442137083166shiftr(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(num,nat,numeral_numeral(nat),N)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(num,nat,numeral_numeral(nat),N)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% shiftr_minus_numeral_numeral
tff(fact_7320_shiftr__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,N: nat] : bit_Sh4282982442137083166shiftr(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(nat,nat,suc,N)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,N)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% shiftr_minus_numeral_Suc
tff(fact_7321_shiftr__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_Sh4282982442137083166shiftr(A,zero_zero(A),N) = zero_zero(A) ) ).

% shiftr_0
tff(fact_7322_shiftr__of__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A3: A] : bit_Sh4282982442137083166shiftr(A,A3,zero_zero(nat)) = A3 ) ).

% shiftr_of_0
tff(fact_7323_shiftr__Suc__0,axiom,
    ! [N: nat] : bit_Sh4282982442137083166shiftr(nat,aa(nat,nat,suc,zero_zero(nat)),N) = aa($o,nat,zero_neq_one_of_bool(nat),N = zero_zero(nat)) ).

% shiftr_Suc_0
tff(fact_7324_shiftr__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N: nat] : bit_Sh4282982442137083166shiftr(A,aa(num,A,numeral_numeral(A),M),aa(nat,nat,suc,N)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,N)),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftr_numeral_Suc
tff(fact_7325_shiftr__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,N: num] : bit_Sh4282982442137083166shiftr(A,aa(num,A,numeral_numeral(A),M),aa(num,nat,numeral_numeral(nat),N)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(num,nat,numeral_numeral(nat),N)),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftr_numeral_numeral
tff(fact_7326_shiftr__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [N: nat] : bit_Sh4282982442137083166shiftr(A,one_one(A),N) = aa($o,A,zero_neq_one_of_bool(A),N = zero_zero(nat)) ) ).

% shiftr_1
tff(fact_7327_shiftr__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Xb: A,N: nat] : bit_Sh4282982442137083166shiftr(A,Xb,N) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),N)) ) ).

% shiftr_eq_div
tff(fact_7328_Ball__comp__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,A4: fun(B,set(C)),Fa: fun(C,$o),G: fun(A,B),X3: A] :
      ( aa(A,$o,comp(B,$o,A,aa(fun(C,$o),fun(B,$o),aTP_Lamp_anw(fun(B,set(C)),fun(fun(C,$o),fun(B,$o)),A4),Fa),G),X3)
    <=> ! [Xa3: C] :
          ( member(C,Xa3,aa(A,set(C),comp(B,set(C),A,A4,G),X3))
         => aa(C,$o,Fa,Xa3) ) ) ).

% Ball_comp_iff
tff(fact_7329_bit__revcast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B),N: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),revcast(B,A,W)),N)
        <=> ( aa(nat,$o,ord_less_eq(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),N)
            & aa(nat,$o,ord_less(nat,N),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),aa(nat,nat,minus_minus(nat,type_len0_len_of(B,type2(B))),type_len0_len_of(A,type2(A))))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B))))) ) ) ) ).

% bit_revcast_iff
tff(fact_7330_revcast__slice1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : slice1(B,A,aa(word(A),nat,size_size(word(A)),revcast(B,A,W)),W) = revcast(B,A,W) ) ).

% revcast_slice1
tff(fact_7331_conj__comp__iff,axiom,
    ! [B: $tType,A: $tType,P: fun(B,$o),Q: fun(B,$o),G: fun(A,B),X3: A] :
      ( aa(A,$o,comp(B,$o,A,aa(fun(B,$o),fun(B,$o),aTP_Lamp_anx(fun(B,$o),fun(fun(B,$o),fun(B,$o)),P),Q),G),X3)
    <=> ( aa(A,$o,comp(B,$o,A,P,G),X3)
        & aa(A,$o,comp(B,$o,A,Q,G),X3) ) ) ).

% conj_comp_iff
tff(fact_7332_empty__natural,axiom,
    ! [C: $tType,B: $tType,D2: $tType,A: $tType,Fa: fun(A,C),G: fun(D2,B)] : comp(C,set(B),A,aTP_Lamp_any(C,set(B)),Fa) = comp(set(D2),set(B),A,image(D2,B,G),aTP_Lamp_anz(A,set(D2))) ).

% empty_natural
tff(fact_7333_Set__filter__fold,axiom,
    ! [A: $tType,A4: set(A),P: fun(A,$o)] :
      ( finite_finite2(A,A4)
     => ( filter3(A,P,A4) = finite_fold(A,set(A),aTP_Lamp_aoa(fun(A,$o),fun(A,fun(set(A),set(A))),P),bot_bot(set(A)),A4) ) ) ).

% Set_filter_fold
tff(fact_7334_lenlex__conv,axiom,
    ! [A: $tType,R2: set(product_prod(A,A))] : lenlex(A,R2) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),$o,aTP_Lamp_aob(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R2))) ).

% lenlex_conv
tff(fact_7335_Set_Ofilter__def,axiom,
    ! [A: $tType,P: fun(A,$o),A4: set(A)] : filter3(A,P,A4) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_aoc(fun(A,$o),fun(set(A),fun(A,$o)),P),A4)) ).

% Set.filter_def
tff(fact_7336_lenlex__irreflexive,axiom,
    ! [A: $tType,R2: set(product_prod(A,A)),Xs: list(A)] :
      ( ! [X2: A] : ~ member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X2),X2),R2)
     => ~ member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs),Xs),lenlex(A,R2)) ) ).

% lenlex_irreflexive
tff(fact_7337_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R2))
     => aa(nat,$o,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns)) ) ).

% lenlex_length
tff(fact_7338_lenlex__append1,axiom,
    ! [A: $tType,Us: list(A),Xs: list(A),R: set(product_prod(A,A)),Vs: list(A),Ys: list(A)] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Us),Xs),lenlex(A,R))
     => ( ( aa(list(A),nat,size_size(list(A)),Vs) = aa(list(A),nat,size_size(list(A)),Ys) )
       => member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),append(A,Us,Vs)),append(A,Xs,Ys)),lenlex(A,R)) ) ) ).

% lenlex_append1
tff(fact_7339_Cons__lenlex__iff,axiom,
    ! [A: $tType,M: A,Ms: list(A),N: A,Ns: list(A),R2: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),cons(A,M),Ms)),aa(list(A),list(A),cons(A,N),Ns)),lenlex(A,R2))
    <=> ( aa(nat,$o,ord_less(nat,aa(list(A),nat,size_size(list(A)),Ms)),aa(list(A),nat,size_size(list(A)),Ns))
        | ( ( aa(list(A),nat,size_size(list(A)),Ms) = aa(list(A),nat,size_size(list(A)),Ns) )
          & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),M),N),R2) )
        | ( ( M = N )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R2)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_7340_word__msb__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] :
          ( most_s684356279273892711sb_msb(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W)))
        <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),W))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) ) ) ).

% word_msb_neg_numeral
tff(fact_7341_Eps__Opt__def,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      eps_Opt(A,P) = $ite(
        ? [X6: A] : aa(A,$o,P,X6),
        aa(A,option(A),some(A),fChoice(A,P)),
        none(A) ) ).

% Eps_Opt_def
tff(fact_7342_some__opt__sym__eq__trivial,axiom,
    ! [A: $tType,Xb: A] : eps_Opt(A,fequal(A,Xb)) = aa(A,option(A),some(A),Xb) ).

% some_opt_sym_eq_trivial
tff(fact_7343_some__opt__eq__trivial,axiom,
    ! [A: $tType,Xb: A] : eps_Opt(A,aTP_Lamp_gc(A,fun(A,$o),Xb)) = aa(A,option(A),some(A),Xb) ).

% some_opt_eq_trivial
tff(fact_7344_some__opt__false__trivial,axiom,
    ! [A: $tType] : eps_Opt(A,aTP_Lamp_dp(A,$o)) = none(A) ).

% some_opt_false_trivial
tff(fact_7345_Eps__Opt__eq__None,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( eps_Opt(A,P) = none(A) )
    <=> ~ ? [X_12: A] : aa(A,$o,P,X_12) ) ).

% Eps_Opt_eq_None
tff(fact_7346_word__msb__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( most_s684356279273892711sb_msb(word(A),one_one(word(A)))
      <=> ( type_len0_len_of(A,type2(A)) = one_one(nat) ) ) ) ).

% word_msb_1
tff(fact_7347_word__msb__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] :
          ( most_s684356279273892711sb_msb(word(A),aa(num,word(A),numeral_numeral(word(A)),W))
        <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) ) ) ).

% word_msb_numeral
tff(fact_7348_msb__word__iff__sless__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> word_sless(A,W,zero_zero(word(A))) ) ) ).

% msb_word_iff_sless_0
tff(fact_7349_word__msb__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ~ most_s684356279273892711sb_msb(word(A),zero_zero(word(A))) ) ).

% word_msb_0
tff(fact_7350_Eps__Opt__eq__Some__implies,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A] :
      ( ( eps_Opt(A,P) = aa(A,option(A),some(A),Xb) )
     => aa(A,$o,P,Xb) ) ).

% Eps_Opt_eq_Some_implies
tff(fact_7351_Eps__Opt__eq__Some,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A] :
      ( ! [X9: A] :
          ( aa(A,$o,P,Xb)
         => ( aa(A,$o,P,X9)
           => ( X9 = Xb ) ) )
     => ( ( eps_Opt(A,P) = aa(A,option(A),some(A),Xb) )
      <=> aa(A,$o,P,Xb) ) ) ).

% Eps_Opt_eq_Some
tff(fact_7352_word__msb__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> aa(int,$o,ord_less(int,ring_1_signed(A,int,W)),zero_zero(int)) ) ) ).

% word_msb_sint
tff(fact_7353_word__sless__msb__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( word_sless(A,Xb,Y)
        <=> ( ( most_s684356279273892711sb_msb(word(A),Y)
             => most_s684356279273892711sb_msb(word(A),Xb) )
            & ( ( most_s684356279273892711sb_msb(word(A),Xb)
                & ~ most_s684356279273892711sb_msb(word(A),Y) )
              | aa(word(A),$o,ord_less(word(A),Xb),Y) ) ) ) ) ).

% word_sless_msb_less
tff(fact_7354_msb__shift,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> ( bit_Sh4282982442137083166shiftr(word(A),W,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) != zero_zero(word(A)) ) ) ) ).

% msb_shift
tff(fact_7355_msb__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) ) ) ).

% msb_word_eq
tff(fact_7356_msb__word__iff__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))) ) ) ).

% msb_word_iff_bit
tff(fact_7357_word__msb__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),W)
        <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) ) ) ).

% word_msb_nth
tff(fact_7358_msb__word__of__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int] :
          ( most_s684356279273892711sb_msb(word(A),aa(int,word(A),ring_1_of_int(word(A)),Xb))
        <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),Xb),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))) ) ) ).

% msb_word_of_int
tff(fact_7359_not__msb__from__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),V),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),one_one(nat))))
         => ~ most_s684356279273892711sb_msb(word(A),V) ) ) ).

% not_msb_from_less
tff(fact_7360_word__sint__msb__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ring_1_signed(A,int,Xb) = aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),Xb)),
            $ite(most_s684356279273892711sb_msb(word(A),Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),nat,size_size(word(A)),Xb)),zero_zero(int))) ) ).

% word_sint_msb_eq
tff(fact_7361_msb__big,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A3: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),A3)
        <=> aa(word(A),$o,ord_less_eq(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))),A3) ) ) ).

% msb_big
tff(fact_7362_word__type__copy__misc__axioms_Ointro,axiom,
    ! [A: $tType,B: $tType] :
      ( ( bit_ri3973907225187159222ations(B)
        & cl_HOL_Oequal(B)
        & linorder(B)
        & type_len(A) )
     => ! [Size: nat,Word_of: fun(B,word(A)),Set_bits_aux: fun(fun(nat,$o),fun(nat,fun(B,B)))] :
          ( ( Size = type_len0_len_of(A,type2(A)) )
         => ( ! [P5: B] :
                ( most_s684356279273892711sb_msb(B,P5)
              <=> most_s684356279273892711sb_msb(word(A),aa(B,word(A),Word_of,P5)) )
           => ( ! [P5: B] :
                  ( aa(B,$o,least_8051144512741203767sb_lsb(B),P5)
                <=> aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),aa(B,word(A),Word_of,P5)) )
             => ( ! [P5: B] : aa(B,nat,size_size(B),P5) = aa(word(A),nat,size_size(word(A)),aa(B,word(A),Word_of,P5))
               => ( ! [P5: B,N3: nat,B2: $o] : aa(B,word(A),Word_of,aa($o,B,aa(nat,fun($o,B),aa(B,fun(nat,fun($o,B)),generi7602027413899671122et_bit(B),P5),N3),(B2))) = aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),aa(B,word(A),Word_of,P5)),N3),(B2))
                 => ( ! [P6: fun(nat,$o)] : aa(B,word(A),Word_of,bit_bi4170147762399595738t_bits(B,P6)) = bit_bi4170147762399595738t_bits(word(A),P6)
                   => ( ! [P6: fun(nat,$o),N3: nat,P5: B] : aa(B,word(A),Word_of,aa(B,B,aa(nat,fun(B,B),aa(fun(nat,$o),fun(nat,fun(B,B)),Set_bits_aux,P6),N3),P5)) = code_T2661198915054445665ts_aux(A,P6,N3,aa(B,word(A),Word_of,P5))
                     => word_T8964210463127689547axioms(B,A,Word_of,Size,Set_bits_aux) ) ) ) ) ) ) ) ) ).

% word_type_copy_misc_axioms.intro
tff(fact_7363_word__type__copy__misc__axioms__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( bit_ri3973907225187159222ations(A)
        & cl_HOL_Oequal(A)
        & linorder(A)
        & type_len(B) )
     => ! [Word_of: fun(A,word(B)),Size: nat,Set_bits_aux: fun(fun(nat,$o),fun(nat,fun(A,A)))] :
          ( word_T8964210463127689547axioms(A,B,Word_of,Size,Set_bits_aux)
        <=> ( ( Size = type_len0_len_of(B,type2(B)) )
            & ! [P7: A] :
                ( most_s684356279273892711sb_msb(A,P7)
              <=> most_s684356279273892711sb_msb(word(B),aa(A,word(B),Word_of,P7)) )
            & ! [P7: A] :
                ( aa(A,$o,least_8051144512741203767sb_lsb(A),P7)
              <=> aa(word(B),$o,least_8051144512741203767sb_lsb(word(B)),aa(A,word(B),Word_of,P7)) )
            & ! [P7: A] : aa(A,nat,size_size(A),P7) = aa(word(B),nat,size_size(word(B)),aa(A,word(B),Word_of,P7))
            & ! [P7: A,N8: nat,B7: $o] : aa(A,word(B),Word_of,aa($o,A,aa(nat,fun($o,A),aa(A,fun(nat,fun($o,A)),generi7602027413899671122et_bit(A),P7),N8),(B7))) = aa($o,word(B),aa(nat,fun($o,word(B)),aa(word(B),fun(nat,fun($o,word(B))),generi7602027413899671122et_bit(word(B)),aa(A,word(B),Word_of,P7)),N8),(B7))
            & ! [P8: fun(nat,$o)] : aa(A,word(B),Word_of,bit_bi4170147762399595738t_bits(A,P8)) = bit_bi4170147762399595738t_bits(word(B),P8)
            & ! [P8: fun(nat,$o),N8: nat,P7: A] : aa(A,word(B),Word_of,aa(A,A,aa(nat,fun(A,A),aa(fun(nat,$o),fun(nat,fun(A,A)),Set_bits_aux,P8),N8),P7)) = code_T2661198915054445665ts_aux(B,P8,N8,aa(A,word(B),Word_of,P7)) ) ) ) ).

% word_type_copy_misc_axioms_def
tff(fact_7364_msb__numeral_I1_J,axiom,
    ! [N: num] : ~ most_s684356279273892711sb_msb(int,aa(num,int,numeral_numeral(int),N)) ).

% msb_numeral(1)
tff(fact_7365_msb__0,axiom,
    ~ most_s684356279273892711sb_msb(int,zero_zero(int)) ).

% msb_0
tff(fact_7366_msb__1,axiom,
    ~ most_s684356279273892711sb_msb(int,one_one(int)) ).

% msb_1
tff(fact_7367_msb__numeral_I2_J,axiom,
    ! [N: num] : most_s684356279273892711sb_msb(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),N))) ).

% msb_numeral(2)
tff(fact_7368_msb__bin__rest,axiom,
    ! [Xb: int] :
      ( most_s684356279273892711sb_msb(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),aa(num,int,numeral_numeral(int),bit0(one2))))
    <=> most_s684356279273892711sb_msb(int,Xb) ) ).

% msb_bin_rest
tff(fact_7369_msb__int__def,axiom,
    ! [Xb: int] :
      ( most_s684356279273892711sb_msb(int,Xb)
    <=> aa(int,$o,ord_less(int,Xb),zero_zero(int)) ) ).

% msb_int_def
tff(fact_7370_msb__integer__code,axiom,
    ! [Xb: code_integer] :
      ( most_s684356279273892711sb_msb(code_integer,Xb)
    <=> aa(code_integer,$o,ord_less(code_integer,Xb),zero_zero(code_integer)) ) ).

% msb_integer_code
tff(fact_7371_uint32__msb__test__bit,axiom,
    ! [Xb: uint32] :
      ( most_s684356279273892711sb_msb(uint32,Xb)
    <=> aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))) ) ).

% uint32_msb_test_bit
tff(fact_7372_word__type__copy__misc_Oset__bits__aux__code,axiom,
    ! [A: $tType,B: $tType] :
      ( ( bit_ri3973907225187159222ations(B)
        & cl_HOL_Oequal(B)
        & linorder(B)
        & type_len(A) )
     => ! [Of_word: fun(word(A),B),Word_of: fun(B,word(A)),Signed_drop_bit: fun(nat,fun(B,B)),Of_nat: fun(nat,B),Nat_of: fun(B,nat),Of_int: fun(int,B),Int_of: fun(B,int),Of_integer: fun(code_integer,B),Integer_of: fun(B,code_integer),Size: nat,Set_bits_aux: fun(fun(nat,$o),fun(nat,fun(B,B))),Fa: fun(nat,$o),N: nat,W: B] :
          ( word_T1870391057261780392y_misc(A,B,Of_word,Word_of,Signed_drop_bit,Of_nat,Nat_of,Of_int,Int_of,Of_integer,Integer_of,Size,Set_bits_aux)
         => ( aa(B,B,aa(nat,fun(B,B),aa(fun(nat,$o),fun(nat,fun(B,B)),Set_bits_aux,Fa),N),W) = $ite(
                N = zero_zero(nat),
                W,
                $let(
                  n: nat,
                  n:= aa(nat,nat,minus_minus(nat,N),one_one(nat)),
                  aa(B,B,aa(nat,fun(B,B),aa(fun(nat,$o),fun(nat,fun(B,B)),Set_bits_aux,Fa),n),
                    aa(B,B,aa(B,fun(B,B),bit_se1065995026697491101ons_or(B),aa(B,B,aa(nat,fun(B,B),bit_se4730199178511100633sh_bit(B),one_one(nat)),W)),
                      $ite(aa(nat,$o,Fa,n),one_one(B),zero_zero(B)))) ) ) ) ) ) ).

% word_type_copy_misc.set_bits_aux_code
tff(fact_7373_msb__uint32__code,axiom,
    ! [Xb: uint32] :
      ( most_s684356279273892711sb_msb(uint32,Xb)
    <=> uint32_test_bit(Xb,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))) ) ).

% msb_uint32_code
tff(fact_7374_word__type__copy__misc_Osize__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( bit_ri3973907225187159222ations(B)
        & cl_HOL_Oequal(B)
        & linorder(B)
        & type_len(A) )
     => ! [Of_word: fun(word(A),B),Word_of: fun(B,word(A)),Signed_drop_bit: fun(nat,fun(B,B)),Of_nat: fun(nat,B),Nat_of: fun(B,nat),Of_int: fun(int,B),Int_of: fun(B,int),Of_integer: fun(code_integer,B),Integer_of: fun(B,code_integer),Size: nat,Set_bits_aux: fun(fun(nat,$o),fun(nat,fun(B,B))),P2: B] :
          ( word_T1870391057261780392y_misc(A,B,Of_word,Word_of,Signed_drop_bit,Of_nat,Nat_of,Of_int,Int_of,Of_integer,Integer_of,Size,Set_bits_aux)
         => ( aa(B,nat,size_size(B),P2) = Size ) ) ) ).

% word_type_copy_misc.size_eq
tff(fact_7375_word__type__copy__misc_Osize__eq__word__of,axiom,
    ! [A: $tType,B: $tType] :
      ( ( bit_ri3973907225187159222ations(B)
        & cl_HOL_Oequal(B)
        & linorder(B)
        & type_len(A) )
     => ! [Of_word: fun(word(A),B),Word_of: fun(B,word(A)),Signed_drop_bit: fun(nat,fun(B,B)),Of_nat: fun(nat,B),Nat_of: fun(B,nat),Of_int: fun(int,B),Int_of: fun(B,int),Of_integer: fun(code_integer,B),Integer_of: fun(B,code_integer),Size: nat,Set_bits_aux: fun(fun(nat,$o),fun(nat,fun(B,B))),P2: B] :
          ( word_T1870391057261780392y_misc(A,B,Of_word,Word_of,Signed_drop_bit,Of_nat,Nat_of,Of_int,Int_of,Of_integer,Integer_of,Size,Set_bits_aux)
         => ( aa(B,nat,size_size(B),P2) = aa(word(A),nat,size_size(word(A)),aa(B,word(A),Word_of,P2)) ) ) ) ).

% word_type_copy_misc.size_eq_word_of
tff(fact_7376_word__type__copy__misc_Oset__bits__code,axiom,
    ! [A: $tType,B: $tType] :
      ( ( bit_ri3973907225187159222ations(B)
        & cl_HOL_Oequal(B)
        & linorder(B)
        & type_len(A) )
     => ! [Of_word: fun(word(A),B),Word_of: fun(B,word(A)),Signed_drop_bit: fun(nat,fun(B,B)),Of_nat: fun(nat,B),Nat_of: fun(B,nat),Of_int: fun(int,B),Int_of: fun(B,int),Of_integer: fun(code_integer,B),Integer_of: fun(B,code_integer),Size: nat,Set_bits_aux: fun(fun(nat,$o),fun(nat,fun(B,B))),P: fun(nat,$o)] :
          ( word_T1870391057261780392y_misc(A,B,Of_word,Word_of,Signed_drop_bit,Of_nat,Nat_of,Of_int,Int_of,Of_integer,Integer_of,Size,Set_bits_aux)
         => ( bit_bi4170147762399595738t_bits(B,P) = aa(B,B,aa(nat,fun(B,B),aa(fun(nat,$o),fun(nat,fun(B,B)),Set_bits_aux,P),Size),zero_zero(B)) ) ) ) ).

% word_type_copy_misc.set_bits_code
tff(fact_7377_uint32__test__bit__def,axiom,
    ! [Xb: uint32,N: code_integer] :
      ( uint32_test_bit(Xb,N)
    <=> $ite(
          ( aa(code_integer,$o,ord_less(code_integer,N),zero_zero(code_integer))
          | aa(code_integer,$o,ord_less(code_integer,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),N) ),
          aa(code_integer,$o,aa(uint32,fun(code_integer,$o),aa(fun(uint32,fun(nat,$o)),fun(uint32,fun(code_integer,$o)),undefined(fun(fun(uint32,fun(nat,$o)),fun(uint32,fun(code_integer,$o)))),bit_se5641148757651400278ts_bit(uint32)),Xb),N),
          aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),Xb),aa(code_integer,nat,code_nat_of_integer,N)) ) ) ).

% uint32_test_bit_def
tff(fact_7378_test__bit__uint32__code,axiom,
    ! [Xb: uint32,N: nat] :
      ( aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),Xb),N)
    <=> ( aa(nat,$o,ord_less(nat,N),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2)))))))
        & uint32_test_bit(Xb,code_integer_of_nat(N)) ) ) ).

% test_bit_uint32_code
tff(fact_7379_of__nat__of__integer,axiom,
    ! [K: code_integer] : aa(nat,code_integer,semiring_1_of_nat(code_integer),aa(code_integer,nat,code_nat_of_integer,K)) = aa(code_integer,code_integer,ord_max(code_integer,zero_zero(code_integer)),K) ).

% of_nat_of_integer
tff(fact_7380_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] : aa(code_integer,nat,code_nat_of_integer,aa(num,code_integer,numeral_numeral(code_integer),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_of_integer_code_post(3)
tff(fact_7381_nat__of__integer__numeral,axiom,
    ! [N: num] : aa(code_integer,nat,code_nat_of_integer,aa(num,code_integer,numeral_numeral(code_integer),N)) = aa(num,nat,numeral_numeral(nat),N) ).

% nat_of_integer_numeral
tff(fact_7382_nat__of__integer__1,axiom,
    aa(code_integer,nat,code_nat_of_integer,one_one(code_integer)) = one_one(nat) ).

% nat_of_integer_1
tff(fact_7383_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa(code_integer,$o,ord_less_eq(code_integer,K),zero_zero(code_integer))
     => ( aa(code_integer,nat,code_nat_of_integer,K) = zero_zero(nat) ) ) ).

% nat_of_integer_non_positive
tff(fact_7384_nat__of__integer__code__post_I1_J,axiom,
    aa(code_integer,nat,code_nat_of_integer,zero_zero(code_integer)) = zero_zero(nat) ).

% nat_of_integer_code_post(1)
tff(fact_7385_integer__of__nat__0,axiom,
    code_integer_of_nat(zero_zero(nat)) = zero_zero(code_integer) ).

% integer_of_nat_0
tff(fact_7386_integer__of__nat__numeral,axiom,
    ! [N: num] : code_integer_of_nat(aa(num,nat,numeral_numeral(nat),N)) = aa(num,code_integer,numeral_numeral(code_integer),N) ).

% integer_of_nat_numeral
tff(fact_7387_integer__of__nat__1,axiom,
    code_integer_of_nat(one_one(nat)) = one_one(code_integer) ).

% integer_of_nat_1
tff(fact_7388_integer__of__nat__less__0__conv,axiom,
    ! [N: nat] : ~ aa(code_integer,$o,ord_less(code_integer,code_integer_of_nat(N)),zero_zero(code_integer)) ).

% integer_of_nat_less_0_conv
tff(fact_7389_nat__of__integer__less__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa(code_integer,$o,ord_less_eq(code_integer,zero_zero(code_integer)),Xb)
     => ( aa(code_integer,$o,ord_less_eq(code_integer,zero_zero(code_integer)),Y)
       => ( aa(nat,$o,ord_less(nat,aa(code_integer,nat,code_nat_of_integer,Xb)),aa(code_integer,nat,code_nat_of_integer,Y))
        <=> aa(code_integer,$o,ord_less(code_integer,Xb),Y) ) ) ) ).

% nat_of_integer_less_iff
tff(fact_7390_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( aa(code_integer,$o,ord_less_eq(code_integer,zero_zero(code_integer)),U)
     => ( set_or7035219750837199246ssThan(code_integer,zero_zero(code_integer),U) = aa(set(nat),set(code_integer),image(nat,code_integer,semiring_1_of_nat(code_integer)),set_ord_lessThan(nat,aa(code_integer,nat,code_nat_of_integer,U))) ) ) ).

% image_atLeastZeroLessThan_integer
tff(fact_7391_integer__set__bit__code,axiom,
    ! [Xb: code_integer,N: code_integer,B3: $o] :
      bits_integer_set_bit(Xb,N,(B3)) = $ite(
        aa(code_integer,$o,ord_less(code_integer,N),zero_zero(code_integer)),
        aa($o,code_integer,aa(code_integer,fun($o,code_integer),aa(code_integer,fun(code_integer,fun($o,code_integer)),undefined(fun(code_integer,fun(code_integer,fun($o,code_integer)))),Xb),N),(B3)),
        $ite((B3),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),Xb),aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),aa(code_integer,nat,code_nat_of_integer,N)),one_one(code_integer))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se5824344872417868541ns_and(code_integer),Xb),aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),aa(code_integer,nat,code_nat_of_integer,N)),one_one(code_integer))))) ) ).

% integer_set_bit_code
tff(fact_7392_uint32__shiftr__def,axiom,
    ! [Xb: uint32,N: code_integer] :
      uint32_shiftr(Xb,N) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,N),zero_zero(code_integer))
        | aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),N) ),
        aa(code_integer,uint32,aa(uint32,fun(code_integer,uint32),aa(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)),undefined(fun(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)))),bit_se4197421643247451524op_bit(uint32)),Xb),N),
        aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4197421643247451524op_bit(uint32),aa(code_integer,nat,code_nat_of_integer,N)),Xb) ) ).

% uint32_shiftr_def
tff(fact_7393_integer__set__bit__def,axiom,
    ! [Xb: code_integer,N: code_integer,B3: $o] :
      bits_integer_set_bit(Xb,N,(B3)) = $ite(aa(code_integer,$o,ord_less(code_integer,N),zero_zero(code_integer)),aa($o,code_integer,aa(code_integer,fun($o,code_integer),aa(code_integer,fun(code_integer,fun($o,code_integer)),undefined(fun(code_integer,fun(code_integer,fun($o,code_integer)))),Xb),N),(B3)),aa($o,code_integer,aa(nat,fun($o,code_integer),aa(code_integer,fun(nat,fun($o,code_integer)),generi7602027413899671122et_bit(code_integer),Xb),aa(code_integer,nat,code_nat_of_integer,N)),(B3))) ).

% integer_set_bit_def
tff(fact_7394_shiftr__uint32__code,axiom,
    ! [N: nat,Xb: uint32] :
      aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4197421643247451524op_bit(uint32),N),Xb) = $ite(aa(nat,$o,ord_less(nat,N),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),uint32_shiftr(Xb,code_integer_of_nat(N)),zero_zero(uint32)) ).

% shiftr_uint32_code
tff(fact_7395_uint32__shiftl__def,axiom,
    ! [Xb: uint32,N: code_integer] :
      uint32_shiftl(Xb,N) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,N),zero_zero(code_integer))
        | aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),N) ),
        aa(code_integer,uint32,aa(uint32,fun(code_integer,uint32),aa(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)),undefined(fun(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)))),bit_se4730199178511100633sh_bit(uint32)),Xb),N),
        aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4730199178511100633sh_bit(uint32),aa(code_integer,nat,code_nat_of_integer,N)),Xb) ) ).

% uint32_shiftl_def
tff(fact_7396_uint32__set__bit__def,axiom,
    ! [Xb: uint32,N: code_integer,B3: $o] :
      uint32_set_bit(Xb,N,(B3)) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,N),zero_zero(code_integer))
        | aa(code_integer,$o,ord_less(code_integer,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),N) ),
        aa($o,uint32,aa(code_integer,fun($o,uint32),aa(uint32,fun(code_integer,fun($o,uint32)),aa(fun(uint32,fun(nat,fun($o,uint32))),fun(uint32,fun(code_integer,fun($o,uint32))),undefined(fun(fun(uint32,fun(nat,fun($o,uint32))),fun(uint32,fun(code_integer,fun($o,uint32))))),generi7602027413899671122et_bit(uint32)),Xb),N),(B3)),
        aa($o,uint32,aa(nat,fun($o,uint32),aa(uint32,fun(nat,fun($o,uint32)),generi7602027413899671122et_bit(uint32),Xb),aa(code_integer,nat,code_nat_of_integer,N)),(B3)) ) ).

% uint32_set_bit_def
tff(fact_7397_set__bit__uint32__code,axiom,
    ! [Xb: uint32,N: nat,B3: $o] :
      aa($o,uint32,aa(nat,fun($o,uint32),aa(uint32,fun(nat,fun($o,uint32)),generi7602027413899671122et_bit(uint32),Xb),N),(B3)) = $ite(aa(nat,$o,ord_less(nat,N),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),uint32_set_bit(Xb,code_integer_of_nat(N),(B3)),Xb) ).

% set_bit_uint32_code
tff(fact_7398_shiftl__uint32__code,axiom,
    ! [N: nat,Xb: uint32] :
      aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4730199178511100633sh_bit(uint32),N),Xb) = $ite(aa(nat,$o,ord_less(nat,N),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),uint32_shiftl(Xb,code_integer_of_nat(N)),zero_zero(uint32)) ).

% shiftl_uint32_code
tff(fact_7399_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,nat,code_nat_of_integer,K) = $ite(aa(code_integer,$o,ord_less_eq(code_integer,K),zero_zero(code_integer)),zero_zero(nat),aa(product_prod(code_integer,code_integer),nat,product_case_prod(code_integer,code_integer,nat,aTP_Lamp_aod(code_integer,fun(code_integer,nat))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ).

% nat_of_integer_code
tff(fact_7400_nth_H__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A3: array(A)] : array_nth2(A,A3) = comp(nat,heap_Time_Heap(A),code_integer,array_nth(A,A3),code_nat_of_integer) ) ).

% nth'_def
tff(fact_7401_num__of__integer__code,axiom,
    ! [K: code_integer] :
      code_num_of_integer(K) = $ite(aa(code_integer,$o,ord_less_eq(code_integer,K),one_one(code_integer)),one2,aa(product_prod(code_integer,code_integer),num,product_case_prod(code_integer,code_integer,num,aTP_Lamp_aoe(code_integer,fun(code_integer,num))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ).

% num_of_integer_code
tff(fact_7402_int__of__integer__code,axiom,
    ! [K: code_integer] :
      code_int_of_integer(K) = $ite(
        aa(code_integer,$o,ord_less(code_integer,K),zero_zero(code_integer)),
        aa(int,int,uminus_uminus(int),code_int_of_integer(aa(code_integer,code_integer,uminus_uminus(code_integer),K))),
        $ite(K = zero_zero(code_integer),zero_zero(int),aa(product_prod(code_integer,code_integer),int,product_case_prod(code_integer,code_integer,int,aTP_Lamp_aof(code_integer,fun(code_integer,int))),code_divmod_integer(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ) ).

% int_of_integer_code
tff(fact_7403_zero__integer_Orep__eq,axiom,
    code_int_of_integer(zero_zero(code_integer)) = zero_zero(int) ).

% zero_integer.rep_eq
tff(fact_7404_int__of__integer__numeral,axiom,
    ! [K: num] : code_int_of_integer(aa(num,code_integer,numeral_numeral(code_integer),K)) = aa(num,int,numeral_numeral(int),K) ).

% int_of_integer_numeral
tff(fact_7405_one__integer_Orep__eq,axiom,
    code_int_of_integer(one_one(code_integer)) = one_one(int) ).

% one_integer.rep_eq
tff(fact_7406_divide__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),Xb),Xaa)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% divide_integer.rep_eq
tff(fact_7407_less__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] :
      ( aa(code_integer,$o,ord_less(code_integer,Xb),Xaa)
    <=> aa(int,$o,ord_less(int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ) ).

% less_integer.rep_eq
tff(fact_7408_integer__less__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa(code_integer,$o,ord_less(code_integer,K),L)
    <=> aa(int,$o,ord_less(int,code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_iff
tff(fact_7409_int__of__integer__less__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa(int,$o,ord_less(int,code_int_of_integer(Xb)),code_int_of_integer(Y))
    <=> aa(code_integer,$o,ord_less(code_integer,Xb),Y) ) ).

% int_of_integer_less_iff
tff(fact_7410_int__of__integer__pow,axiom,
    ! [Xb: code_integer,N: nat] : code_int_of_integer(aa(nat,code_integer,aa(code_integer,fun(nat,code_integer),power_power(code_integer),Xb),N)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),code_int_of_integer(Xb)),N) ).

% int_of_integer_pow
tff(fact_7411_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa(code_integer,$o,ord_less_eq(code_integer,K),L)
    <=> aa(int,$o,ord_less_eq(int,code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_eq_iff
tff(fact_7412_less__eq__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] :
      ( aa(code_integer,$o,ord_less_eq(code_integer,Xb),Xaa)
    <=> aa(int,$o,ord_less_eq(int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ) ).

% less_eq_integer.rep_eq
tff(fact_7413_bin__last__integer_Orep__eq,axiom,
    ! [Xb: code_integer] :
      ( bits_b8758750999018896077nteger(Xb)
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),code_int_of_integer(Xb)) ) ).

% bin_last_integer.rep_eq
tff(fact_7414_bin__rest__integer_Orep__eq,axiom,
    ! [Xb: code_integer] : code_int_of_integer(bits_b2549910563261871055nteger(Xb)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),code_int_of_integer(Xb)),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% bin_rest_integer.rep_eq
tff(fact_7415_Bit__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: $o] : code_int_of_integer(bits_Bit_integer(Xb,(Xaa))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($o,int,zero_neq_one_of_bool(int),(Xaa))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),code_int_of_integer(Xb))) ).

% Bit_integer.rep_eq
tff(fact_7416_uint32__shiftr__code,axiom,
    ! [W: uint32,N: code_integer] :
      rep_uint322(uint32_shiftr(W,N)) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,N),zero_zero(code_integer))
        | aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),N) ),
        rep_uint322(aa(code_integer,uint32,aa(uint32,fun(code_integer,uint32),aa(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)),undefined(fun(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)))),bit_se4197421643247451524op_bit(uint32)),W),N)),
        aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(nat,fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),bit_se4197421643247451524op_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(code_integer,nat,code_nat_of_integer,N)),rep_uint322(W)) ) ).

% uint32_shiftr_code
tff(fact_7417_uint32__shiftl__code,axiom,
    ! [W: uint32,N: code_integer] :
      rep_uint322(uint32_shiftl(W,N)) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,N),zero_zero(code_integer))
        | aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),N) ),
        rep_uint322(aa(code_integer,uint32,aa(uint32,fun(code_integer,uint32),aa(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)),undefined(fun(fun(nat,fun(uint32,uint32)),fun(uint32,fun(code_integer,uint32)))),bit_se4730199178511100633sh_bit(uint32)),W),N)),
        aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(nat,fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),bit_se4730199178511100633sh_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(code_integer,nat,code_nat_of_integer,N)),rep_uint322(W)) ) ).

% uint32_shiftl_code
tff(fact_7418_zero__uint32_Orep__eq,axiom,
    rep_uint322(zero_zero(uint32)) = zero_zero(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))) ).

% zero_uint32.rep_eq
tff(fact_7419_uint32_Oeven__iff__word__of,axiom,
    ! [P2: uint32] :
      ( aa(uint32,$o,dvd_dvd(uint32,aa(num,uint32,numeral_numeral(uint32),bit0(one2))),P2)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,dvd_dvd(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(num,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),numeral_numeral(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),bit0(one2))),rep_uint322(P2)) ) ).

% uint32.even_iff_word_of
tff(fact_7420_size__uint32_Orep__eq,axiom,
    ! [Xb: uint32] : aa(uint32,nat,size_size(uint32),Xb) = aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),nat,size_size(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint322(Xb)) ).

% size_uint32.rep_eq
tff(fact_7421_uint32_Osize__eq__word__of,axiom,
    ! [P2: uint32] : aa(uint32,nat,size_size(uint32),P2) = aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),nat,size_size(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint322(P2)) ).

% uint32.size_eq_word_of
tff(fact_7422_uint32_Oless__iff__word__of,axiom,
    ! [P2: uint32,Q2: uint32] :
      ( aa(uint32,$o,ord_less(uint32,P2),Q2)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,ord_less(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322(P2)),rep_uint322(Q2)) ) ).

% uint32.less_iff_word_of
tff(fact_7423_less__uint32_Orep__eq,axiom,
    ! [Xb: uint32,Xaa: uint32] :
      ( aa(uint32,$o,ord_less(uint32,Xb),Xaa)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,ord_less(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322(Xb)),rep_uint322(Xaa)) ) ).

% less_uint32.rep_eq
tff(fact_7424_uint32__test__bit__code,axiom,
    ! [W: uint32,N: code_integer] :
      ( uint32_test_bit(W,N)
    <=> $ite(
          ( aa(code_integer,$o,ord_less(code_integer,N),zero_zero(code_integer))
          | aa(code_integer,$o,ord_less(code_integer,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),N) ),
          aa(code_integer,$o,aa(uint32,fun(code_integer,$o),aa(fun(uint32,fun(nat,$o)),fun(uint32,fun(code_integer,$o)),undefined(fun(fun(uint32,fun(nat,$o)),fun(uint32,fun(code_integer,$o)))),bit_se5641148757651400278ts_bit(uint32)),W),N),
          aa(nat,$o,aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(nat,$o),bit_se5641148757651400278ts_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint322(W)),aa(code_integer,nat,code_nat_of_integer,N)) ) ) ).

% uint32_test_bit_code
tff(fact_7425_uint32__set__bit__code,axiom,
    ! [W: uint32,N: code_integer,B3: $o] :
      rep_uint322(uint32_set_bit(W,N,(B3))) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,N),zero_zero(code_integer))
        | aa(code_integer,$o,ord_less(code_integer,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),N) ),
        rep_uint322(aa($o,uint32,aa(code_integer,fun($o,uint32),aa(uint32,fun(code_integer,fun($o,uint32)),aa(fun(uint32,fun(nat,fun($o,uint32))),fun(uint32,fun(code_integer,fun($o,uint32))),undefined(fun(fun(uint32,fun(nat,fun($o,uint32))),fun(uint32,fun(code_integer,fun($o,uint32))))),generi7602027413899671122et_bit(uint32)),W),N),(B3))),
        aa($o,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(nat,fun($o,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(nat,fun($o,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))))),generi7602027413899671122et_bit(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint322(W)),aa(code_integer,nat,code_nat_of_integer,N)),(B3)) ) ).

% uint32_set_bit_code
tff(fact_7426_Uint32__signed__code,axiom,
    ! [I: code_integer] :
      rep_uint322(uint32_signed(I)) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,I),aa(code_integer,code_integer,uminus_uminus(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))))))
        | aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))))),I) ),
        rep_uint322(aa(code_integer,uint32,aa(fun(code_integer,uint32),fun(code_integer,uint32),undefined(fun(fun(code_integer,uint32),fun(code_integer,uint32))),uint322),I)),
        aa(int,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),ring_1_of_int(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),code_I935103866777955880mbolic(I)) ) ).

% Uint32_signed_code
tff(fact_7427_uint32_Oset__bits__aux__code,axiom,
    ! [Fa: fun(nat,$o),N: nat,W: uint32] :
      set_bits_aux_uint32(Fa,N,W) = $ite(
        N = zero_zero(nat),
        W,
        $let(
          n: nat,
          n:= aa(nat,nat,minus_minus(nat,N),one_one(nat)),
          set_bits_aux_uint32(Fa,n,
            aa(uint32,uint32,aa(uint32,fun(uint32,uint32),bit_se1065995026697491101ons_or(uint32),aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4730199178511100633sh_bit(uint32),one_one(nat)),W)),
              $ite(aa(nat,$o,Fa,n),one_one(uint32),zero_zero(uint32)))) ) ) ).

% uint32.set_bits_aux_code
tff(fact_7428_zero__uint32_Orsp,axiom,
    zero_zero(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))) = zero_zero(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))) ).

% zero_uint32.rsp
tff(fact_7429_uint32_Oset__bits__code,axiom,
    ! [P: fun(nat,$o)] : bit_bi4170147762399595738t_bits(uint32,P) = set_bits_aux_uint32(P,aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2)))))),zero_zero(uint32)) ).

% uint32.set_bits_code
tff(fact_7430_int__of__integer__symbolic__aux__code_I1_J,axiom,
    code_I935103866777955880mbolic(zero_zero(code_integer)) = zero_zero(int) ).

% int_of_integer_symbolic_aux_code(1)
tff(fact_7431_uint32__divmod__code,axiom,
    ! [Xb: uint32,Y: uint32] :
      uint32_divmod(Xb,Y) = $ite(
        aa(uint32,$o,ord_less_eq(uint32,aa(num,uint32,numeral_numeral(uint32),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))))),Y),
        $ite(aa(uint32,$o,ord_less(uint32,Xb),Y),aa(uint32,product_prod(uint32,uint32),aa(uint32,fun(uint32,product_prod(uint32,uint32)),product_Pair(uint32,uint32),zero_zero(uint32)),Xb),aa(uint32,product_prod(uint32,uint32),aa(uint32,fun(uint32,product_prod(uint32,uint32)),product_Pair(uint32,uint32),one_one(uint32)),aa(uint32,uint32,minus_minus(uint32,Xb),Y))),
        $ite(
          Y = zero_zero(uint32),
          aa(uint32,product_prod(uint32,uint32),aa(uint32,fun(uint32,product_prod(uint32,uint32)),product_Pair(uint32,uint32),div0_uint32(Xb)),mod0_uint32(Xb)),
          $let(
            q: uint32,
            q:= aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4730199178511100633sh_bit(uint32),one_one(nat)),uint32_sdiv(aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4197421643247451524op_bit(uint32),one_one(nat)),Xb),Y)),
            $let(
              r: uint32,
              r:= aa(uint32,uint32,minus_minus(uint32,Xb),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),q),Y)),
              $ite(aa(uint32,$o,ord_less_eq(uint32,Y),r),aa(uint32,product_prod(uint32,uint32),aa(uint32,fun(uint32,product_prod(uint32,uint32)),product_Pair(uint32,uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),q),one_one(uint32))),aa(uint32,uint32,minus_minus(uint32,r),Y)),aa(uint32,product_prod(uint32,uint32),aa(uint32,fun(uint32,product_prod(uint32,uint32)),product_Pair(uint32,uint32),q),r)) ) ) ) ) ).

% uint32_divmod_code
tff(fact_7432_uint32__divmod__def,axiom,
    ! [Xb: uint32,Y: uint32] :
      uint32_divmod(Xb,Y) = $ite(Y = zero_zero(uint32),aa(uint32,product_prod(uint32,uint32),aa(uint32,fun(uint32,product_prod(uint32,uint32)),product_Pair(uint32,uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),aa(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)),undefined(fun(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)))),divide_divide(uint32)),Xb),zero_zero(uint32))),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),aa(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)),undefined(fun(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)))),modulo_modulo(uint32)),Xb),zero_zero(uint32))),aa(uint32,product_prod(uint32,uint32),aa(uint32,fun(uint32,product_prod(uint32,uint32)),product_Pair(uint32,uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),divide_divide(uint32),Xb),Y)),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),modulo_modulo(uint32),Xb),Y))) ).

% uint32_divmod_def
tff(fact_7433_div0__uint32__def,axiom,
    ! [Xb: uint32] : div0_uint32(Xb) = aa(uint32,uint32,aa(uint32,fun(uint32,uint32),aa(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)),undefined(fun(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)))),divide_divide(uint32)),Xb),zero_zero(uint32)) ).

% div0_uint32_def
tff(fact_7434_mod0__uint32__def,axiom,
    ! [Xb: uint32] : mod0_uint32(Xb) = aa(uint32,uint32,aa(uint32,fun(uint32,uint32),aa(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)),undefined(fun(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)))),modulo_modulo(uint32)),Xb),zero_zero(uint32)) ).

% mod0_uint32_def
tff(fact_7435_uint32__sdiv__code,axiom,
    ! [Xb: uint32,Y: uint32] :
      rep_uint322(uint32_sdiv(Xb,Y)) = $ite(Y = zero_zero(uint32),rep_uint322(aa(uint32,uint32,aa(uint32,fun(uint32,uint32),aa(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)),undefined(fun(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)))),divide_divide(uint32)),Xb),zero_zero(uint32))),signed7115095781618012415divide(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322(Xb),rep_uint322(Y))) ).

% uint32_sdiv_code
tff(fact_7436_sshiftr__uint32__code,axiom,
    ! [N: nat,Xb: uint32] :
      aa(uint32,uint32,aa(nat,fun(uint32,uint32),signed489701013188660438uint32,N),Xb) = $ite(
        aa(nat,$o,ord_less(nat,N),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),
        uint32_sshiftr(Xb,code_integer_of_nat(N)),
        $ite(aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),Xb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(uint32,uint32,uminus_uminus(uint32),one_one(uint32)),zero_zero(uint32)) ) ).

% sshiftr_uint32_code
tff(fact_7437_integer__shiftl__def,axiom,
    ! [Xb: code_integer,N: code_integer] :
      bits_integer_shiftl(Xb,N) = $ite(aa(code_integer,$o,ord_less(code_integer,N),zero_zero(code_integer)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),undefined(fun(code_integer,fun(code_integer,code_integer))),Xb),N),aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),aa(code_integer,nat,code_nat_of_integer,N)),Xb)) ).

% integer_shiftl_def
tff(fact_7438_integer__shiftl__code_I2_J,axiom,
    ! [Xb: code_integer] : bits_integer_shiftl(Xb,zero_zero(code_integer)) = Xb ).

% integer_shiftl_code(2)
tff(fact_7439_uint32__sshiftr__def,axiom,
    ! [Xb: uint32,N: code_integer] :
      uint32_sshiftr(Xb,N) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,N),zero_zero(code_integer))
        | aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),N) ),
        aa(uint32,uint32,aa(code_integer,fun(uint32,uint32),aa(fun(nat,fun(uint32,uint32)),fun(code_integer,fun(uint32,uint32)),undefined(fun(fun(nat,fun(uint32,uint32)),fun(code_integer,fun(uint32,uint32)))),signed489701013188660438uint32),N),Xb),
        aa(uint32,uint32,aa(nat,fun(uint32,uint32),signed489701013188660438uint32,aa(code_integer,nat,code_nat_of_integer,N)),Xb) ) ).

% uint32_sshiftr_def
tff(fact_7440_uint32__sshiftr__code,axiom,
    ! [W: uint32,N: code_integer] :
      rep_uint322(uint32_sshiftr(W,N)) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,N),zero_zero(code_integer))
        | aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),N) ),
        rep_uint322(aa(uint32,uint32,aa(code_integer,fun(uint32,uint32),aa(fun(nat,fun(uint32,uint32)),fun(code_integer,fun(uint32,uint32)),undefined(fun(fun(nat,fun(uint32,uint32)),fun(code_integer,fun(uint32,uint32)))),signed489701013188660438uint32),N),W)),
        signed_drop_bit(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))),aa(code_integer,nat,code_nat_of_integer,N),rep_uint322(W)) ) ).

% uint32_sshiftr_code
tff(fact_7441_div__uint32__code,axiom,
    ! [Xb: uint32,Y: uint32] :
      aa(uint32,uint32,aa(uint32,fun(uint32,uint32),divide_divide(uint32),Xb),Y) = $ite(Y = zero_zero(uint32),zero_zero(uint32),uint32_div(Xb,Y)) ).

% div_uint32_code
tff(fact_7442_mod__uint32__code,axiom,
    ! [Xb: uint32,Y: uint32] :
      aa(uint32,uint32,aa(uint32,fun(uint32,uint32),modulo_modulo(uint32),Xb),Y) = $ite(Y = zero_zero(uint32),Xb,uint32_mod(Xb,Y)) ).

% mod_uint32_code
tff(fact_7443_integer__shiftr__def,axiom,
    ! [Xb: code_integer,N: code_integer] :
      bits_integer_shiftr(Xb,N) = $ite(aa(code_integer,$o,ord_less(code_integer,N),zero_zero(code_integer)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),undefined(fun(code_integer,fun(code_integer,code_integer))),Xb),N),aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4197421643247451524op_bit(code_integer),aa(code_integer,nat,code_nat_of_integer,N)),Xb)) ).

% integer_shiftr_def
tff(fact_7444_lift__def,axiom,
    ! [B: $tType,A: $tType,Fa: fun(A,B)] : heap_Time_lift(A,B,Fa) = comp(B,heap_Time_Heap(B),A,heap_Time_return(B),Fa) ).

% lift_def
tff(fact_7445_lift__collapse,axiom,
    ! [A: $tType,B: $tType,Fa: fun(B,A),Xb: B] : aa(B,heap_Time_Heap(A),heap_Time_lift(B,A,Fa),Xb) = aa(A,heap_Time_Heap(A),heap_Time_return(A),aa(B,A,Fa,Xb)) ).

% lift_collapse
tff(fact_7446_integer__shiftr__code_I2_J,axiom,
    ! [Xb: code_integer] : bits_integer_shiftr(Xb,zero_zero(code_integer)) = Xb ).

% integer_shiftr_code(2)
tff(fact_7447_bind__lift,axiom,
    ! [A: $tType,B: $tType,Fa: heap_Time_Heap(B),G: fun(B,A)] : heap_Time_bind(B,A,Fa,heap_Time_lift(B,A,G)) = heap_Time_bind(B,A,Fa,aTP_Lamp_aog(fun(B,A),fun(B,heap_Time_Heap(A)),G)) ).

% bind_lift
tff(fact_7448_divmod__integer__eq__cases,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_zero(code_integer),
        aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)),
        $ite(
          L = zero_zero(code_integer),
          aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K),
          aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),comp(code_integer,fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,comp(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),code_integer,product_apsnd(code_integer,code_integer,code_integer),times_times(code_integer)),sgn_sgn(code_integer)),L),
            $ite(aa(code_integer,code_integer,sgn_sgn(code_integer),K) = aa(code_integer,code_integer,sgn_sgn(code_integer),L),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_aoh(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ).

% divmod_integer_eq_cases
tff(fact_7449_integer__of__uint32__code,axiom,
    ! [N: uint32] :
      aa(uint32,code_integer,integer_of_uint32,N) = $ite(aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),bit_se1065995026697491101ons_or(code_integer),intege5370686899274169573signed(aa(uint32,uint32,aa(uint32,fun(uint32,uint32),bit_se5824344872417868541ns_and(uint32),N),aa(num,uint32,numeral_numeral(uint32),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))))))))))))))))))))))))))))))),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))))))))))))))))))))))))),intege5370686899274169573signed(N)) ).

% integer_of_uint32_code
tff(fact_7450_apsnd__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,Fa: fun(C,B),Xb: A,Y: C] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),Fa),aa(C,product_prod(A,C),aa(A,fun(C,product_prod(A,C)),product_Pair(A,C),Xb),Y)) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Xb),aa(C,B,Fa,Y)) ).

% apsnd_conv
tff(fact_7451_fst__apsnd,axiom,
    ! [B: $tType,C: $tType,A: $tType,Fa: fun(C,B),Xb: product_prod(A,C)] : aa(product_prod(A,B),A,product_fst(A,B),aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),Fa),Xb)) = aa(product_prod(A,C),A,product_fst(A,C),Xb) ).

% fst_apsnd
tff(fact_7452_snd__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,Fa: fun(C,A),Xb: product_prod(B,C)] : aa(product_prod(B,A),A,product_snd(B,A),aa(product_prod(B,C),product_prod(B,A),aa(fun(C,A),fun(product_prod(B,C),product_prod(B,A)),product_apsnd(C,A,B),Fa),Xb)) = aa(C,A,Fa,aa(product_prod(B,C),C,product_snd(B,C),Xb)) ).

% snd_apsnd
tff(fact_7453_apsnd__eq__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,Fa: fun(C,B),Xb: product_prod(A,C),G: fun(C,B)] :
      ( ( aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),Fa),Xb) = aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),G),Xb) )
    <=> ( aa(C,B,Fa,aa(product_prod(A,C),C,product_snd(A,C),Xb)) = aa(C,B,G,aa(product_prod(A,C),C,product_snd(A,C),Xb)) ) ) ).

% apsnd_eq_conv
tff(fact_7454_fst__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,Fa: fun(B,C)] : comp(product_prod(A,C),A,product_prod(A,B),product_fst(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),Fa)) = product_fst(A,B) ).

% fst_comp_apsnd
tff(fact_7455_snd__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,Fa: fun(B,C)] : comp(product_prod(A,C),C,product_prod(A,B),product_snd(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),Fa)) = comp(B,C,product_prod(A,B),Fa,product_snd(A,B)) ).

% snd_comp_apsnd
tff(fact_7456_apsnd__compose,axiom,
    ! [C: $tType,B: $tType,D2: $tType,A: $tType,Fa: fun(C,B),G: fun(D2,C),Xb: product_prod(A,D2)] : aa(product_prod(A,C),product_prod(A,B),aa(fun(C,B),fun(product_prod(A,C),product_prod(A,B)),product_apsnd(C,B,A),Fa),aa(product_prod(A,D2),product_prod(A,C),aa(fun(D2,C),fun(product_prod(A,D2),product_prod(A,C)),product_apsnd(D2,C,A),G),Xb)) = aa(product_prod(A,D2),product_prod(A,B),aa(fun(D2,B),fun(product_prod(A,D2),product_prod(A,B)),product_apsnd(D2,B,A),comp(C,B,D2,Fa,G)),Xb) ).

% apsnd_compose
tff(fact_7457_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] : code_divmod_abs(zero_zero(code_integer),J) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).

% divmod_abs_code(6)
tff(fact_7458_integer__of__uint32__signed__def,axiom,
    ! [N: uint32] :
      intege5370686899274169573signed(N) = $ite(aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(uint32,code_integer,aa(fun(uint32,code_integer),fun(uint32,code_integer),undefined(fun(fun(uint32,code_integer),fun(uint32,code_integer))),integer_of_uint32),N),aa(uint32,code_integer,integer_of_uint32,N)) ).

% integer_of_uint32_signed_def
tff(fact_7459_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] : code_divmod_abs(J,zero_zero(code_integer)) = aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),abs_abs(code_integer,J)) ).

% divmod_abs_code(5)
tff(fact_7460_divmod__integer__code,axiom,
    ! [K: code_integer,L: code_integer] :
      code_divmod_integer(K,L) = $ite(
        K = zero_zero(code_integer),
        aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),zero_zero(code_integer)),
        $ite(
          aa(code_integer,$o,ord_less(code_integer,zero_zero(code_integer)),L),
          $ite(aa(code_integer,$o,ord_less(code_integer,zero_zero(code_integer)),K),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_aoi(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L))),
          $ite(
            L = zero_zero(code_integer),
            aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),zero_zero(code_integer)),K),
            aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),aa(fun(code_integer,code_integer),fun(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer)),product_apsnd(code_integer,code_integer,code_integer),uminus_uminus(code_integer)),
              $ite(aa(code_integer,$o,ord_less(code_integer,K),zero_zero(code_integer)),code_divmod_abs(K,L),aa(product_prod(code_integer,code_integer),product_prod(code_integer,code_integer),product_case_prod(code_integer,code_integer,product_prod(code_integer,code_integer),aTP_Lamp_aoj(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),code_divmod_abs(K,L)))) ) ) ) ).

% divmod_integer_code
tff(fact_7461_integer__of__uint32__signed__code,axiom,
    ! [N: uint32] :
      intege5370686899274169573signed(N) = $ite(aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),N),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),aa(uint32,code_integer,aa(fun(uint32,code_integer),fun(uint32,code_integer),undefined(fun(fun(uint32,code_integer),fun(uint32,code_integer))),integer_of_uint32),N),code_integer_of_int(aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),int,semiring_1_unsigned(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))),int),rep_uint32(N)))) ).

% integer_of_uint32_signed_code
tff(fact_7462_bit__cut__integer__code,axiom,
    ! [K: code_integer] :
      code_bit_cut_integer(K) = $ite(K = zero_zero(code_integer),aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),zero_zero(code_integer)),$false),aa(product_prod(code_integer,code_integer),product_prod(code_integer,$o),product_case_prod(code_integer,code_integer,product_prod(code_integer,$o),aTP_Lamp_aok(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ).

% bit_cut_integer_code
tff(fact_7463_Rep__uint32_H__code,axiom,
    ! [Xb: uint32] : rep_uint32(Xb) = bit_bi4170147762399595738t_bits(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),Xb)) ).

% Rep_uint32'_code
tff(fact_7464_bit__cut__integer__def,axiom,
    ! [K: code_integer] : code_bit_cut_integer(K) = aa($o,product_prod(code_integer,$o),aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)))),~ aa(code_integer,$o,dvd_dvd(code_integer,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),K)) ).

% bit_cut_integer_def
tff(fact_7465_char__of__integer__code,axiom,
    ! [K: code_integer] : char_of_integer(K) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aTP_Lamp_aos(code_integer,fun($o,char))),code_bit_cut_integer(K)) ).

% char_of_integer_code
tff(fact_7466_uint32__sdiv__def,axiom,
    ! [Xb: uint32,Y: uint32] :
      uint32_sdiv(Xb,Y) = $ite(Y = zero_zero(uint32),aa(uint32,uint32,aa(uint32,fun(uint32,uint32),aa(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)),undefined(fun(fun(uint32,fun(uint32,uint32)),fun(uint32,fun(uint32,uint32)))),divide_divide(uint32)),Xb),zero_zero(uint32)),abs_uint32(signed7115095781618012415divide(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322(Xb),rep_uint322(Y)))) ).

% uint32_sdiv_def
tff(fact_7467_less__uint32_Oabs__eq,axiom,
    ! [Xaa: word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),Xb: word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))] :
      ( aa(uint32,$o,ord_less(uint32,abs_uint32(Xaa)),abs_uint32(Xb))
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,ord_less(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),Xaa),Xb) ) ).

% less_uint32.abs_eq
tff(fact_7468_size__uint32_Oabs__eq,axiom,
    ! [Xb: word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))] : aa(uint32,nat,size_size(uint32),abs_uint32(Xb)) = aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),nat,size_size(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),Xb) ).

% size_uint32.abs_eq
tff(fact_7469_zero__uint32__def,axiom,
    zero_zero(uint32) = abs_uint32(zero_zero(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))))) ).

% zero_uint32_def
tff(fact_7470_inj__on__word__of__int,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => inj_on(int,word(A),ring_1_of_int(word(A)),set_or7035219750837199246ssThan(int,zero_zero(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% inj_on_word_of_int
tff(fact_7471_inj__on__word__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => inj_on(nat,word(A),semiring_1_of_nat(word(A)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),type_len0_len_of(A,type2(A))))) ) ).

% inj_on_word_of_nat
tff(fact_7472_inj__mult__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A3: A] :
          ( inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),top_top(set(A)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% inj_mult_left
tff(fact_7473_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A3: A] :
          ( inj_on(A,A,aTP_Lamp_aot(A,fun(A,A),A3),top_top(set(A)))
        <=> ( A3 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_7474_inj__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,Fa: fun(B,C)] :
      ( inj_on(product_prod(A,B),product_prod(A,C),aa(fun(B,C),fun(product_prod(A,B),product_prod(A,C)),product_apsnd(B,C,A),Fa),top_top(set(product_prod(A,B))))
    <=> inj_on(B,C,Fa,top_top(set(B))) ) ).

% inj_apsnd
tff(fact_7475_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A4: set(A),Fa: fun(A,B)] :
          ( ! [X2: A,Y2: A] :
              ( aa(A,$o,ord_less(A,X2),Y2)
             => ( member(A,X2,A4)
               => ( member(A,Y2,A4)
                 => ( aa(A,B,Fa,X2) != aa(A,B,Fa,Y2) ) ) ) )
         => ( ! [X2: A,Y2: A] :
                ( member(A,X2,A4)
               => ( member(A,Y2,A4)
                 => ( aa(A,$o,ord_less_eq(A,X2),Y2)
                    | aa(A,$o,ord_less_eq(A,Y2),X2) ) ) )
           => inj_on(A,B,Fa,A4) ) ) ) ).

% linorder_inj_onI
tff(fact_7476_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A] : inj_on(A,A,aTP_Lamp_ahe(A,fun(A,A),A3),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_7477_prod_Oimage__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(A,B),A4: set(A)] :
          ( inj_on(A,B,G,A4)
         => ( groups7121269368397514597t_prod(B,B,aTP_Lamp_aou(B,B),aa(set(A),set(B),image(A,B,G),A4)) = groups7121269368397514597t_prod(A,B,G,A4) ) ) ) ).

% prod.image_eq
tff(fact_7478_sum_Oimage__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(A,B),A4: set(A)] :
          ( inj_on(A,B,G,A4)
         => ( aa(set(B),B,groups7311177749621191930dd_sum(B,B,aTP_Lamp_aov(B,B)),aa(set(A),set(B),image(A,B,G),A4)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A4) ) ) ) ).

% sum.image_eq
tff(fact_7479_inj__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => inj_on(nat,A,semiring_1_of_nat(A),top_top(set(nat))) ) ).

% inj_of_nat
tff(fact_7480_inj__on__id2,axiom,
    ! [A: $tType,A4: set(A)] : inj_on(A,A,aTP_Lamp_ct(A,A),A4) ).

% inj_on_id2
tff(fact_7481_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A3: A,A4: set(A)] : inj_on(A,A,aTP_Lamp_aow(A,fun(A,A),A3),A4) ) ).

% inj_on_add'
tff(fact_7482_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A4: set(A),Fa: fun(B,A),D5: set(B)] :
      ( finite_finite2(A,A4)
     => ( inj_on(B,A,Fa,D5)
       => finite_finite2(B,collect(B,aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_aox(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),A4),Fa),D5))) ) ) ).

% finite_inverse_image_gen
tff(fact_7483_finite__Collect,axiom,
    ! [A: $tType,B: $tType,S3: set(A),Fa: fun(B,A)] :
      ( finite_finite2(A,S3)
     => ( inj_on(B,A,Fa,top_top(set(B)))
       => finite_finite2(B,collect(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_aoy(set(A),fun(fun(B,A),fun(B,$o)),S3),Fa))) ) ) ).

% finite_Collect
tff(fact_7484_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A4: set(A),Fa: fun(B,A)] :
      ( finite_finite2(A,A4)
     => ( inj_on(B,A,Fa,top_top(set(B)))
       => finite_finite2(B,collect(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_aoy(set(A),fun(fun(B,A),fun(B,$o)),A4),Fa))) ) ) ).

% finite_inverse_image
tff(fact_7485_inj__fun,axiom,
    ! [B: $tType,C: $tType,A: $tType,Fa: fun(A,B)] :
      ( inj_on(A,B,Fa,top_top(set(A)))
     => inj_on(A,fun(C,B),aTP_Lamp_aoz(fun(A,B),fun(A,fun(C,B)),Fa),top_top(set(A))) ) ).

% inj_fun
tff(fact_7486_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_akt(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_7487_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A3: A,A4: set(A)] :
          ( ( A3 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A3),A4) ) ) ).

% inj_on_mult
tff(fact_7488_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [Fa: fun(A,B)] :
          ( ! [X2: A,Y2: A] :
              ( aa(A,$o,ord_less(A,X2),Y2)
             => ( aa(A,B,Fa,X2) != aa(A,B,Fa,Y2) ) )
         => inj_on(A,B,Fa,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_7489_inj__on__strict__subset,axiom,
    ! [B: $tType,A: $tType,Fa: fun(A,B),B6: set(A),A4: set(A)] :
      ( inj_on(A,B,Fa,B6)
     => ( aa(set(A),$o,ord_less(set(A),A4),B6)
       => aa(set(B),$o,ord_less(set(B),aa(set(A),set(B),image(A,B,Fa),A4)),aa(set(A),set(B),image(A,B,Fa),B6)) ) ) ).

% inj_on_strict_subset
tff(fact_7490_injective__scaleR,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [C2: real] :
          ( ( C2 != zero_zero(real) )
         => inj_on(A,A,real_V8093663219630862766scaleR(A,C2),top_top(set(A))) ) ) ).

% injective_scaleR
tff(fact_7491_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A4: set(A)] :
      ( finite_finite2(A,A4)
     => ? [N3: nat,F4: fun(nat,A)] :
          ( ( A4 = aa(set(nat),set(A),image(nat,A,F4),collect(nat,aTP_Lamp_sx(nat,fun(nat,$o),N3))) )
          & inj_on(nat,A,F4,collect(nat,aTP_Lamp_sx(nat,fun(nat,$o),N3))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_7492_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,Fa: fun(A,B),Y: A,Xs: list(A)] :
      ( inj_on(A,B,Fa,aa(set(A),set(A),insert(A,Y),set2(A,Xs)))
     => ( filter2(A,aa(A,fun(A,$o),aTP_Lamp_apa(fun(A,B),fun(A,fun(A,$o)),Fa),Y),Xs) = filter2(A,fequal(A,Y),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_7493_DERIV__real__root__generic,axiom,
    ! [N: nat,Xb: real,D5: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( ( Xb != zero_zero(real) )
       => ( ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
           => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
             => ( D5 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),Xb)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))) ) ) )
         => ( ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
             => ( aa(real,$o,ord_less(real,Xb),zero_zero(real))
               => ( D5 = aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),Xb)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat))))))) ) ) )
           => ( ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
               => ( D5 = aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),Xb)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))) ) )
             => has_field_derivative(real,root(N),D5,topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ) ) ).

% DERIV_real_root_generic
tff(fact_7494_DERIV__even__real__root,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
       => ( aa(real,$o,ord_less(real,Xb),zero_zero(real))
         => has_field_derivative(real,root(N),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),Xb)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_even_real_root
tff(fact_7495_swap__inj__on,axiom,
    ! [B: $tType,A: $tType,A4: set(product_prod(A,B))] : inj_on(product_prod(A,B),product_prod(B,A),product_case_prod(A,B,product_prod(B,A),aTP_Lamp_tq(A,fun(B,product_prod(B,A)))),A4) ).

% swap_inj_on
tff(fact_7496_inj__split__Cons,axiom,
    ! [A: $tType,X5: set(product_prod(list(A),A))] : inj_on(product_prod(list(A),A),list(A),product_case_prod(list(A),A,list(A),aTP_Lamp_alj(list(A),fun(A,list(A)))),X5) ).

% inj_split_Cons
tff(fact_7497_inj__graph,axiom,
    ! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_apc(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).

% inj_graph
tff(fact_7498_inj__on__diff__nat,axiom,
    ! [N2: set(nat),K: nat] :
      ( ! [N3: nat] :
          ( member(nat,N3,N2)
         => aa(nat,$o,ord_less_eq(nat,K),N3) )
     => inj_on(nat,nat,aTP_Lamp_aiq(nat,fun(nat,nat),K),N2) ) ).

% inj_on_diff_nat
tff(fact_7499_inj__singleton,axiom,
    ! [A: $tType,A4: set(A)] : inj_on(A,set(A),aTP_Lamp_apd(A,set(A)),A4) ).

% inj_singleton
tff(fact_7500_inj__Some,axiom,
    ! [A: $tType,A4: set(A)] : inj_on(A,option(A),some(A),A4) ).

% inj_Some
tff(fact_7501_inj__Suc,axiom,
    ! [N2: set(nat)] : inj_on(nat,nat,suc,N2) ).

% inj_Suc
tff(fact_7502_signed_Osorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => inj_on(word(A),word(A),aTP_Lamp_ape(word(A),word(A)),top_top(set(word(A)))) ) ).

% signed.sorted_list_of_set.inj_on
tff(fact_7503_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,Fa: fun(A,B),X5: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_apf(fun(A,B),fun(A,product_prod(A,B)),Fa),X5) ).

% inj_on_convol_ident
tff(fact_7504_inj__Pair_I2_J,axiom,
    ! [B: $tType,A: $tType,C2: fun(A,B),S3: set(A)] : inj_on(A,product_prod(B,A),aTP_Lamp_apg(fun(A,B),fun(A,product_prod(B,A)),C2),S3) ).

% inj_Pair(2)
tff(fact_7505_inj__Pair_I1_J,axiom,
    ! [B: $tType,A: $tType,C2: fun(A,B),S3: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_apf(fun(A,B),fun(A,product_prod(A,B)),C2),S3) ).

% inj_Pair(1)
tff(fact_7506_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A,S: set(A),G: fun(A,A),E6: A] :
          ( has_field_derivative(A,Fa,D5,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E6,topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,A,G,Xb) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aph(fun(A,A),fun(fun(A,A),fun(A,A)),Fa),G),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(A,A,G,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Fa,Xb)),E6))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,G,Xb)),aa(A,A,G,Xb))),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% DERIV_divide
tff(fact_7507_DERIV__cdivide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A,S: set(A),C2: A] :
          ( has_field_derivative(A,Fa,D5,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_api(fun(A,A),fun(A,fun(A,A)),Fa),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),D5),C2),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_cdivide
tff(fact_7508_DERIV__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,Fa,D5,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aTP_Lamp_apj(fun(A,A),fun(A,A),Fa),aa(A,A,uminus_uminus(A),D5),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_minus
tff(fact_7509_field__differentiable__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),F3: A,F5: filter(A)] :
          ( has_field_derivative(A,Fa,F3,F5)
         => has_field_derivative(A,aTP_Lamp_apj(fun(A,A),fun(A,A),Fa),aa(A,A,uminus_uminus(A),F3),F5) ) ) ).

% field_differentiable_minus
tff(fact_7510_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K: A,Xaa: A] : has_field_derivative(A,aTP_Lamp_apk(A,fun(A,A),K),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xaa),K))),topolo174197925503356063within(A,Xaa,top_top(set(A)))) ) ).

% DERIV_cos_add
tff(fact_7511_DERIV__fun__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,Xb: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_apl(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),sin(A,aa(A,A,G,Xb)))),M),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_fun_cos
tff(fact_7512_DERIV__mirror,axiom,
    ! [Fa: fun(real,real),Y: real,Xb: real] :
      ( has_field_derivative(real,Fa,Y,topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),Xb),top_top(set(real))))
    <=> has_field_derivative(real,aTP_Lamp_apm(fun(real,real),fun(real,real),Fa),aa(real,real,uminus_uminus(real),Y),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_mirror
tff(fact_7513_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xb: A,S: set(A)] :
          ( ( Xb != zero_zero(A) )
         => has_field_derivative(A,inverse_inverse(A),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_inverse
tff(fact_7514_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,Fa,D5,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,A,Fa,Xb) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_apn(fun(A,A),fun(A,A),Fa),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(A,A,Fa,Xb))),D5)),aa(A,A,inverse_inverse(A),aa(A,A,Fa,Xb)))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_inverse'
tff(fact_7515_DERIV__ln__divide,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => has_field_derivative(real,ln_ln(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),Xb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_ln_divide
tff(fact_7516_DERIV__const__average,axiom,
    ! [A3: real,B3: real,V: fun(real,real),K: real] :
      ( ( A3 != B3 )
     => ( ! [X2: real] : has_field_derivative(real,V,K,topolo174197925503356063within(real,X2,top_top(set(real))))
       => ( aa(real,real,V,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),A3),B3)),aa(num,real,numeral_numeral(real),bit0(one2)))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,V,A3)),aa(real,real,V,B3))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ) ).

% DERIV_const_average
tff(fact_7517_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A,S: set(A),N: nat] :
          ( has_field_derivative(A,Fa,D5,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_apo(fun(A,A),fun(nat,fun(A,A)),Fa),N),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),N))),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Fa,Xb)),N))),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_power_Suc
tff(fact_7518_at__within__Icc__at,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,Xb: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),Xb)
         => ( aa(A,$o,ord_less(A,Xb),B3)
           => ( topolo174197925503356063within(A,Xb,set_or1337092689740270186AtMost(A,A3,B3)) = topolo174197925503356063within(A,Xb,top_top(set(A))) ) ) ) ) ).

% at_within_Icc_at
tff(fact_7519_DERIV__ident,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F5: filter(A)] : has_field_derivative(A,aTP_Lamp_app(A,A),one_one(A),F5) ) ).

% DERIV_ident
tff(fact_7520_DERIV__ln,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => has_field_derivative(real,ln_ln(real),aa(real,real,inverse_inverse(real),Xb),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_ln
tff(fact_7521_DERIV__neg__dec__right,axiom,
    ! [Fa: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,Fa,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D3)
            & ! [H3: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H3)
               => ( aa(real,$o,ord_less(real,H3),D3)
                 => aa(real,$o,ord_less(real,aa(real,real,Fa,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H3))),aa(real,real,Fa,Xb)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_7522_DERIV__pos__inc__right,axiom,
    ! [Fa: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,Fa,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D3)
            & ! [H3: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H3)
               => ( aa(real,$o,ord_less(real,H3),D3)
                 => aa(real,$o,ord_less(real,aa(real,real,Fa,Xb)),aa(real,real,Fa,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H3))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_7523_DERIV__isconst__all,axiom,
    ! [Fa: fun(real,real),Xb: real,Y: real] :
      ( ! [X2: real] : has_field_derivative(real,Fa,zero_zero(real),topolo174197925503356063within(real,X2,top_top(set(real))))
     => ( aa(real,real,Fa,Xb) = aa(real,real,Fa,Y) ) ) ).

% DERIV_isconst_all
tff(fact_7524_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K: A,F5: filter(A)] : has_field_derivative(A,aTP_Lamp_apq(A,fun(A,A),K),zero_zero(A),F5) ) ).

% DERIV_const
tff(fact_7525_has__real__derivative__neg__dec__right,axiom,
    ! [Fa: fun(real,real),L: real,Xb: real,S3: set(real)] :
      ( has_field_derivative(real,Fa,L,topolo174197925503356063within(real,Xb,S3))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D3)
            & ! [H3: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H3)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H3),S3)
                 => ( aa(real,$o,ord_less(real,H3),D3)
                   => aa(real,$o,ord_less(real,aa(real,real,Fa,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H3))),aa(real,real,Fa,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_7526_has__real__derivative__pos__inc__right,axiom,
    ! [Fa: fun(real,real),L: real,Xb: real,S3: set(real)] :
      ( has_field_derivative(real,Fa,L,topolo174197925503356063within(real,Xb,S3))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D3)
            & ! [H3: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H3)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H3),S3)
                 => ( aa(real,$o,ord_less(real,H3),D3)
                   => aa(real,$o,ord_less(real,aa(real,real,Fa,Xb)),aa(real,real,Fa,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_7527_at__within__Icc__at__left,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ( topolo174197925503356063within(A,B3,set_or1337092689740270186AtMost(A,A3,B3)) = topolo174197925503356063within(A,B3,set_ord_lessThan(A,B3)) ) ) ) ).

% at_within_Icc_at_left
tff(fact_7528_DERIV__fun__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,Xb: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_apr(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,exp(A),aa(A,A,G,Xb))),M),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_fun_exp
tff(fact_7529_DERIV__chain_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A,S: set(A),G: fun(A,A),E6: A] :
          ( has_field_derivative(A,Fa,D5,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E6,topolo174197925503356063within(A,aa(A,A,Fa,Xb),top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aps(fun(A,A),fun(fun(A,A),fun(A,A)),Fa),G),aa(A,A,aa(A,fun(A,A),times_times(A),E6),D5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_chain'
tff(fact_7530_DERIV__chain2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),Da: A,G: fun(A,A),Xb: A,Db: A,S: set(A)] :
          ( has_field_derivative(A,Fa,Da,topolo174197925503356063within(A,aa(A,A,G,Xb),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_apt(fun(A,A),fun(fun(A,A),fun(A,A)),Fa),G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_chain2
tff(fact_7531_DERIV__chain3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [G: fun(A,A),G5: fun(A,A),Fa: fun(A,A),F3: A,Xb: A] :
          ( ! [X2: A] : has_field_derivative(A,G,aa(A,A,G5,X2),topolo174197925503356063within(A,X2,top_top(set(A))))
         => ( has_field_derivative(A,Fa,F3,topolo174197925503356063within(A,Xb,top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_apt(fun(A,A),fun(fun(A,A),fun(A,A)),G),Fa),aa(A,A,aa(A,fun(A,A),times_times(A),F3),aa(A,A,G5,aa(A,A,Fa,Xb))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% DERIV_chain3
tff(fact_7532_DERIV__chain__s,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [S: set(A),G: fun(A,A),G5: fun(A,A),Fa: fun(A,A),F3: A,Xb: A] :
          ( ! [X2: A] :
              ( member(A,X2,S)
             => has_field_derivative(A,G,aa(A,A,G5,X2),topolo174197925503356063within(A,X2,top_top(set(A)))) )
         => ( has_field_derivative(A,Fa,F3,topolo174197925503356063within(A,Xb,top_top(set(A))))
           => ( member(A,aa(A,A,Fa,Xb),S)
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_apt(fun(A,A),fun(fun(A,A),fun(A,A)),G),Fa),aa(A,A,aa(A,fun(A,A),times_times(A),F3),aa(A,A,G5,aa(A,A,Fa,Xb))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ) ).

% DERIV_chain_s
tff(fact_7533_DERIV__fun__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),M: A,Xb: A] :
          ( has_field_derivative(A,G,M,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => has_field_derivative(A,aTP_Lamp_apu(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,aa(A,A,G,Xb))),M),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_fun_sin
tff(fact_7534_DERIV__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),Y: A,Xb: A,Z: A] :
          ( has_field_derivative(A,Fa,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z),top_top(set(A))))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_apv(fun(A,A),fun(A,fun(A,A)),Fa),Z),Y,topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_shift
tff(fact_7535_field__differentiable__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),F3: A,F5: filter(A),G: fun(A,A),G5: A] :
          ( has_field_derivative(A,Fa,F3,F5)
         => ( has_field_derivative(A,G,G5,F5)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_apw(fun(A,A),fun(fun(A,A),fun(A,A)),Fa),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),F3),G5),F5) ) ) ) ).

% field_differentiable_add
tff(fact_7536_DERIV__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A,S: set(A),G: fun(A,A),E6: A] :
          ( has_field_derivative(A,Fa,D5,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E6,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_apw(fun(A,A),fun(fun(A,A),fun(A,A)),Fa),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),D5),E6),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_add
tff(fact_7537_has__field__derivative__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,F5: filter(A),C2: real] :
          ( has_field_derivative(A,Fa,D5,F5)
         => has_field_derivative(A,aa(real,fun(A,A),aTP_Lamp_apx(fun(A,A),fun(real,fun(A,A)),Fa),C2),aa(A,A,real_V8093663219630862766scaleR(A,C2),D5),F5) ) ) ).

% has_field_derivative_scaleR_right
tff(fact_7538_DERIV__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [S3: set(A),Fa: fun(B,fun(A,B)),F3: fun(C,fun(A,B)),Xb: C,F5: filter(B)] :
          ( ! [N3: A] :
              ( member(A,N3,S3)
             => has_field_derivative(B,aa(A,fun(B,B),aTP_Lamp_apy(fun(B,fun(A,B)),fun(A,fun(B,B)),Fa),N3),aa(A,B,aa(C,fun(A,B),F3,Xb),N3),F5) )
         => has_field_derivative(B,aa(fun(B,fun(A,B)),fun(B,B),aTP_Lamp_apz(set(A),fun(fun(B,fun(A,B)),fun(B,B)),S3),Fa),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(C,fun(A,B),F3,Xb)),S3),F5) ) ) ).

% DERIV_sum
tff(fact_7539_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),Da: A,Xb: A,S: set(A),G: fun(A,A),Db: A] :
          ( has_field_derivative(A,Fa,Da,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aqa(fun(A,A),fun(fun(A,A),fun(A,A)),Fa),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Da),aa(A,A,G,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),Db),aa(A,A,Fa,Xb))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_mult
tff(fact_7540_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A,S: set(A),G: fun(A,A),E6: A] :
          ( has_field_derivative(A,Fa,D5,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E6,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aqa(fun(A,A),fun(fun(A,A),fun(A,A)),Fa),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Fa,Xb)),E6)),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(A,A,G,Xb))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_mult'
tff(fact_7541_DERIV__cmult__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A,S: set(A),C2: A] :
          ( has_field_derivative(A,Fa,D5,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_aqb(fun(A,A),fun(A,fun(A,A)),Fa),C2),aa(A,A,aa(A,fun(A,A),times_times(A),D5),C2),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_cmult_right
tff(fact_7542_DERIV__cmult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A,S: set(A),C2: A] :
          ( has_field_derivative(A,Fa,D5,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_aqc(fun(A,A),fun(A,fun(A,A)),Fa),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D5),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_cmult
tff(fact_7543_has__field__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aTP_Lamp_aqd(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,aa(A,A,G,Xb))),Db),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_field_derivative_cosh
tff(fact_7544_has__field__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aTP_Lamp_aqe(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,aa(A,A,G,Xb))),Db),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_field_derivative_sinh
tff(fact_7545_termdiffs__strong__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xb: A] :
          ( ! [Y2: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_xc(fun(nat,A),fun(A,fun(nat,A)),C2),Y2))
         => has_field_derivative(A,aTP_Lamp_aqf(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_xd(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% termdiffs_strong_converges_everywhere
tff(fact_7546_DERIV__at__within__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),Y: A,Z: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,Fa,Y,topolo174197925503356063within(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),Xb),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),Z)),S3)))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_aqg(fun(A,A),fun(A,fun(A,A)),Fa),Z),Y,topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_at_within_shift
tff(fact_7547_field__differentiable__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),F3: A,F5: filter(A),G: fun(A,A),G5: A] :
          ( has_field_derivative(A,Fa,F3,F5)
         => ( has_field_derivative(A,G,G5,F5)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aqh(fun(A,A),fun(fun(A,A),fun(A,A)),Fa),G),aa(A,A,minus_minus(A,F3),G5),F5) ) ) ) ).

% field_differentiable_diff
tff(fact_7548_DERIV__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A,S: set(A),G: fun(A,A),E6: A] :
          ( has_field_derivative(A,Fa,D5,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E6,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aqh(fun(A,A),fun(fun(A,A),fun(A,A)),Fa),G),aa(A,A,minus_minus(A,D5),E6),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_diff
tff(fact_7549_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A,S: set(A),N: nat] :
          ( has_field_derivative(A,Fa,D5,topolo174197925503356063within(A,Xb,S))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_aqi(fun(A,A),fun(nat,fun(A,A)),Fa),N),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),N)),aa(A,A,aa(A,fun(A,A),times_times(A),D5),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Fa,Xb)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,Xb,S)) ) ) ).

% DERIV_power
tff(fact_7550_DERIV__pow,axiom,
    ! [N: nat,Xb: real,S: set(real)] : has_field_derivative(real,aTP_Lamp_aqj(nat,fun(real,real),N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(real,Xb,S)) ).

% DERIV_pow
tff(fact_7551_has__real__derivative__pos__inc__left,axiom,
    ! [Fa: fun(real,real),L: real,Xb: real,S3: set(real)] :
      ( has_field_derivative(real,Fa,L,topolo174197925503356063within(real,Xb,S3))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D3)
            & ! [H3: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H3)
               => ( member(real,aa(real,real,minus_minus(real,Xb),H3),S3)
                 => ( aa(real,$o,ord_less(real,H3),D3)
                   => aa(real,$o,ord_less(real,aa(real,real,Fa,aa(real,real,minus_minus(real,Xb),H3))),aa(real,real,Fa,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_7552_has__real__derivative__neg__dec__left,axiom,
    ! [Fa: fun(real,real),L: real,Xb: real,S3: set(real)] :
      ( has_field_derivative(real,Fa,L,topolo174197925503356063within(real,Xb,S3))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D3)
            & ! [H3: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H3)
               => ( member(real,aa(real,real,minus_minus(real,Xb),H3),S3)
                 => ( aa(real,$o,ord_less(real,H3),D3)
                   => aa(real,$o,ord_less(real,aa(real,real,Fa,Xb)),aa(real,real,Fa,aa(real,real,minus_minus(real,Xb),H3))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_7553_deriv__nonneg__imp__mono,axiom,
    ! [A3: real,B3: real,G: fun(real,real),G5: fun(real,real)] :
      ( ! [X2: real] :
          ( member(real,X2,set_or1337092689740270186AtMost(real,A3,B3))
         => has_field_derivative(real,G,aa(real,real,G5,X2),topolo174197925503356063within(real,X2,top_top(set(real)))) )
     => ( ! [X2: real] :
            ( member(real,X2,set_or1337092689740270186AtMost(real,A3,B3))
           => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,G5,X2)) )
       => ( aa(real,$o,ord_less_eq(real,A3),B3)
         => aa(real,$o,ord_less_eq(real,aa(real,real,G,A3)),aa(real,real,G,B3)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_7554_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A3: real,B3: real,Fa: fun(real,real)] :
      ( aa(real,$o,ord_less_eq(real,A3),B3)
     => ( ! [X2: real] :
            ( aa(real,$o,ord_less_eq(real,A3),X2)
           => ( aa(real,$o,ord_less_eq(real,X2),B3)
             => ? [Y4: real] :
                  ( has_field_derivative(real,Fa,Y4,topolo174197925503356063within(real,X2,top_top(set(real))))
                  & aa(real,$o,ord_less_eq(real,zero_zero(real)),Y4) ) ) )
       => aa(real,$o,ord_less_eq(real,aa(real,real,Fa,A3)),aa(real,real,Fa,B3)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_7555_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A3: real,B3: real,Fa: fun(real,real)] :
      ( aa(real,$o,ord_less_eq(real,A3),B3)
     => ( ! [X2: real] :
            ( aa(real,$o,ord_less_eq(real,A3),X2)
           => ( aa(real,$o,ord_less_eq(real,X2),B3)
             => ? [Y4: real] :
                  ( has_field_derivative(real,Fa,Y4,topolo174197925503356063within(real,X2,top_top(set(real))))
                  & aa(real,$o,ord_less_eq(real,Y4),zero_zero(real)) ) ) )
       => aa(real,$o,ord_less_eq(real,aa(real,real,Fa,B3)),aa(real,real,Fa,A3)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_7556_DERIV__local__const,axiom,
    ! [Fa: fun(real,real),L: real,Xb: real,D: real] :
      ( has_field_derivative(real,Fa,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),D)
       => ( ! [Y2: real] :
              ( aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,Xb),Y2))),D)
             => ( aa(real,real,Fa,Xb) = aa(real,real,Fa,Y2) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_7557_DERIV__pos__inc__left,axiom,
    ! [Fa: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,Fa,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D3)
            & ! [H3: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H3)
               => ( aa(real,$o,ord_less(real,H3),D3)
                 => aa(real,$o,ord_less(real,aa(real,real,Fa,aa(real,real,minus_minus(real,Xb),H3))),aa(real,real,Fa,Xb)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_7558_DERIV__neg__dec__left,axiom,
    ! [Fa: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,Fa,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D3)
            & ! [H3: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H3)
               => ( aa(real,$o,ord_less(real,H3),D3)
                 => aa(real,$o,ord_less(real,aa(real,real,Fa,Xb)),aa(real,real,Fa,aa(real,real,minus_minus(real,Xb),H3))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_7559_DERIV__local__min,axiom,
    ! [Fa: fun(real,real),L: real,Xb: real,D: real] :
      ( has_field_derivative(real,Fa,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),D)
       => ( ! [Y2: real] :
              ( aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,Xb),Y2))),D)
             => aa(real,$o,ord_less_eq(real,aa(real,real,Fa,Xb)),aa(real,real,Fa,Y2)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_min
tff(fact_7560_DERIV__local__max,axiom,
    ! [Fa: fun(real,real),L: real,Xb: real,D: real] :
      ( has_field_derivative(real,Fa,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),D)
       => ( ! [Y2: real] :
              ( aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,Xb),Y2))),D)
             => aa(real,$o,ord_less_eq(real,aa(real,real,Fa,Y2)),aa(real,real,Fa,Xb)) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_max
tff(fact_7561_DERIV__pos__imp__increasing,axiom,
    ! [A3: real,B3: real,Fa: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( ! [X2: real] :
            ( aa(real,$o,ord_less_eq(real,A3),X2)
           => ( aa(real,$o,ord_less_eq(real,X2),B3)
             => ? [Y4: real] :
                  ( has_field_derivative(real,Fa,Y4,topolo174197925503356063within(real,X2,top_top(set(real))))
                  & aa(real,$o,ord_less(real,zero_zero(real)),Y4) ) ) )
       => aa(real,$o,ord_less(real,aa(real,real,Fa,A3)),aa(real,real,Fa,B3)) ) ) ).

% DERIV_pos_imp_increasing
tff(fact_7562_DERIV__neg__imp__decreasing,axiom,
    ! [A3: real,B3: real,Fa: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( ! [X2: real] :
            ( aa(real,$o,ord_less_eq(real,A3),X2)
           => ( aa(real,$o,ord_less_eq(real,X2),B3)
             => ? [Y4: real] :
                  ( has_field_derivative(real,Fa,Y4,topolo174197925503356063within(real,X2,top_top(set(real))))
                  & aa(real,$o,ord_less(real,Y4),zero_zero(real)) ) ) )
       => aa(real,$o,ord_less(real,aa(real,real,Fa,B3)),aa(real,real,Fa,A3)) ) ) ).

% DERIV_neg_imp_decreasing
tff(fact_7563_MVT2,axiom,
    ! [A3: real,B3: real,Fa: fun(real,real),F3: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( ! [X2: real] :
            ( aa(real,$o,ord_less_eq(real,A3),X2)
           => ( aa(real,$o,ord_less_eq(real,X2),B3)
             => has_field_derivative(real,Fa,aa(real,real,F3,X2),topolo174197925503356063within(real,X2,top_top(set(real)))) ) )
       => ? [Z2: real] :
            ( aa(real,$o,ord_less(real,A3),Z2)
            & aa(real,$o,ord_less(real,Z2),B3)
            & ( aa(real,real,minus_minus(real,aa(real,real,Fa,B3)),aa(real,real,Fa,A3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,B3),A3)),aa(real,real,F3,Z2)) ) ) ) ) ).

% MVT2
tff(fact_7564_DERIV__fun__pow,axiom,
    ! [G: fun(real,real),M: real,Xb: real,N: nat] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_aqk(fun(real,real),fun(nat,fun(real,real)),G),N),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,G,Xb)),aa(nat,nat,minus_minus(nat,N),one_one(nat))))),M),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_fun_pow
tff(fact_7565_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D: A,Xb: A,S: set(A),G: fun(A,A),E3: A] :
          ( has_field_derivative(A,Fa,D,topolo174197925503356063within(A,Xb,S))
         => ( has_field_derivative(A,G,E3,topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,A,G,Xb) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aph(fun(A,A),fun(fun(A,A),fun(A,A)),Fa),G),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),D),aa(A,A,G,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),E3),aa(A,A,Fa,Xb)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,G,Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% DERIV_quotient
tff(fact_7566_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D: A,Xb: A,S: set(A)] :
          ( has_field_derivative(A,Fa,D,topolo174197925503356063within(A,Xb,S))
         => ( ( aa(A,A,Fa,Xb) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_apn(fun(A,A),fun(A,A),Fa),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Fa,Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_7567_finite__imp__inj__to__nat__seg_H,axiom,
    ! [A: $tType,A4: set(A)] :
      ( finite_finite2(A,A4)
     => ~ ! [F4: fun(A,nat)] :
            ( ? [N3: nat] : aa(set(A),set(nat),image(A,nat,F4),A4) = collect(nat,aTP_Lamp_sx(nat,fun(nat,$o),N3))
           => ~ inj_on(A,nat,F4,A4) ) ) ).

% finite_imp_inj_to_nat_seg'
tff(fact_7568_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A4: set(A)] :
      ( finite_finite2(A,A4)
     => ? [F4: fun(A,nat),N3: nat] :
          ( ( aa(set(A),set(nat),image(A,nat,F4),A4) = collect(nat,aTP_Lamp_sx(nat,fun(nat,$o),N3)) )
          & inj_on(A,nat,F4,A4) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_7569_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C2: fun(nat,A),Fa: fun(A,A),F3: A,Z: A] :
          ( ! [Z2: A] :
              ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z2)),K5)
             => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_xc(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)),aa(A,A,Fa,Z2)) )
         => ( has_field_derivative(A,Fa,F3,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),K5)
             => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_xd(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),F3) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_7570_has__real__derivative__powr,axiom,
    ! [Z: real,R2: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Z)
     => has_field_derivative(real,aTP_Lamp_aql(real,fun(real,real),R2),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,Z,aa(real,real,minus_minus(real,R2),one_one(real)))),topolo174197925503356063within(real,Z,top_top(set(real)))) ) ).

% has_real_derivative_powr
tff(fact_7571_termdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_xc(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_xd(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
           => ( summable(A,aa(A,fun(nat,A),aTP_Lamp_aqm(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
             => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K5))
               => has_field_derivative(A,aTP_Lamp_aqf(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_xd(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ) ) ).

% termdiffs
tff(fact_7572_termdiffs__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_xc(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K5))
           => has_field_derivative(A,aTP_Lamp_aqf(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_xd(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% termdiffs_strong
tff(fact_7573_termdiffs__strong_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K5: real,C2: fun(nat,A),Z: A] :
          ( ! [Z2: A] :
              ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z2)),K5)
             => summable(A,aa(A,fun(nat,A),aTP_Lamp_xc(fun(nat,A),fun(A,fun(nat,A)),C2),Z2)) )
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),K5)
           => has_field_derivative(A,aTP_Lamp_aqf(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_xd(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),topolo174197925503356063within(A,Z,top_top(set(A)))) ) ) ) ).

% termdiffs_strong'
tff(fact_7574_summable__reindex,axiom,
    ! [Fa: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,Fa)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X2: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,Fa,X2))
         => summable(real,comp(nat,real,nat,Fa,G)) ) ) ) ).

% summable_reindex
tff(fact_7575_DERIV__fun__powr,axiom,
    ! [G: fun(real,real),M: real,Xb: real,R2: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,G,Xb))
       => has_field_derivative(real,aa(real,fun(real,real),aTP_Lamp_aqn(fun(real,real),fun(real,fun(real,real)),G),R2),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),powr(real,aa(real,real,G,Xb),aa(real,real,minus_minus(real,R2),aa(nat,real,semiring_1_of_nat(real),one_one(nat)))))),M),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_fun_powr
tff(fact_7576_DERIV__log,axiom,
    ! [Xb: real,B3: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => has_field_derivative(real,log(B3),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,ln_ln(real),B3)),Xb)),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_log
tff(fact_7577_DERIV__powr,axiom,
    ! [G: fun(real,real),M: real,Xb: real,Fa: fun(real,real),R2: real] :
      ( has_field_derivative(real,G,M,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,G,Xb))
       => ( has_field_derivative(real,Fa,R2,topolo174197925503356063within(real,Xb,top_top(set(real))))
         => has_field_derivative(real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqo(fun(real,real),fun(fun(real,real),fun(real,real)),G),Fa),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(real,real,G,Xb),aa(real,real,Fa,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),R2),aa(real,real,ln_ln(real),aa(real,real,G,Xb)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),M),aa(real,real,Fa,Xb))),aa(real,real,G,Xb)))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_powr
tff(fact_7578_DERIV__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => has_field_derivative(A,tan(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),cos(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_tan
tff(fact_7579_DERIV__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => has_field_derivative(real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb))),aa(num,real,numeral_numeral(real),bit0(one2))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_real_sqrt
tff(fact_7580_DERIV__arctan,axiom,
    ! [Xb: real] : has_field_derivative(real,arctan,aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ).

% DERIV_arctan
tff(fact_7581_arsinh__real__has__field__derivative,axiom,
    ! [Xb: real,A4: set(real)] : has_field_derivative(real,arsinh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,Xb,A4)) ).

% arsinh_real_has_field_derivative
tff(fact_7582_DERIV__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sin(A,Xb) != zero_zero(A) )
         => has_field_derivative(A,cot(A),aa(A,A,uminus_uminus(A),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),sin(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_cot
tff(fact_7583_has__field__derivative__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Xb: A,Db: A,S: set(A)] :
          ( ( cosh(A,aa(A,A,G,Xb)) != zero_zero(A) )
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S))
           => has_field_derivative(A,aTP_Lamp_aqp(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,tanh(A),aa(A,A,G,Xb))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Db),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_field_derivative_tanh
tff(fact_7584_DERIV__real__sqrt__generic,axiom,
    ! [Xb: real,D5: real] :
      ( ( Xb != zero_zero(real) )
     => ( ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( D5 = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb))),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
       => ( ( aa(real,$o,ord_less(real,Xb),zero_zero(real))
           => ( D5 = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb)))),aa(num,real,numeral_numeral(real),bit0(one2))) ) )
         => has_field_derivative(real,sqrt,D5,topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_real_sqrt_generic
tff(fact_7585_arcosh__real__has__field__derivative,axiom,
    ! [Xb: real,A4: set(real)] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xb)
     => has_field_derivative(real,arcosh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,sqrt,aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(real)))),topolo174197925503356063within(real,Xb,A4)) ) ).

% arcosh_real_has_field_derivative
tff(fact_7586_artanh__real__has__field__derivative,axiom,
    ! [Xb: real,A4: set(real)] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xb)),one_one(real))
     => has_field_derivative(real,artanh(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))),topolo174197925503356063within(real,Xb,A4)) ) ).

% artanh_real_has_field_derivative
tff(fact_7587_suminf__reindex__mono,axiom,
    ! [Fa: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,Fa)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X2: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,Fa,X2))
         => aa(real,$o,ord_less_eq(real,suminf(real,comp(nat,real,nat,Fa,G))),suminf(real,Fa)) ) ) ) ).

% suminf_reindex_mono
tff(fact_7588_DERIV__real__root,axiom,
    ! [N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => has_field_derivative(real,root(N),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),Xb)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_real_root
tff(fact_7589_DERIV__arccos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),one_one(real))
       => has_field_derivative(real,arccos,aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_arccos
tff(fact_7590_DERIV__arcsin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),one_one(real))
       => has_field_derivative(real,arcsin,aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_arcsin
tff(fact_7591_inj__on__char__of__nat,axiom,
    inj_on(nat,char,unique5772411509450598832har_of(nat),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) ).

% inj_on_char_of_nat
tff(fact_7592_suminf__reindex,axiom,
    ! [Fa: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,Fa)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X2: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,Fa,X2))
         => ( ! [X2: nat] :
                ( ~ member(nat,X2,aa(set(nat),set(nat),image(nat,nat,G),top_top(set(nat))))
               => ( aa(nat,real,Fa,X2) = zero_zero(real) ) )
           => ( suminf(real,comp(nat,real,nat,Fa,G)) = suminf(real,Fa) ) ) ) ) ) ).

% suminf_reindex
tff(fact_7593_Maclaurin__all__le__objl,axiom,
    ! [Diff: fun(nat,fun(real,real)),Fa: fun(real,real),Xb: real,N: nat] :
      ( ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = Fa )
        & ! [M3: nat,X2: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),X2),topolo174197925503356063within(real,X2,top_top(set(real)))) )
     => ? [T2: real] :
          ( aa(real,$o,ord_less_eq(real,abs_abs(real,T2)),abs_abs(real,Xb))
          & ( aa(real,real,Fa,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_aqq(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T2)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N))) ) ) ) ).

% Maclaurin_all_le_objl
tff(fact_7594_Maclaurin__all__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),Fa: fun(real,real),Xb: real,N: nat] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = Fa )
     => ( ! [M3: nat,X2: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),X2),topolo174197925503356063within(real,X2,top_top(set(real))))
       => ? [T2: real] :
            ( aa(real,$o,ord_less_eq(real,abs_abs(real,T2)),abs_abs(real,Xb))
            & ( aa(real,real,Fa,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_aqq(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T2)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N))) ) ) ) ) ).

% Maclaurin_all_le
tff(fact_7595_DERIV__odd__real__root,axiom,
    ! [N: nat,Xb: real] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N)
     => ( ( Xb != zero_zero(real) )
       => has_field_derivative(real,root(N),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(N),Xb)),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ).

% DERIV_odd_real_root
tff(fact_7596_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: fun(nat,fun(real,real)),Fa: fun(real,real)] :
      ( aa(real,$o,ord_less(real,H),zero_zero(real))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = Fa )
         => ( ! [M3: nat,T2: real] :
                ( ( aa(nat,$o,ord_less(nat,M3),N)
                  & aa(real,$o,ord_less_eq(real,H),T2)
                  & aa(real,$o,ord_less_eq(real,T2),zero_zero(real)) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
           => ? [T2: real] :
                ( aa(real,$o,ord_less(real,H),T2)
                & aa(real,$o,ord_less(real,T2),zero_zero(real))
                & ( aa(real,real,Fa,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_aqr(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T2)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ) ).

% Maclaurin_minus
tff(fact_7597_Maclaurin2,axiom,
    ! [H: real,Diff: fun(nat,fun(real,real)),Fa: fun(real,real),N: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),H)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = Fa )
       => ( ! [M3: nat,T2: real] :
              ( ( aa(nat,$o,ord_less(nat,M3),N)
                & aa(real,$o,ord_less_eq(real,zero_zero(real)),T2)
                & aa(real,$o,ord_less_eq(real,T2),H) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
         => ? [T2: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),T2)
              & aa(real,$o,ord_less_eq(real,T2),H)
              & ( aa(real,real,Fa,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_aqr(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T2)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ).

% Maclaurin2
tff(fact_7598_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: fun(nat,fun(real,real)),Fa: fun(real,real)] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),H)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
       => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = Fa )
         => ( ! [M3: nat,T2: real] :
                ( ( aa(nat,$o,ord_less(nat,M3),N)
                  & aa(real,$o,ord_less_eq(real,zero_zero(real)),T2)
                  & aa(real,$o,ord_less_eq(real,T2),H) )
               => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
           => ? [T2: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),T2)
                & aa(real,$o,ord_less(real,T2),H)
                & ( aa(real,real,Fa,H) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_aqr(real,fun(fun(nat,fun(real,real)),fun(nat,real)),H),Diff)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T2)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),H),N))) ) ) ) ) ) ) ).

% Maclaurin
tff(fact_7599_Maclaurin__all__lt,axiom,
    ! [Diff: fun(nat,fun(real,real)),Fa: fun(real,real),N: nat,Xb: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = Fa )
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
       => ( ( Xb != zero_zero(real) )
         => ( ! [M3: nat,X2: real] : has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),X2),topolo174197925503356063within(real,X2,top_top(set(real))))
           => ? [T2: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),abs_abs(real,T2))
                & aa(real,$o,ord_less(real,abs_abs(real,T2)),abs_abs(real,Xb))
                & ( aa(real,real,Fa,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_aqq(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T2)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N))) ) ) ) ) ) ) ).

% Maclaurin_all_lt
tff(fact_7600_Maclaurin__bi__le,axiom,
    ! [Diff: fun(nat,fun(real,real)),Fa: fun(real,real),N: nat,Xb: real] :
      ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = Fa )
     => ( ! [M3: nat,T2: real] :
            ( ( aa(nat,$o,ord_less(nat,M3),N)
              & aa(real,$o,ord_less_eq(real,abs_abs(real,T2)),abs_abs(real,Xb)) )
           => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
       => ? [T2: real] :
            ( aa(real,$o,ord_less_eq(real,abs_abs(real,T2)),abs_abs(real,Xb))
            & ( aa(real,real,Fa,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aTP_Lamp_aqq(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Diff),Xb)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T2)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N))) ) ) ) ) ).

% Maclaurin_bi_le
tff(fact_7601_Taylor,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),Fa: fun(real,real),A3: real,B3: real,C2: real,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = Fa )
       => ( ! [M3: nat,T2: real] :
              ( ( aa(nat,$o,ord_less(nat,M3),N)
                & aa(real,$o,ord_less_eq(real,A3),T2)
                & aa(real,$o,ord_less_eq(real,T2),B3) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
         => ( aa(real,$o,ord_less_eq(real,A3),C2)
           => ( aa(real,$o,ord_less_eq(real,C2),B3)
             => ( aa(real,$o,ord_less_eq(real,A3),Xb)
               => ( aa(real,$o,ord_less_eq(real,Xb),B3)
                 => ( ( Xb != C2 )
                   => ? [T2: real] :
                        ( $ite(
                            aa(real,$o,ord_less(real,Xb),C2),
                            ( aa(real,$o,ord_less(real,Xb),T2)
                            & aa(real,$o,ord_less(real,T2),C2) ),
                            ( aa(real,$o,ord_less(real,C2),T2)
                            & aa(real,$o,ord_less(real,T2),Xb) ) )
                        & ( aa(real,real,Fa,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_aqs(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),C2),Xb)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T2)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,Xb),C2)),N))) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
tff(fact_7602_Taylor__up,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),Fa: fun(real,real),A3: real,B3: real,C2: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = Fa )
       => ( ! [M3: nat,T2: real] :
              ( ( aa(nat,$o,ord_less(nat,M3),N)
                & aa(real,$o,ord_less_eq(real,A3),T2)
                & aa(real,$o,ord_less_eq(real,T2),B3) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
         => ( aa(real,$o,ord_less_eq(real,A3),C2)
           => ( aa(real,$o,ord_less(real,C2),B3)
             => ? [T2: real] :
                  ( aa(real,$o,ord_less(real,C2),T2)
                  & aa(real,$o,ord_less(real,T2),B3)
                  & ( aa(real,real,Fa,B3) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_aqt(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),B3),C2)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T2)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,B3),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_up
tff(fact_7603_Taylor__down,axiom,
    ! [N: nat,Diff: fun(nat,fun(real,real)),Fa: fun(real,real),A3: real,B3: real,C2: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( ( aa(nat,fun(real,real),Diff,zero_zero(nat)) = Fa )
       => ( ! [M3: nat,T2: real] :
              ( ( aa(nat,$o,ord_less(nat,M3),N)
                & aa(real,$o,ord_less_eq(real,A3),T2)
                & aa(real,$o,ord_less_eq(real,T2),B3) )
             => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
         => ( aa(real,$o,ord_less(real,A3),C2)
           => ( aa(real,$o,ord_less_eq(real,C2),B3)
             => ? [T2: real] :
                  ( aa(real,$o,ord_less(real,A3),T2)
                  & aa(real,$o,ord_less(real,T2),C2)
                  & ( aa(real,real,Fa,A3) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_aqt(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Diff),A3),C2)),set_ord_lessThan(nat,N))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Diff,N),T2)),semiring_char_0_fact(real,N))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,A3),C2)),N))) ) ) ) ) ) ) ) ).

% Taylor_down
tff(fact_7604_inj__sgn__power,axiom,
    ! [N: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => inj_on(real,real,aTP_Lamp_afq(nat,fun(real,real),N),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_7605_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: fun(nat,fun(real,real)),K: nat,B6: real] :
      ( ! [M3: nat,T2: real] :
          ( ( aa(nat,$o,ord_less(nat,M3),N)
            & aa(real,$o,ord_less_eq(real,zero_zero(real)),T2)
            & aa(real,$o,ord_less_eq(real,T2),H) )
         => has_field_derivative(real,aa(nat,fun(real,real),Diff,M3),aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M3)),T2),topolo174197925503356063within(real,T2,top_top(set(real)))) )
     => ( ( N = aa(nat,nat,suc,K) )
       => ! [M2: nat,T7: real] :
            ( ( aa(nat,$o,ord_less(nat,M2),N)
              & aa(real,$o,ord_less_eq(real,zero_zero(real)),T7)
              & aa(real,$o,ord_less_eq(real,T7),H) )
           => has_field_derivative(real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_aqv(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),N),Diff),B6),M2),aa(real,real,minus_minus(real,aa(real,real,aa(nat,fun(real,real),Diff,aa(nat,nat,suc,M2)),T7)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_aqw(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Diff),M2),T7)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,M2))))),aa(real,real,aa(real,fun(real,real),times_times(real),B6),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),T7),aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,M2)))),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,N),aa(nat,nat,suc,M2))))))),topolo174197925503356063within(real,T7,top_top(set(real)))) ) ) ) ).

% Maclaurin_lemma2
tff(fact_7606_DERIV__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,Xb)),one_one(real))
     => has_field_derivative(real,aTP_Lamp_aqx(real,real),suminf(real,aTP_Lamp_aqy(real,fun(nat,real),Xb)),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_arctan_series
tff(fact_7607_DERIV__power__series_H,axiom,
    ! [R: real,Fa: fun(nat,real),X0: real] :
      ( ! [X2: real] :
          ( member(real,X2,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R))
         => summable(real,aa(real,fun(nat,real),aTP_Lamp_aqz(fun(nat,real),fun(real,fun(nat,real)),Fa),X2)) )
     => ( member(real,X0,set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),R),R))
       => ( aa(real,$o,ord_less(real,zero_zero(real)),R)
         => has_field_derivative(real,aTP_Lamp_arb(fun(nat,real),fun(real,real),Fa),suminf(real,aa(real,fun(nat,real),aTP_Lamp_aqz(fun(nat,real),fun(real,fun(nat,real)),Fa),X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ).

% DERIV_power_series'
tff(fact_7608_has__derivative__arcsin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G5: fun(A,real),S: set(A)] :
          ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,Xb))
         => ( aa(real,$o,ord_less(real,aa(A,real,G,Xb)),one_one(real))
           => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S))
             => has_derivative(A,real,aTP_Lamp_arc(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_ard(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% has_derivative_arcsin
tff(fact_7609_greaterThanLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or5935395276787703475ssThan(A,L,U))
        <=> ( aa(A,$o,ord_less(A,L),I)
            & aa(A,$o,ord_less(A,I),U) ) ) ) ).

% greaterThanLessThan_iff
tff(fact_7610_greaterThanLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ( bot_bot(set(A)) = set_or5935395276787703475ssThan(A,A3,B3) )
        <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ).

% greaterThanLessThan_empty_iff2
tff(fact_7611_greaterThanLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ( set_or5935395276787703475ssThan(A,A3,B3) = bot_bot(set(A)) )
        <=> aa(A,$o,ord_less_eq(A,B3),A3) ) ) ).

% greaterThanLessThan_empty_iff
tff(fact_7612_greaterThanLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,ord_less_eq(A,L),K)
         => ( set_or5935395276787703475ssThan(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanLessThan_empty
tff(fact_7613_infinite__Ioo__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ finite_finite2(A,set_or5935395276787703475ssThan(A,A3,B3))
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% infinite_Ioo_iff
tff(fact_7614_has__derivative__zero__unique,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [F5: fun(A,B),Xb: A] :
          ( has_derivative(A,B,aTP_Lamp_are(A,B),F5,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => ! [X3: A] : aa(A,B,F5,X3) = zero_zero(B) ) ) ).

% has_derivative_zero_unique
tff(fact_7615_has__derivative__compose,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),F3: fun(A,B),Xb: A,S: set(A),G: fun(B,C),G5: fun(B,C)] :
          ( has_derivative(A,B,Fa,F3,topolo174197925503356063within(A,Xb,S))
         => ( has_derivative(B,C,G,G5,topolo174197925503356063within(B,aa(A,B,Fa,Xb),top_top(set(B))))
           => has_derivative(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_arf(fun(A,B),fun(fun(B,C),fun(A,C)),Fa),G),aa(fun(B,C),fun(A,C),aTP_Lamp_arf(fun(A,B),fun(fun(B,C),fun(A,C)),F3),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_compose
tff(fact_7616_has__derivative__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,real),F3: fun(A,real),Xb: A,S: set(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,real,Fa,F3,topolo174197925503356063within(A,Xb,S))
         => ( has_derivative(A,B,G,G5,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_arg(fun(A,real),fun(fun(A,B),fun(A,B)),Fa),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_arh(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Fa),F3),Xb),G),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_scaleR
tff(fact_7617_has__derivative__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Fa: fun(A,B),F3: fun(A,B),Xb: A,S: set(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,Fa,F3,topolo174197925503356063within(A,Xb,S))
         => ( has_derivative(A,B,G,G5,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ari(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_arj(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Fa),F3),Xb),G),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_mult
tff(fact_7618_has__derivative__in__compose,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),F3: fun(A,B),Xb: A,S: set(A),G: fun(B,C),G5: fun(B,C)] :
          ( has_derivative(A,B,Fa,F3,topolo174197925503356063within(A,Xb,S))
         => ( has_derivative(B,C,G,G5,topolo174197925503356063within(B,aa(A,B,Fa,Xb),aa(set(A),set(B),image(A,B,Fa),S)))
           => has_derivative(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_arf(fun(A,B),fun(fun(B,C),fun(A,C)),Fa),G),aa(fun(B,C),fun(A,C),aTP_Lamp_arf(fun(A,B),fun(fun(B,C),fun(A,C)),F3),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_in_compose
tff(fact_7619_has__derivative__in__compose2,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Tb: set(A),G: fun(A,B),G5: fun(A,fun(A,B)),Fa: fun(C,A),S: set(C),Xb: C,F3: fun(C,A)] :
          ( ! [X2: A] :
              ( member(A,X2,Tb)
             => has_derivative(A,B,G,aa(A,fun(A,B),G5,X2),topolo174197925503356063within(A,X2,Tb)) )
         => ( aa(set(A),$o,ord_less_eq(set(A),aa(set(C),set(A),image(C,A,Fa),S)),Tb)
           => ( member(C,Xb,S)
             => ( has_derivative(C,A,Fa,F3,topolo174197925503356063within(C,Xb,S))
               => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ark(fun(A,B),fun(fun(C,A),fun(C,B)),G),Fa),aa(fun(C,A),fun(C,B),aa(C,fun(fun(C,A),fun(C,B)),aa(fun(C,A),fun(C,fun(fun(C,A),fun(C,B))),aTP_Lamp_arl(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),G5),Fa),Xb),F3),topolo174197925503356063within(C,Xb,S)) ) ) ) ) ) ).

% has_derivative_in_compose2
tff(fact_7620_has__derivative__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),F3: fun(A,B),F5: filter(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,Fa,F3,F5)
         => ( has_derivative(A,B,G,G5,F5)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_arm(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),aa(fun(A,B),fun(A,B),aTP_Lamp_arm(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G5),F5) ) ) ) ).

% has_derivative_diff
tff(fact_7621_has__derivative__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G: fun(A,B),G5: fun(A,B),F5: filter(A),Xb: B] :
          ( has_derivative(A,B,G,G5,F5)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_arn(fun(A,B),fun(B,fun(A,B)),G),Xb),aa(B,fun(A,B),aTP_Lamp_arn(fun(A,B),fun(B,fun(A,B)),G5),Xb),F5) ) ) ).

% has_derivative_mult_right
tff(fact_7622_has__derivative__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [G: fun(A,B),G5: fun(A,B),F5: filter(A),Y: B] :
          ( has_derivative(A,B,G,G5,F5)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_aro(fun(A,B),fun(B,fun(A,B)),G),Y),aa(B,fun(A,B),aTP_Lamp_aro(fun(A,B),fun(B,fun(A,B)),G5),Y),F5) ) ) ).

% has_derivative_mult_left
tff(fact_7623_has__derivative__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [I5: set(A),Fa: fun(A,fun(B,C)),F3: fun(A,fun(B,C)),F5: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => has_derivative(B,C,aa(A,fun(B,C),Fa,I2),aa(A,fun(B,C),F3,I2),F5) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_arq(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),Fa),aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_arq(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),F3),F5) ) ) ).

% has_derivative_sum
tff(fact_7624_has__derivative__scaleR__right,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,B),G5: fun(A,B),F5: filter(A),R2: real] :
          ( has_derivative(A,B,G,G5,F5)
         => has_derivative(A,B,aa(real,fun(A,B),aTP_Lamp_arr(fun(A,B),fun(real,fun(A,B)),G),R2),aa(real,fun(A,B),aTP_Lamp_arr(fun(A,B),fun(real,fun(A,B)),G5),R2),F5) ) ) ).

% has_derivative_scaleR_right
tff(fact_7625_has__derivative__scaleR__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),G5: fun(A,real),F5: filter(A),Xb: B] :
          ( has_derivative(A,real,G,G5,F5)
         => has_derivative(A,B,aa(B,fun(A,B),aTP_Lamp_ars(fun(A,real),fun(B,fun(A,B)),G),Xb),aa(B,fun(A,B),aTP_Lamp_ars(fun(A,real),fun(B,fun(A,B)),G5),Xb),F5) ) ) ).

% has_derivative_scaleR_left
tff(fact_7626_has__derivative__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),G5: fun(A,real),F5: filter(A)] :
          ( has_derivative(A,real,G,G5,F5)
         => has_derivative(A,B,aTP_Lamp_art(fun(A,real),fun(A,B),G),aTP_Lamp_art(fun(A,real),fun(A,B),G5),F5) ) ) ).

% has_derivative_of_real
tff(fact_7627_has__derivative__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F5: filter(A)] : has_derivative(A,A,aTP_Lamp_aru(A,A),aTP_Lamp_aru(A,A),F5) ) ).

% has_derivative_ident
tff(fact_7628_has__derivative__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),F3: fun(A,B),F5: filter(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,Fa,F3,F5)
         => ( has_derivative(A,B,G,G5,F5)
           => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_arv(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),aa(fun(A,B),fun(A,B),aTP_Lamp_arv(fun(A,B),fun(fun(A,B),fun(A,B)),F3),G5),F5) ) ) ) ).

% has_derivative_add
tff(fact_7629_has__derivative__const,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [C2: B,F5: filter(A)] : has_derivative(A,B,aTP_Lamp_arw(B,fun(A,B),C2),aTP_Lamp_are(A,B),F5) ) ).

% has_derivative_const
tff(fact_7630_infinite__Ioo,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ~ finite_finite2(A,set_or5935395276787703475ssThan(A,A3,B3)) ) ) ).

% infinite_Ioo
tff(fact_7631_has__derivative__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),F3: fun(A,B),F5: filter(A)] :
          ( has_derivative(A,B,Fa,F3,F5)
         => has_derivative(A,B,aTP_Lamp_arx(fun(A,B),fun(A,B),Fa),aTP_Lamp_arx(fun(A,B),fun(A,B),F3),F5) ) ) ).

% has_derivative_minus
tff(fact_7632_has__derivative__exp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G5: fun(A,real),Xb: A,S: set(A)] :
          ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S))
         => has_derivative(A,real,aTP_Lamp_ary(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_arz(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G5),Xb),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_derivative_exp
tff(fact_7633_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or5935395276787703475ssThan(A,A3,B3)),set_or5935395276787703475ssThan(A,C2,D))
        <=> ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less_eq(A,B3),D) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanLessThan
tff(fact_7634_has__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xb: A,S: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,Xb,S))
         => has_derivative(A,A,aTP_Lamp_aqe(fun(A,A),fun(A,A),G),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,aa(A,A,G,Xb))),Db)),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_derivative_sinh
tff(fact_7635_has__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xb: A,S: set(A)] :
          ( has_derivative(A,A,G,aa(A,fun(A,A),times_times(A),Db),topolo174197925503356063within(A,Xb,S))
         => has_derivative(A,A,aTP_Lamp_aqd(fun(A,A),fun(A,A),G),aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,aa(A,A,G,Xb))),Db)),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_derivative_cosh
tff(fact_7636_has__derivative__sin,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G5: fun(A,real),Xb: A,S: set(A)] :
          ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S))
         => has_derivative(A,real,aTP_Lamp_asa(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_asb(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G5),Xb),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_derivative_sin
tff(fact_7637_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or5935395276787703475ssThan(A,A3,B3)),set_or1337092689740270186AtMost(A,C2,D))
        <=> ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less_eq(A,B3),D) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastAtMost_iff
tff(fact_7638_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or5935395276787703475ssThan(A,A3,B3)),set_or7035219750837199246ssThan(A,C2,D))
        <=> ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less_eq(A,B3),D) ) ) ) ) ).

% greaterThanLessThan_subseteq_atLeastLessThan_iff
tff(fact_7639_has__derivative__divide_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Fa: fun(A,B),F3: fun(A,B),Xb: A,S3: set(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,Fa,F3,topolo174197925503356063within(A,Xb,S3))
         => ( has_derivative(A,B,G,G5,topolo174197925503356063within(A,Xb,S3))
           => ( ( aa(A,B,G,Xb) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_asc(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_asd(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Fa),F3),Xb),G),G5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ).

% has_derivative_divide'
tff(fact_7640_has__derivative__inverse,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Fa: fun(B,A),Xb: B,F3: fun(B,A),S3: set(B)] :
          ( ( aa(B,A,Fa,Xb) != zero_zero(A) )
         => ( has_derivative(B,A,Fa,F3,topolo174197925503356063within(B,Xb,S3))
           => has_derivative(B,A,aTP_Lamp_ase(fun(B,A),fun(B,A),Fa),aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_asf(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),Fa),Xb),F3),topolo174197925503356063within(B,Xb,S3)) ) ) ) ).

% has_derivative_inverse
tff(fact_7641_has__derivative__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A,S3: set(A)] :
          ( ( Xb != zero_zero(A) )
         => has_derivative(A,A,inverse_inverse(A),aTP_Lamp_asg(A,fun(A,A),Xb),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_inverse'
tff(fact_7642_DERIV__compose__FDERIV,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(real,real),F3: real,G: fun(A,real),Xb: A,G5: fun(A,real),S: set(A)] :
          ( has_field_derivative(real,Fa,F3,topolo174197925503356063within(real,aa(A,real,G,Xb),top_top(set(real))))
         => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ash(fun(real,real),fun(fun(A,real),fun(A,real)),Fa),G),aa(fun(A,real),fun(A,real),aTP_Lamp_asi(real,fun(fun(A,real),fun(A,real)),F3),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% DERIV_compose_FDERIV
tff(fact_7643_has__derivative__cos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G5: fun(A,real),Xb: A,S: set(A)] :
          ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S))
         => has_derivative(A,real,aTP_Lamp_asj(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ask(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G5),Xb),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_derivative_cos
tff(fact_7644_DERIV__isconst3,axiom,
    ! [A3: real,B3: real,Xb: real,Y: real,Fa: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( member(real,Xb,set_or5935395276787703475ssThan(real,A3,B3))
       => ( member(real,Y,set_or5935395276787703475ssThan(real,A3,B3))
         => ( ! [X2: real] :
                ( member(real,X2,set_or5935395276787703475ssThan(real,A3,B3))
               => has_field_derivative(real,Fa,zero_zero(real),topolo174197925503356063within(real,X2,top_top(set(real)))) )
           => ( aa(real,real,Fa,Xb) = aa(real,real,Fa,Y) ) ) ) ) ) ).

% DERIV_isconst3
tff(fact_7645_has__derivative__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Fa: fun(A,B),F3: fun(A,B),Xb: A,S3: set(A),N: nat] :
          ( has_derivative(A,B,Fa,F3,topolo174197925503356063within(A,Xb,S3))
         => has_derivative(A,B,aa(nat,fun(A,B),aTP_Lamp_asl(fun(A,B),fun(nat,fun(A,B)),Fa),N),aa(nat,fun(A,B),aa(A,fun(nat,fun(A,B)),aa(fun(A,B),fun(A,fun(nat,fun(A,B))),aTP_Lamp_asm(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),Fa),F3),Xb),N),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_derivative_power
tff(fact_7646_has__derivative__ln,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G5: fun(A,real),S: set(A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,Xb))
         => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,real,aTP_Lamp_asn(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_aso(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_ln
tff(fact_7647_has__derivative__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Fa: fun(A,B),F3: fun(A,B),Xb: A,S3: set(A),G: fun(A,B),G5: fun(A,B)] :
          ( has_derivative(A,B,Fa,F3,topolo174197925503356063within(A,Xb,S3))
         => ( has_derivative(A,B,G,G5,topolo174197925503356063within(A,Xb,S3))
           => ( ( aa(A,B,G,Xb) != zero_zero(B) )
             => has_derivative(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_asp(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_asq(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Fa),F3),Xb),G),G5),topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ).

% has_derivative_divide
tff(fact_7648_has__derivative__prod,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [I5: set(A),Fa: fun(A,fun(B,C)),F3: fun(A,fun(B,C)),Xb: B,S3: set(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => has_derivative(B,C,aa(A,fun(B,C),Fa,I2),aa(A,fun(B,C),F3,I2),topolo174197925503356063within(B,Xb,S3)) )
         => has_derivative(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ass(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),Fa),aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_asu(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),I5),Fa),F3),Xb),topolo174197925503356063within(B,Xb,S3)) ) ) ).

% has_derivative_prod
tff(fact_7649_has__derivative__powr,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G5: fun(A,real),Xb: A,X5: set(A),Fa: fun(A,real),F3: fun(A,real)] :
          ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,X5))
         => ( has_derivative(A,real,Fa,F3,topolo174197925503356063within(A,Xb,X5))
           => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,Xb))
             => ( member(A,Xb,X5)
               => has_derivative(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_asv(fun(A,real),fun(fun(A,real),fun(A,real)),G),Fa),aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_asw(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),G),G5),Xb),Fa),F3),topolo174197925503356063within(A,Xb,X5)) ) ) ) ) ) ).

% has_derivative_powr
tff(fact_7650_has__derivative__real__sqrt,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G5: fun(A,real),S: set(A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,Xb))
         => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,real,aTP_Lamp_asx(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_asy(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_real_sqrt
tff(fact_7651_has__derivative__arctan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),G5: fun(A,real),Xb: A,S: set(A)] :
          ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S))
         => has_derivative(A,real,aTP_Lamp_asz(fun(A,real),fun(A,real),G),aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ata(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),G),G5),Xb),topolo174197925503356063within(A,Xb,S)) ) ) ).

% has_derivative_arctan
tff(fact_7652_has__derivative__tan,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G5: fun(A,real),S: set(A)] :
          ( ( cos(real,aa(A,real,G,Xb)) != zero_zero(real) )
         => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,real,aTP_Lamp_atb(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_atc(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_tan
tff(fact_7653_DERIV__series_H,axiom,
    ! [Fa: fun(real,fun(nat,real)),F3: fun(real,fun(nat,real)),X0: real,A3: real,B3: real,L5: fun(nat,real)] :
      ( ! [N3: nat] : has_field_derivative(real,aa(nat,fun(real,real),aTP_Lamp_atd(fun(real,fun(nat,real)),fun(nat,fun(real,real)),Fa),N3),aa(nat,real,aa(real,fun(nat,real),F3,X0),N3),topolo174197925503356063within(real,X0,top_top(set(real))))
     => ( ! [X2: real] :
            ( member(real,X2,set_or5935395276787703475ssThan(real,A3,B3))
           => summable(real,aa(real,fun(nat,real),Fa,X2)) )
       => ( member(real,X0,set_or5935395276787703475ssThan(real,A3,B3))
         => ( summable(real,aa(real,fun(nat,real),F3,X0))
           => ( summable(real,L5)
             => ( ! [N3: nat,X2: real,Y2: real] :
                    ( member(real,X2,set_or5935395276787703475ssThan(real,A3,B3))
                   => ( member(real,Y2,set_or5935395276787703475ssThan(real,A3,B3))
                     => aa(real,$o,ord_less_eq(real,abs_abs(real,aa(real,real,minus_minus(real,aa(nat,real,aa(real,fun(nat,real),Fa,X2),N3)),aa(nat,real,aa(real,fun(nat,real),Fa,Y2),N3)))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,L5,N3)),abs_abs(real,aa(real,real,minus_minus(real,X2),Y2)))) ) )
               => has_field_derivative(real,aTP_Lamp_ate(fun(real,fun(nat,real)),fun(real,real),Fa),suminf(real,aa(real,fun(nat,real),F3,X0)),topolo174197925503356063within(real,X0,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_series'
tff(fact_7654_has__derivative__arccos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [G: fun(A,real),Xb: A,G5: fun(A,real),S: set(A)] :
          ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),aa(A,real,G,Xb))
         => ( aa(real,$o,ord_less(real,aa(A,real,G,Xb)),one_one(real))
           => ( has_derivative(A,real,G,G5,topolo174197925503356063within(A,Xb,S))
             => has_derivative(A,real,aTP_Lamp_atf(fun(A,real),fun(A,real),G),aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_atg(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),G),Xb),G5),topolo174197925503356063within(A,Xb,S)) ) ) ) ) ).

% has_derivative_arccos
tff(fact_7655_has__derivative__floor,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [G: fun(B,real),Xb: B,Fa: fun(real,A),G5: fun(B,real),S: set(B)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,aa(B,real,G,Xb),top_top(set(real))),Fa)
         => ( ~ member(A,aa(real,A,Fa,aa(B,real,G,Xb)),ring_1_Ints(A))
           => ( has_derivative(B,real,G,G5,topolo174197925503356063within(B,Xb,S))
             => has_derivative(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_ath(fun(B,real),fun(fun(real,A),fun(B,real)),G),Fa),aTP_Lamp_ati(fun(B,real),fun(B,real),G5),topolo174197925503356063within(B,Xb,S)) ) ) ) ) ).

% has_derivative_floor
tff(fact_7656_termdiffs__aux,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_aqm(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K5))
           => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_atk(fun(nat,A),fun(A,fun(A,A)),C2),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% termdiffs_aux
tff(fact_7657_tendsto__const,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [K: B,F5: filter(A)] : filterlim(A,B,aTP_Lamp_atl(B,fun(A,B),K),topolo7230453075368039082e_nhds(B,K),F5) ) ).

% tendsto_const
tff(fact_7658_tendsto__ident__at,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [A3: A,S: set(A)] : filterlim(A,A,aTP_Lamp_atm(A,A),topolo7230453075368039082e_nhds(A,A3),topolo174197925503356063within(A,A3,S)) ) ).

% tendsto_ident_at
tff(fact_7659_tendsto__mult__right__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,Fa: fun(B,A),L: A,F5: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_atn(A,fun(fun(B,A),fun(B,A)),C2),Fa),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),L),C2)),F5)
          <=> filterlim(B,A,Fa,topolo7230453075368039082e_nhds(A,L),F5) ) ) ) ).

% tendsto_mult_right_iff
tff(fact_7660_tendsto__mult__left__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,Fa: fun(B,A),L: A,F5: filter(B)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ato(A,fun(fun(B,A),fun(B,A)),C2),Fa),topolo7230453075368039082e_nhds(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),L)),F5)
          <=> filterlim(B,A,Fa,topolo7230453075368039082e_nhds(A,L),F5) ) ) ) ).

% tendsto_mult_left_iff
tff(fact_7661_power__tendsto__0__iff,axiom,
    ! [A: $tType,N: nat,Fa: fun(A,real),F5: filter(A)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_atp(nat,fun(fun(A,real),fun(A,real)),N),Fa),topolo7230453075368039082e_nhds(real,zero_zero(real)),F5)
      <=> filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,zero_zero(real)),F5) ) ) ).

% power_tendsto_0_iff
tff(fact_7662_isCont__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,Fa: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => ( topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_atq(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Fa),G)) ) ) ) ).

% isCont_Pair
tff(fact_7663_DERIV__LIM__iff,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Fa: fun(A,A),A3: A,D5: A] :
          ( filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_atr(fun(A,A),fun(A,fun(A,A)),Fa),A3),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_ats(fun(A,A),fun(A,fun(A,A)),Fa),A3),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% DERIV_LIM_iff
tff(fact_7664_has__field__derivativeD,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,Fa,D5,topolo174197925503356063within(A,Xb,S3))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_att(fun(A,A),fun(A,fun(A,A)),Fa),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_field_derivativeD
tff(fact_7665_has__field__derivative__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,Fa,D5,topolo174197925503356063within(A,Xb,S3))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_att(fun(A,A),fun(A,fun(A,A)),Fa),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% has_field_derivative_iff
tff(fact_7666_isCont__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_atu(fun(A,B),fun(A,B),Fa)) ) ) ).

% isCont_minus
tff(fact_7667_continuous__at__within__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,S: set(A),Fa: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S),Fa)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S),G)
           => ( ( aa(A,B,G,A3) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S),aa(fun(A,B),fun(A,B),aTP_Lamp_atv(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G)) ) ) ) ) ).

% continuous_at_within_divide
tff(fact_7668_LIM__not__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo8386298272705272623_space(B)
        & zero(A)
        & topological_t2_space(A) )
     => ! [K: A,A3: B] :
          ( ( K != zero_zero(A) )
         => ~ filterlim(B,A,aTP_Lamp_atw(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,zero_zero(A)),topolo174197925503356063within(B,A3,top_top(set(B)))) ) ) ).

% LIM_not_zero
tff(fact_7669_LIM__offset__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Fa: fun(A,B),A3: A,L5: B] :
          ( filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_atx(fun(A,B),fun(A,fun(A,B)),Fa),A3),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIM_offset_zero_cancel
tff(fact_7670_LIM__offset__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Fa: fun(A,B),L5: B,A3: A] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_atx(fun(A,B),fun(A,fun(A,B)),Fa),A3),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_offset_zero
tff(fact_7671_LIM__isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Fa: fun(A,B),A3: A] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,aa(A,B,Fa,A3)),topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_atx(fun(A,B),fun(A,fun(A,B)),Fa),A3),topolo7230453075368039082e_nhds(B,aa(A,B,Fa,A3)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% LIM_isCont_iff
tff(fact_7672_isCont__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Xb: A,Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,top_top(set(A))),Fa)
        <=> filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aty(A,fun(fun(A,B),fun(A,B)),Xb),Fa),topolo7230453075368039082e_nhds(B,aa(A,B,Fa,Xb)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% isCont_iff
tff(fact_7673_continuous__at__within__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [A3: A,S: set(A),Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S),Fa)
         => ( ( aa(A,B,Fa,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S),aTP_Lamp_atz(fun(A,B),fun(A,B),Fa)) ) ) ) ).

% continuous_at_within_inverse
tff(fact_7674_continuous__at__within__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,S: set(A),Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S),Fa)
         => ( ( aa(A,B,Fa,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,S),aTP_Lamp_aua(fun(A,B),fun(A,B),Fa)) ) ) ) ).

% continuous_at_within_sgn
tff(fact_7675_isCont__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [A3: A,Fa: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_aub(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G)) ) ) ) ).

% isCont_mult
tff(fact_7676_isCont__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [A4: set(A),A3: B,Fa: fun(A,fun(B,C))] :
          ( ! [X2: A] :
              ( member(A,X2,A4)
             => topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,A3,top_top(set(B))),aa(A,fun(B,C),Fa,X2)) )
         => topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,A3,top_top(set(B))),aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aud(set(A),fun(fun(A,fun(B,C)),fun(B,C)),A4),Fa)) ) ) ).

% isCont_sum
tff(fact_7677_isCont__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,Fa: fun(A,real),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_aue(fun(A,real),fun(fun(A,B),fun(A,B)),Fa),G)) ) ) ) ).

% isCont_scaleR
tff(fact_7678_isCont__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),G)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_auf(fun(A,real),fun(A,B),G)) ) ) ).

% isCont_of_real
tff(fact_7679_filterlim__at__If,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Fa: fun(A,B),G4: filter(B),Xb: A,P: fun(A,$o),G: fun(A,B)] :
          ( filterlim(A,B,Fa,G4,topolo174197925503356063within(A,Xb,collect(A,P)))
         => ( filterlim(A,B,G,G4,topolo174197925503356063within(A,Xb,collect(A,aTP_Lamp_aug(fun(A,$o),fun(A,$o),P))))
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_auh(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),Fa),P),G),G4,topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% filterlim_at_If
tff(fact_7680_isCont__o2,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4958980785337419405_space(C)
        & topological_t2_space(A) )
     => ! [A3: A,Fa: fun(A,B),G: fun(B,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => ( topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,aa(A,B,Fa,A3),top_top(set(B))),G)
           => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(B,C),fun(A,C),aTP_Lamp_aui(fun(A,B),fun(fun(B,C),fun(A,C)),Fa),G)) ) ) ) ).

% isCont_o2
tff(fact_7681_LIM__const__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo8386298272705272623_space(A) )
     => ! [K: B,L5: B,A3: A] :
          ( filterlim(A,B,aTP_Lamp_auj(B,fun(A,B),K),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( K = L5 ) ) ) ).

% LIM_const_eq
tff(fact_7682_tendsto__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [G: fun(A,B),L: A,Fa: fun(C,A),F5: filter(C)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),topolo174197925503356063within(A,L,top_top(set(A))))
         => ( filterlim(C,A,Fa,topolo7230453075368039082e_nhds(A,L),F5)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_auk(fun(A,B),fun(fun(C,A),fun(C,B)),G),Fa),topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),F5) ) ) ) ).

% tendsto_compose
tff(fact_7683_LIM__const__not__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo8386298272705272623_space(B)
        & topological_t2_space(A) )
     => ! [K: A,L5: A,A3: B] :
          ( ( K != L5 )
         => ~ filterlim(B,A,aTP_Lamp_aul(A,fun(B,A),K),topolo7230453075368039082e_nhds(A,L5),topolo174197925503356063within(B,A3,top_top(set(B)))) ) ) ).

% LIM_const_not_eq
tff(fact_7684_isCont__tendsto__compose,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [L: A,G: fun(A,B),Fa: fun(C,A),F5: filter(C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,L,top_top(set(A))),G)
         => ( filterlim(C,A,Fa,topolo7230453075368039082e_nhds(A,L),F5)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aum(fun(A,B),fun(fun(C,A),fun(C,B)),G),Fa),topolo7230453075368039082e_nhds(B,aa(A,B,G,L)),F5) ) ) ) ).

% isCont_tendsto_compose
tff(fact_7685_continuous__within__compose3,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topological_t2_space(C) )
     => ! [Fa: fun(C,A),Xb: C,G: fun(A,B),S: set(C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,aa(C,A,Fa,Xb),top_top(set(A))),G)
         => ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,Xb,S),Fa)
           => topolo3448309680560233919inuous(C,B,topolo174197925503356063within(C,Xb,S),aa(fun(A,B),fun(C,B),aTP_Lamp_aun(fun(C,A),fun(fun(A,B),fun(C,B)),Fa),G)) ) ) ) ).

% continuous_within_compose3
tff(fact_7686_isCont__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_auo(fun(A,B),fun(A,real),Fa)) ) ) ).

% isCont_norm
tff(fact_7687_isCont__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [A3: A,Fa: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_aup(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G)) ) ) ) ).

% isCont_add
tff(fact_7688_isCont__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [A3: A,Fa: fun(A,B),N: nat] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(nat,fun(A,B),aTP_Lamp_auq(fun(A,B),fun(nat,fun(A,B)),Fa),N)) ) ) ).

% isCont_power
tff(fact_7689_isCont__fst,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A3: A,Fa: fun(A,product_prod(B,C))] :
          ( topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_aur(fun(A,product_prod(B,C)),fun(A,B),Fa)) ) ) ).

% isCont_fst
tff(fact_7690_isCont__snd,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A3: A,Fa: fun(A,product_prod(B,C))] :
          ( topolo3448309680560233919inuous(A,product_prod(B,C),topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_aus(fun(A,product_prod(B,C)),fun(A,C),Fa)) ) ) ).

% isCont_snd
tff(fact_7691_continuous__ident,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xb: A,S3: set(A)] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S3),aTP_Lamp_aut(A,A)) ) ).

% continuous_ident
tff(fact_7692_continuous__within__compose2,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4958980785337419405_space(C)
        & topological_t2_space(A) )
     => ! [Xb: A,S: set(A),Fa: fun(A,B),G: fun(B,C)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,S),Fa)
         => ( topolo3448309680560233919inuous(B,C,topolo174197925503356063within(B,aa(A,B,Fa,Xb),aa(set(A),set(B),image(A,B,Fa),S)),G)
           => topolo3448309680560233919inuous(A,C,topolo174197925503356063within(A,Xb,S),aa(fun(B,C),fun(A,C),aTP_Lamp_aui(fun(A,B),fun(fun(B,C),fun(A,C)),Fa),G)) ) ) ) ).

% continuous_within_compose2
tff(fact_7693_IVT2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [Fa: fun(B,A),B3: B,Y: A,A3: B] :
          ( aa(A,$o,ord_less_eq(A,aa(B,A,Fa,B3)),Y)
         => ( aa(A,$o,ord_less_eq(A,Y),aa(B,A,Fa,A3))
           => ( aa(B,$o,ord_less_eq(B,A3),B3)
             => ( ! [X2: B] :
                    ( ( aa(B,$o,ord_less_eq(B,A3),X2)
                      & aa(B,$o,ord_less_eq(B,X2),B3) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X2,top_top(set(B))),Fa) )
               => ? [X2: B] :
                    ( aa(B,$o,ord_less_eq(B,A3),X2)
                    & aa(B,$o,ord_less_eq(B,X2),B3)
                    & ( aa(B,A,Fa,X2) = Y ) ) ) ) ) ) ) ).

% IVT2
tff(fact_7694_IVT,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & topolo8458572112393995274pology(B) )
     => ! [Fa: fun(B,A),A3: B,Y: A,B3: B] :
          ( aa(A,$o,ord_less_eq(A,aa(B,A,Fa,A3)),Y)
         => ( aa(A,$o,ord_less_eq(A,Y),aa(B,A,Fa,B3))
           => ( aa(B,$o,ord_less_eq(B,A3),B3)
             => ( ! [X2: B] :
                    ( ( aa(B,$o,ord_less_eq(B,A3),X2)
                      & aa(B,$o,ord_less_eq(B,X2),B3) )
                   => topolo3448309680560233919inuous(B,A,topolo174197925503356063within(B,X2,top_top(set(B))),Fa) )
               => ? [X2: B] :
                    ( aa(B,$o,ord_less_eq(B,A3),X2)
                    & aa(B,$o,ord_less_eq(B,X2),B3)
                    & ( aa(B,A,Fa,X2) = Y ) ) ) ) ) ) ) ).

% IVT
tff(fact_7695_isCont__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,Fa: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_auu(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G)) ) ) ) ).

% isCont_diff
tff(fact_7696_LIM__offset,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Fa: fun(A,B),L5: B,A3: A,K: A] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
         => filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_auv(fun(A,B),fun(A,fun(A,B)),Fa),K),topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,aa(A,A,minus_minus(A,A3),K),top_top(set(A)))) ) ) ).

% LIM_offset
tff(fact_7697_real__LIM__sandwich__zero,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Fa: fun(A,real),A3: A,G: fun(A,real)] :
          ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( ! [X2: A] :
                ( ( X2 != A3 )
               => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(A,real,G,X2)) )
           => ( ! [X2: A] :
                  ( ( X2 != A3 )
                 => aa(real,$o,ord_less_eq(real,aa(A,real,G,X2)),aa(A,real,Fa,X2)) )
             => filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% real_LIM_sandwich_zero
tff(fact_7698_LIM__fun__less__zero,axiom,
    ! [Fa: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,Fa,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [R4: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),R4)
            & ! [X3: real] :
                ( ( ( X3 != C2 )
                  & aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,C2),X3))),R4) )
               => aa(real,$o,ord_less(real,aa(real,real,Fa,X3)),zero_zero(real)) ) ) ) ) ).

% LIM_fun_less_zero
tff(fact_7699_LIM__fun__not__zero,axiom,
    ! [Fa: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,Fa,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( ( L != zero_zero(real) )
       => ? [R4: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),R4)
            & ! [X3: real] :
                ( ( ( X3 != C2 )
                  & aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,C2),X3))),R4) )
               => ( aa(real,real,Fa,X3) != zero_zero(real) ) ) ) ) ) ).

% LIM_fun_not_zero
tff(fact_7700_LIM__fun__gt__zero,axiom,
    ! [Fa: fun(real,real),L: real,C2: real] :
      ( filterlim(real,real,Fa,topolo7230453075368039082e_nhds(real,L),topolo174197925503356063within(real,C2,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [R4: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),R4)
            & ! [X3: real] :
                ( ( ( X3 != C2 )
                  & aa(real,$o,ord_less(real,abs_abs(real,aa(real,real,minus_minus(real,C2),X3))),R4) )
               => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,Fa,X3)) ) ) ) ) ).

% LIM_fun_gt_zero
tff(fact_7701_LIM__D,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),L5: B,A3: A,R2: real] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
           => ? [S2: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),S2)
                & ! [X3: A] :
                    ( ( ( X3 != A3 )
                      & aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X3),A3))),S2) )
                   => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,Fa,X3)),L5))),R2) ) ) ) ) ) ).

% LIM_D
tff(fact_7702_LIM__I,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,Fa: fun(A,B),L5: B] :
          ( ! [R4: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R4)
             => ? [S7: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),S7)
                  & ! [X2: A] :
                      ( ( ( X2 != A3 )
                        & aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X2),A3))),S7) )
                     => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,Fa,X2)),L5))),R4) ) ) )
         => filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIM_I
tff(fact_7703_LIM__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),L5: B,A3: A] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> ! [R6: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R6)
             => ? [S8: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),S8)
                  & ! [X: A] :
                      ( ( ( X != A3 )
                        & aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X),A3))),S8) )
                     => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,Fa,X)),L5))),R6) ) ) ) ) ) ).

% LIM_eq
tff(fact_7704_LIM__equal2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [R: real,A3: A,Fa: fun(A,B),G: fun(A,B),L: B] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),R)
         => ( ! [X2: A] :
                ( ( X2 != A3 )
               => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X2),A3))),R)
                 => ( aa(A,B,Fa,X2) = aa(A,B,G,X2) ) ) )
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A))))
             => filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% LIM_equal2
tff(fact_7705_LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Fa: fun(A,B),B3: B,A3: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,B3),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B3,top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),D6)
                  & ! [X2: A] :
                      ( ( ( X2 != A3 )
                        & aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X2),A3))),D6) )
                     => ( aa(A,B,Fa,X2) != B3 ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_auw(fun(A,B),fun(fun(B,C),fun(A,C)),Fa),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% LIM_compose2
tff(fact_7706_isCont__LIM__compose2,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [A3: A,Fa: fun(A,B),G: fun(B,C),L: C] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(B,aa(A,B,Fa,A3),top_top(set(B))))
           => ( ? [D6: real] :
                  ( aa(real,$o,ord_less(real,zero_zero(real)),D6)
                  & ! [X2: A] :
                      ( ( ( X2 != A3 )
                        & aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,X2),A3))),D6) )
                     => ( aa(A,B,Fa,X2) != aa(A,B,Fa,A3) ) ) )
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_auw(fun(A,B),fun(fun(B,C),fun(A,C)),Fa),G),topolo7230453075368039082e_nhds(C,L),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% isCont_LIM_compose2
tff(fact_7707_isCont__cos_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_aux(fun(A,B),fun(A,B),Fa)) ) ) ).

% isCont_cos'
tff(fact_7708_isCont__sin_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_auy(fun(A,B),fun(A,B),Fa)) ) ) ).

% isCont_sin'
tff(fact_7709_isCont__exp_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_auz(fun(A,B),fun(A,B),Fa)) ) ) ).

% isCont_exp'
tff(fact_7710_continuous__Pair,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [F5: filter(A),Fa: fun(A,B),G: fun(A,C)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => ( topolo3448309680560233919inuous(A,C,F5,G)
           => topolo3448309680560233919inuous(A,product_prod(B,C),F5,aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_atq(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Fa),G)) ) ) ) ).

% continuous_Pair
tff(fact_7711_tendsto__Pair,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Fa: fun(A,B),A3: B,F5: filter(A),G: fun(A,C),B3: C] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => ( filterlim(A,C,G,topolo7230453075368039082e_nhds(C,B3),F5)
           => filterlim(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ava(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Fa),G),topolo7230453075368039082e_nhds(product_prod(B,C),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A3),B3)),F5) ) ) ) ).

% tendsto_Pair
tff(fact_7712_tendsto__minus__cancel__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Fa: fun(A,B),Y: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),Y)),F5)
        <=> filterlim(A,B,aTP_Lamp_avb(fun(A,B),fun(A,B),Fa),topolo7230453075368039082e_nhds(B,Y),F5) ) ) ).

% tendsto_minus_cancel_left
tff(fact_7713_tendsto__minus__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Fa: fun(A,B),A3: B,F5: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_avb(fun(A,B),fun(A,B),Fa),topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),A3)),F5)
         => filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5) ) ) ).

% tendsto_minus_cancel
tff(fact_7714_tendsto__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Fa: fun(A,B),A3: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => filterlim(A,B,aTP_Lamp_avb(fun(A,B),fun(A,B),Fa),topolo7230453075368039082e_nhds(B,aa(B,B,uminus_uminus(B),A3)),F5) ) ) ).

% tendsto_minus
tff(fact_7715_continuous__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F5: filter(A),Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => topolo3448309680560233919inuous(A,B,F5,aTP_Lamp_avc(fun(A,B),fun(A,B),Fa)) ) ) ).

% continuous_minus
tff(fact_7716_tendsto__divide__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Fa: fun(A,B),F5: filter(A),C2: B] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_avd(fun(A,B),fun(B,fun(A,B)),Fa),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5) ) ) ).

% tendsto_divide_zero
tff(fact_7717_tendsto__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Fa: fun(A,B),A3: B,F5: filter(A),G: fun(A,B),B3: B] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B3),F5)
           => ( ( B3 != zero_zero(B) )
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ave(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),divide_divide(B),A3),B3)),F5) ) ) ) ) ).

% tendsto_divide
tff(fact_7718_tendsto__artanh,axiom,
    ! [A: $tType,Fa: fun(A,real),A3: real,F5: filter(A)] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,A3),F5)
     => ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),A3)
       => ( aa(real,$o,ord_less(real,A3),one_one(real))
         => filterlim(A,real,aTP_Lamp_avf(fun(A,real),fun(A,real),Fa),topolo7230453075368039082e_nhds(real,aa(real,real,artanh(real),A3)),F5) ) ) ) ).

% tendsto_artanh
tff(fact_7719_tendsto__arcosh,axiom,
    ! [A: $tType,Fa: fun(A,real),A3: real,F5: filter(A)] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,A3),F5)
     => ( aa(real,$o,ord_less(real,one_one(real)),A3)
       => filterlim(A,real,aTP_Lamp_avg(fun(A,real),fun(A,real),Fa),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A3)),F5) ) ) ).

% tendsto_arcosh
tff(fact_7720_tendsto__mult__one,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Fa: fun(A,B),F5: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,one_one(B)),F5)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,one_one(B)),F5)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_avh(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),topolo7230453075368039082e_nhds(B,one_one(B)),F5) ) ) ) ).

% tendsto_mult_one
tff(fact_7721_tendsto__one__prod_H,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(A),Fa: fun(B,fun(A,C)),F5: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_avi(fun(B,fun(A,C)),fun(A,fun(B,C)),Fa),I2),topolo7230453075368039082e_nhds(C,one_one(C)),F5) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_avj(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),Fa),topolo7230453075368039082e_nhds(C,one_one(C)),F5) ) ) ).

% tendsto_one_prod'
tff(fact_7722_tendsto__of__int__floor,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( ring_1(C)
        & topolo4958980785337419405_space(C)
        & archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Fa: fun(A,B),L: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),F5)
         => ( ~ member(B,L,ring_1_Ints(B))
           => filterlim(A,C,aTP_Lamp_avk(fun(A,B),fun(A,C),Fa),topolo7230453075368039082e_nhds(C,aa(int,C,ring_1_of_int(C),archim6421214686448440834_floor(B,L))),F5) ) ) ) ).

% tendsto_of_int_floor
tff(fact_7723_tendsto__of__int__ceiling,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( ring_1(C)
        & topolo4958980785337419405_space(C)
        & archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Fa: fun(A,B),L: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),F5)
         => ( ~ member(B,L,ring_1_Ints(B))
           => filterlim(A,C,aTP_Lamp_avl(fun(A,B),fun(A,C),Fa),topolo7230453075368039082e_nhds(C,aa(int,C,ring_1_of_int(C),archimedean_ceiling(B,L))),F5) ) ) ) ).

% tendsto_of_int_ceiling
tff(fact_7724_tendsto__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Fa: fun(A,B),A3: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => ( ( A3 != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_avm(fun(A,B),fun(A,B),Fa),topolo7230453075368039082e_nhds(B,aa(B,B,inverse_inverse(B),A3)),F5) ) ) ) ).

% tendsto_inverse
tff(fact_7725_tendsto__add__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Fa: fun(A,B),F5: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_avn(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5) ) ) ) ).

% tendsto_add_zero
tff(fact_7726_tendsto__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Fa: fun(A,A),A3: A,F5: filter(A)] :
          ( filterlim(A,A,Fa,topolo7230453075368039082e_nhds(A,A3),F5)
         => ( ( sin(A,A3) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_avo(fun(A,A),fun(A,A),Fa),topolo7230453075368039082e_nhds(A,aa(A,A,cot(A),A3)),F5) ) ) ) ).

% tendsto_cot
tff(fact_7727_tendsto__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Fa: fun(A,B),A3: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => ( ( cosh(B,A3) != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_avp(fun(A,B),fun(A,B),Fa),topolo7230453075368039082e_nhds(B,aa(B,B,tanh(B),A3)),F5) ) ) ) ).

% tendsto_tanh
tff(fact_7728_tendsto__null__sum,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),Fa: fun(B,fun(A,C)),F5: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => filterlim(B,C,aa(A,fun(B,C),aTP_Lamp_avq(fun(B,fun(A,C)),fun(A,fun(B,C)),Fa),I2),topolo7230453075368039082e_nhds(C,zero_zero(C)),F5) )
         => filterlim(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_avr(set(A),fun(fun(B,fun(A,C)),fun(B,C)),I5),Fa),topolo7230453075368039082e_nhds(C,zero_zero(C)),F5) ) ) ).

% tendsto_null_sum
tff(fact_7729_tendsto__powr,axiom,
    ! [A: $tType,Fa: fun(A,real),A3: real,F5: filter(A),G: fun(A,real),B3: real] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,A3),F5)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B3),F5)
       => ( ( A3 != zero_zero(real) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_avs(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G),topolo7230453075368039082e_nhds(real,powr(real,A3,B3)),F5) ) ) ) ).

% tendsto_powr
tff(fact_7730_tendsto__rabs__zero,axiom,
    ! [A: $tType,Fa: fun(A,real),F5: filter(A)] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,zero_zero(real)),F5)
     => filterlim(A,real,aTP_Lamp_avt(fun(A,real),fun(A,real),Fa),topolo7230453075368039082e_nhds(real,zero_zero(real)),F5) ) ).

% tendsto_rabs_zero
tff(fact_7731_tendsto__rabs__zero__iff,axiom,
    ! [A: $tType,Fa: fun(A,real),F5: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_avt(fun(A,real),fun(A,real),Fa),topolo7230453075368039082e_nhds(real,zero_zero(real)),F5)
    <=> filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,zero_zero(real)),F5) ) ).

% tendsto_rabs_zero_iff
tff(fact_7732_tendsto__rabs__zero__cancel,axiom,
    ! [A: $tType,Fa: fun(A,real),F5: filter(A)] :
      ( filterlim(A,real,aTP_Lamp_avt(fun(A,real),fun(A,real),Fa),topolo7230453075368039082e_nhds(real,zero_zero(real)),F5)
     => filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,zero_zero(real)),F5) ) ).

% tendsto_rabs_zero_cancel
tff(fact_7733_tendsto__ln,axiom,
    ! [A: $tType,Fa: fun(A,real),A3: real,F5: filter(A)] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,A3),F5)
     => ( ( A3 != zero_zero(real) )
       => filterlim(A,real,aTP_Lamp_zc(fun(A,real),fun(A,real),Fa),topolo7230453075368039082e_nhds(real,aa(real,real,ln_ln(real),A3)),F5) ) ) ).

% tendsto_ln
tff(fact_7734_tendsto__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Fa: fun(A,A),A3: A,F5: filter(A)] :
          ( filterlim(A,A,Fa,topolo7230453075368039082e_nhds(A,A3),F5)
         => ( ( cos(A,A3) != zero_zero(A) )
           => filterlim(A,A,aTP_Lamp_avu(fun(A,A),fun(A,A),Fa),topolo7230453075368039082e_nhds(A,aa(A,A,tan(A),A3)),F5) ) ) ) ).

% tendsto_tan
tff(fact_7735_tendsto__norm__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,B),F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
         => filterlim(A,real,aTP_Lamp_avv(fun(A,B),fun(A,real),Fa),topolo7230453075368039082e_nhds(real,zero_zero(real)),F5) ) ) ).

% tendsto_norm_zero
tff(fact_7736_tendsto__norm__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,B),F5: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_avv(fun(A,B),fun(A,real),Fa),topolo7230453075368039082e_nhds(real,zero_zero(real)),F5)
        <=> filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,zero_zero(B)),F5) ) ) ).

% tendsto_norm_zero_iff
tff(fact_7737_tendsto__norm__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,B),F5: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_avv(fun(A,B),fun(A,real),Fa),topolo7230453075368039082e_nhds(real,zero_zero(real)),F5)
         => filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,zero_zero(B)),F5) ) ) ).

% tendsto_norm_zero_cancel
tff(fact_7738_tendsto__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,B),L: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),F5)
         => ( ( L != zero_zero(B) )
           => filterlim(A,B,aTP_Lamp_avw(fun(A,B),fun(A,B),Fa),topolo7230453075368039082e_nhds(B,aa(B,B,sgn_sgn(B),L)),F5) ) ) ) ).

% tendsto_sgn
tff(fact_7739_tendsto__mult__right__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Fa: fun(A,B),F5: filter(A),C2: B] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_avx(fun(A,B),fun(B,fun(A,B)),Fa),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5) ) ) ).

% tendsto_mult_right_zero
tff(fact_7740_tendsto__mult__left__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Fa: fun(A,B),F5: filter(A),C2: B] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_avy(fun(A,B),fun(B,fun(A,B)),Fa),C2),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5) ) ) ).

% tendsto_mult_left_zero
tff(fact_7741_tendsto__mult__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Fa: fun(A,B),F5: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_avz(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5) ) ) ) ).

% tendsto_mult_zero
tff(fact_7742_tendsto__null__power,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Fa: fun(A,B),F5: filter(A),N: nat] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_awa(fun(A,B),fun(nat,fun(A,B)),Fa),N),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5) ) ) ) ).

% tendsto_null_power
tff(fact_7743_tendsto__log,axiom,
    ! [A: $tType,Fa: fun(A,real),A3: real,F5: filter(A),G: fun(A,real),B3: real] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,A3),F5)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B3),F5)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
         => ( ( A3 != one_one(real) )
           => ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
             => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_awb(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G),topolo7230453075368039082e_nhds(real,aa(real,real,log(A3),B3)),F5) ) ) ) ) ) ).

% tendsto_log
tff(fact_7744_continuous__snd,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F5: filter(A),Fa: fun(A,product_prod(B,C))] :
          ( topolo3448309680560233919inuous(A,product_prod(B,C),F5,Fa)
         => topolo3448309680560233919inuous(A,C,F5,aTP_Lamp_aus(fun(A,product_prod(B,C)),fun(A,C),Fa)) ) ) ).

% continuous_snd
tff(fact_7745_continuous__fst,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [F5: filter(A),Fa: fun(A,product_prod(B,C))] :
          ( topolo3448309680560233919inuous(A,product_prod(B,C),F5,Fa)
         => topolo3448309680560233919inuous(A,B,F5,aTP_Lamp_aur(fun(A,product_prod(B,C)),fun(A,B),Fa)) ) ) ).

% continuous_fst
tff(fact_7746_tendsto__snd,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Fa: fun(A,product_prod(B,C)),A3: product_prod(B,C),F5: filter(A)] :
          ( filterlim(A,product_prod(B,C),Fa,topolo7230453075368039082e_nhds(product_prod(B,C),A3),F5)
         => filterlim(A,C,aTP_Lamp_awc(fun(A,product_prod(B,C)),fun(A,C),Fa),topolo7230453075368039082e_nhds(C,aa(product_prod(B,C),C,product_snd(B,C),A3)),F5) ) ) ).

% tendsto_snd
tff(fact_7747_tendsto__fst,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Fa: fun(A,product_prod(B,C)),A3: product_prod(B,C),F5: filter(A)] :
          ( filterlim(A,product_prod(B,C),Fa,topolo7230453075368039082e_nhds(product_prod(B,C),A3),F5)
         => filterlim(A,B,aTP_Lamp_awd(fun(A,product_prod(B,C)),fun(A,B),Fa),topolo7230453075368039082e_nhds(B,aa(product_prod(B,C),B,product_fst(B,C),A3)),F5) ) ) ).

% tendsto_fst
tff(fact_7748_tendsto__add__const__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [C2: B,Fa: fun(A,B),D: B,F5: filter(A)] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_awe(B,fun(fun(A,B),fun(A,B)),C2),Fa),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),D)),F5)
        <=> filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,D),F5) ) ) ).

% tendsto_add_const_iff
tff(fact_7749_tendsto__add,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Fa: fun(A,B),A3: B,F5: filter(A),G: fun(A,B),B3: B] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B3),F5)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_avn(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A3),B3)),F5) ) ) ) ).

% tendsto_add
tff(fact_7750_continuous__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [F5: filter(A),Fa: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => ( topolo3448309680560233919inuous(A,B,F5,G)
           => topolo3448309680560233919inuous(A,B,F5,aa(fun(A,B),fun(A,B),aTP_Lamp_aup(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G)) ) ) ) ).

% continuous_add
tff(fact_7751_continuous__prod_H,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [I5: set(A),F5: filter(B),Fa: fun(A,fun(B,C))] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => topolo3448309680560233919inuous(B,C,F5,aa(A,fun(B,C),Fa,I2)) )
         => topolo3448309680560233919inuous(B,C,F5,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_awg(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),Fa)) ) ) ).

% continuous_prod'
tff(fact_7752_continuous__prod,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [S3: set(A),F5: filter(B),Fa: fun(A,fun(B,C))] :
          ( ! [I2: A] :
              ( member(A,I2,S3)
             => topolo3448309680560233919inuous(B,C,F5,aa(A,fun(B,C),Fa,I2)) )
         => topolo3448309680560233919inuous(B,C,F5,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_awi(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S3),Fa)) ) ) ).

% continuous_prod
tff(fact_7753_tendsto__prod_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [I5: set(A),Fa: fun(A,fun(B,C)),A3: fun(A,C),F5: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => filterlim(B,C,aa(A,fun(B,C),Fa,I2),topolo7230453075368039082e_nhds(C,aa(A,C,A3,I2)),F5) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_awk(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),Fa),topolo7230453075368039082e_nhds(C,groups7121269368397514597t_prod(A,C,A3,I5)),F5) ) ) ).

% tendsto_prod'
tff(fact_7754_tendsto__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [S3: set(A),Fa: fun(A,fun(B,C)),L5: fun(A,C),F5: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,S3)
             => filterlim(B,C,aa(A,fun(B,C),Fa,I2),topolo7230453075368039082e_nhds(C,aa(A,C,L5,I2)),F5) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_awm(set(A),fun(fun(A,fun(B,C)),fun(B,C)),S3),Fa),topolo7230453075368039082e_nhds(C,groups7121269368397514597t_prod(A,C,L5,S3)),F5) ) ) ).

% tendsto_prod
tff(fact_7755_continuous__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F5: filter(A),Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => topolo3448309680560233919inuous(A,real,F5,aTP_Lamp_auo(fun(A,B),fun(A,real),Fa)) ) ) ).

% continuous_norm
tff(fact_7756_tendsto__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,B),A3: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => filterlim(A,real,aTP_Lamp_avv(fun(A,B),fun(A,real),Fa),topolo7230453075368039082e_nhds(real,real_V7770717601297561774m_norm(B,A3)),F5) ) ) ).

% tendsto_norm
tff(fact_7757_tendsto__Complex,axiom,
    ! [A: $tType,Fa: fun(A,real),A3: real,F5: filter(A),G: fun(A,real),B3: real] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,A3),F5)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B3),F5)
       => filterlim(A,complex,aa(fun(A,real),fun(A,complex),aTP_Lamp_awn(fun(A,real),fun(fun(A,real),fun(A,complex)),Fa),G),topolo7230453075368039082e_nhds(complex,complex2(A3,B3)),F5) ) ) ).

% tendsto_Complex
tff(fact_7758_continuous__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F5: filter(A),Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => topolo3448309680560233919inuous(A,B,F5,aTP_Lamp_aux(fun(A,B),fun(A,B),Fa)) ) ) ).

% continuous_cos
tff(fact_7759_continuous__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F5: filter(A),Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => topolo3448309680560233919inuous(A,B,F5,aTP_Lamp_auy(fun(A,B),fun(A,B),Fa)) ) ) ).

% continuous_sin
tff(fact_7760_tendsto__arsinh,axiom,
    ! [A: $tType,Fa: fun(A,real),A3: real,F5: filter(A)] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,A3),F5)
     => filterlim(A,real,aTP_Lamp_awo(fun(A,real),fun(A,real),Fa),topolo7230453075368039082e_nhds(real,aa(real,real,arsinh(real),A3)),F5) ) ).

% tendsto_arsinh
tff(fact_7761_tendsto__sin,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Fa: fun(A,B),A3: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => filterlim(A,B,aTP_Lamp_auy(fun(A,B),fun(A,B),Fa),topolo7230453075368039082e_nhds(B,sin(B,A3)),F5) ) ) ).

% tendsto_sin
tff(fact_7762_tendsto__cos,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Fa: fun(A,B),A3: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => filterlim(A,B,aTP_Lamp_aux(fun(A,B),fun(A,B),Fa),topolo7230453075368039082e_nhds(B,cos(B,A3)),F5) ) ) ).

% tendsto_cos
tff(fact_7763_tendsto__arctan,axiom,
    ! [A: $tType,Fa: fun(A,real),Xb: real,F5: filter(A)] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,Xb),F5)
     => filterlim(A,real,aTP_Lamp_awp(fun(A,real),fun(A,real),Fa),topolo7230453075368039082e_nhds(real,aa(real,real,arctan,Xb)),F5) ) ).

% tendsto_arctan
tff(fact_7764_tendsto__real__sqrt,axiom,
    ! [A: $tType,Fa: fun(A,real),Xb: real,F5: filter(A)] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,Xb),F5)
     => filterlim(A,real,aTP_Lamp_awq(fun(A,real),fun(A,real),Fa),topolo7230453075368039082e_nhds(real,aa(real,real,sqrt,Xb)),F5) ) ).

% tendsto_real_sqrt
tff(fact_7765_continuous__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [F5: filter(A),C2: B] : topolo3448309680560233919inuous(A,B,F5,aTP_Lamp_awr(B,fun(A,B),C2)) ) ).

% continuous_const
tff(fact_7766_tendsto__of__real__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Fa: fun(A,real),C2: real,F5: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_aws(fun(A,real),fun(A,B),Fa),topolo7230453075368039082e_nhds(B,aa(real,B,real_Vector_of_real(B),C2)),F5)
        <=> filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,C2),F5) ) ) ).

% tendsto_of_real_iff
tff(fact_7767_continuous__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [F5: filter(A),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F5,G)
         => topolo3448309680560233919inuous(A,B,F5,aTP_Lamp_auf(fun(A,real),fun(A,B),G)) ) ) ).

% continuous_of_real
tff(fact_7768_tendsto__of__real,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [G: fun(A,real),A3: real,F5: filter(A)] :
          ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,A3),F5)
         => filterlim(A,B,aTP_Lamp_awt(fun(A,real),fun(A,B),G),topolo7230453075368039082e_nhds(B,aa(real,B,real_Vector_of_real(B),A3)),F5) ) ) ).

% tendsto_of_real
tff(fact_7769_tendsto__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,real),A3: real,F5: filter(A),G: fun(A,B),B3: B] :
          ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,A3),F5)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B3),F5)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_awu(fun(A,real),fun(fun(A,B),fun(A,B)),Fa),G),topolo7230453075368039082e_nhds(B,aa(B,B,real_V8093663219630862766scaleR(B,A3),B3)),F5) ) ) ) ).

% tendsto_scaleR
tff(fact_7770_continuous__scaleR,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F5: filter(A),Fa: fun(A,real),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,real,F5,Fa)
         => ( topolo3448309680560233919inuous(A,B,F5,G)
           => topolo3448309680560233919inuous(A,B,F5,aa(fun(A,B),fun(A,B),aTP_Lamp_aue(fun(A,real),fun(fun(A,B),fun(A,B)),Fa),G)) ) ) ) ).

% continuous_scaleR
tff(fact_7771_tendsto__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [I5: set(A),Fa: fun(A,fun(B,C)),A3: fun(A,C),F5: filter(B)] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => filterlim(B,C,aa(A,fun(B,C),Fa,I2),topolo7230453075368039082e_nhds(C,aa(A,C,A3,I2)),F5) )
         => filterlim(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aww(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),Fa),topolo7230453075368039082e_nhds(C,aa(set(A),C,groups7311177749621191930dd_sum(A,C,A3),I5)),F5) ) ) ).

% tendsto_sum
tff(fact_7772_continuous__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [I5: set(A),F5: filter(B),Fa: fun(A,fun(B,C))] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => topolo3448309680560233919inuous(B,C,F5,aa(A,fun(B,C),Fa,I2)) )
         => topolo3448309680560233919inuous(B,C,F5,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aud(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),Fa)) ) ) ).

% continuous_sum
tff(fact_7773_continuous__sinh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F5: filter(A),Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => topolo3448309680560233919inuous(A,B,F5,aTP_Lamp_awx(fun(A,B),fun(A,B),Fa)) ) ) ).

% continuous_sinh
tff(fact_7774_continuous__cosh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F5: filter(A),Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => topolo3448309680560233919inuous(A,B,F5,aTP_Lamp_awy(fun(A,B),fun(A,B),Fa)) ) ) ).

% continuous_cosh
tff(fact_7775_tendsto__sinh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Fa: fun(A,B),A3: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => filterlim(A,B,aTP_Lamp_awz(fun(A,B),fun(A,B),Fa),topolo7230453075368039082e_nhds(B,sinh(B,A3)),F5) ) ) ).

% tendsto_sinh
tff(fact_7776_tendsto__cosh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Fa: fun(A,B),A3: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => filterlim(A,B,aTP_Lamp_axa(fun(A,B),fun(A,B),Fa),topolo7230453075368039082e_nhds(B,cosh(B,A3)),F5) ) ) ).

% tendsto_cosh
tff(fact_7777_tendsto__exp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Fa: fun(A,B),A3: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => filterlim(A,B,aTP_Lamp_axb(fun(A,B),fun(A,B),Fa),topolo7230453075368039082e_nhds(B,aa(B,B,exp(B),A3)),F5) ) ) ).

% tendsto_exp
tff(fact_7778_continuous__exp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F5: filter(A),Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => topolo3448309680560233919inuous(A,B,F5,aTP_Lamp_auz(fun(A,B),fun(A,B),Fa)) ) ) ).

% continuous_exp
tff(fact_7779_tendsto__real__root,axiom,
    ! [A: $tType,Fa: fun(A,real),Xb: real,F5: filter(A),N: nat] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,Xb),F5)
     => filterlim(A,real,aa(nat,fun(A,real),aTP_Lamp_axc(fun(A,real),fun(nat,fun(A,real)),Fa),N),topolo7230453075368039082e_nhds(real,aa(real,real,root(N),Xb)),F5) ) ).

% tendsto_real_root
tff(fact_7780_tendsto__rabs,axiom,
    ! [A: $tType,Fa: fun(A,real),L: real,F5: filter(A)] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,L),F5)
     => filterlim(A,real,aTP_Lamp_avt(fun(A,real),fun(A,real),Fa),topolo7230453075368039082e_nhds(real,abs_abs(real,L)),F5) ) ).

% tendsto_rabs
tff(fact_7781_continuous__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [F5: filter(A),Fa: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => ( topolo3448309680560233919inuous(A,B,F5,G)
           => topolo3448309680560233919inuous(A,B,F5,aa(fun(A,B),fun(A,B),aTP_Lamp_aub(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G)) ) ) ) ).

% continuous_mult
tff(fact_7782_continuous__mult_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [F5: filter(A),Fa: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => ( topolo3448309680560233919inuous(A,B,F5,G)
           => topolo3448309680560233919inuous(A,B,F5,aa(fun(A,B),fun(A,B),aTP_Lamp_axd(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G)) ) ) ) ).

% continuous_mult'
tff(fact_7783_continuous__mult__left,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F5: filter(A),Fa: fun(A,B),C2: B] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => topolo3448309680560233919inuous(A,B,F5,aa(B,fun(A,B),aTP_Lamp_axe(fun(A,B),fun(B,fun(A,B)),Fa),C2)) ) ) ).

% continuous_mult_left
tff(fact_7784_continuous__mult__right,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F5: filter(A),Fa: fun(A,B),C2: B] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => topolo3448309680560233919inuous(A,B,F5,aa(B,fun(A,B),aTP_Lamp_axf(fun(A,B),fun(B,fun(A,B)),Fa),C2)) ) ) ).

% continuous_mult_right
tff(fact_7785_tendsto__mult,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Fa: fun(A,B),A3: B,F5: filter(A),G: fun(A,B),B3: B] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B3),F5)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axg(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),A3),B3)),F5) ) ) ) ).

% tendsto_mult
tff(fact_7786_tendsto__mult__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Fa: fun(A,B),L: B,F5: filter(A),C2: B] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),F5)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_axh(fun(A,B),fun(B,fun(A,B)),Fa),C2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),C2),L)),F5) ) ) ).

% tendsto_mult_left
tff(fact_7787_tendsto__mult__right,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Fa: fun(A,B),L: B,F5: filter(A),C2: B] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),F5)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_axi(fun(A,B),fun(B,fun(A,B)),Fa),C2),topolo7230453075368039082e_nhds(B,aa(B,B,aa(B,fun(B,B),times_times(B),L),C2)),F5) ) ) ).

% tendsto_mult_right
tff(fact_7788_tendsto__power,axiom,
    ! [B: $tType,A: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [Fa: fun(A,B),A3: B,F5: filter(A),N: nat] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_axj(fun(A,B),fun(nat,fun(A,B)),Fa),N),topolo7230453075368039082e_nhds(B,aa(nat,B,aa(B,fun(nat,B),power_power(B),A3),N)),F5) ) ) ).

% tendsto_power
tff(fact_7789_tendsto__power__strong,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Fa: fun(A,B),A3: B,F5: filter(A),G: fun(A,nat),B3: nat] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => ( filterlim(A,nat,G,topolo7230453075368039082e_nhds(nat,B3),F5)
           => filterlim(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_axk(fun(A,B),fun(fun(A,nat),fun(A,B)),Fa),G),topolo7230453075368039082e_nhds(B,aa(nat,B,aa(B,fun(nat,B),power_power(B),A3),B3)),F5) ) ) ) ).

% tendsto_power_strong
tff(fact_7790_continuous__power_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [F5: filter(A),Fa: fun(A,B),G: fun(A,nat)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => ( topolo3448309680560233919inuous(A,nat,F5,G)
           => topolo3448309680560233919inuous(A,B,F5,aa(fun(A,nat),fun(A,B),aTP_Lamp_axl(fun(A,B),fun(fun(A,nat),fun(A,B)),Fa),G)) ) ) ) ).

% continuous_power'
tff(fact_7791_continuous__power,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [F5: filter(A),Fa: fun(A,B),N: nat] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => topolo3448309680560233919inuous(A,B,F5,aa(nat,fun(A,B),aTP_Lamp_auq(fun(A,B),fun(nat,fun(A,B)),Fa),N)) ) ) ).

% continuous_power
tff(fact_7792_tendsto__min,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [X5: fun(A,B),Xb: B,Net: filter(A),Y6: fun(A,B),Y: B] :
          ( filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,Xb),Net)
         => ( filterlim(A,B,Y6,topolo7230453075368039082e_nhds(B,Y),Net)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axm(fun(A,B),fun(fun(A,B),fun(A,B)),X5),Y6),topolo7230453075368039082e_nhds(B,ord_min(B,Xb,Y)),Net) ) ) ) ).

% tendsto_min
tff(fact_7793_continuous__min,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F5: filter(A),Fa: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => ( topolo3448309680560233919inuous(A,B,F5,G)
           => topolo3448309680560233919inuous(A,B,F5,aa(fun(A,B),fun(A,B),aTP_Lamp_axn(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G)) ) ) ) ).

% continuous_min
tff(fact_7794_tendsto__const__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [F5: filter(A),A3: B,B3: B] :
          ( ( F5 != bot_bot(filter(A)) )
         => ( filterlim(A,B,aTP_Lamp_axo(B,fun(A,B),A3),topolo7230453075368039082e_nhds(B,B3),F5)
          <=> ( A3 = B3 ) ) ) ) ).

% tendsto_const_iff
tff(fact_7795_tendsto__max,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [X5: fun(A,B),Xb: B,Net: filter(A),Y6: fun(A,B),Y: B] :
          ( filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,Xb),Net)
         => ( filterlim(A,B,Y6,topolo7230453075368039082e_nhds(B,Y),Net)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axp(fun(A,B),fun(fun(A,B),fun(A,B)),X5),Y6),topolo7230453075368039082e_nhds(B,aa(B,B,ord_max(B,Xb),Y)),Net) ) ) ) ).

% tendsto_max
tff(fact_7796_continuous__max,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [F5: filter(A),Fa: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => ( topolo3448309680560233919inuous(A,B,F5,G)
           => topolo3448309680560233919inuous(A,B,F5,aa(fun(A,B),fun(A,B),aTP_Lamp_axq(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G)) ) ) ) ).

% continuous_max
tff(fact_7797_continuous__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [F5: filter(A),Fa: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => ( topolo3448309680560233919inuous(A,B,F5,G)
           => topolo3448309680560233919inuous(A,B,F5,aa(fun(A,B),fun(A,B),aTP_Lamp_axr(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G)) ) ) ) ).

% continuous_diff
tff(fact_7798_tendsto__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Fa: fun(A,B),A3: B,F5: filter(A),G: fun(A,B),B3: B] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,B3),F5)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axs(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),topolo7230453075368039082e_nhds(B,aa(B,B,minus_minus(B,A3),B3)),F5) ) ) ) ).

% tendsto_diff
tff(fact_7799_LIM__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,B),L: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),F5)
         => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_axt(fun(A,B),fun(B,fun(A,B)),Fa),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5) ) ) ).

% LIM_zero
tff(fact_7800_LIM__zero__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,B),L: B,F5: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_axt(fun(A,B),fun(B,fun(A,B)),Fa),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
        <=> filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),F5) ) ) ).

% LIM_zero_iff
tff(fact_7801_Lim__transform,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [G: fun(A,B),A3: B,F5: filter(A),Fa: fun(A,B)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A3),F5)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axu(fun(A,B),fun(fun(A,B),fun(A,B)),G),Fa),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
           => filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5) ) ) ) ).

% Lim_transform
tff(fact_7802_Lim__transform2,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,B),A3: B,F5: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axv(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A3),F5) ) ) ) ).

% Lim_transform2
tff(fact_7803_LIM__zero__cancel,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,B),L: B,F5: filter(A)] :
          ( filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_axt(fun(A,B),fun(B,fun(A,B)),Fa),L),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
         => filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),F5) ) ) ).

% LIM_zero_cancel
tff(fact_7804_Lim__transform__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,B),G: fun(A,B),F5: filter(A),A3: B] :
          ( filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axv(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
         => ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,A3),F5)
          <=> filterlim(A,B,G,topolo7230453075368039082e_nhds(B,A3),F5) ) ) ) ).

% Lim_transform_eq
tff(fact_7805_isCont__pochhammer,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Z: A,N: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Z,top_top(set(A))),aTP_Lamp_axw(nat,fun(A,A),N)) ) ).

% isCont_pochhammer
tff(fact_7806_DERIV__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A] :
          ( has_field_derivative(A,Fa,D5,topolo174197925503356063within(A,Xb,top_top(set(A))))
         => filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_axx(fun(A,A),fun(A,fun(A,A)),Fa),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_D
tff(fact_7807_DERIV__def,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A] :
          ( has_field_derivative(A,Fa,D5,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_axx(fun(A,A),fun(A,fun(A,A)),Fa),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% DERIV_def
tff(fact_7808_lim__exp__minus__1,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => filterlim(A,A,aTP_Lamp_axy(A,A),topolo7230453075368039082e_nhds(A,one_one(A)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% lim_exp_minus_1
tff(fact_7809_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : set_or7035219750837199246ssThan(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),L),one_one(int)),U) = set_or5935395276787703475ssThan(int,L,U) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
tff(fact_7810_lemma__termdiff4,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [K: real,Fa: fun(A,B),K5: real] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),K)
         => ( ! [H5: A] :
                ( ( H5 != zero_zero(A) )
               => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,H5)),K)
                 => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,Fa,H5))),aa(real,real,aa(real,fun(real,real),times_times(real),K5),real_V7770717601297561774m_norm(A,H5))) ) )
           => filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% lemma_termdiff4
tff(fact_7811_isCont__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B3: real,Fa: fun(real,A)] :
          ( aa(real,$o,ord_less_eq(real,A3),B3)
         => ( ! [X2: real] :
                ( ( aa(real,$o,ord_less_eq(real,A3),X2)
                  & aa(real,$o,ord_less_eq(real,X2),B3) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X2,top_top(set(real))),Fa) )
           => ? [M10: A] :
              ! [X3: real] :
                ( ( aa(real,$o,ord_less_eq(real,A3),X3)
                  & aa(real,$o,ord_less_eq(real,X3),B3) )
               => aa(A,$o,ord_less_eq(A,aa(real,A,Fa,X3)),M10) ) ) ) ) ).

% isCont_bounded
tff(fact_7812_isCont__eq__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B3: real,Fa: fun(real,A)] :
          ( aa(real,$o,ord_less_eq(real,A3),B3)
         => ( ! [X2: real] :
                ( ( aa(real,$o,ord_less_eq(real,A3),X2)
                  & aa(real,$o,ord_less_eq(real,X2),B3) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X2,top_top(set(real))),Fa) )
           => ? [M10: A] :
                ( ! [X3: real] :
                    ( ( aa(real,$o,ord_less_eq(real,A3),X3)
                      & aa(real,$o,ord_less_eq(real,X3),B3) )
                   => aa(A,$o,ord_less_eq(A,aa(real,A,Fa,X3)),M10) )
                & ? [X2: real] :
                    ( aa(real,$o,ord_less_eq(real,A3),X2)
                    & aa(real,$o,ord_less_eq(real,X2),B3)
                    & ( aa(real,A,Fa,X2) = M10 ) ) ) ) ) ) ).

% isCont_eq_Ub
tff(fact_7813_isCont__eq__Lb,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B3: real,Fa: fun(real,A)] :
          ( aa(real,$o,ord_less_eq(real,A3),B3)
         => ( ! [X2: real] :
                ( ( aa(real,$o,ord_less_eq(real,A3),X2)
                  & aa(real,$o,ord_less_eq(real,X2),B3) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X2,top_top(set(real))),Fa) )
           => ? [M10: A] :
                ( ! [X3: real] :
                    ( ( aa(real,$o,ord_less_eq(real,A3),X3)
                      & aa(real,$o,ord_less_eq(real,X3),B3) )
                   => aa(A,$o,ord_less_eq(A,M10),aa(real,A,Fa,X3)) )
                & ? [X2: real] :
                    ( aa(real,$o,ord_less_eq(real,A3),X2)
                    & aa(real,$o,ord_less_eq(real,X2),B3)
                    & ( aa(real,A,Fa,X2) = M10 ) ) ) ) ) ) ).

% isCont_eq_Lb
tff(fact_7814_isCont__inverse__function2,axiom,
    ! [A3: real,Xb: real,B3: real,G: fun(real,real),Fa: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),Xb)
     => ( aa(real,$o,ord_less(real,Xb),B3)
       => ( ! [Z2: real] :
              ( aa(real,$o,ord_less_eq(real,A3),Z2)
             => ( aa(real,$o,ord_less_eq(real,Z2),B3)
               => ( aa(real,real,G,aa(real,real,Fa,Z2)) = Z2 ) ) )
         => ( ! [Z2: real] :
                ( aa(real,$o,ord_less_eq(real,A3),Z2)
               => ( aa(real,$o,ord_less_eq(real,Z2),B3)
                 => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),Fa) ) )
           => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,Fa,Xb),top_top(set(real))),G) ) ) ) ) ).

% isCont_inverse_function2
tff(fact_7815_field__has__derivative__at,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),D5: A,Xb: A] :
          ( has_derivative(A,A,Fa,aa(A,fun(A,A),times_times(A),D5),topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> filterlim(A,A,aa(A,fun(A,A),aTP_Lamp_axx(fun(A,A),fun(A,fun(A,A)),Fa),Xb),topolo7230453075368039082e_nhds(A,D5),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% field_has_derivative_at
tff(fact_7816_isCont__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [A3: A,Fa: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => ( ( aa(A,B,G,A3) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,B),fun(A,B),aTP_Lamp_atv(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G)) ) ) ) ) ).

% isCont_divide
tff(fact_7817_isCont__ln,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero(real) )
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),ln_ln(real)) ) ).

% isCont_ln
tff(fact_7818_isCont__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [A3: A,Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => ( ( aa(A,B,Fa,A3) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_aua(fun(A,B),fun(A,B),Fa)) ) ) ) ).

% isCont_sgn
tff(fact_7819_filterlim__at__to__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(A,B),F5: filter(B),A3: A] :
          ( filterlim(A,B,Fa,F5,topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> filterlim(A,B,aa(A,fun(A,B),aTP_Lamp_axz(fun(A,B),fun(A,fun(A,B)),Fa),A3),F5,topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% filterlim_at_to_0
tff(fact_7820_continuous__within__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,S: set(A),Fa: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),Fa)
         => ( ( cos(A,aa(A,A,Fa,Xb)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),aTP_Lamp_avu(fun(A,A),fun(A,A),Fa)) ) ) ) ).

% continuous_within_tan
tff(fact_7821_continuous__within__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,S: set(A),Fa: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),Fa)
         => ( ( sin(A,aa(A,A,Fa,Xb)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,S),aTP_Lamp_avo(fun(A,A),fun(A,A),Fa)) ) ) ) ).

% continuous_within_cot
tff(fact_7822_continuous__at__within__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Xb: A,A4: set(A),Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,A4),Fa)
         => ( ( cosh(B,aa(A,B,Fa,Xb)) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,A4),aTP_Lamp_aya(fun(A,B),fun(A,B),Fa)) ) ) ) ).

% continuous_at_within_tanh
tff(fact_7823_isCont__has__Ub,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: real,B3: real,Fa: fun(real,A)] :
          ( aa(real,$o,ord_less_eq(real,A3),B3)
         => ( ! [X2: real] :
                ( ( aa(real,$o,ord_less_eq(real,A3),X2)
                  & aa(real,$o,ord_less_eq(real,X2),B3) )
               => topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,X2,top_top(set(real))),Fa) )
           => ? [M10: A] :
                ( ! [X3: real] :
                    ( ( aa(real,$o,ord_less_eq(real,A3),X3)
                      & aa(real,$o,ord_less_eq(real,X3),B3) )
                   => aa(A,$o,ord_less_eq(A,aa(real,A,Fa,X3)),M10) )
                & ! [N10: A] :
                    ( aa(A,$o,ord_less(A,N10),M10)
                   => ? [X2: real] :
                        ( aa(real,$o,ord_less_eq(real,A3),X2)
                        & aa(real,$o,ord_less_eq(real,X2),B3)
                        & aa(A,$o,ord_less(A,N10),aa(real,A,Fa,X2)) ) ) ) ) ) ) ).

% isCont_has_Ub
tff(fact_7824_isCont__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cos(A,Xb) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),tan(A)) ) ) ).

% isCont_tan
tff(fact_7825_isCont__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sin(A,Xb) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),cot(A)) ) ) ).

% isCont_cot
tff(fact_7826_isCont__tanh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( cosh(A,Xb) != zero_zero(A) )
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),tanh(A)) ) ) ).

% isCont_tanh
tff(fact_7827_powser__limit__0__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A3: fun(nat,A),Fa: fun(A,A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),S)
         => ( ! [X2: A] :
                ( ( X2 != zero_zero(A) )
               => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X2)),S)
                 => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_xc(fun(nat,A),fun(A,fun(nat,A)),A3),X2)),aa(A,A,Fa,X2)) ) )
           => filterlim(A,A,Fa,topolo7230453075368039082e_nhds(A,aa(nat,A,A3,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0_strong
tff(fact_7828_powser__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [S: real,A3: fun(nat,A),Fa: fun(A,A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),S)
         => ( ! [X2: A] :
                ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X2)),S)
               => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_xc(fun(nat,A),fun(A,fun(nat,A)),A3),X2)),aa(A,A,Fa,X2)) )
           => filterlim(A,A,Fa,topolo7230453075368039082e_nhds(A,aa(nat,A,A3,zero_zero(nat))),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% powser_limit_0
tff(fact_7829_lemma__termdiff5,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [K: real,Fa: fun(nat,real),G: fun(A,fun(nat,B))] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),K)
         => ( summable(real,Fa)
           => ( ! [H5: A,N3: nat] :
                  ( ( H5 != zero_zero(A) )
                 => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,H5)),K)
                   => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(nat,B,aa(A,fun(nat,B),G,H5),N3))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Fa,N3)),real_V7770717601297561774m_norm(A,H5))) ) )
             => filterlim(A,B,aTP_Lamp_ayb(fun(A,fun(nat,B)),fun(A,B),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% lemma_termdiff5
tff(fact_7830_isCont__tan_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: A,Fa: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => ( ( cos(A,aa(A,A,Fa,A3)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_avu(fun(A,A),fun(A,A),Fa)) ) ) ) ).

% isCont_tan'
tff(fact_7831_isCont__arcosh,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xb)
     => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),arcosh(real)) ) ).

% isCont_arcosh
tff(fact_7832_LIM__cos__div__sin,axiom,
    filterlim(real,real,aTP_Lamp_ayc(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))),top_top(set(real)))) ).

% LIM_cos_div_sin
tff(fact_7833_isCont__cot_H,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [A3: A,Fa: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => ( ( sin(A,aa(A,A,Fa,A3)) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_avo(fun(A,A),fun(A,A),Fa)) ) ) ) ).

% isCont_cot'
tff(fact_7834_DERIV__inverse__function,axiom,
    ! [Fa: fun(real,real),D5: real,G: fun(real,real),Xb: real,A3: real,B3: real] :
      ( has_field_derivative(real,Fa,D5,topolo174197925503356063within(real,aa(real,real,G,Xb),top_top(set(real))))
     => ( ( D5 != zero_zero(real) )
       => ( aa(real,$o,ord_less(real,A3),Xb)
         => ( aa(real,$o,ord_less(real,Xb),B3)
           => ( ! [Y2: real] :
                  ( aa(real,$o,ord_less(real,A3),Y2)
                 => ( aa(real,$o,ord_less(real,Y2),B3)
                   => ( aa(real,real,Fa,aa(real,real,G,Y2)) = Y2 ) ) )
             => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),G)
               => has_field_derivative(real,G,aa(real,real,inverse_inverse(real),D5),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ) ) ) ).

% DERIV_inverse_function
tff(fact_7835_isCont__polynom,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A3: A,C2: fun(nat,A),N: nat] : topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,A3,top_top(set(A))),aa(nat,fun(A,A),aTP_Lamp_ayd(fun(nat,A),fun(nat,fun(A,A)),C2),N)) ) ).

% isCont_polynom
tff(fact_7836_isCont__powser__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xb: A] :
          ( ! [Y2: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_xc(fun(nat,A),fun(A,fun(nat,A)),C2),Y2))
         => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),aTP_Lamp_aqf(fun(nat,A),fun(A,A),C2)) ) ) ).

% isCont_powser_converges_everywhere
tff(fact_7837_isCont__arccos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),arccos) ) ) ).

% isCont_arccos
tff(fact_7838_isCont__arcsin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),arcsin) ) ) ).

% isCont_arcsin
tff(fact_7839_LIM__less__bound,axiom,
    ! [B3: real,Xb: real,Fa: fun(real,real)] :
      ( aa(real,$o,ord_less(real,B3),Xb)
     => ( ! [X2: real] :
            ( member(real,X2,set_or5935395276787703475ssThan(real,B3,Xb))
           => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,Fa,X2)) )
       => ( topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),Fa)
         => aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(real,real,Fa,Xb)) ) ) ) ).

% LIM_less_bound
tff(fact_7840_isCont__artanh,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,ord_less(real,Xb),one_one(real))
       => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Xb,top_top(set(real))),artanh(real)) ) ) ).

% isCont_artanh
tff(fact_7841_greaterThanLessThan__upto,axiom,
    ! [I: int,J: int] : set_or5935395276787703475ssThan(int,I,J) = set2(int,upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),I),one_one(int)),aa(int,int,minus_minus(int,J),one_one(int)))) ).

% greaterThanLessThan_upto
tff(fact_7842_isCont__inverse__function,axiom,
    ! [D: real,Xb: real,G: fun(real,real),Fa: fun(real,real)] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),D)
     => ( ! [Z2: real] :
            ( aa(real,$o,ord_less_eq(real,abs_abs(real,aa(real,real,minus_minus(real,Z2),Xb))),D)
           => ( aa(real,real,G,aa(real,real,Fa,Z2)) = Z2 ) )
       => ( ! [Z2: real] :
              ( aa(real,$o,ord_less_eq(real,abs_abs(real,aa(real,real,minus_minus(real,Z2),Xb))),D)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),Fa) )
         => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,aa(real,real,Fa,Xb),top_top(set(real))),G) ) ) ) ).

% isCont_inverse_function
tff(fact_7843_GMVT_H,axiom,
    ! [A3: real,B3: real,Fa: fun(real,real),G: fun(real,real),G5: fun(real,real),F3: fun(real,real)] :
      ( aa(real,$o,ord_less(real,A3),B3)
     => ( ! [Z2: real] :
            ( aa(real,$o,ord_less_eq(real,A3),Z2)
           => ( aa(real,$o,ord_less_eq(real,Z2),B3)
             => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),Fa) ) )
       => ( ! [Z2: real] :
              ( aa(real,$o,ord_less_eq(real,A3),Z2)
             => ( aa(real,$o,ord_less_eq(real,Z2),B3)
               => topolo3448309680560233919inuous(real,real,topolo174197925503356063within(real,Z2,top_top(set(real))),G) ) )
         => ( ! [Z2: real] :
                ( aa(real,$o,ord_less(real,A3),Z2)
               => ( aa(real,$o,ord_less(real,Z2),B3)
                 => has_field_derivative(real,G,aa(real,real,G5,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
           => ( ! [Z2: real] :
                  ( aa(real,$o,ord_less(real,A3),Z2)
                 => ( aa(real,$o,ord_less(real,Z2),B3)
                   => has_field_derivative(real,Fa,aa(real,real,F3,Z2),topolo174197925503356063within(real,Z2,top_top(set(real)))) ) )
             => ? [C3: real] :
                  ( aa(real,$o,ord_less(real,A3),C3)
                  & aa(real,$o,ord_less(real,C3),B3)
                  & ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,Fa,B3)),aa(real,real,Fa,A3))),aa(real,real,G5,C3)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,aa(real,real,G,B3)),aa(real,real,G,A3))),aa(real,real,F3,C3)) ) ) ) ) ) ) ) ).

% GMVT'
tff(fact_7844_floor__has__real__derivative,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Xb: real,Fa: fun(real,A)] :
          ( topolo3448309680560233919inuous(real,A,topolo174197925503356063within(real,Xb,top_top(set(real))),Fa)
         => ( ~ member(A,aa(real,A,Fa,Xb),ring_1_Ints(A))
           => has_field_derivative(real,aTP_Lamp_aye(fun(real,A),fun(real,real),Fa),zero_zero(real),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% floor_has_real_derivative
tff(fact_7845_isCont__powser,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),K5: A,Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_xc(fun(nat,A),fun(A,fun(nat,A)),C2),K5))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,K5))
           => topolo3448309680560233919inuous(A,A,topolo174197925503356063within(A,Xb,top_top(set(A))),aTP_Lamp_aqf(fun(nat,A),fun(A,A),C2)) ) ) ) ).

% isCont_powser
tff(fact_7846_isCont__powser_H,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [A3: A,Fa: fun(A,B),C2: fun(nat,B),K5: B] :
          ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => ( summable(B,aa(B,fun(nat,B),aTP_Lamp_ayf(fun(nat,B),fun(B,fun(nat,B)),C2),K5))
           => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,aa(A,B,Fa,A3))),real_V7770717601297561774m_norm(B,K5))
             => topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(nat,B),fun(A,B),aTP_Lamp_ayh(fun(A,B),fun(fun(nat,B),fun(A,B)),Fa),C2)) ) ) ) ) ).

% isCont_powser'
tff(fact_7847_summable__Leibniz_I3_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => ( aa(real,$o,ord_less(real,aa(nat,real,A3,zero_zero(nat))),zero_zero(real))
         => ! [N4: nat] : member(real,suminf(real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3)),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)),one_one(nat)))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4))))) ) ) ) ).

% summable_Leibniz(3)
tff(fact_7848_summable__Leibniz_I2_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(nat,real,A3,zero_zero(nat)))
         => ! [N4: nat] : member(real,suminf(real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3)),set_or1337092689740270186AtMost(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)),one_one(nat)))))) ) ) ) ).

% summable_Leibniz(2)
tff(fact_7849_tendsto__zero__mult__right__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A3: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ayj(A,fun(fun(nat,A),fun(nat,A)),C2),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_right_iff
tff(fact_7850_tendsto__zero__mult__left__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A3: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ayk(A,fun(fun(nat,A),fun(nat,A)),C2),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_mult_left_iff
tff(fact_7851_tendsto__zero__divide__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [C2: A,A3: fun(nat,A)] :
          ( ( C2 != zero_zero(A) )
         => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ayl(A,fun(fun(nat,A),fun(nat,A)),C2),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
          <=> filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ) ).

% tendsto_zero_divide_iff
tff(fact_7852_isCont__rabs,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,Fa: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),aTP_Lamp_aym(fun(A,real),fun(A,real),Fa)) ) ) ).

% isCont_rabs
tff(fact_7853_approx__from__above__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
         => ? [U4: fun(nat,A)] :
              ( ! [N4: nat] : aa(A,$o,ord_less(A,Xb),aa(nat,A,U4,N4))
              & filterlim(nat,A,U4,topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ) ).

% approx_from_above_dense_linorder
tff(fact_7854_approx__from__below__dense__linorder,axiom,
    ! [A: $tType] :
      ( ( dense_linorder(A)
        & topolo3112930676232923870pology(A)
        & topolo1944317154257567458pology(A) )
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,ord_less(A,Y),Xb)
         => ? [U4: fun(nat,A)] :
              ( ! [N4: nat] : aa(A,$o,ord_less(A,aa(nat,A,U4,N4)),Xb)
              & filterlim(nat,A,U4,topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ) ).

% approx_from_below_dense_linorder
tff(fact_7855_summable__LIMSEQ__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A)] :
          ( summable(A,Fa)
         => filterlim(nat,A,Fa,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% summable_LIMSEQ_zero
tff(fact_7856_LIMSEQ__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Fa: fun(nat,A),A3: A,K: nat] :
          ( filterlim(nat,A,Fa,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
         => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ayn(fun(nat,A),fun(nat,fun(nat,A)),Fa),K),topolo7230453075368039082e_nhds(A,A3),at_top(nat)) ) ) ).

% LIMSEQ_ignore_initial_segment
tff(fact_7857_LIMSEQ__offset,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Fa: fun(nat,A),K: nat,A3: A] :
          ( filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ayn(fun(nat,A),fun(nat,fun(nat,A)),Fa),K),topolo7230453075368039082e_nhds(A,A3),at_top(nat))
         => filterlim(nat,A,Fa,topolo7230453075368039082e_nhds(A,A3),at_top(nat)) ) ) ).

% LIMSEQ_offset
tff(fact_7858_continuous__arctan,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F5: filter(A),Fa: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F5,Fa)
         => topolo3448309680560233919inuous(A,real,F5,aTP_Lamp_ayo(fun(A,real),fun(A,real),Fa)) ) ) ).

% continuous_arctan
tff(fact_7859_LIMSEQ__const__iff,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [K: A,L: A] :
          ( filterlim(nat,A,aTP_Lamp_ayp(A,fun(nat,A),K),topolo7230453075368039082e_nhds(A,L),at_top(nat))
        <=> ( K = L ) ) ) ).

% LIMSEQ_const_iff
tff(fact_7860_LIMSEQ__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Fa: fun(nat,A),L: A] :
          ( filterlim(nat,A,Fa,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,aTP_Lamp_ayq(fun(nat,A),fun(nat,A),Fa),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_Suc
tff(fact_7861_LIMSEQ__imp__Suc,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Fa: fun(nat,A),L: A] :
          ( filterlim(nat,A,aTP_Lamp_ayq(fun(nat,A),fun(nat,A),Fa),topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,Fa,topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% LIMSEQ_imp_Suc
tff(fact_7862_continuous__real__sqrt,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F5: filter(A),Fa: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F5,Fa)
         => topolo3448309680560233919inuous(A,real,F5,aTP_Lamp_ayr(fun(A,real),fun(A,real),Fa)) ) ) ).

% continuous_real_sqrt
tff(fact_7863_continuous__arsinh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F5: filter(A),Fa: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F5,Fa)
         => topolo3448309680560233919inuous(A,real,F5,aTP_Lamp_ays(fun(A,real),fun(A,real),Fa)) ) ) ).

% continuous_arsinh
tff(fact_7864_continuous__real__root,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F5: filter(A),Fa: fun(A,real),N: nat] :
          ( topolo3448309680560233919inuous(A,real,F5,Fa)
         => topolo3448309680560233919inuous(A,real,F5,aa(nat,fun(A,real),aTP_Lamp_ayt(fun(A,real),fun(nat,fun(A,real)),Fa),N)) ) ) ).

% continuous_real_root
tff(fact_7865_continuous__rabs,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F5: filter(A),Fa: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F5,Fa)
         => topolo3448309680560233919inuous(A,real,F5,aTP_Lamp_aym(fun(A,real),fun(A,real),Fa)) ) ) ).

% continuous_rabs
tff(fact_7866_seq__offset__neg,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Fa: fun(nat,A),L: A,K: nat] :
          ( filterlim(nat,A,Fa,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ayu(fun(nat,A),fun(nat,fun(nat,A)),Fa),K),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% seq_offset_neg
tff(fact_7867_lim__mono,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [N2: nat,X5: fun(nat,A),Y6: fun(nat,A),Xb: A,Y: A] :
          ( ! [N3: nat] :
              ( aa(nat,$o,ord_less_eq(nat,N2),N3)
             => aa(A,$o,ord_less_eq(A,aa(nat,A,X5,N3)),aa(nat,A,Y6,N3)) )
         => ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
           => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
             => aa(A,$o,ord_less_eq(A,Xb),Y) ) ) ) ) ).

% lim_mono
tff(fact_7868_LIMSEQ__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),Xb: A,Y6: fun(nat,A),Y: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( filterlim(nat,A,Y6,topolo7230453075368039082e_nhds(A,Y),at_top(nat))
           => ( ? [N10: nat] :
                ! [N3: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,N10),N3)
                 => aa(A,$o,ord_less_eq(A,aa(nat,A,X5,N3)),aa(nat,A,Y6,N3)) )
             => aa(A,$o,ord_less_eq(A,Xb),Y) ) ) ) ) ).

% LIMSEQ_le
tff(fact_7869_Lim__bounded,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Fa: fun(nat,A),L: A,M8: nat,C5: A] :
          ( filterlim(nat,A,Fa,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N3: nat] :
                ( aa(nat,$o,ord_less_eq(nat,M8),N3)
               => aa(A,$o,ord_less_eq(A,aa(nat,A,Fa,N3)),C5) )
           => aa(A,$o,ord_less_eq(A,L),C5) ) ) ) ).

% Lim_bounded
tff(fact_7870_Lim__bounded2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Fa: fun(nat,A),L: A,N2: nat,C5: A] :
          ( filterlim(nat,A,Fa,topolo7230453075368039082e_nhds(A,L),at_top(nat))
         => ( ! [N3: nat] :
                ( aa(nat,$o,ord_less_eq(nat,N2),N3)
               => aa(A,$o,ord_less_eq(A,C5),aa(nat,A,Fa,N3)) )
           => aa(A,$o,ord_less_eq(A,C5),L) ) ) ) ).

% Lim_bounded2
tff(fact_7871_LIMSEQ__le__const,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),Xb: A,A3: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( ? [N10: nat] :
              ! [N3: nat] :
                ( aa(nat,$o,ord_less_eq(nat,N10),N3)
               => aa(A,$o,ord_less_eq(A,A3),aa(nat,A,X5,N3)) )
           => aa(A,$o,ord_less_eq(A,A3),Xb) ) ) ) ).

% LIMSEQ_le_const
tff(fact_7872_LIMSEQ__le__const2,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [X5: fun(nat,A),Xb: A,A3: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( ? [N10: nat] :
              ! [N3: nat] :
                ( aa(nat,$o,ord_less_eq(nat,N10),N3)
               => aa(A,$o,ord_less_eq(A,aa(nat,A,X5,N3)),A3) )
           => aa(A,$o,ord_less_eq(A,Xb),A3) ) ) ) ).

% LIMSEQ_le_const2
tff(fact_7873_continuous__at__within__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,S: set(A),Fa: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,S),Fa)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,S),G)
           => ( ( aa(A,real,Fa,A3) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,S),aa(fun(A,real),fun(A,real),aTP_Lamp_ayv(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G)) ) ) ) ) ).

% continuous_at_within_powr
tff(fact_7874_continuous__within__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xb: A,S: set(A),Fa: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Xb,S),Fa)
         => ( ( aa(A,real,Fa,Xb) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Xb,S),aTP_Lamp_ayw(fun(A,real),fun(A,real),Fa)) ) ) ) ).

% continuous_within_ln
tff(fact_7875_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => filterlim(nat,nat,aa(nat,fun(nat,nat),times_times(nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_left_at_top
tff(fact_7876_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => filterlim(nat,nat,aTP_Lamp_ayx(nat,fun(nat,nat),C2),at_top(nat),at_top(nat)) ) ).

% mult_nat_right_at_top
tff(fact_7877_LIMSEQ__lessThan__iff__atMost,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Fa: fun(set(nat),A),Xb: A] :
          ( filterlim(nat,A,aTP_Lamp_ayy(fun(set(nat),A),fun(nat,A),Fa),topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
        <=> filterlim(nat,A,aTP_Lamp_ayz(fun(set(nat),A),fun(nat,A),Fa),topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ).

% LIMSEQ_lessThan_iff_atMost
tff(fact_7878_LIMSEQ__root,axiom,
    filterlim(nat,real,aTP_Lamp_aza(nat,real),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ).

% LIMSEQ_root
tff(fact_7879_isCont__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,Fa: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => ( ( aa(A,real,Fa,A3) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_ayv(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G)) ) ) ) ) ).

% isCont_powr
tff(fact_7880_isCont__ln_H,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xb: A,Fa: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Xb,top_top(set(A))),Fa)
         => ( ( aa(A,real,Fa,Xb) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,Xb,top_top(set(A))),aTP_Lamp_ayw(fun(A,real),fun(A,real),Fa)) ) ) ) ).

% isCont_ln'
tff(fact_7881_monoseq__le,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [A3: fun(nat,A),Xb: A] :
          ( topological_monoseq(A,A3)
         => ( filterlim(nat,A,A3,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
           => ( ( ! [N4: nat] : aa(A,$o,ord_less_eq(A,aa(nat,A,A3,N4)),Xb)
                & ! [M2: nat,N4: nat] :
                    ( aa(nat,$o,ord_less_eq(nat,M2),N4)
                   => aa(A,$o,ord_less_eq(A,aa(nat,A,A3,M2)),aa(nat,A,A3,N4)) ) )
              | ( ! [N4: nat] : aa(A,$o,ord_less_eq(A,Xb),aa(nat,A,A3,N4))
                & ! [M2: nat,N4: nat] :
                    ( aa(nat,$o,ord_less_eq(nat,M2),N4)
                   => aa(A,$o,ord_less_eq(A,aa(nat,A,A3,N4)),aa(nat,A,A3,M2)) ) ) ) ) ) ) ).

% monoseq_le
tff(fact_7882_lim__const__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A3: A] : filterlim(nat,A,aTP_Lamp_azb(A,fun(nat,A),A3),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_const_over_n
tff(fact_7883_LIMSEQ__linear,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [X5: fun(nat,A),Xb: A,L: nat] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,Xb),at_top(nat))
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),L)
           => filterlim(nat,A,aa(nat,fun(nat,A),aTP_Lamp_azc(fun(nat,A),fun(nat,fun(nat,A)),X5),L),topolo7230453075368039082e_nhds(A,Xb),at_top(nat)) ) ) ) ).

% LIMSEQ_linear
tff(fact_7884_lim__inverse__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_azd(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_inverse_n
tff(fact_7885_LIMSEQ__SEQ__conv2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,Fa: fun(A,B),L: B] :
          ( ! [S5: fun(nat,A)] :
              ( ( ! [N4: nat] : aa(nat,A,S5,N4) != A3
                & filterlim(nat,A,S5,topolo7230453075368039082e_nhds(A,A3),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aze(fun(A,B),fun(fun(nat,A),fun(nat,B)),Fa),S5),topolo7230453075368039082e_nhds(B,L),at_top(nat)) )
         => filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIMSEQ_SEQ_conv2
tff(fact_7886_LIMSEQ__SEQ__conv1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Fa: fun(A,B),L: B,A3: A] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ! [S9: fun(nat,A)] :
              ( ( ! [N3: nat] : aa(nat,A,S9,N3) != A3
                & filterlim(nat,A,S9,topolo7230453075368039082e_nhds(A,A3),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_azf(fun(A,B),fun(fun(nat,A),fun(nat,B)),Fa),S9),topolo7230453075368039082e_nhds(B,L),at_top(nat)) ) ) ) ).

% LIMSEQ_SEQ_conv1
tff(fact_7887_LIMSEQ__SEQ__conv,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [A3: A,X5: fun(A,B),L5: B] :
          ( ! [S10: fun(nat,A)] :
              ( ( ! [N8: nat] : aa(nat,A,S10,N8) != A3
                & filterlim(nat,A,S10,topolo7230453075368039082e_nhds(A,A3),at_top(nat)) )
             => filterlim(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aze(fun(A,B),fun(fun(nat,A),fun(nat,B)),X5),S10),topolo7230453075368039082e_nhds(B,L5),at_top(nat)) )
        <=> filterlim(A,B,X5,topolo7230453075368039082e_nhds(B,L5),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% LIMSEQ_SEQ_conv
tff(fact_7888_telescope__summable,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),C2: A] :
          ( filterlim(nat,A,Fa,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => summable(A,aTP_Lamp_azg(fun(nat,A),fun(nat,A),Fa)) ) ) ).

% telescope_summable
tff(fact_7889_telescope__summable_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),C2: A] :
          ( filterlim(nat,A,Fa,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => summable(A,aTP_Lamp_azh(fun(nat,A),fun(nat,A),Fa)) ) ) ).

% telescope_summable'
tff(fact_7890_nested__sequence__unique,axiom,
    ! [Fa: fun(nat,real),G: fun(nat,real)] :
      ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,Fa,N3)),aa(nat,real,Fa,aa(nat,nat,suc,N3)))
     => ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,G,aa(nat,nat,suc,N3))),aa(nat,real,G,N3))
       => ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,Fa,N3)),aa(nat,real,G,N3))
         => ( filterlim(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_azi(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Fa),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => ? [L2: real] :
                ( ! [N4: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,Fa,N4)),L2)
                & filterlim(nat,real,Fa,topolo7230453075368039082e_nhds(real,L2),at_top(nat))
                & ! [N4: nat] : aa(real,$o,ord_less_eq(real,L2),aa(nat,real,G,N4))
                & filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ) ).

% nested_sequence_unique
tff(fact_7891_LIMSEQ__inverse__zero,axiom,
    ! [X5: fun(nat,real)] :
      ( ! [R4: real] :
        ? [N10: nat] :
        ! [N3: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N10),N3)
         => aa(real,$o,ord_less(real,R4),aa(nat,real,X5,N3)) )
     => filterlim(nat,real,aTP_Lamp_azj(fun(nat,real),fun(nat,real),X5),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_zero
tff(fact_7892_lim__inverse__n_H,axiom,
    filterlim(nat,real,aTP_Lamp_azk(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% lim_inverse_n'
tff(fact_7893_LIMSEQ__inverse__real__of__nat,axiom,
    filterlim(nat,real,aTP_Lamp_azl(nat,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat
tff(fact_7894_LIMSEQ__root__const,axiom,
    ! [C2: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
     => filterlim(nat,real,aTP_Lamp_azm(real,fun(nat,real),C2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat)) ) ).

% LIMSEQ_root_const
tff(fact_7895_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_azn(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add
tff(fact_7896_sums__def,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Fa: fun(nat,A),S: A] :
          ( aa(A,$o,sums(A,Fa),S)
        <=> filterlim(nat,A,aTP_Lamp_azo(fun(nat,A),fun(nat,A),Fa),topolo7230453075368039082e_nhds(A,S),at_top(nat)) ) ) ).

% sums_def
tff(fact_7897_sums__def__le,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Fa: fun(nat,A),S: A] :
          ( aa(A,$o,sums(A,Fa),S)
        <=> filterlim(nat,A,aTP_Lamp_azp(fun(nat,A),fun(nat,A),Fa),topolo7230453075368039082e_nhds(A,S),at_top(nat)) ) ) ).

% sums_def_le
tff(fact_7898_continuous__at__within__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,S: set(A),Fa: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,S),Fa)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,S),G)
           => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,Fa,A3))
             => ( ( aa(A,real,Fa,A3) != one_one(real) )
               => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,A3))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,S),aa(fun(A,real),fun(A,real),aTP_Lamp_azq(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G)) ) ) ) ) ) ) ).

% continuous_at_within_log
tff(fact_7899_increasing__LIMSEQ,axiom,
    ! [Fa: fun(nat,real),L: real] :
      ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,Fa,N3)),aa(nat,real,Fa,aa(nat,nat,suc,N3)))
     => ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,Fa,N3)),L)
       => ( ! [E2: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E2)
             => ? [N4: nat] : aa(real,$o,ord_less_eq(real,L),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,Fa,N4)),E2)) )
         => filterlim(nat,real,Fa,topolo7230453075368039082e_nhds(real,L),at_top(nat)) ) ) ) ).

% increasing_LIMSEQ
tff(fact_7900_lim__1__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_azr(nat,A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ).

% lim_1_over_n
tff(fact_7901_LIMSEQ__n__over__Suc__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_azs(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_n_over_Suc_n
tff(fact_7902_LIMSEQ__Suc__n__over__n,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => filterlim(nat,A,aTP_Lamp_azt(nat,A),topolo7230453075368039082e_nhds(A,one_one(A)),at_top(nat)) ) ).

% LIMSEQ_Suc_n_over_n
tff(fact_7903_telescope__sums_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),C2: A] :
          ( filterlim(nat,A,Fa,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => aa(A,$o,sums(A,aTP_Lamp_azh(fun(nat,A),fun(nat,A),Fa)),aa(A,A,minus_minus(A,aa(nat,A,Fa,zero_zero(nat))),C2)) ) ) ).

% telescope_sums'
tff(fact_7904_telescope__sums,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),C2: A] :
          ( filterlim(nat,A,Fa,topolo7230453075368039082e_nhds(A,C2),at_top(nat))
         => aa(A,$o,sums(A,aTP_Lamp_azg(fun(nat,A),fun(nat,A),Fa)),aa(A,A,minus_minus(A,C2),aa(nat,A,Fa,zero_zero(nat)))) ) ) ).

% telescope_sums
tff(fact_7905_LIMSEQ__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),one_one(real))
       => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ) ).

% LIMSEQ_realpow_zero
tff(fact_7906_LIMSEQ__divide__realpow__zero,axiom,
    ! [Xb: real,A3: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xb)
     => filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_azu(real,fun(real,fun(nat,real)),Xb),A3),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_divide_realpow_zero
tff(fact_7907_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C2: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,C2)),one_one(real))
     => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),C2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero2
tff(fact_7908_LIMSEQ__abs__realpow__zero,axiom,
    ! [C2: real] :
      ( aa(real,$o,ord_less(real,abs_abs(real,C2)),one_one(real))
     => filterlim(nat,real,aa(real,fun(nat,real),power_power(real),abs_abs(real,C2)),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_abs_realpow_zero
tff(fact_7909_LIMSEQ__inverse__realpow__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xb)
     => filterlim(nat,real,aTP_Lamp_azv(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% LIMSEQ_inverse_realpow_zero
tff(fact_7910_sums__def_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Fa: fun(nat,A),S: A] :
          ( aa(A,$o,sums(A,Fa),S)
        <=> filterlim(nat,A,aTP_Lamp_azw(fun(nat,A),fun(nat,A),Fa),topolo7230453075368039082e_nhds(A,S),at_top(nat)) ) ) ).

% sums_def'
tff(fact_7911_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_azx(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus
tff(fact_7912_root__test__convergence,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Fa: fun(nat,A),Xb: real] :
          ( filterlim(nat,real,aTP_Lamp_azy(fun(nat,A),fun(nat,real),Fa),topolo7230453075368039082e_nhds(real,Xb),at_top(nat))
         => ( aa(real,$o,ord_less(real,Xb),one_one(real))
           => summable(A,Fa) ) ) ) ).

% root_test_convergence
tff(fact_7913_summable__LIMSEQ,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Fa: fun(nat,A)] :
          ( summable(A,Fa)
         => filterlim(nat,A,aTP_Lamp_azz(fun(nat,A),fun(nat,A),Fa),topolo7230453075368039082e_nhds(A,suminf(A,Fa)),at_top(nat)) ) ) ).

% summable_LIMSEQ
tff(fact_7914_summable__LIMSEQ_H,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Fa: fun(nat,A)] :
          ( summable(A,Fa)
         => filterlim(nat,A,aTP_Lamp_baa(fun(nat,A),fun(nat,A),Fa),topolo7230453075368039082e_nhds(A,suminf(A,Fa)),at_top(nat)) ) ) ).

% summable_LIMSEQ'
tff(fact_7915_isCont__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A,Fa: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),Fa)
         => ( topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),G)
           => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,Fa,A3))
             => ( ( aa(A,real,Fa,A3) != one_one(real) )
               => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,A3))
                 => topolo3448309680560233919inuous(A,real,topolo174197925503356063within(A,A3,top_top(set(A))),aa(fun(A,real),fun(A,real),aTP_Lamp_azq(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G)) ) ) ) ) ) ) ).

% isCont_log
tff(fact_7916_LIMSEQ__D,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A,R2: real] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
           => ? [No: nat] :
              ! [N4: nat] :
                ( aa(nat,$o,ord_less_eq(nat,No),N4)
               => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,N4)),L5))),R2) ) ) ) ) ).

% LIMSEQ_D
tff(fact_7917_LIMSEQ__I,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( ! [R4: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R4)
             => ? [No2: nat] :
                ! [N3: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,No2),N3)
                 => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,N3)),L5))),R4) ) )
         => filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat)) ) ) ).

% LIMSEQ_I
tff(fact_7918_LIMSEQ__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),L5: A] :
          ( filterlim(nat,A,X5,topolo7230453075368039082e_nhds(A,L5),at_top(nat))
        <=> ! [R6: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),R6)
             => ? [No3: nat] :
                ! [N8: nat] :
                  ( aa(nat,$o,ord_less_eq(nat,No3),N8)
                 => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,N8)),L5))),R6) ) ) ) ) ).

% LIMSEQ_iff
tff(fact_7919_LIMSEQ__power__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xb: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xb)),one_one(real))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_power_zero
tff(fact_7920_tendsto__exp__limit__sequentially,axiom,
    ! [Xb: real] : filterlim(nat,real,aTP_Lamp_bab(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),Xb)),at_top(nat)) ).

% tendsto_exp_limit_sequentially
tff(fact_7921_tendsto__power__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Fa: fun(A,nat),F5: filter(A),Xb: B] :
          ( filterlim(A,nat,Fa,at_top(nat),F5)
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,Xb)),one_one(real))
           => filterlim(A,B,aa(B,fun(A,B),aTP_Lamp_bac(fun(A,nat),fun(B,fun(A,B)),Fa),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5) ) ) ) ).

% tendsto_power_zero
tff(fact_7922_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] : filterlim(nat,real,aTP_Lamp_bad(real,fun(nat,real),R2),topolo7230453075368039082e_nhds(real,R2),at_top(nat)) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
tff(fact_7923_LIMSEQ__norm__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A)] :
          ( ! [N3: nat] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(nat,A,Fa,N3))),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N3))))
         => filterlim(nat,A,Fa,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% LIMSEQ_norm_0
tff(fact_7924_summable__Leibniz_I1_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => summable(real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3)) ) ) ).

% summable_Leibniz(1)
tff(fact_7925_field__derivative__lim__unique,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),Df: A,Z: A,S: fun(nat,A),A3: A] :
          ( has_field_derivative(A,Fa,Df,topolo174197925503356063within(A,Z,top_top(set(A))))
         => ( filterlim(nat,A,S,topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat))
           => ( ! [N3: nat] : aa(nat,A,S,N3) != zero_zero(A)
             => ( filterlim(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_bae(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),Fa),Z),S),topolo7230453075368039082e_nhds(A,A3),at_top(nat))
               => ( Df = A3 ) ) ) ) ) ) ).

% field_derivative_lim_unique
tff(fact_7926_powser__times__n__limit__0,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Xb: A] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xb)),one_one(real))
         => filterlim(nat,A,aTP_Lamp_baf(A,fun(nat,A),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% powser_times_n_limit_0
tff(fact_7927_lim__n__over__pown,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( aa(real,$o,ord_less(real,one_one(real)),real_V7770717601297561774m_norm(A,Xb))
         => filterlim(nat,A,aTP_Lamp_bag(A,fun(nat,A),Xb),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_top(nat)) ) ) ).

% lim_n_over_pown
tff(fact_7928_summable,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A3,N3))
       => ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A3,aa(nat,nat,suc,N3))),aa(nat,real,A3,N3))
         => summable(real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3)) ) ) ) ).

% summable
tff(fact_7929_cos__diff__limit__1,axiom,
    ! [Theta: fun(nat,real),Theta2: real] :
      ( filterlim(nat,real,aa(real,fun(nat,real),aTP_Lamp_bah(fun(nat,real),fun(real,fun(nat,real)),Theta),Theta2),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ~ ! [K2: fun(nat,int)] : ~ filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_bai(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K2),topolo7230453075368039082e_nhds(real,Theta2),at_top(nat)) ) ).

% cos_diff_limit_1
tff(fact_7930_cos__limit__1,axiom,
    ! [Theta: fun(nat,real)] :
      ( filterlim(nat,real,aTP_Lamp_baj(fun(nat,real),fun(nat,real),Theta),topolo7230453075368039082e_nhds(real,one_one(real)),at_top(nat))
     => ? [K2: fun(nat,int)] : filterlim(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_bai(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Theta),K2),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% cos_limit_1
tff(fact_7931_summable__Leibniz_I4_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => filterlim(nat,real,aTP_Lamp_bak(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ).

% summable_Leibniz(4)
tff(fact_7932_zeroseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less_eq(real,abs_abs(real,Xb)),one_one(real))
     => filterlim(nat,real,aTP_Lamp_mj(real,fun(nat,real),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat)) ) ).

% zeroseq_arctan_series
tff(fact_7933_summable__Leibniz_H_I2_J,axiom,
    ! [A3: fun(nat,real),N: nat] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A3,N3))
       => ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A3,aa(nat,nat,suc,N3))),aa(nat,real,A3,N3))
         => aa(real,$o,ord_less_eq(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)))),suminf(real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3))) ) ) ) ).

% summable_Leibniz'(2)
tff(fact_7934_summable__Leibniz_H_I3_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A3,N3))
       => ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A3,aa(nat,nat,suc,N3))),aa(nat,real,A3,N3))
         => filterlim(nat,real,aTP_Lamp_bak(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(3)
tff(fact_7935_sums__alternating__upper__lower,axiom,
    ! [A3: fun(nat,real)] :
      ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A3,aa(nat,nat,suc,N3))),aa(nat,real,A3,N3))
     => ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A3,N3))
       => ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
         => ? [L2: real] :
              ( ! [N4: nat] : aa(real,$o,ord_less_eq(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)))),L2)
              & filterlim(nat,real,aTP_Lamp_bak(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,L2),at_top(nat))
              & ! [N4: nat] : aa(real,$o,ord_less_eq(real,L2),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N4)),one_one(nat)))))
              & filterlim(nat,real,aTP_Lamp_bal(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,L2),at_top(nat)) ) ) ) ) ).

% sums_alternating_upper_lower
tff(fact_7936_summable__Leibniz_I5_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( topological_monoseq(real,A3)
       => filterlim(nat,real,aTP_Lamp_bal(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ).

% summable_Leibniz(5)
tff(fact_7937_summable__Leibniz_H_I5_J,axiom,
    ! [A3: fun(nat,real)] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A3,N3))
       => ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A3,aa(nat,nat,suc,N3))),aa(nat,real,A3,N3))
         => filterlim(nat,real,aTP_Lamp_bal(fun(nat,real),fun(nat,real),A3),topolo7230453075368039082e_nhds(real,suminf(real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3))),at_top(nat)) ) ) ) ).

% summable_Leibniz'(5)
tff(fact_7938_summable__Leibniz_H_I4_J,axiom,
    ! [A3: fun(nat,real),N: nat] :
      ( filterlim(nat,real,A3,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
     => ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(nat,real,A3,N3))
       => ( ! [N3: nat] : aa(real,$o,ord_less_eq(real,aa(nat,real,A3,aa(nat,nat,suc,N3))),aa(nat,real,A3,N3))
         => aa(real,$o,ord_less_eq(real,suminf(real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),A3)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N)),one_one(nat))))) ) ) ) ).

% summable_Leibniz'(4)
tff(fact_7939_filterlim__sequentially__Suc,axiom,
    ! [A: $tType,Fa: fun(nat,A),F5: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_ajk(fun(nat,A),fun(nat,A),Fa),F5,at_top(nat))
    <=> filterlim(nat,A,Fa,F5,at_top(nat)) ) ).

% filterlim_sequentially_Suc
tff(fact_7940_has__derivative__at2,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),F3: fun(A,B),Xb: A] :
          ( has_derivative(A,B,Fa,F3,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F3)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_bam(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Fa),F3),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% has_derivative_at2
tff(fact_7941_bounded__linear_Otendsto__zero,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),G: fun(C,A),F5: filter(C)] :
          ( real_V3181309239436604168linear(A,B,Fa)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,zero_zero(A)),F5)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ban(fun(A,B),fun(fun(C,A),fun(C,B)),Fa),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5) ) ) ) ).

% bounded_linear.tendsto_zero
tff(fact_7942_bounded__linear_Ocontinuous,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),F5: filter(C),G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,Fa)
         => ( topolo3448309680560233919inuous(C,A,F5,G)
           => topolo3448309680560233919inuous(C,B,F5,aa(fun(C,A),fun(C,B),aTP_Lamp_bao(fun(A,B),fun(fun(C,A),fun(C,B)),Fa),G)) ) ) ) ).

% bounded_linear.continuous
tff(fact_7943_bounded__linear_Otendsto,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Fa: fun(A,B),G: fun(C,A),A3: A,F5: filter(C)] :
          ( real_V3181309239436604168linear(A,B,Fa)
         => ( filterlim(C,A,G,topolo7230453075368039082e_nhds(A,A3),F5)
           => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ban(fun(A,B),fun(fun(C,A),fun(C,B)),Fa),G),topolo7230453075368039082e_nhds(B,aa(A,B,Fa,A3)),F5) ) ) ) ).

% bounded_linear.tendsto
tff(fact_7944_bounded__linear_OisCont,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),A3: C,G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,Fa)
         => ( topolo3448309680560233919inuous(C,A,topolo174197925503356063within(C,A3,top_top(set(C))),G)
           => topolo3448309680560233919inuous(C,B,topolo174197925503356063within(C,A3,top_top(set(C))),aa(fun(C,A),fun(C,B),aTP_Lamp_bao(fun(A,B),fun(fun(C,A),fun(C,B)),Fa),G)) ) ) ) ).

% bounded_linear.isCont
tff(fact_7945_bounded__linear__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_bap(A,fun(A,A),Y)) ) ).

% bounded_linear_divide
tff(fact_7946_bounded__linear__minus,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,Fa)
         => real_V3181309239436604168linear(A,B,aTP_Lamp_arx(fun(A,B),fun(A,B),Fa)) ) ) ).

% bounded_linear_minus
tff(fact_7947_bounded__linear__zero,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => real_V3181309239436604168linear(A,B,aTP_Lamp_are(A,B)) ) ).

% bounded_linear_zero
tff(fact_7948_bounded__linear_OCauchy,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),X5: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,Fa)
         => ( topolo3814608138187158403Cauchy(A,X5)
           => topolo3814608138187158403Cauchy(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_baq(fun(A,B),fun(fun(nat,A),fun(nat,B)),Fa),X5)) ) ) ) ).

% bounded_linear.Cauchy
tff(fact_7949_bounded__linear_Osuminf,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),X5: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,Fa)
         => ( summable(A,X5)
           => ( aa(A,B,Fa,suminf(A,X5)) = suminf(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_baq(fun(A,B),fun(fun(nat,A),fun(nat,B)),Fa),X5)) ) ) ) ) ).

% bounded_linear.suminf
tff(fact_7950_bounded__linear__add,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),G: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,Fa)
         => ( real_V3181309239436604168linear(A,B,G)
           => real_V3181309239436604168linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_arv(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G)) ) ) ) ).

% bounded_linear_add
tff(fact_7951_bounded__linear__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => real_V3181309239436604168linear(real,A,real_Vector_of_real(A)) ) ).

% bounded_linear_of_real
tff(fact_7952_bounded__linear_Osummable,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),X5: fun(nat,A)] :
          ( real_V3181309239436604168linear(A,B,Fa)
         => ( summable(A,X5)
           => summable(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_baq(fun(A,B),fun(fun(nat,A),fun(nat,B)),Fa),X5)) ) ) ) ).

% bounded_linear.summable
tff(fact_7953_bounded__linear__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),G: fun(C,A)] :
          ( real_V3181309239436604168linear(A,B,Fa)
         => ( real_V3181309239436604168linear(C,A,G)
           => real_V3181309239436604168linear(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ark(fun(A,B),fun(fun(C,A),fun(C,B)),Fa),G)) ) ) ) ).

% bounded_linear_compose
tff(fact_7954_bounded__linear__ident,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => real_V3181309239436604168linear(A,A,aTP_Lamp_aru(A,A)) ) ).

% bounded_linear_ident
tff(fact_7955_bounded__linear_Osums,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),X5: fun(nat,A),A3: A] :
          ( real_V3181309239436604168linear(A,B,Fa)
         => ( aa(A,$o,sums(A,X5),A3)
           => aa(B,$o,sums(B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_baq(fun(A,B),fun(fun(nat,A),fun(nat,B)),Fa),X5)),aa(A,B,Fa,A3)) ) ) ) ).

% bounded_linear.sums
tff(fact_7956_bounded__linear__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] : real_V3181309239436604168linear(real,A,aTP_Lamp_bar(A,fun(real,A),Xb)) ) ).

% bounded_linear_scaleR_left
tff(fact_7957_bounded__linear__const__scaleR,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),R2: real] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(real,fun(A,B),aTP_Lamp_arr(fun(A,B),fun(real,fun(A,B)),G),R2)) ) ) ).

% bounded_linear_const_scaleR
tff(fact_7958_bounded__linear__scaleR__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,real),Xb: B] :
          ( real_V3181309239436604168linear(A,real,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_ars(fun(A,real),fun(B,fun(A,B)),G),Xb)) ) ) ).

% bounded_linear_scaleR_const
tff(fact_7959_bounded__linear__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R2: real] : real_V3181309239436604168linear(A,A,real_V8093663219630862766scaleR(A,R2)) ) ).

% bounded_linear_scaleR_right
tff(fact_7960_bounded__linear__sum,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [I5: set(A),Fa: fun(A,fun(B,C))] :
          ( ! [I2: A] :
              ( member(A,I2,I5)
             => real_V3181309239436604168linear(B,C,aa(A,fun(B,C),Fa,I2)) )
         => real_V3181309239436604168linear(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_arq(set(A),fun(fun(A,fun(B,C)),fun(B,C)),I5),Fa)) ) ) ).

% bounded_linear_sum
tff(fact_7961_bounded__linear__mult__right,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xb: A] : real_V3181309239436604168linear(A,A,aa(A,fun(A,A),times_times(A),Xb)) ) ).

% bounded_linear_mult_right
tff(fact_7962_bounded__linear__mult__const,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),Y: B] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_aro(fun(A,B),fun(B,fun(A,B)),G),Y)) ) ) ).

% bounded_linear_mult_const
tff(fact_7963_bounded__linear__const__mult,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [G: fun(A,B),Xb: B] :
          ( real_V3181309239436604168linear(A,B,G)
         => real_V3181309239436604168linear(A,B,aa(B,fun(A,B),aTP_Lamp_arn(fun(A,B),fun(B,fun(A,B)),G),Xb)) ) ) ).

% bounded_linear_const_mult
tff(fact_7964_bounded__linear__mult__left,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Y: A] : real_V3181309239436604168linear(A,A,aTP_Lamp_bas(A,fun(A,A),Y)) ) ).

% bounded_linear_mult_left
tff(fact_7965_bounded__linear__sub,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),G: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,Fa)
         => ( real_V3181309239436604168linear(A,B,G)
           => real_V3181309239436604168linear(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_arm(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G)) ) ) ) ).

% bounded_linear_sub
tff(fact_7966_bounded__linear_Ohas__derivative,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Fa: fun(A,B),G: fun(C,A),G5: fun(C,A),F5: filter(C)] :
          ( real_V3181309239436604168linear(A,B,Fa)
         => ( has_derivative(C,A,G,G5,F5)
           => has_derivative(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ark(fun(A,B),fun(fun(C,A),fun(C,B)),Fa),G),aa(fun(C,A),fun(C,B),aTP_Lamp_ark(fun(A,B),fun(fun(C,A),fun(C,B)),Fa),G5),F5) ) ) ) ).

% bounded_linear.has_derivative
tff(fact_7967_bounded__linear_Ononneg__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,Fa)
         => ? [K7: real] :
              ( aa(real,$o,ord_less_eq(real,zero_zero(real)),K7)
              & ! [X3: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,Fa,X3))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X3)),K7)) ) ) ) ).

% bounded_linear.nonneg_bounded
tff(fact_7968_filterlim__ident,axiom,
    ! [A: $tType,F5: filter(A)] : filterlim(A,A,aTP_Lamp_ct(A,A),F5,F5) ).

% filterlim_ident
tff(fact_7969_filterlim__compose,axiom,
    ! [B: $tType,A: $tType,C: $tType,G: fun(A,B),F33: filter(B),F23: filter(A),Fa: fun(C,A),F13: filter(C)] :
      ( filterlim(A,B,G,F33,F23)
     => ( filterlim(C,A,Fa,F23,F13)
       => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_bat(fun(A,B),fun(fun(C,A),fun(C,B)),G),Fa),F33,F13) ) ) ).

% filterlim_compose
tff(fact_7970_bounded__linear_Opos__bounded,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B)] :
          ( real_V3181309239436604168linear(A,B,Fa)
         => ? [K7: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K7)
              & ! [X3: A] : aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(B,aa(A,B,Fa,X3))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,X3)),K7)) ) ) ) ).

% bounded_linear.pos_bounded
tff(fact_7971_has__derivative__iff__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),F3: fun(A,B),Xb: A,S: set(A)] :
          ( has_derivative(A,B,Fa,F3,topolo174197925503356063within(A,Xb,S))
        <=> ( real_V3181309239436604168linear(A,B,F3)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_bau(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Fa),F3),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_iff_norm
tff(fact_7972_has__derivativeI,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [F3: fun(A,B),Xb: A,Fa: fun(A,B),S: set(A)] :
          ( real_V3181309239436604168linear(A,B,F3)
         => ( filterlim(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_bav(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),F3),Xb),Fa),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S))
           => has_derivative(A,B,Fa,F3,topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivativeI
tff(fact_7973_has__derivative__at__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),F3: fun(A,B),Xb: A,S: set(A)] :
          ( has_derivative(A,B,Fa,F3,topolo174197925503356063within(A,Xb,S))
        <=> ( real_V3181309239436604168linear(A,B,F3)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_baw(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Fa),F3),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_at_within
tff(fact_7974_has__derivative__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),F3: fun(A,B),Xb: A] :
          ( has_derivative(A,B,Fa,F3,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,F3)
            & ? [E5: fun(A,B)] :
                ( ! [H6: A] : aa(A,B,Fa,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),H6)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Fa,Xb)),aa(A,B,F3,H6))),aa(A,B,E5,H6))
                & filterlim(A,real,aTP_Lamp_bax(fun(A,B),fun(A,real),E5),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ).

% has_derivative_iff_Ex
tff(fact_7975_has__derivative__within,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),F3: fun(A,B),Xb: A,S: set(A)] :
          ( has_derivative(A,B,Fa,F3,topolo174197925503356063within(A,Xb,S))
        <=> ( real_V3181309239436604168linear(A,B,F3)
            & filterlim(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_bam(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Fa),F3),Xb),topolo7230453075368039082e_nhds(B,zero_zero(B)),topolo174197925503356063within(A,Xb,S)) ) ) ) ).

% has_derivative_within
tff(fact_7976_has__derivative__at,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),D5: fun(A,B),Xb: A] :
          ( has_derivative(A,B,Fa,D5,topolo174197925503356063within(A,Xb,top_top(set(A))))
        <=> ( real_V3181309239436604168linear(A,B,D5)
            & filterlim(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_bay(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Fa),D5),Xb),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ).

% has_derivative_at
tff(fact_7977_has__derivative__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),F3: fun(A,B),F5: filter(A)] :
          ( has_derivative(A,B,Fa,F3,F5)
        <=> ( real_V3181309239436604168linear(A,B,F3)
            & filterlim(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_baz(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),Fa),F3),F5),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5) ) ) ) ).

% has_derivative_def
tff(fact_7978_summable__Cauchy_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Fa: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_bba(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Fa),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,Fa) ) ) ) ).

% summable_Cauchy'
tff(fact_7979_lim__const,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [A3: A] : topolo3827282254853284352ce_Lim(nat,A,at_top(nat),aTP_Lamp_ayp(A,fun(nat,A),A3)) = A3 ) ).

% lim_const
tff(fact_7980_eventually__const,axiom,
    ! [A: $tType,F5: filter(A),P: $o] :
      ( ( F5 != bot_bot(filter(A)) )
     => ( eventually(A,aTP_Lamp_afb($o,fun(A,$o),(P)),F5)
      <=> (P) ) ) ).

% eventually_const
tff(fact_7981_eventually__sequentially__Suc,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,aTP_Lamp_anh(fun(nat,$o),fun(nat,$o),P),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_Suc
tff(fact_7982_eventually__sequentially__seg,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_bbb(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat))
    <=> eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentially_seg
tff(fact_7983_sequentially__imp__eventually__at,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [A3: A,P: fun(A,$o)] :
          ( ! [F4: fun(nat,A)] :
              ( ( ! [N4: nat] : aa(nat,A,F4,N4) != A3
                & filterlim(nat,A,F4,topolo7230453075368039082e_nhds(A,A3),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_bbc(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F4),at_top(nat)) )
         => eventually(A,P,topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ).

% sequentially_imp_eventually_at
tff(fact_7984_filterlim__at__within__not__equal,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t2_space(B)
     => ! [Fa: fun(A,B),A3: B,S: set(B),F5: filter(A),B3: B] :
          ( filterlim(A,B,Fa,topolo174197925503356063within(B,A3,S),F5)
         => eventually(A,aa(B,fun(A,$o),aa(set(B),fun(B,fun(A,$o)),aTP_Lamp_bbd(fun(A,B),fun(set(B),fun(B,fun(A,$o))),Fa),S),B3),F5) ) ) ).

% filterlim_at_within_not_equal
tff(fact_7985_sequentially__imp__eventually__within,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [S: set(A),A3: A,P: fun(A,$o)] :
          ( ! [F4: fun(nat,A)] :
              ( ( ! [N4: nat] :
                    ( member(A,aa(nat,A,F4,N4),S)
                    & ( aa(nat,A,F4,N4) != A3 ) )
                & filterlim(nat,A,F4,topolo7230453075368039082e_nhds(A,A3),at_top(nat)) )
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_bbc(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F4),at_top(nat)) )
         => eventually(A,P,topolo174197925503356063within(A,A3,S)) ) ) ).

% sequentially_imp_eventually_within
tff(fact_7986_filterlim__at,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Fa: fun(A,B),B3: B,S: set(B),F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo174197925503356063within(B,B3,S),F5)
        <=> ( eventually(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_bbe(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Fa),B3),S),F5)
            & filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,B3),F5) ) ) ) ).

% filterlim_at
tff(fact_7987_filterlim__at__top__dense,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Fa: fun(A,B),F5: filter(A)] :
          ( filterlim(A,B,Fa,at_top(B),F5)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_bbf(fun(A,B),fun(B,fun(A,$o)),Fa),Z7),F5) ) ) ).

% filterlim_at_top_dense
tff(fact_7988_order__tendsto__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Fa: fun(A,B),Xb: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,Xb),F5)
        <=> ( ! [L4: B] :
                ( aa(B,$o,ord_less(B,L4),Xb)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_bbg(fun(A,B),fun(B,fun(A,$o)),Fa),L4),F5) )
            & ! [U5: B] :
                ( aa(B,$o,ord_less(B,Xb),U5)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_bbh(fun(A,B),fun(B,fun(A,$o)),Fa),U5),F5) ) ) ) ) ).

% order_tendsto_iff
tff(fact_7989_order__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Y: A,Fa: fun(B,A),F5: filter(B)] :
          ( ! [A2: A] :
              ( aa(A,$o,ord_less(A,A2),Y)
             => eventually(B,aa(A,fun(B,$o),aTP_Lamp_bbi(fun(B,A),fun(A,fun(B,$o)),Fa),A2),F5) )
         => ( ! [A2: A] :
                ( aa(A,$o,ord_less(A,Y),A2)
               => eventually(B,aa(A,fun(B,$o),aTP_Lamp_bbj(fun(B,A),fun(A,fun(B,$o)),Fa),A2),F5) )
           => filterlim(B,A,Fa,topolo7230453075368039082e_nhds(A,Y),F5) ) ) ) ).

% order_tendstoI
tff(fact_7990_order__tendstoD_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Fa: fun(A,B),Y: B,F5: filter(A),A3: B] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,Y),F5)
         => ( aa(B,$o,ord_less(B,A3),Y)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_bbg(fun(A,B),fun(B,fun(A,$o)),Fa),A3),F5) ) ) ) ).

% order_tendstoD(1)
tff(fact_7991_order__tendstoD_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Fa: fun(A,B),Y: B,F5: filter(A),A3: B] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,Y),F5)
         => ( aa(B,$o,ord_less(B,Y),A3)
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_bbh(fun(A,B),fun(B,fun(A,$o)),Fa),A3),F5) ) ) ) ).

% order_tendstoD(2)
tff(fact_7992_filterlim__iff,axiom,
    ! [B: $tType,A: $tType,Fa: fun(A,B),F23: filter(B),F13: filter(A)] :
      ( filterlim(A,B,Fa,F23,F13)
    <=> ! [P8: fun(B,$o)] :
          ( eventually(B,P8,F23)
         => eventually(A,aa(fun(B,$o),fun(A,$o),aTP_Lamp_bbk(fun(A,B),fun(fun(B,$o),fun(A,$o)),Fa),P8),F13) ) ) ).

% filterlim_iff
tff(fact_7993_filterlim__cong,axiom,
    ! [A: $tType,B: $tType,F13: filter(A),F14: filter(A),F23: filter(B),F24: filter(B),Fa: fun(B,A),G: fun(B,A)] :
      ( ( F13 = F14 )
     => ( ( F23 = F24 )
       => ( eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_bbl(fun(B,A),fun(fun(B,A),fun(B,$o)),Fa),G),F23)
         => ( filterlim(B,A,Fa,F13,F23)
          <=> filterlim(B,A,G,F14,F24) ) ) ) ) ).

% filterlim_cong
tff(fact_7994_eventually__compose__filterlim,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F5: filter(A),Fa: fun(B,A),G4: filter(B)] :
      ( eventually(A,P,F5)
     => ( filterlim(B,A,Fa,F5,G4)
       => eventually(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_bbm(fun(A,$o),fun(fun(B,A),fun(B,$o)),P),Fa),G4) ) ) ).

% eventually_compose_filterlim
tff(fact_7995_Lim__transform__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Fa: fun(A,B),L: B,F5: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),F5)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bbn(fun(A,B),fun(fun(A,B),fun(A,$o)),Fa),G),F5)
           => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,L),F5) ) ) ) ).

% Lim_transform_eventually
tff(fact_7996_eventually__nhds__iff__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [P: fun(A,$o),A3: A] :
          ( eventually(A,P,topolo7230453075368039082e_nhds(A,A3))
        <=> ! [F7: fun(nat,A)] :
              ( filterlim(nat,A,F7,topolo7230453075368039082e_nhds(A,A3),at_top(nat))
             => eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_bbc(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),P),F7),at_top(nat)) ) ) ) ).

% eventually_nhds_iff_sequentially
tff(fact_7997_tendsto__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Fa: fun(A,B),L: B,Net: filter(A)] :
          ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_bbo(fun(A,B),fun(B,fun(A,$o)),Fa),L),Net)
         => filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),Net) ) ) ).

% tendsto_eventually
tff(fact_7998_tendsto__imp__eventually__ne,axiom,
    ! [B: $tType,A: $tType] :
      ( topological_t1_space(B)
     => ! [Fa: fun(A,B),C2: B,F5: filter(A),C6: B] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,C2),F5)
         => ( ( C2 != C6 )
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_bbp(fun(A,B),fun(B,fun(A,$o)),Fa),C6),F5) ) ) ) ).

% tendsto_imp_eventually_ne
tff(fact_7999_tendsto__discrete,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo8865339358273720382pology(B)
     => ! [Fa: fun(A,B),Y: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,Y),F5)
        <=> eventually(A,aa(B,fun(A,$o),aTP_Lamp_bbq(fun(A,B),fun(B,fun(A,$o)),Fa),Y),F5) ) ) ).

% tendsto_discrete
tff(fact_8000_tendsto__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Fa: fun(A,B),G: fun(A,B),F5: filter(A),C2: B] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bbn(fun(A,B),fun(fun(A,B),fun(A,$o)),Fa),G),F5)
         => ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,C2),F5)
          <=> filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F5) ) ) ) ).

% tendsto_cong
tff(fact_8001_filterlim__at__top__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( ( linorder(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),Fa: fun(A,B),P: fun(B,$o),G: fun(B,A)] :
          ( ! [X2: A,Y2: A] :
              ( aa(A,$o,Q,X2)
             => ( aa(A,$o,Q,Y2)
               => ( aa(A,$o,ord_less_eq(A,X2),Y2)
                 => aa(B,$o,ord_less_eq(B,aa(A,B,Fa,X2)),aa(A,B,Fa,Y2)) ) ) )
         => ( ! [X2: B] :
                ( aa(B,$o,P,X2)
               => ( aa(A,B,Fa,aa(B,A,G,X2)) = X2 ) )
           => ( ! [X2: B] :
                  ( aa(B,$o,P,X2)
                 => aa(A,$o,Q,aa(B,A,G,X2)) )
             => ( eventually(A,Q,at_top(A))
               => ( eventually(B,P,at_top(B))
                 => filterlim(A,B,Fa,at_top(B),at_top(A)) ) ) ) ) ) ) ).

% filterlim_at_top_at_top
tff(fact_8002_filterlim__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Fa: fun(A,B),F5: filter(A)] :
          ( filterlim(A,B,Fa,at_top(B),F5)
        <=> ! [Z7: B] : eventually(A,aa(B,fun(A,$o),aTP_Lamp_bbr(fun(A,B),fun(B,fun(A,$o)),Fa),Z7),F5) ) ) ).

% filterlim_at_top
tff(fact_8003_filterlim__at__top__ge,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Fa: fun(A,B),F5: filter(A),C2: B] :
          ( filterlim(A,B,Fa,at_top(B),F5)
        <=> ! [Z7: B] :
              ( aa(B,$o,ord_less_eq(B,C2),Z7)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_bbr(fun(A,B),fun(B,fun(A,$o)),Fa),Z7),F5) ) ) ) ).

% filterlim_at_top_ge
tff(fact_8004_filterlim__at__top__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Fa: fun(A,B),F5: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,Fa,at_top(B),F5)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bbs(fun(A,B),fun(fun(A,B),fun(A,$o)),Fa),G),F5)
           => filterlim(A,B,G,at_top(B),F5) ) ) ) ).

% filterlim_at_top_mono
tff(fact_8005_filterlim__mono__eventually,axiom,
    ! [B: $tType,A: $tType,Fa: fun(A,B),F5: filter(B),G4: filter(A),F6: filter(B),G6: filter(A),F3: fun(A,B)] :
      ( filterlim(A,B,Fa,F5,G4)
     => ( aa(filter(B),$o,ord_less_eq(filter(B),F5),F6)
       => ( aa(filter(A),$o,ord_less_eq(filter(A),G6),G4)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bbt(fun(A,B),fun(fun(A,B),fun(A,$o)),Fa),F3),G6)
           => filterlim(A,B,F3,F6,G6) ) ) ) ) ).

% filterlim_mono_eventually
tff(fact_8006_real__tendsto__sandwich,axiom,
    ! [A: $tType,Fa: fun(A,real),G: fun(A,real),Net: filter(A),H: fun(A,real),C2: real] :
      ( eventually(A,aa(fun(A,real),fun(A,$o),aTP_Lamp_bbu(fun(A,real),fun(fun(A,real),fun(A,$o)),Fa),G),Net)
     => ( eventually(A,aa(fun(A,real),fun(A,$o),aTP_Lamp_bbu(fun(A,real),fun(fun(A,real),fun(A,$o)),G),H),Net)
       => ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,C2),Net)
         => ( filterlim(A,real,H,topolo7230453075368039082e_nhds(real,C2),Net)
           => filterlim(A,real,G,topolo7230453075368039082e_nhds(real,C2),Net) ) ) ) ) ).

% real_tendsto_sandwich
tff(fact_8007_tendsto__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Fa: fun(A,B),G: fun(A,B),Net: filter(A),H: fun(A,B),C2: B] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bbv(fun(A,B),fun(fun(A,B),fun(A,$o)),Fa),G),Net)
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bbv(fun(A,B),fun(fun(A,B),fun(A,$o)),G),H),Net)
           => ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,C2),Net)
             => ( filterlim(A,B,H,topolo7230453075368039082e_nhds(B,C2),Net)
               => filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),Net) ) ) ) ) ) ).

% tendsto_sandwich
tff(fact_8008_eventually__all__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
         => eventually(A,aTP_Lamp_bbw(fun(A,$o),fun(A,$o),P),at_top(A)) ) ) ).

% eventually_all_ge_at_top
tff(fact_8009_eventually__ge__at__top,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A] : eventually(A,ord_less_eq(A,C2),at_top(A)) ) ).

% eventually_ge_at_top
tff(fact_8010_le__sequentially,axiom,
    ! [F5: filter(nat)] :
      ( aa(filter(nat),$o,ord_less_eq(filter(nat),F5),at_top(nat))
    <=> ! [N9: nat] : eventually(nat,ord_less_eq(nat,N9),F5) ) ).

% le_sequentially
tff(fact_8011_eventually__sequentially,axiom,
    ! [P: fun(nat,$o)] :
      ( eventually(nat,P,at_top(nat))
    <=> ? [N9: nat] :
        ! [N8: nat] :
          ( aa(nat,$o,ord_less_eq(nat,N9),N8)
         => aa(nat,$o,P,N8) ) ) ).

% eventually_sequentially
tff(fact_8012_eventually__sequentiallyI,axiom,
    ! [C2: nat,P: fun(nat,$o)] :
      ( ! [X2: nat] :
          ( aa(nat,$o,ord_less_eq(nat,C2),X2)
         => aa(nat,$o,P,X2) )
     => eventually(nat,P,at_top(nat)) ) ).

% eventually_sequentiallyI
tff(fact_8013_eventually__at__top__linorder,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N9: A] :
            ! [N8: A] :
              ( aa(A,$o,ord_less_eq(A,N9),N8)
             => aa(A,$o,P,N8) ) ) ) ).

% eventually_at_top_linorder
tff(fact_8014_eventually__at__top__linorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,P: fun(A,$o)] :
          ( ! [X2: A] :
              ( aa(A,$o,ord_less_eq(A,C2),X2)
             => aa(A,$o,P,X2) )
         => eventually(A,P,at_top(A)) ) ) ).

% eventually_at_top_linorderI
tff(fact_8015_False__imp__not__eventually,axiom,
    ! [A: $tType,P: fun(A,$o),Net: filter(A)] :
      ( ! [X2: A] : ~ aa(A,$o,P,X2)
     => ( ( Net != bot_bot(filter(A)) )
       => ~ eventually(A,P,Net) ) ) ).

% False_imp_not_eventually
tff(fact_8016_eventually__const__iff,axiom,
    ! [A: $tType,P: $o,F5: filter(A)] :
      ( eventually(A,aTP_Lamp_afb($o,fun(A,$o),(P)),F5)
    <=> ( (P)
        | ( F5 = bot_bot(filter(A)) ) ) ) ).

% eventually_const_iff
tff(fact_8017_trivial__limit__def,axiom,
    ! [A: $tType,F5: filter(A)] :
      ( ( F5 = bot_bot(filter(A)) )
    <=> eventually(A,aTP_Lamp_dp(A,$o),F5) ) ).

% trivial_limit_def
tff(fact_8018_eventually__False__sequentially,axiom,
    ~ eventually(nat,aTP_Lamp_adh(nat,$o),at_top(nat)) ).

% eventually_False_sequentially
tff(fact_8019_eventually__at__top__not__equal,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,aTP_Lamp_bbx(A,fun(A,$o),C2),at_top(A)) ) ).

% eventually_at_top_not_equal
tff(fact_8020_eventually__gt__at__top,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [C2: A] : eventually(A,ord_less(A,C2),at_top(A)) ) ).

% eventually_gt_at_top
tff(fact_8021_eventually__at__top__dense,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [P: fun(A,$o)] :
          ( eventually(A,P,at_top(A))
        <=> ? [N9: A] :
            ! [N8: A] :
              ( aa(A,$o,ord_less(A,N9),N8)
             => aa(A,$o,P,N8) ) ) ) ).

% eventually_at_top_dense
tff(fact_8022_eventually__ball__finite,axiom,
    ! [A: $tType,B: $tType,A4: set(A),P: fun(B,fun(A,$o)),Net: filter(B)] :
      ( finite_finite2(A,A4)
     => ( ! [X2: A] :
            ( member(A,X2,A4)
           => eventually(B,aa(A,fun(B,$o),aTP_Lamp_aiv(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X2),Net) )
       => eventually(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_anb(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A4),P),Net) ) ) ).

% eventually_ball_finite
tff(fact_8023_eventually__ball__finite__distrib,axiom,
    ! [A: $tType,B: $tType,A4: set(A),P: fun(B,fun(A,$o)),Net: filter(B)] :
      ( finite_finite2(A,A4)
     => ( eventually(B,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_anb(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A4),P),Net)
      <=> ! [X: A] :
            ( member(A,X,A4)
           => eventually(B,aa(A,fun(B,$o),aTP_Lamp_aiv(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),X),Net) ) ) ) ).

% eventually_ball_finite_distrib
tff(fact_8024_eventually__frequently__const__simps_I6_J,axiom,
    ! [A: $tType,C5: $o,P: fun(A,$o),F5: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bby($o,fun(fun(A,$o),fun(A,$o)),(C5)),P),F5)
    <=> ( (C5)
       => eventually(A,P,F5) ) ) ).

% eventually_frequently_const_simps(6)
tff(fact_8025_eventually__frequently__const__simps_I4_J,axiom,
    ! [A: $tType,C5: $o,P: fun(A,$o),F5: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bbz($o,fun(fun(A,$o),fun(A,$o)),(C5)),P),F5)
    <=> ( (C5)
        | eventually(A,P,F5) ) ) ).

% eventually_frequently_const_simps(4)
tff(fact_8026_eventually__frequently__const__simps_I3_J,axiom,
    ! [A: $tType,P: fun(A,$o),C5: $o,F5: filter(A)] :
      ( eventually(A,aa($o,fun(A,$o),aTP_Lamp_bca(fun(A,$o),fun($o,fun(A,$o)),P),(C5)),F5)
    <=> ( eventually(A,P,F5)
        | (C5) ) ) ).

% eventually_frequently_const_simps(3)
tff(fact_8027_eventually__mp,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F5: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bcb(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F5)
     => ( eventually(A,P,F5)
       => eventually(A,Q,F5) ) ) ).

% eventually_mp
tff(fact_8028_eventually__True,axiom,
    ! [A: $tType,F5: filter(A)] : eventually(A,aTP_Lamp_do(A,$o),F5) ).

% eventually_True
tff(fact_8029_eventually__conj,axiom,
    ! [A: $tType,P: fun(A,$o),F5: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F5)
     => ( eventually(A,Q,F5)
       => eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ta(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F5) ) ) ).

% eventually_conj
tff(fact_8030_eventually__elim2,axiom,
    ! [A: $tType,P: fun(A,$o),F5: filter(A),Q: fun(A,$o),R: fun(A,$o)] :
      ( eventually(A,P,F5)
     => ( eventually(A,Q,F5)
       => ( ! [I2: A] :
              ( aa(A,$o,P,I2)
             => ( aa(A,$o,Q,I2)
               => aa(A,$o,R,I2) ) )
         => eventually(A,R,F5) ) ) ) ).

% eventually_elim2
tff(fact_8031_eventually__subst,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F5: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bcc(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F5)
     => ( eventually(A,P,F5)
      <=> eventually(A,Q,F5) ) ) ).

% eventually_subst
tff(fact_8032_eventually__rev__mp,axiom,
    ! [A: $tType,P: fun(A,$o),F5: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F5)
     => ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bcb(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F5)
       => eventually(A,Q,F5) ) ) ).

% eventually_rev_mp
tff(fact_8033_eventually__conj__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),F5: filter(A)] :
      ( eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ta(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F5)
    <=> ( eventually(A,P,F5)
        & eventually(A,Q,F5) ) ) ).

% eventually_conj_iff
tff(fact_8034_not__eventually__impI,axiom,
    ! [A: $tType,P: fun(A,$o),F5: filter(A),Q: fun(A,$o)] :
      ( eventually(A,P,F5)
     => ( ~ eventually(A,Q,F5)
       => ~ eventually(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bcb(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),F5) ) ) ).

% not_eventually_impI
tff(fact_8035_eventually__all__finite,axiom,
    ! [A: $tType,B: $tType] :
      ( finite_finite(A)
     => ! [P: fun(B,fun(A,$o)),Net: filter(B)] :
          ( ! [Y2: A] : eventually(B,aa(A,fun(B,$o),aTP_Lamp_bcd(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),Y2),Net)
         => eventually(B,aTP_Lamp_bce(fun(B,fun(A,$o)),fun(B,$o),P),Net) ) ) ).

% eventually_all_finite
tff(fact_8036_eventually__ex,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),F5: filter(A)] :
      ( eventually(A,aTP_Lamp_amw(fun(A,fun(B,$o)),fun(A,$o),P),F5)
    <=> ? [Y7: fun(A,B)] : eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bcf(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),P),Y7),F5) ) ).

% eventually_ex
tff(fact_8037_Lim__ident__at,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Xb: A,S: set(A)] :
          ( ( topolo174197925503356063within(A,Xb,S) != bot_bot(filter(A)) )
         => ( topolo3827282254853284352ce_Lim(A,A,topolo174197925503356063within(A,Xb,S),aTP_Lamp_aut(A,A)) = Xb ) ) ) ).

% Lim_ident_at
tff(fact_8038_has__derivative__transform__eventually,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),F3: fun(A,B),Xb: A,S: set(A),G: fun(A,B)] :
          ( has_derivative(A,B,Fa,F3,topolo174197925503356063within(A,Xb,S))
         => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bcg(fun(A,B),fun(fun(A,B),fun(A,$o)),Fa),G),topolo174197925503356063within(A,Xb,S))
           => ( ( aa(A,B,Fa,Xb) = aa(A,B,G,Xb) )
             => ( member(A,Xb,S)
               => has_derivative(A,B,G,F3,topolo174197925503356063within(A,Xb,S)) ) ) ) ) ) ).

% has_derivative_transform_eventually
tff(fact_8039_eventually__Lim__ident__at,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [P: fun(A,fun(A,$o)),Xb: A,X5: set(A)] :
          ( eventually(A,aa(A,fun(A,$o),P,topolo3827282254853284352ce_Lim(A,A,topolo174197925503356063within(A,Xb,X5),aTP_Lamp_aut(A,A))),topolo174197925503356063within(A,Xb,X5))
        <=> eventually(A,aa(A,fun(A,$o),P,Xb),topolo174197925503356063within(A,Xb,X5)) ) ) ).

% eventually_Lim_ident_at
tff(fact_8040_has__field__derivative__cong__eventually,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),G: fun(A,A),Xb: A,S3: set(A),U: A] :
          ( eventually(A,aa(fun(A,A),fun(A,$o),aTP_Lamp_bch(fun(A,A),fun(fun(A,A),fun(A,$o)),Fa),G),topolo174197925503356063within(A,Xb,S3))
         => ( ( aa(A,A,Fa,Xb) = aa(A,A,G,Xb) )
           => ( has_field_derivative(A,Fa,U,topolo174197925503356063within(A,Xb,S3))
            <=> has_field_derivative(A,G,U,topolo174197925503356063within(A,Xb,S3)) ) ) ) ) ).

% has_field_derivative_cong_eventually
tff(fact_8041_eventually__at__left,axiom,
    ! [A: $tType] :
      ( topolo1944317154257567458pology(A)
     => ! [Y: A,Xb: A,P: fun(A,$o)] :
          ( aa(A,$o,ord_less(A,Y),Xb)
         => ( eventually(A,P,topolo174197925503356063within(A,Xb,set_ord_lessThan(A,Xb)))
          <=> ? [B7: A] :
                ( aa(A,$o,ord_less(A,B7),Xb)
                & ! [Y3: A] :
                    ( aa(A,$o,ord_less(A,B7),Y3)
                   => ( aa(A,$o,ord_less(A,Y3),Xb)
                     => aa(A,$o,P,Y3) ) ) ) ) ) ) ).

% eventually_at_left
tff(fact_8042_eventually__at__left__field,axiom,
    ! [A: $tType] :
      ( ( linordered_field(A)
        & topolo1944317154257567458pology(A) )
     => ! [P: fun(A,$o),Xb: A] :
          ( eventually(A,P,topolo174197925503356063within(A,Xb,set_ord_lessThan(A,Xb)))
        <=> ? [B7: A] :
              ( aa(A,$o,ord_less(A,B7),Xb)
              & ! [Y3: A] :
                  ( aa(A,$o,ord_less(A,B7),Y3)
                 => ( aa(A,$o,ord_less(A,Y3),Xb)
                   => aa(A,$o,P,Y3) ) ) ) ) ) ).

% eventually_at_left_field
tff(fact_8043_sequentially__offset,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( eventually(nat,P,at_top(nat))
     => eventually(nat,aa(nat,fun(nat,$o),aTP_Lamp_bbb(fun(nat,$o),fun(nat,fun(nat,$o)),P),K),at_top(nat)) ) ).

% sequentially_offset
tff(fact_8044_summable__cong,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Fa: fun(nat,A),G: fun(nat,A)] :
          ( eventually(nat,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_bci(fun(nat,A),fun(fun(nat,A),fun(nat,$o)),Fa),G),at_top(nat))
         => ( summable(A,Fa)
          <=> summable(A,G) ) ) ) ).

% summable_cong
tff(fact_8045_t1__space__nhds,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb != Y )
         => eventually(A,aTP_Lamp_bcj(A,fun(A,$o),Y),topolo7230453075368039082e_nhds(A,Xb)) ) ) ).

% t1_space_nhds
tff(fact_8046_eventually__eventually,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),Xb: A] :
          ( eventually(A,aTP_Lamp_bck(fun(A,$o),fun(A,$o),P),topolo7230453075368039082e_nhds(A,Xb))
        <=> eventually(A,P,topolo7230453075368039082e_nhds(A,Xb)) ) ) ).

% eventually_eventually
tff(fact_8047_eventually__nhds__top,axiom,
    ! [A: $tType] :
      ( ( order_top(A)
        & topolo1944317154257567458pology(A) )
     => ! [B3: A,P: fun(A,$o)] :
          ( aa(A,$o,ord_less(A,B3),top_top(A))
         => ( eventually(A,P,topolo7230453075368039082e_nhds(A,top_top(A)))
          <=> ? [B7: A] :
                ( aa(A,$o,ord_less(A,B7),top_top(A))
                & ! [Z3: A] :
                    ( aa(A,$o,ord_less(A,B7),Z3)
                   => aa(A,$o,P,Z3) ) ) ) ) ) ).

% eventually_nhds_top
tff(fact_8048_eventually__at__filter,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [P: fun(A,$o),A3: A,S: set(A)] :
          ( eventually(A,P,topolo174197925503356063within(A,A3,S))
        <=> eventually(A,aa(set(A),fun(A,$o),aa(A,fun(set(A),fun(A,$o)),aTP_Lamp_bcl(fun(A,$o),fun(A,fun(set(A),fun(A,$o))),P),A3),S),topolo7230453075368039082e_nhds(A,A3)) ) ) ).

% eventually_at_filter
tff(fact_8049_has__field__derivative__cong__ev,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xb: A,Y: A,S3: set(A),Fa: fun(A,A),G: fun(A,A),U: A,V: A,Tb: set(A)] :
          ( ( Xb = Y )
         => ( eventually(A,aa(fun(A,A),fun(A,$o),aa(fun(A,A),fun(fun(A,A),fun(A,$o)),aTP_Lamp_bcm(set(A),fun(fun(A,A),fun(fun(A,A),fun(A,$o))),S3),Fa),G),topolo7230453075368039082e_nhds(A,Xb))
           => ( ( U = V )
             => ( ( S3 = Tb )
               => ( member(A,Xb,S3)
                 => ( has_field_derivative(A,Fa,U,topolo174197925503356063within(A,Xb,S3))
                  <=> has_field_derivative(A,G,V,topolo174197925503356063within(A,Y,Tb)) ) ) ) ) ) ) ) ).

% has_field_derivative_cong_ev
tff(fact_8050_eventually__at__leftI,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A3: A,B3: A,P: fun(A,$o)] :
          ( ! [X2: A] :
              ( member(A,X2,set_or5935395276787703475ssThan(A,A3,B3))
             => aa(A,$o,P,X2) )
         => ( aa(A,$o,ord_less(A,A3),B3)
           => eventually(A,P,topolo174197925503356063within(A,B3,set_ord_lessThan(A,B3))) ) ) ) ).

% eventually_at_leftI
tff(fact_8051_eventually__at__to__0,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P: fun(A,$o),A3: A] :
          ( eventually(A,P,topolo174197925503356063within(A,A3,top_top(set(A))))
        <=> eventually(A,aa(A,fun(A,$o),aTP_Lamp_bcn(fun(A,$o),fun(A,fun(A,$o)),P),A3),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% eventually_at_to_0
tff(fact_8052_decreasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [L: B,Fa: fun(A,B),F5: filter(A)] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bco(B,fun(fun(A,B),fun(A,$o)),L),Fa),F5)
         => ( ! [X2: B] :
                ( aa(B,$o,ord_less(B,L),X2)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_bbh(fun(A,B),fun(B,fun(A,$o)),Fa),X2),F5) )
           => filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),F5) ) ) ) ).

% decreasing_tendsto
tff(fact_8053_increasing__tendsto,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Fa: fun(A,B),L: B,F5: filter(A)] :
          ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_bcp(fun(A,B),fun(B,fun(A,$o)),Fa),L),F5)
         => ( ! [X2: B] :
                ( aa(B,$o,ord_less(B,X2),L)
               => eventually(A,aa(B,fun(A,$o),aTP_Lamp_bbg(fun(A,B),fun(B,fun(A,$o)),Fa),X2),F5) )
           => filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),F5) ) ) ) ).

% increasing_tendsto
tff(fact_8054_filterlim__at__top__gt,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Fa: fun(A,B),F5: filter(A),C2: B] :
          ( filterlim(A,B,Fa,at_top(B),F5)
        <=> ! [Z7: B] :
              ( aa(B,$o,ord_less(B,C2),Z7)
             => eventually(A,aa(B,fun(A,$o),aTP_Lamp_bcq(fun(A,B),fun(B,fun(A,$o)),Fa),Z7),F5) ) ) ) ).

% filterlim_at_top_gt
tff(fact_8055_tendsto__compose__eventually,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [G: fun(A,B),M: B,L: A,Fa: fun(C,A),F5: filter(C)] :
          ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,M),topolo174197925503356063within(A,L,top_top(set(A))))
         => ( filterlim(C,A,Fa,topolo7230453075368039082e_nhds(A,L),F5)
           => ( eventually(C,aa(fun(C,A),fun(C,$o),aTP_Lamp_bcr(A,fun(fun(C,A),fun(C,$o)),L),Fa),F5)
             => filterlim(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_auk(fun(A,B),fun(fun(C,A),fun(C,B)),G),Fa),topolo7230453075368039082e_nhds(B,M),F5) ) ) ) ) ).

% tendsto_compose_eventually
tff(fact_8056_LIM__compose__eventually,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Fa: fun(A,B),B3: B,A3: A,G: fun(B,C),C2: C] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,B3),topolo174197925503356063within(A,A3,top_top(set(A))))
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(B,B3,top_top(set(B))))
           => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_bcs(fun(A,B),fun(B,fun(A,$o)),Fa),B3),topolo174197925503356063within(A,A3,top_top(set(A))))
             => filterlim(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_bct(fun(A,B),fun(fun(B,C),fun(A,C)),Fa),G),topolo7230453075368039082e_nhds(C,C2),topolo174197925503356063within(A,A3,top_top(set(A)))) ) ) ) ) ).

% LIM_compose_eventually
tff(fact_8057_filterlim__atI,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Fa: fun(A,B),C2: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,C2),F5)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_bcu(fun(A,B),fun(B,fun(A,$o)),Fa),C2),F5)
           => filterlim(A,B,Fa,topolo174197925503356063within(B,C2,top_top(set(B))),F5) ) ) ) ).

% filterlim_atI
tff(fact_8058_isCont__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Fa: fun(A,B),G: fun(A,B),Xb: A] :
          ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bcv(fun(A,B),fun(fun(A,B),fun(A,$o)),Fa),G),topolo7230453075368039082e_nhds(A,Xb))
         => ( topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,top_top(set(A))),Fa)
          <=> topolo3448309680560233919inuous(A,B,topolo174197925503356063within(A,Xb,top_top(set(A))),G) ) ) ) ).

% isCont_cong
tff(fact_8059_DERIV__cong__ev,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xb: A,Y: A,Fa: fun(A,A),G: fun(A,A),U: A,V: A] :
          ( ( Xb = Y )
         => ( eventually(A,aa(fun(A,A),fun(A,$o),aTP_Lamp_bch(fun(A,A),fun(fun(A,A),fun(A,$o)),Fa),G),topolo7230453075368039082e_nhds(A,Xb))
           => ( ( U = V )
             => ( has_field_derivative(A,Fa,U,topolo174197925503356063within(A,Xb,top_top(set(A))))
              <=> has_field_derivative(A,G,V,topolo174197925503356063within(A,Y,top_top(set(A)))) ) ) ) ) ) ).

% DERIV_cong_ev
tff(fact_8060_tendsto__le,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [F5: filter(A),Fa: fun(A,B),Xb: B,G: fun(A,B),Y: B] :
          ( ( F5 != bot_bot(filter(A)) )
         => ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,Xb),F5)
           => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,Y),F5)
             => ( eventually(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_bcw(fun(A,B),fun(fun(A,B),fun(A,$o)),Fa),G),F5)
               => aa(B,$o,ord_less_eq(B,Y),Xb) ) ) ) ) ) ).

% tendsto_le
tff(fact_8061_tendsto__lowerbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Fa: fun(A,B),Xb: B,F5: filter(A),A3: B] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,Xb),F5)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_bcx(fun(A,B),fun(B,fun(A,$o)),Fa),A3),F5)
           => ( ( F5 != bot_bot(filter(A)) )
             => aa(B,$o,ord_less_eq(B,A3),Xb) ) ) ) ) ).

% tendsto_lowerbound
tff(fact_8062_tendsto__upperbound,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Fa: fun(A,B),Xb: B,F5: filter(A),A3: B] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,Xb),F5)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_bcy(fun(A,B),fun(B,fun(A,$o)),Fa),A3),F5)
           => ( ( F5 != bot_bot(filter(A)) )
             => aa(B,$o,ord_less_eq(B,Xb),A3) ) ) ) ) ).

% tendsto_upperbound
tff(fact_8063_eventually__floor__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Fa: fun(A,B),L: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),F5)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_bcz(fun(A,B),fun(B,fun(A,$o)),Fa),L),F5) ) ) ) ).

% eventually_floor_eq
tff(fact_8064_eventually__ceiling__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Fa: fun(A,B),L: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),F5)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_bda(fun(A,B),fun(B,fun(A,$o)),Fa),L),F5) ) ) ) ).

% eventually_ceiling_eq
tff(fact_8065_continuous__divide,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [F5: filter(A),Fa: fun(A,B),G: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => ( topolo3448309680560233919inuous(A,B,F5,G)
           => ( ( aa(A,B,G,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_aut(A,A))) != zero_zero(B) )
             => topolo3448309680560233919inuous(A,B,F5,aa(fun(A,B),fun(A,B),aTP_Lamp_atv(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G)) ) ) ) ) ).

% continuous_divide
tff(fact_8066_continuous__inverse,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [F5: filter(A),Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => ( ( aa(A,B,Fa,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_aut(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F5,aTP_Lamp_atz(fun(A,B),fun(A,B),Fa)) ) ) ) ).

% continuous_inverse
tff(fact_8067_continuous__arcosh__strong,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F5: filter(A),Fa: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F5,Fa)
         => ( eventually(A,aTP_Lamp_bdb(fun(A,real),fun(A,$o),Fa),F5)
           => topolo3448309680560233919inuous(A,real,F5,aTP_Lamp_bdc(fun(A,real),fun(A,real),Fa)) ) ) ) ).

% continuous_arcosh_strong
tff(fact_8068_continuous__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [F5: filter(A),Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
        <=> filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,aa(A,B,Fa,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_aut(A,A)))),F5) ) ) ).

% continuous_def
tff(fact_8069_continuous__sgn,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [F5: filter(A),Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => ( ( aa(A,B,Fa,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_aut(A,A))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F5,aTP_Lamp_aua(fun(A,B),fun(A,B),Fa)) ) ) ) ).

% continuous_sgn
tff(fact_8070_t2__space__class_OLim__def,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(A)
     => ! [A4: filter(B),Fa: fun(B,A)] : topolo3827282254853284352ce_Lim(B,A,A4,Fa) = the(A,aa(fun(B,A),fun(A,$o),aTP_Lamp_bdd(filter(B),fun(fun(B,A),fun(A,$o)),A4),Fa)) ) ).

% t2_space_class.Lim_def
tff(fact_8071_continuous__powr,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F5: filter(A),Fa: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F5,Fa)
         => ( topolo3448309680560233919inuous(A,real,F5,G)
           => ( ( aa(A,real,Fa,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_aut(A,A))) != zero_zero(real) )
             => topolo3448309680560233919inuous(A,real,F5,aa(fun(A,real),fun(A,real),aTP_Lamp_ayv(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G)) ) ) ) ) ).

% continuous_powr
tff(fact_8072_continuous__ln,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F5: filter(A),Fa: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F5,Fa)
         => ( ( aa(A,real,Fa,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_aut(A,A))) != zero_zero(real) )
           => topolo3448309680560233919inuous(A,real,F5,aTP_Lamp_ayw(fun(A,real),fun(A,real),Fa)) ) ) ) ).

% continuous_ln
tff(fact_8073_tendsto__compose__at,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Fa: fun(A,B),Y: B,F5: filter(A),G: fun(B,C),Z: C] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,Y),F5)
         => ( filterlim(B,C,G,topolo7230453075368039082e_nhds(C,Z),topolo174197925503356063within(B,Y,top_top(set(B))))
           => ( eventually(A,aa(C,fun(A,$o),aa(fun(B,C),fun(C,fun(A,$o)),aa(B,fun(fun(B,C),fun(C,fun(A,$o))),aTP_Lamp_bde(fun(A,B),fun(B,fun(fun(B,C),fun(C,fun(A,$o)))),Fa),Y),G),Z),F5)
             => filterlim(A,C,comp(B,C,A,G,Fa),topolo7230453075368039082e_nhds(C,Z),F5) ) ) ) ) ).

% tendsto_compose_at
tff(fact_8074_tendsto__imp__filterlim__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Fa: fun(A,B),L5: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L5),F5)
         => ( eventually(A,aa(B,fun(A,$o),aTP_Lamp_bbh(fun(A,B),fun(B,fun(A,$o)),Fa),L5),F5)
           => filterlim(A,B,Fa,topolo174197925503356063within(B,L5,set_ord_lessThan(B,L5)),F5) ) ) ) ).

% tendsto_imp_filterlim_at_left
tff(fact_8075_summable__comparison__test__ev,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Fa: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_bdf(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Fa),G),at_top(nat))
         => ( summable(real,G)
           => summable(A,Fa) ) ) ) ).

% summable_comparison_test_ev
tff(fact_8076_tendsto__arcosh__strong,axiom,
    ! [A: $tType,Fa: fun(A,real),A3: real,F5: filter(A)] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,A3),F5)
     => ( aa(real,$o,ord_less_eq(real,one_one(real)),A3)
       => ( eventually(A,aTP_Lamp_bdg(fun(A,real),fun(A,$o),Fa),F5)
         => filterlim(A,real,aTP_Lamp_avg(fun(A,real),fun(A,real),Fa),topolo7230453075368039082e_nhds(real,aa(real,real,arcosh(real),A3)),F5) ) ) ) ).

% tendsto_arcosh_strong
tff(fact_8077_filterlim__at__top__at__left,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo1944317154257567458pology(A)
        & linorder(B) )
     => ! [Q: fun(A,$o),Fa: fun(A,B),P: fun(B,$o),G: fun(B,A),A3: A] :
          ( ! [X2: A,Y2: A] :
              ( aa(A,$o,Q,X2)
             => ( aa(A,$o,Q,Y2)
               => ( aa(A,$o,ord_less_eq(A,X2),Y2)
                 => aa(B,$o,ord_less_eq(B,aa(A,B,Fa,X2)),aa(A,B,Fa,Y2)) ) ) )
         => ( ! [X2: B] :
                ( aa(B,$o,P,X2)
               => ( aa(A,B,Fa,aa(B,A,G,X2)) = X2 ) )
           => ( ! [X2: B] :
                  ( aa(B,$o,P,X2)
                 => aa(A,$o,Q,aa(B,A,G,X2)) )
             => ( eventually(A,Q,topolo174197925503356063within(A,A3,set_ord_lessThan(A,A3)))
               => ( ! [B2: A] :
                      ( aa(A,$o,Q,B2)
                     => aa(A,$o,ord_less(A,B2),A3) )
                 => ( eventually(B,P,at_top(B))
                   => filterlim(A,B,Fa,at_top(B),topolo174197925503356063within(A,A3,set_ord_lessThan(A,A3))) ) ) ) ) ) ) ) ).

% filterlim_at_top_at_left
tff(fact_8078_suminf__eq__lim,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Fa: fun(nat,A)] : suminf(A,Fa) = topolo3827282254853284352ce_Lim(nat,A,at_top(nat),aTP_Lamp_azz(fun(nat,A),fun(nat,A),Fa)) ) ).

% suminf_eq_lim
tff(fact_8079_Topological__Spaces_Olim__def,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [X5: fun(nat,A)] : topolo3827282254853284352ce_Lim(nat,A,at_top(nat),X5) = the(A,aTP_Lamp_bdh(fun(nat,A),fun(A,$o),X5)) ) ).

% Topological_Spaces.lim_def
tff(fact_8080_filterlim__at__withinI,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Fa: fun(A,B),C2: B,F5: filter(A),A4: set(B)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,C2),F5)
         => ( eventually(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_bdi(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Fa),C2),A4),F5)
           => filterlim(A,B,Fa,topolo174197925503356063within(B,C2,A4),F5) ) ) ) ).

% filterlim_at_withinI
tff(fact_8081_tendsto__0__le,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [Fa: fun(A,B),F5: filter(A),G: fun(A,C),K5: real] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,zero_zero(B)),F5)
         => ( eventually(A,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_bdj(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),Fa),G),K5),F5)
           => filterlim(A,C,G,topolo7230453075368039082e_nhds(C,zero_zero(C)),F5) ) ) ) ).

% tendsto_0_le
tff(fact_8082_continuous__tan,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F5: filter(A),Fa: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F5,Fa)
         => ( ( cos(A,aa(A,A,Fa,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_bdk(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F5,aTP_Lamp_avu(fun(A,A),fun(A,A),Fa)) ) ) ) ).

% continuous_tan
tff(fact_8083_eventually__floor__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Fa: fun(A,B),L: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),F5)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_bdl(fun(A,B),fun(B,fun(A,$o)),Fa),L),F5) ) ) ) ).

% eventually_floor_less
tff(fact_8084_eventually__less__ceiling,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Fa: fun(A,B),L: B,F5: filter(A)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),F5)
         => ( ~ member(B,L,ring_1_Ints(B))
           => eventually(A,aa(B,fun(A,$o),aTP_Lamp_bdm(fun(A,B),fun(B,fun(A,$o)),Fa),L),F5) ) ) ) ).

% eventually_less_ceiling
tff(fact_8085_tendsto__zero__powrI,axiom,
    ! [A: $tType,Fa: fun(A,real),F5: filter(A),G: fun(A,real),B3: real] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,zero_zero(real)),F5)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B3),F5)
       => ( eventually(A,aTP_Lamp_bdn(fun(A,real),fun(A,$o),Fa),F5)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_avs(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F5) ) ) ) ) ).

% tendsto_zero_powrI
tff(fact_8086_tendsto__powr2,axiom,
    ! [A: $tType,Fa: fun(A,real),A3: real,F5: filter(A),G: fun(A,real),B3: real] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,A3),F5)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B3),F5)
       => ( eventually(A,aTP_Lamp_bdn(fun(A,real),fun(A,$o),Fa),F5)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_avs(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G),topolo7230453075368039082e_nhds(real,powr(real,A3,B3)),F5) ) ) ) ) ).

% tendsto_powr2
tff(fact_8087_tendsto__powr_H,axiom,
    ! [A: $tType,Fa: fun(A,real),A3: real,F5: filter(A),G: fun(A,real),B3: real] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,A3),F5)
     => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,B3),F5)
       => ( ( ( A3 != zero_zero(real) )
            | ( aa(real,$o,ord_less(real,zero_zero(real)),B3)
              & eventually(A,aTP_Lamp_bdn(fun(A,real),fun(A,$o),Fa),F5) ) )
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_avs(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G),topolo7230453075368039082e_nhds(real,powr(real,A3,B3)),F5) ) ) ) ).

% tendsto_powr'
tff(fact_8088_continuous__cot,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [F5: filter(A),Fa: fun(A,A)] :
          ( topolo3448309680560233919inuous(A,A,F5,Fa)
         => ( ( sin(A,aa(A,A,Fa,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_bdk(A,A)))) != zero_zero(A) )
           => topolo3448309680560233919inuous(A,A,F5,aTP_Lamp_avo(fun(A,A),fun(A,A),Fa)) ) ) ) ).

% continuous_cot
tff(fact_8089_continuous__tanh,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [F5: filter(A),Fa: fun(A,B)] :
          ( topolo3448309680560233919inuous(A,B,F5,Fa)
         => ( ( cosh(B,aa(A,B,Fa,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_aut(A,A)))) != zero_zero(B) )
           => topolo3448309680560233919inuous(A,B,F5,aTP_Lamp_aya(fun(A,B),fun(A,B),Fa)) ) ) ) ).

% continuous_tanh
tff(fact_8090_continuous__arcosh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F5: filter(A),Fa: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F5,Fa)
         => ( aa(real,$o,ord_less(real,one_one(real)),aa(A,real,Fa,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_aut(A,A))))
           => topolo3448309680560233919inuous(A,real,F5,aTP_Lamp_bdc(fun(A,real),fun(A,real),Fa)) ) ) ) ).

% continuous_arcosh
tff(fact_8091_continuous__log,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F5: filter(A),Fa: fun(A,real),G: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F5,Fa)
         => ( topolo3448309680560233919inuous(A,real,F5,G)
           => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,Fa,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_aut(A,A))))
             => ( ( aa(A,real,Fa,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_aut(A,A))) != one_one(real) )
               => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,G,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_aut(A,A))))
                 => topolo3448309680560233919inuous(A,real,F5,aa(fun(A,real),fun(A,real),aTP_Lamp_azq(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G)) ) ) ) ) ) ) ).

% continuous_log
tff(fact_8092_continuous__artanh,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [F5: filter(A),Fa: fun(A,real)] :
          ( topolo3448309680560233919inuous(A,real,F5,Fa)
         => ( member(real,aa(A,real,Fa,topolo3827282254853284352ce_Lim(A,A,F5,aTP_Lamp_aut(A,A))),set_or5935395276787703475ssThan(real,aa(real,real,uminus_uminus(real),one_one(real)),one_one(real)))
           => topolo3448309680560233919inuous(A,real,F5,aTP_Lamp_bdo(fun(A,real),fun(A,real),Fa)) ) ) ) ).

% continuous_artanh
tff(fact_8093_polyfun__extremal,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [C2: fun(nat,A),K: nat,N: nat,B6: real] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,ord_less_eq(nat,one_one(nat)),K)
           => ( aa(nat,$o,ord_less_eq(nat,K),N)
             => eventually(A,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_bdp(fun(nat,A),fun(nat,fun(real,fun(A,$o))),C2),N),B6),at_infinity(A)) ) ) ) ) ).

% polyfun_extremal
tff(fact_8094_summable__bounded__partials,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Fa: fun(nat,A),G: fun(nat,real)] :
          ( eventually(nat,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_bdq(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Fa),G),at_top(nat))
         => ( filterlim(nat,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(nat))
           => summable(A,Fa) ) ) ) ).

% summable_bounded_partials
tff(fact_8095_greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or3652927894154168847AtMost(A,L,U))
        <=> ( aa(A,$o,ord_less(A,L),I)
            & aa(A,$o,ord_less_eq(A,I),U) ) ) ) ).

% greaterThanAtMost_iff
tff(fact_8096_greaterThanAtMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,K: A] :
          ( aa(A,$o,ord_less_eq(A,L),K)
         => ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) ) ) ) ).

% greaterThanAtMost_empty
tff(fact_8097_greaterThanAtMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( set_or3652927894154168847AtMost(A,K,L) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,ord_less(A,K),L) ) ) ).

% greaterThanAtMost_empty_iff
tff(fact_8098_greaterThanAtMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A,L: A] :
          ( ( bot_bot(set(A)) = set_or3652927894154168847AtMost(A,K,L) )
        <=> ~ aa(A,$o,ord_less(A,K),L) ) ) ).

% greaterThanAtMost_empty_iff2
tff(fact_8099_infinite__Ioc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( ~ finite_finite2(A,set_or3652927894154168847AtMost(A,A3,B3))
        <=> aa(A,$o,ord_less(A,A3),B3) ) ) ).

% infinite_Ioc_iff
tff(fact_8100_lhopital__left__at__top__at__top,axiom,
    ! [Fa: fun(real,real),A3: real,G: fun(real,real),F3: fun(real,real),G5: fun(real,real)] :
      ( filterlim(real,real,Fa,at_top(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_bdr(fun(real,real),fun(fun(real,real),fun(real,$o)),Fa),F3),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_bdr(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_bds(fun(real,real),fun(fun(real,real),fun(real,real)),F3),G5),at_top(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_bds(fun(real,real),fun(fun(real,real),fun(real,real)),Fa),G),at_top(real),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3))) ) ) ) ) ) ).

% lhopital_left_at_top_at_top
tff(fact_8101_lhopital__at__top__at__top,axiom,
    ! [Fa: fun(real,real),A3: real,G: fun(real,real),F3: fun(real,real),G5: fun(real,real)] :
      ( filterlim(real,real,Fa,at_top(real),topolo174197925503356063within(real,A3,top_top(set(real))))
     => ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,A3,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_bdr(fun(real,real),fun(fun(real,real),fun(real,$o)),Fa),F3),topolo174197925503356063within(real,A3,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_bdr(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,A3,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_bds(fun(real,real),fun(fun(real,real),fun(real,real)),F3),G5),at_top(real),topolo174197925503356063within(real,A3,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_bds(fun(real,real),fun(fun(real,real),fun(real,real)),Fa),G),at_top(real),topolo174197925503356063within(real,A3,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top_at_top
tff(fact_8102_filterlim__at__infinity__imp__filterlim__at__top,axiom,
    ! [A: $tType,Fa: fun(A,real),F5: filter(A)] :
      ( filterlim(A,real,Fa,at_infinity(real),F5)
     => ( eventually(A,aTP_Lamp_bdt(fun(A,real),fun(A,$o),Fa),F5)
       => filterlim(A,real,Fa,at_top(real),F5) ) ) ).

% filterlim_at_infinity_imp_filterlim_at_top
tff(fact_8103_filterlim__at__infinity__imp__eventually__ne,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,B),F5: filter(A),C2: B] :
          ( filterlim(A,B,Fa,at_infinity(B),F5)
         => eventually(A,aa(B,fun(A,$o),aTP_Lamp_bdu(fun(A,B),fun(B,fun(A,$o)),Fa),C2),F5) ) ) ).

% filterlim_at_infinity_imp_eventually_ne
tff(fact_8104_eventually__not__equal__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A3: A] : eventually(A,aTP_Lamp_bdv(A,fun(A,$o),A3),at_infinity(A)) ) ).

% eventually_not_equal_at_infinity
tff(fact_8105_infinite__Ioc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A] :
          ( aa(A,$o,ord_less(A,A3),B3)
         => ~ finite_finite2(A,set_or3652927894154168847AtMost(A,A3,B3)) ) ) ).

% infinite_Ioc
tff(fact_8106_Ioc__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or3652927894154168847AtMost(A,A3,B3)),set_or3652927894154168847AtMost(A,C2,D))
        <=> ( aa(A,$o,ord_less_eq(A,B3),A3)
            | ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less_eq(A,B3),D) ) ) ) ) ).

% Ioc_subset_iff
tff(fact_8107_Ioc__inj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( ( set_or3652927894154168847AtMost(A,A3,B3) = set_or3652927894154168847AtMost(A,C2,D) )
        <=> ( ( aa(A,$o,ord_less_eq(A,B3),A3)
              & aa(A,$o,ord_less_eq(A,D),C2) )
            | ( ( A3 = C2 )
              & ( B3 = D ) ) ) ) ) ).

% Ioc_inj
tff(fact_8108_less__filter__def,axiom,
    ! [A: $tType,F5: filter(A),F6: filter(A)] :
      ( aa(filter(A),$o,ord_less(filter(A),F5),F6)
    <=> ( aa(filter(A),$o,ord_less_eq(filter(A),F5),F6)
        & ~ aa(filter(A),$o,ord_less_eq(filter(A),F6),F5) ) ) ).

% less_filter_def
tff(fact_8109_filterlim__at__top__mult__at__top,axiom,
    ! [A: $tType,Fa: fun(A,real),F5: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,Fa,at_top(real),F5)
     => ( filterlim(A,real,G,at_top(real),F5)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_bdw(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G),at_top(real),F5) ) ) ).

% filterlim_at_top_mult_at_top
tff(fact_8110_exp__at__top,axiom,
    filterlim(real,real,exp(real),at_top(real),at_top(real)) ).

% exp_at_top
tff(fact_8111_sqrt__at__top,axiom,
    filterlim(real,real,sqrt,at_top(real),at_top(real)) ).

% sqrt_at_top
tff(fact_8112_ln__at__top,axiom,
    filterlim(real,real,ln_ln(real),at_top(real),at_top(real)) ).

% ln_at_top
tff(fact_8113_filterlim__at__infinity__imp__norm__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,B),F5: filter(A)] :
          ( filterlim(A,B,Fa,at_infinity(B),F5)
         => filterlim(A,real,aTP_Lamp_avv(fun(A,B),fun(A,real),Fa),at_top(real),F5) ) ) ).

% filterlim_at_infinity_imp_norm_at_top
tff(fact_8114_filterlim__norm__at__top__imp__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,B),F5: filter(A)] :
          ( filterlim(A,real,aTP_Lamp_avv(fun(A,B),fun(A,real),Fa),at_top(real),F5)
         => filterlim(A,B,Fa,at_infinity(B),F5) ) ) ).

% filterlim_norm_at_top_imp_at_infinity
tff(fact_8115_filterlim__at__infinity__conv__norm__at__top,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,B),G4: filter(A)] :
          ( filterlim(A,B,Fa,at_infinity(B),G4)
        <=> filterlim(A,real,aTP_Lamp_avv(fun(A,B),fun(A,real),Fa),at_top(real),G4) ) ) ).

% filterlim_at_infinity_conv_norm_at_top
tff(fact_8116_filterlim__at__top__add__at__top,axiom,
    ! [A: $tType,Fa: fun(A,real),F5: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,Fa,at_top(real),F5)
     => ( filterlim(A,real,G,at_top(real),F5)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_bdx(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G),at_top(real),F5) ) ) ).

% filterlim_at_top_add_at_top
tff(fact_8117_eventually__at__left__real,axiom,
    ! [B3: real,A3: real] :
      ( aa(real,$o,ord_less(real,B3),A3)
     => eventually(real,aa(real,fun(real,$o),aTP_Lamp_bdy(real,fun(real,fun(real,$o)),B3),A3),topolo174197925503356063within(real,A3,set_ord_lessThan(real,A3))) ) ).

% eventually_at_left_real
tff(fact_8118_filterlim__tendsto__add__at__top,axiom,
    ! [A: $tType,Fa: fun(A,real),C2: real,F5: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,C2),F5)
     => ( filterlim(A,real,G,at_top(real),F5)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_bdx(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G),at_top(real),F5) ) ) ).

% filterlim_tendsto_add_at_top
tff(fact_8119_lhopital__left__at__top,axiom,
    ! [G: fun(real,real),Xb: real,G5: fun(real,real),Fa: fun(real,real),F3: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,Xb,set_ord_lessThan(real,Xb)))
     => ( eventually(real,aTP_Lamp_bdz(fun(real,real),fun(real,$o),G5),topolo174197925503356063within(real,Xb,set_ord_lessThan(real,Xb)))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_bdr(fun(real,real),fun(fun(real,real),fun(real,$o)),Fa),F3),topolo174197925503356063within(real,Xb,set_ord_lessThan(real,Xb)))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_bdr(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,Xb,set_ord_lessThan(real,Xb)))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_bea(fun(real,real),fun(fun(real,real),fun(real,real)),G5),F3),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xb,set_ord_lessThan(real,Xb)))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_bea(fun(real,real),fun(fun(real,real),fun(real,real)),G),Fa),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xb,set_ord_lessThan(real,Xb))) ) ) ) ) ) ).

% lhopital_left_at_top
tff(fact_8120_lhospital__at__top__at__top,axiom,
    ! [G: fun(real,real),G5: fun(real,real),Fa: fun(real,real),F3: fun(real,real),Xb: real] :
      ( filterlim(real,real,G,at_top(real),at_top(real))
     => ( eventually(real,aTP_Lamp_bdz(fun(real,real),fun(real,$o),G5),at_top(real))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_bdr(fun(real,real),fun(fun(real,real),fun(real,$o)),Fa),F3),at_top(real))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_bdr(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),at_top(real))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_bea(fun(real,real),fun(fun(real,real),fun(real,real)),G5),F3),topolo7230453075368039082e_nhds(real,Xb),at_top(real))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_bea(fun(real,real),fun(fun(real,real),fun(real,real)),G),Fa),topolo7230453075368039082e_nhds(real,Xb),at_top(real)) ) ) ) ) ) ).

% lhospital_at_top_at_top
tff(fact_8121_lhopital__at__top,axiom,
    ! [G: fun(real,real),Xb: real,G5: fun(real,real),Fa: fun(real,real),F3: fun(real,real),Y: real] :
      ( filterlim(real,real,G,at_top(real),topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( eventually(real,aTP_Lamp_bdz(fun(real,real),fun(real,$o),G5),topolo174197925503356063within(real,Xb,top_top(set(real))))
       => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_bdr(fun(real,real),fun(fun(real,real),fun(real,$o)),Fa),F3),topolo174197925503356063within(real,Xb,top_top(set(real))))
         => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_bdr(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,Xb,top_top(set(real))))
           => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_bea(fun(real,real),fun(fun(real,real),fun(real,real)),G5),F3),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xb,top_top(set(real))))
             => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_bea(fun(real,real),fun(fun(real,real),fun(real,real)),G),Fa),topolo7230453075368039082e_nhds(real,Y),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ) ) ).

% lhopital_at_top
tff(fact_8122_filterlim__real__sequentially,axiom,
    filterlim(nat,real,semiring_1_of_nat(real),at_top(real),at_top(nat)) ).

% filterlim_real_sequentially
tff(fact_8123_sum_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,M)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).

% sum.head
tff(fact_8124_prod_Ohead,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,N: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less_eq(nat,M),N)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,N)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,G,set_or3652927894154168847AtMost(nat,M,N))) ) ) ) ).

% prod.head
tff(fact_8125_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or3652927894154168847AtMost(A,A3,B3)),set_or1337092689740270186AtMost(A,C2,D))
        <=> ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less_eq(A,B3),D) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastAtMost_iff
tff(fact_8126_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or3652927894154168847AtMost(A,A3,B3)),set_or7035219750837199246ssThan(A,C2,D))
        <=> ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less(A,B3),D) ) ) ) ) ).

% greaterThanAtMost_subseteq_atLeastLessThan_iff
tff(fact_8127_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A3: A,B3: A,C2: A,D: A] :
          ( aa(set(A),$o,ord_less_eq(set(A),set_or5935395276787703475ssThan(A,A3,B3)),set_or3652927894154168847AtMost(A,C2,D))
        <=> ( aa(A,$o,ord_less(A,A3),B3)
           => ( aa(A,$o,ord_less_eq(A,C2),A3)
              & aa(A,$o,ord_less_eq(A,B3),D) ) ) ) ) ).

% greaterThanLessThan_subseteq_greaterThanAtMost_iff
tff(fact_8128_filterlim__pow__at__top,axiom,
    ! [A: $tType,N: nat,Fa: fun(A,real),F5: filter(A)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
     => ( filterlim(A,real,Fa,at_top(real),F5)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_atp(nat,fun(fun(A,real),fun(A,real)),N),Fa),at_top(real),F5) ) ) ).

% filterlim_pow_at_top
tff(fact_8129_tendsto__add__filterlim__at__infinity_H,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,B),F5: filter(A),G: fun(A,B),C2: B] :
          ( filterlim(A,B,Fa,at_infinity(B),F5)
         => ( filterlim(A,B,G,topolo7230453075368039082e_nhds(B,C2),F5)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_beb(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),at_infinity(B),F5) ) ) ) ).

% tendsto_add_filterlim_at_infinity'
tff(fact_8130_tendsto__add__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Fa: fun(A,B),C2: B,F5: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,C2),F5)
         => ( filterlim(A,B,G,at_infinity(B),F5)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_beb(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),at_infinity(B),F5) ) ) ) ).

% tendsto_add_filterlim_at_infinity
tff(fact_8131_real__tendsto__divide__at__top,axiom,
    ! [A: $tType,Fa: fun(A,real),C2: real,F5: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,C2),F5)
     => ( filterlim(A,real,G,at_top(real),F5)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_bec(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G),topolo7230453075368039082e_nhds(real,zero_zero(real)),F5) ) ) ).

% real_tendsto_divide_at_top
tff(fact_8132_tendsto__inverse__0__at__top,axiom,
    ! [A: $tType,Fa: fun(A,real),F5: filter(A)] :
      ( filterlim(A,real,Fa,at_top(real),F5)
     => filterlim(A,real,aTP_Lamp_bed(fun(A,real),fun(A,real),Fa),topolo7230453075368039082e_nhds(real,zero_zero(real)),F5) ) ).

% tendsto_inverse_0_at_top
tff(fact_8133_filterlim__int__of__nat__at__topD,axiom,
    ! [A: $tType,Fa: fun(int,A),F5: filter(A)] :
      ( filterlim(nat,A,aTP_Lamp_bee(fun(int,A),fun(nat,A),Fa),F5,at_top(nat))
     => filterlim(int,A,Fa,F5,at_top(int)) ) ).

% filterlim_int_of_nat_at_topD
tff(fact_8134_filterlim__sequentially__iff__filterlim__real,axiom,
    ! [A: $tType,Fa: fun(A,nat),F5: filter(A)] :
      ( filterlim(A,nat,Fa,at_top(nat),F5)
    <=> filterlim(A,real,aTP_Lamp_go(fun(A,nat),fun(A,real),Fa),at_top(real),F5) ) ).

% filterlim_sequentially_iff_filterlim_real
tff(fact_8135_tendsto__inverse__0,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => filterlim(A,A,inverse_inverse(A),topolo7230453075368039082e_nhds(A,zero_zero(A)),at_infinity(A)) ) ).

% tendsto_inverse_0
tff(fact_8136_filterlim__at__top__mult__tendsto__pos,axiom,
    ! [A: $tType,Fa: fun(A,real),C2: real,F5: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,C2),F5)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F5)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_bef(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G),at_top(real),F5) ) ) ) ).

% filterlim_at_top_mult_tendsto_pos
tff(fact_8137_filterlim__tendsto__pos__mult__at__top,axiom,
    ! [A: $tType,Fa: fun(A,real),C2: real,F5: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,C2),F5)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
       => ( filterlim(A,real,G,at_top(real),F5)
         => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_bdw(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G),at_top(real),F5) ) ) ) ).

% filterlim_tendsto_pos_mult_at_top
tff(fact_8138_tendsto__neg__powr,axiom,
    ! [A: $tType,S: real,Fa: fun(A,real),F5: filter(A)] :
      ( aa(real,$o,ord_less(real,S),zero_zero(real))
     => ( filterlim(A,real,Fa,at_top(real),F5)
       => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_beg(real,fun(fun(A,real),fun(A,real)),S),Fa),topolo7230453075368039082e_nhds(real,zero_zero(real)),F5) ) ) ).

% tendsto_neg_powr
tff(fact_8139_tendsto__mult__filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Fa: fun(A,B),C2: B,F5: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,C2),F5)
         => ( ( C2 != zero_zero(B) )
           => ( filterlim(A,B,G,at_infinity(B),F5)
             => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_beh(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),at_infinity(B),F5) ) ) ) ) ).

% tendsto_mult_filterlim_at_infinity
tff(fact_8140_filterlim__power__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Fa: fun(A,B),F5: filter(A),N: nat] :
          ( filterlim(A,B,Fa,at_infinity(B),F5)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N)
           => filterlim(A,B,aa(nat,fun(A,B),aTP_Lamp_bei(fun(A,B),fun(nat,fun(A,B)),Fa),N),at_infinity(B),F5) ) ) ) ).

% filterlim_power_at_infinity
tff(fact_8141_tendsto__divide__0,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Fa: fun(A,B),C2: B,F5: filter(A),G: fun(A,B)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,C2),F5)
         => ( filterlim(A,B,G,at_infinity(B),F5)
           => filterlim(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bej(fun(A,B),fun(fun(A,B),fun(A,B)),Fa),G),topolo7230453075368039082e_nhds(B,zero_zero(B)),F5) ) ) ) ).

% tendsto_divide_0
tff(fact_8142_ln__x__over__x__tendsto__0,axiom,
    filterlim(real,real,aTP_Lamp_bek(real,real),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% ln_x_over_x_tendsto_0
tff(fact_8143_tendsto__at__topI__sequentially,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Fa: fun(real,A),Y: A] :
          ( ! [X10: fun(nat,real)] :
              ( filterlim(nat,real,X10,at_top(real),at_top(nat))
             => filterlim(nat,A,aa(fun(nat,real),fun(nat,A),aTP_Lamp_bel(fun(real,A),fun(fun(nat,real),fun(nat,A)),Fa),X10),topolo7230453075368039082e_nhds(A,Y),at_top(nat)) )
         => filterlim(real,A,Fa,topolo7230453075368039082e_nhds(A,Y),at_top(real)) ) ) ).

% tendsto_at_topI_sequentially
tff(fact_8144_filterlim__inverse__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => filterlim(A,A,inverse_inverse(A),at_infinity(A),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ).

% filterlim_inverse_at_infinity
tff(fact_8145_LIM__at__top__divide,axiom,
    ! [A: $tType,Fa: fun(A,real),A3: real,F5: filter(A),G: fun(A,real)] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,A3),F5)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A3)
       => ( filterlim(A,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),F5)
         => ( eventually(A,aTP_Lamp_bdt(fun(A,real),fun(A,$o),G),F5)
           => filterlim(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_bec(fun(A,real),fun(fun(A,real),fun(A,real)),Fa),G),at_top(real),F5) ) ) ) ) ).

% LIM_at_top_divide
tff(fact_8146_filterlim__inverse__at__top__iff,axiom,
    ! [A: $tType,Fa: fun(A,real),F5: filter(A)] :
      ( eventually(A,aTP_Lamp_bdt(fun(A,real),fun(A,$o),Fa),F5)
     => ( filterlim(A,real,aTP_Lamp_bed(fun(A,real),fun(A,real),Fa),at_top(real),F5)
      <=> filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,zero_zero(real)),F5) ) ) ).

% filterlim_inverse_at_top_iff
tff(fact_8147_filterlim__inverse__at__top,axiom,
    ! [A: $tType,Fa: fun(A,real),F5: filter(A)] :
      ( filterlim(A,real,Fa,topolo7230453075368039082e_nhds(real,zero_zero(real)),F5)
     => ( eventually(A,aTP_Lamp_bdt(fun(A,real),fun(A,$o),Fa),F5)
       => filterlim(A,real,aTP_Lamp_bed(fun(A,real),fun(A,real),Fa),at_top(real),F5) ) ) ).

% filterlim_inverse_at_top
tff(fact_8148_filterlim__at__top__iff__inverse__0,axiom,
    ! [A: $tType,Fa: fun(A,real),F5: filter(A)] :
      ( eventually(A,aTP_Lamp_bdt(fun(A,real),fun(A,$o),Fa),F5)
     => ( filterlim(A,real,Fa,at_top(real),F5)
      <=> filterlim(A,real,comp(real,real,A,inverse_inverse(real),Fa),topolo7230453075368039082e_nhds(real,zero_zero(real)),F5) ) ) ).

% filterlim_at_top_iff_inverse_0
tff(fact_8149_eventually__at__infinity__pos,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [P2: fun(A,$o)] :
          ( eventually(A,P2,at_infinity(A))
        <=> ? [B7: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),B7)
              & ! [X: A] :
                  ( aa(real,$o,ord_less_eq(real,B7),real_V7770717601297561774m_norm(A,X))
                 => aa(A,$o,P2,X) ) ) ) ) ).

% eventually_at_infinity_pos
tff(fact_8150_tendsto__power__div__exp__0,axiom,
    ! [K: nat] : filterlim(real,real,aTP_Lamp_bem(nat,fun(real,real),K),topolo7230453075368039082e_nhds(real,zero_zero(real)),at_top(real)) ).

% tendsto_power_div_exp_0
tff(fact_8151_lim__infinity__imp__sequentially,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Fa: fun(real,A),L: A] :
          ( filterlim(real,A,Fa,topolo7230453075368039082e_nhds(A,L),at_infinity(real))
         => filterlim(nat,A,aTP_Lamp_ben(fun(real,A),fun(nat,A),Fa),topolo7230453075368039082e_nhds(A,L),at_top(nat)) ) ) ).

% lim_infinity_imp_sequentially
tff(fact_8152_filterlim__inverse__at__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [G: fun(A,B),F5: filter(A)] :
          ( filterlim(A,B,aTP_Lamp_avm(fun(A,B),fun(A,B),G),topolo174197925503356063within(B,zero_zero(B),top_top(set(B))),F5)
        <=> filterlim(A,B,G,at_infinity(B),F5) ) ) ).

% filterlim_inverse_at_iff
tff(fact_8153_lhopital,axiom,
    ! [Fa: fun(real,real),Xb: real,G: fun(real,real),G5: fun(real,real),F3: fun(real,real),F5: filter(real)] :
      ( filterlim(real,real,Fa,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,Xb,top_top(set(real))))
       => ( eventually(real,aTP_Lamp_bdz(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,Xb,top_top(set(real))))
         => ( eventually(real,aTP_Lamp_bdz(fun(real,real),fun(real,$o),G5),topolo174197925503356063within(real,Xb,top_top(set(real))))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_bdr(fun(real,real),fun(fun(real,real),fun(real,$o)),Fa),F3),topolo174197925503356063within(real,Xb,top_top(set(real))))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_bdr(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,Xb,top_top(set(real))))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_bea(fun(real,real),fun(fun(real,real),fun(real,real)),G5),F3),F5,topolo174197925503356063within(real,Xb,top_top(set(real))))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_bds(fun(real,real),fun(fun(real,real),fun(real,real)),Fa),G),F5,topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ) ) ) ) ).

% lhopital
tff(fact_8154_lhopital__left,axiom,
    ! [Fa: fun(real,real),Xb: real,G: fun(real,real),G5: fun(real,real),F3: fun(real,real),F5: filter(real)] :
      ( filterlim(real,real,Fa,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,Xb,set_ord_lessThan(real,Xb)))
     => ( filterlim(real,real,G,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(real,Xb,set_ord_lessThan(real,Xb)))
       => ( eventually(real,aTP_Lamp_bdz(fun(real,real),fun(real,$o),G),topolo174197925503356063within(real,Xb,set_ord_lessThan(real,Xb)))
         => ( eventually(real,aTP_Lamp_bdz(fun(real,real),fun(real,$o),G5),topolo174197925503356063within(real,Xb,set_ord_lessThan(real,Xb)))
           => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_bdr(fun(real,real),fun(fun(real,real),fun(real,$o)),Fa),F3),topolo174197925503356063within(real,Xb,set_ord_lessThan(real,Xb)))
             => ( eventually(real,aa(fun(real,real),fun(real,$o),aTP_Lamp_bdr(fun(real,real),fun(fun(real,real),fun(real,$o)),G),G5),topolo174197925503356063within(real,Xb,set_ord_lessThan(real,Xb)))
               => ( filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_bea(fun(real,real),fun(fun(real,real),fun(real,real)),G5),F3),F5,topolo174197925503356063within(real,Xb,set_ord_lessThan(real,Xb)))
                 => filterlim(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_bds(fun(real,real),fun(fun(real,real),fun(real,real)),Fa),G),F5,topolo174197925503356063within(real,Xb,set_ord_lessThan(real,Xb))) ) ) ) ) ) ) ) ).

% lhopital_left
tff(fact_8155_tendsto__exp__limit__at__top,axiom,
    ! [Xb: real] : filterlim(real,real,aTP_Lamp_beo(real,fun(real,real),Xb),topolo7230453075368039082e_nhds(real,aa(real,real,exp(real),Xb)),at_top(real)) ).

% tendsto_exp_limit_at_top
tff(fact_8156_filterlim__divide__at__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),C2: A,F5: filter(A),G: fun(A,A)] :
          ( filterlim(A,A,Fa,topolo7230453075368039082e_nhds(A,C2),F5)
         => ( filterlim(A,A,G,topolo174197925503356063within(A,zero_zero(A),top_top(set(A))),F5)
           => ( ( C2 != zero_zero(A) )
             => filterlim(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_aph(fun(A,A),fun(fun(A,A),fun(A,A)),Fa),G),at_infinity(A),F5) ) ) ) ) ).

% filterlim_divide_at_infinity
tff(fact_8157_filterlim__tan__at__left,axiom,
    filterlim(real,real,tan(real),at_top(real),topolo174197925503356063within(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))),set_ord_lessThan(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))))) ).

% filterlim_tan_at_left
tff(fact_8158_tendsto__arctan__at__top,axiom,
    filterlim(real,real,arctan,topolo7230453075368039082e_nhds(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))),at_top(real)) ).

% tendsto_arctan_at_top
tff(fact_8159_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B3: real,Fa: fun(real,real),Flim: real] :
      ( ! [X2: real] :
          ( aa(real,$o,ord_less_eq(real,B3),X2)
         => ? [Y4: real] :
              ( has_field_derivative(real,Fa,Y4,topolo174197925503356063within(real,X2,top_top(set(real))))
              & aa(real,$o,ord_less(real,Y4),zero_zero(real)) ) )
     => ( filterlim(real,real,Fa,topolo7230453075368039082e_nhds(real,Flim),at_top(real))
       => aa(real,$o,ord_less(real,Flim),aa(real,real,Fa,B3)) ) ) ).

% DERIV_neg_imp_decreasing_at_top
tff(fact_8160_filterlim__at__infinity,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [C2: real,Fa: fun(A,B),F5: filter(A)] :
          ( aa(real,$o,ord_less_eq(real,zero_zero(real)),C2)
         => ( filterlim(A,B,Fa,at_infinity(B),F5)
          <=> ! [R6: real] :
                ( aa(real,$o,ord_less(real,C2),R6)
               => eventually(A,aa(real,fun(A,$o),aTP_Lamp_bep(fun(A,B),fun(real,fun(A,$o)),Fa),R6),F5) ) ) ) ) ).

% filterlim_at_infinity
tff(fact_8161_filterlim__realpow__sequentially__gt1,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A] :
          ( aa(real,$o,ord_less(real,one_one(real)),real_V7770717601297561774m_norm(A,Xb))
         => filterlim(nat,A,aa(A,fun(nat,A),power_power(A),Xb),at_infinity(A),at_top(nat)) ) ) ).

% filterlim_realpow_sequentially_gt1
tff(fact_8162_lim__at__infinity__0,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),L: A] :
          ( filterlim(A,A,Fa,topolo7230453075368039082e_nhds(A,L),at_infinity(A))
        <=> filterlim(A,A,comp(A,A,A,Fa,inverse_inverse(A)),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ).

% lim_at_infinity_0
tff(fact_8163_lim__zero__infinity,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Fa: fun(A,A),L: A] :
          ( filterlim(A,A,aTP_Lamp_beq(fun(A,A),fun(A,A),Fa),topolo7230453075368039082e_nhds(A,L),topolo174197925503356063within(A,zero_zero(A),top_top(set(A))))
         => filterlim(A,A,Fa,topolo7230453075368039082e_nhds(A,L),at_infinity(A)) ) ) ).

% lim_zero_infinity
tff(fact_8164_has__derivative__at__within__iff__Ex,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Xb: A,S3: set(A),Fa: fun(A,B),F3: fun(A,B)] :
          ( member(A,Xb,S3)
         => ( topolo1002775350975398744n_open(A,S3)
           => ( has_derivative(A,B,Fa,F3,topolo174197925503356063within(A,Xb,S3))
            <=> ( real_V3181309239436604168linear(A,B,F3)
                & ? [E5: fun(A,B)] :
                    ( ! [H6: A] :
                        ( member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),H6),S3)
                       => ( aa(A,B,Fa,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),H6)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Fa,Xb)),aa(A,B,F3,H6))),aa(A,B,E5,H6)) ) )
                    & filterlim(A,real,aTP_Lamp_bax(fun(A,B),fun(A,real),E5),topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,zero_zero(A),top_top(set(A)))) ) ) ) ) ) ) ).

% has_derivative_at_within_iff_Ex
tff(fact_8165_has__derivativeI__sandwich,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [E3: real,F3: fun(A,B),S: set(A),Xb: A,Fa: fun(A,B),H9: fun(A,real)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
         => ( real_V3181309239436604168linear(A,B,F3)
           => ( ! [Y2: A] :
                  ( member(A,Y2,S)
                 => ( ( Y2 != Xb )
                   => ( aa(real,$o,ord_less(real,real_V557655796197034286t_dist(A,Y2,Xb)),E3)
                     => aa(real,$o,ord_less_eq(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Fa,Y2)),aa(A,B,Fa,Xb))),aa(A,B,F3,aa(A,A,minus_minus(A,Y2),Xb))))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y2),Xb)))),aa(A,real,H9,Y2)) ) ) )
             => ( filterlim(A,real,H9,topolo7230453075368039082e_nhds(real,zero_zero(real)),topolo174197925503356063within(A,Xb,S))
               => has_derivative(A,B,Fa,F3,topolo174197925503356063within(A,Xb,S)) ) ) ) ) ) ).

% has_derivativeI_sandwich
tff(fact_8166_dist__eq__0__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Y: A] :
          ( ( real_V557655796197034286t_dist(A,Xb,Y) = zero_zero(real) )
        <=> ( Xb = Y ) ) ) ).

% dist_eq_0_iff
tff(fact_8167_dist__self,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A] : real_V557655796197034286t_dist(A,Xb,Xb) = zero_zero(real) ) ).

% dist_self
tff(fact_8168_dist__0__norm,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] : real_V557655796197034286t_dist(A,zero_zero(A),Xb) = real_V7770717601297561774m_norm(A,Xb) ) ).

% dist_0_norm
tff(fact_8169_zero__less__dist__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Y: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),real_V557655796197034286t_dist(A,Xb,Y))
        <=> ( Xb != Y ) ) ) ).

% zero_less_dist_iff
tff(fact_8170_dist__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V7819770556892013058_space(A)
     => ! [Xb: A,Y: A] :
          ( aa(real,$o,ord_less_eq(real,real_V557655796197034286t_dist(A,Xb,Y)),zero_zero(real))
        <=> ( Xb = Y ) ) ) ).

% dist_le_zero_iff
tff(fact_8171_topological__tendstoI,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [L: A,Fa: fun(B,A),F5: filter(B)] :
          ( ! [S5: set(A)] :
              ( topolo1002775350975398744n_open(A,S5)
             => ( member(A,L,S5)
               => eventually(B,aa(set(A),fun(B,$o),aTP_Lamp_ber(fun(B,A),fun(set(A),fun(B,$o)),Fa),S5),F5) ) )
         => filterlim(B,A,Fa,topolo7230453075368039082e_nhds(A,L),F5) ) ) ).

% topological_tendstoI
tff(fact_8172_topological__tendstoD,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Fa: fun(A,B),L: B,F5: filter(A),S3: set(B)] :
          ( filterlim(A,B,Fa,topolo7230453075368039082e_nhds(B,L),F5)
         => ( topolo1002775350975398744n_open(B,S3)
           => ( member(B,L,S3)
             => eventually(A,aa(set(B),fun(A,$o),aTP_Lamp_bes(fun(A,B),fun(set(B),fun(A,$o)),Fa),S3),F5) ) ) ) ) ).

% topological_tendstoD
tff(fact_8173_ATP_Olambda__1,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_md(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uu)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))) ).

% ATP.lambda_1
tff(fact_8174_ATP_Olambda__2,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_axy(A,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,exp(A),Uu)),one_one(A))),Uu) ) ).

% ATP.lambda_2
tff(fact_8175_ATP_Olambda__3,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_wn(nat,real),Uu) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),bit0(one2)))),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_3
tff(fact_8176_ATP_Olambda__4,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_adn(real,$o),Uu)
    <=> ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Uu)
        & aa(real,$o,ord_less_eq(real,Uu),aa(num,real,numeral_numeral(real),bit0(one2)))
        & ( cos(real,Uu) = zero_zero(real) ) ) ) ).

% ATP.lambda_4
tff(fact_8177_ATP_Olambda__5,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_azt(nat,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu))),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_5
tff(fact_8178_ATP_Olambda__6,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_ww(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uu)),aa(nat,real,aa(real,fun(nat,real),power_power(real),zero_zero(real)),Uu)) ).

% ATP.lambda_6
tff(fact_8179_ATP_Olambda__7,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_azs(nat,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uu)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uu))) ) ).

% ATP.lambda_7
tff(fact_8180_ATP_Olambda__8,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_ayc(real,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),cos(real,Uu)),sin(real,Uu)) ).

% ATP.lambda_8
tff(fact_8181_ATP_Olambda__9,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_bek(real,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Uu)),Uu) ).

% ATP.lambda_9
tff(fact_8182_ATP_Olambda__10,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Uu: word(B)] :
          ( aa(word(B),$o,aTP_Lamp_amh(word(B),$o),Uu)
        <=> aa(word(B),$o,ord_less(word(B),Uu),aa(nat,word(B),aa(word(B),fun(nat,word(B)),power_power(word(B)),aa(num,word(B),numeral_numeral(word(B)),bit0(one2))),type_len0_len_of(A,type2(A)))) ) ) ).

% ATP.lambda_10
tff(fact_8183_ATP_Olambda__11,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_aza(nat,real),Uu) = aa(real,real,root(Uu),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_11
tff(fact_8184_ATP_Olambda__12,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_ajh(nat,nat),Uu) = aa(nat,nat,minus_minus(nat,Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_12
tff(fact_8185_ATP_Olambda__13,axiom,
    ! [B: $tType,Uu: B] : aa(B,product_prod(B,B),aTP_Lamp_anr(B,product_prod(B,B)),Uu) = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),Uu),Uu) ).

% ATP.lambda_13
tff(fact_8186_ATP_Olambda__14,axiom,
    ! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_anq(A,product_prod(A,A)),Uu) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),Uu) ).

% ATP.lambda_14
tff(fact_8187_ATP_Olambda__15,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_xn(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_15
tff(fact_8188_ATP_Olambda__16,axiom,
    ! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_aiz(A,list(A)),Uu) = aa(list(A),list(A),cons(A,Uu),nil(A)) ).

% ATP.lambda_16
tff(fact_8189_ATP_Olambda__17,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_apd(A,set(A)),Uu) = aa(set(A),set(A),insert(A,Uu),bot_bot(set(A))) ).

% ATP.lambda_17
tff(fact_8190_ATP_Olambda__18,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_azk(nat,real),Uu) = aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Uu)) ).

% ATP.lambda_18
tff(fact_8191_ATP_Olambda__19,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_azr(nat,A),Uu) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_19
tff(fact_8192_ATP_Olambda__20,axiom,
    ! [Uu: int] :
      ( aa(int,$o,aTP_Lamp_afy(int,$o),Uu)
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Uu) ) ).

% ATP.lambda_20
tff(fact_8193_ATP_Olambda__21,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_ajx(list(B),fun(nat,nat)),Uu) = ord_max(nat,aa(nat,nat,minus_minus(nat,aa(list(B),nat,size_size(list(B)),Uu)),aa(nat,nat,suc,zero_zero(nat)))) ).

% ATP.lambda_21
tff(fact_8194_ATP_Olambda__22,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_ajy(list(B),$o),Uu)
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_22
tff(fact_8195_ATP_Olambda__23,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_alb(list(A),$o),Uu)
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_23
tff(fact_8196_ATP_Olambda__24,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(product_prod(heap_ext(product_unit),nat),A),aTP_Lamp_akj(A,fun(product_prod(heap_ext(product_unit),nat),A)),Uu) = product_case_prod(heap_ext(product_unit),nat,A,aTP_Lamp_aki(A,fun(heap_ext(product_unit),fun(nat,A)),Uu)) ).

% ATP.lambda_24
tff(fact_8197_ATP_Olambda__25,axiom,
    ! [Uu: real] : aa(real,real,aTP_Lamp_aqx(real,real),Uu) = suminf(real,aTP_Lamp_mk(real,fun(nat,real),Uu)) ).

% ATP.lambda_25
tff(fact_8198_ATP_Olambda__26,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_apc(fun(A,B),set(product_prod(A,B))),Uu) = collect(product_prod(A,B),product_case_prod(A,B,$o,aTP_Lamp_apb(fun(A,B),fun(A,fun(B,$o)),Uu))) ).

% ATP.lambda_26
tff(fact_8199_ATP_Olambda__27,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_azl(nat,real),Uu) = aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uu))) ).

% ATP.lambda_27
tff(fact_8200_ATP_Olambda__28,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_za(nat,A),Uu) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uu)) ) ).

% ATP.lambda_28
tff(fact_8201_ATP_Olambda__29,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_azd(nat,A),Uu) = aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_29
tff(fact_8202_ATP_Olambda__30,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_ajw(list(B),fun(nat,nat)),Uu) = ord_max(nat,aa(list(B),nat,size_size(list(B)),Uu)) ).

% ATP.lambda_30
tff(fact_8203_ATP_Olambda__31,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_ala(list(A),fun(nat,nat)),Uu) = ord_max(nat,aa(list(A),nat,size_size(list(A)),Uu)) ).

% ATP.lambda_31
tff(fact_8204_ATP_Olambda__32,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat] : aa(nat,heap_Time_Heap(A),aTP_Lamp_aky(nat,heap_Time_Heap(A)),Uu) = aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),aa(nat,A,semiring_1_of_nat(A),Uu)) ) ).

% ATP.lambda_32
tff(fact_8205_ATP_Olambda__33,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_acv(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).

% ATP.lambda_33
tff(fact_8206_ATP_Olambda__34,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_act(num,option(num)),Uu) = aa(num,option(num),some(num),bit0(Uu)) ).

% ATP.lambda_34
tff(fact_8207_ATP_Olambda__35,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ago(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),Uu)) ).

% ATP.lambda_35
tff(fact_8208_ATP_Olambda__36,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_agr(int,fun(int,product_prod(int,int))),Uu) = aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),abs_abs(int,Uu)) ).

% ATP.lambda_36
tff(fact_8209_ATP_Olambda__37,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_gn(nat,fun(nat,product_prod(nat,nat))),Uu) = aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,suc,Uu)) ).

% ATP.lambda_37
tff(fact_8210_ATP_Olambda__38,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat)))] :
      ( aa(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),$o,aTP_Lamp_yx(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),$o),Uu)
    <=> ! [X: vEBT_VEBT,Xa3: vEBT_VEBTi,Xb4: nat] : refine_Imp_refines(option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_vebt_predi,Xa3),Xb4),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),product_curry(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),product_curry(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(option(nat)),Uu)),X),Xa3),Xb4)) ) ).

% ATP.lambda_38
tff(fact_8211_ATP_Olambda__39,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit)),aTP_Lamp_ako(A,fun(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit))),Uu) = product_case_prod(heap_ext(product_unit),nat,heap_ext(product_unit),aTP_Lamp_akn(heap_ext(product_unit),fun(nat,heap_ext(product_unit)))) ).

% ATP.lambda_39
tff(fact_8212_ATP_Olambda__40,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(product_prod(heap_ext(product_unit),nat),nat),aTP_Lamp_ake(A,fun(product_prod(heap_ext(product_unit),nat),nat)),Uu) = product_case_prod(heap_ext(product_unit),nat,nat,aTP_Lamp_akd(heap_ext(product_unit),fun(nat,nat))) ).

% ATP.lambda_40
tff(fact_8213_ATP_Olambda__41,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,heap_Time_Heap(B),aTP_Lamp_mh(A,heap_Time_Heap(B)),Uu) = heap_Time_Heap2(B,aTP_Lamp_mg(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))) ).

% ATP.lambda_41
tff(fact_8214_ATP_Olambda__42,axiom,
    ! [Uu: nat] : aa(nat,option(num),aTP_Lamp_acy(nat,option(num)),Uu) = aa(num,option(num),some(num),one2) ).

% ATP.lambda_42
tff(fact_8215_ATP_Olambda__43,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_adc(num,fun(nat,option(num)),Uu),Uua) = case_num(option(num),aa(num,option(num),some(num),one2),aTP_Lamp_ada(nat,fun(num,option(num)),Uua),aTP_Lamp_adb(nat,fun(num,option(num)),Uua),Uu) ).

% ATP.lambda_43
tff(fact_8216_ATP_Olambda__44,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_aci(A,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_44
tff(fact_8217_ATP_Olambda__45,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_aaw(nat,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_45
tff(fact_8218_ATP_Olambda__46,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_ach(A,fun(nat,A),Uu),Uua) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_46
tff(fact_8219_ATP_Olambda__47,axiom,
    ! [Uu: fun(nat,real),Uua: nat] :
      aa(nat,real,aTP_Lamp_wp(fun(nat,real),fun(nat,real),Uu),Uua) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),zero_zero(real),aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,Uua),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_47
tff(fact_8220_ATP_Olambda__48,axiom,
    ! [Uu: $o,Uua: $o] :
      aa($o,heap_Time_Heap(option(nat)),aa($o,fun($o,heap_Time_Heap(option(nat))),aTP_Lamp_aj($o,fun($o,heap_Time_Heap(option(nat)))),(Uu)),(Uua)) = $ite(
        aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),xa),
        $ite(
          (Uua),
          aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),one_one(nat))),
          $ite((Uu),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),zero_zero(nat))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))) ),
        $ite(
          xa = one_one(nat),
          $ite((Uu),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),zero_zero(nat))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))),
          aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)) ) ) ).

% ATP.lambda_48
tff(fact_8221_ATP_Olambda__49,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] :
      aa(nat,option(A),aTP_Lamp_agl(list(A),fun(nat,option(A)),Uu),Uua) = $ite(aa(nat,$o,ord_less(nat,Uua),aa(list(A),nat,size_size(list(A)),Uu)),aa(A,option(A),some(A),aa(nat,A,nth(A,Uu),Uua)),none(A)) ).

% ATP.lambda_49
tff(fact_8222_ATP_Olambda__50,axiom,
    ! [Uu: int,Uua: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_agm(int,fun(int,product_prod(int,int))),Uu),Uua) = $ite(Uu = zero_zero(int),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),one_one(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),Uu)),Uua)),abs_abs(int,Uu))) ).

% ATP.lambda_50
tff(fact_8223_ATP_Olambda__51,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_aav(nat,fun(nat,A),Uu),Uua) = $ite(~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua)),zero_zero(A)) ) ).

% ATP.lambda_51
tff(fact_8224_ATP_Olambda__52,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(nat,A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_bdh(fun(nat,A),fun(A,$o),Uu),Uua)
        <=> filterlim(nat,A,Uu,topolo7230453075368039082e_nhds(A,Uua),at_top(nat)) ) ) ).

% ATP.lambda_52
tff(fact_8225_ATP_Olambda__53,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_dn(option(A),fun(A,$o),Uu),Uua)
        <=> case_option($o,A,$true,aTP_Lamp_dm(A,fun(A,$o),Uua),Uu) ) ) ).

% ATP.lambda_53
tff(fact_8226_ATP_Olambda__54,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_add(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_adc(num,fun(nat,option(num)),Uua),Uu) ).

% ATP.lambda_54
tff(fact_8227_ATP_Olambda__55,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_dl(option(A),fun(A,$o),Uu),Uua)
        <=> case_option($o,A,$false,ord_less_eq(A,Uua),Uu) ) ) ).

% ATP.lambda_55
tff(fact_8228_ATP_Olambda__56,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_ada(nat,fun(num,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_act(num,option(num)),bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_56
tff(fact_8229_ATP_Olambda__57,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_acz(num,fun(nat,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_act(num,option(num)),bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_57
tff(fact_8230_ATP_Olambda__58,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_adq(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),Uu),Uua)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)),aa(product_prod(A,B),B,product_snd(A,B),Uua)) ) ).

% ATP.lambda_58
tff(fact_8231_ATP_Olambda__59,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_alf(list(list(A)),fun(nat,list(A)),Uu),Uua) = map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ale(list(list(A)),fun(nat,fun(nat,A)),Uu),Uua),upt(zero_zero(nat),aa(list(list(A)),nat,size_size(list(list(A))),Uu))) ).

% ATP.lambda_59
tff(fact_8232_ATP_Olambda__60,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_aau(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aat(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_60
tff(fact_8233_ATP_Olambda__61,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_aas(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aar(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_61
tff(fact_8234_ATP_Olambda__62,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_lt(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,Uu),one_one(real))),aa(nat,nat,suc,Uua))) ).

% ATP.lambda_62
tff(fact_8235_ATP_Olambda__63,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_wz(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_63
tff(fact_8236_ATP_Olambda__64,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_xi(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ).

% ATP.lambda_64
tff(fact_8237_ATP_Olambda__65,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_zz(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),Uua))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_65
tff(fact_8238_ATP_Olambda__66,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_aca(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua))),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_66
tff(fact_8239_ATP_Olambda__67,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_ado(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,ord_less_eq(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Uua)
        & aa(real,$o,ord_less_eq(real,Uua),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( sin(real,Uua) = Uu ) ) ) ).

% ATP.lambda_67
tff(fact_8240_ATP_Olambda__68,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_ade(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),Uua)
        & aa(real,$o,ord_less(real,Uua),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
        & ( aa(real,real,tan(real),Uua) = Uu ) ) ) ).

% ATP.lambda_68
tff(fact_8241_ATP_Olambda__69,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_aof(code_integer,fun(code_integer,int)),Uu),Uua) = $let(
        l2: int,
        l2:= aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),code_int_of_integer(Uu)),
        $ite(Uua = zero_zero(code_integer),l2,aa(int,int,aa(int,fun(int,int),plus_plus(int),l2),one_one(int))) ) ).

% ATP.lambda_69
tff(fact_8242_ATP_Olambda__70,axiom,
    ! [Uu: nat,Uua: nat] :
      aa(nat,a,aa(nat,fun(nat,a),aTP_Lamp_gm(nat,fun(nat,a)),Uu),Uua) = $let(
        m2: a,
        m2:= aa(a,a,aa(a,fun(a,a),times_times(a),aa(num,a,numeral_numeral(a),bit0(one2))),aa(nat,a,semiring_1_of_nat(a),Uu)),
        $ite(Uua = zero_zero(nat),m2,aa(a,a,aa(a,fun(a,a),plus_plus(a),m2),one_one(a))) ) ).

% ATP.lambda_70
tff(fact_8243_ATP_Olambda__71,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa(real,$o,aTP_Lamp_ael(complex,fun(real,$o),Uu),Uua)
    <=> ( ( aa(complex,complex,sgn_sgn(complex),Uu) = cis(Uua) )
        & aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),Uua)
        & aa(real,$o,ord_less_eq(real,Uua),pi) ) ) ).

% ATP.lambda_71
tff(fact_8244_ATP_Olambda__72,axiom,
    ! [Uu: real,Uua: int] :
      ( aa(int,$o,aTP_Lamp_ads(real,fun(int,$o),Uu),Uua)
    <=> ( aa(real,$o,ord_less_eq(real,aa(int,real,ring_1_of_int(real),Uua)),Uu)
        & aa(real,$o,ord_less(real,Uu),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_72
tff(fact_8245_ATP_Olambda__73,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_adu(rat,fun(int,$o),Uu),Uua)
    <=> ( aa(rat,$o,ord_less_eq(rat,aa(int,rat,ring_1_of_int(rat),Uua)),Uu)
        & aa(rat,$o,ord_less(rat,Uu),aa(int,rat,ring_1_of_int(rat),aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),one_one(int)))) ) ) ).

% ATP.lambda_73
tff(fact_8246_ATP_Olambda__74,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_mk(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))) ).

% ATP.lambda_74
tff(fact_8247_ATP_Olambda__75,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_aqy(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_75
tff(fact_8248_ATP_Olambda__76,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Uua)),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_76
tff(fact_8249_ATP_Olambda__77,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_aab(nat,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uu),Uua))) ) ).

% ATP.lambda_77
tff(fact_8250_ATP_Olambda__78,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aap(A,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))),Uua) ) ).

% ATP.lambda_78
tff(fact_8251_ATP_Olambda__79,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_yv(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_79
tff(fact_8252_ATP_Olambda__80,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_abo(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,Uu),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Uua)) ) ).

% ATP.lambda_80
tff(fact_8253_ATP_Olambda__81,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_zx(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,binomial(Uu),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% ATP.lambda_81
tff(fact_8254_ATP_Olambda__82,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_xk(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)),semiring_char_0_fact(real,Uua)) ).

% ATP.lambda_82
tff(fact_8255_ATP_Olambda__83,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_bem(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu)),aa(real,real,exp(real),Uua)) ).

% ATP.lambda_83
tff(fact_8256_ATP_Olambda__84,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_zp(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uu) ).

% ATP.lambda_84
tff(fact_8257_ATP_Olambda__85,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_zo(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),Uua)),Uua) ).

% ATP.lambda_85
tff(fact_8258_ATP_Olambda__86,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa(complex,$o,aTP_Lamp_wd(nat,fun(complex,$o),Uu),Uua)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_86
tff(fact_8259_ATP_Olambda__87,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( aa(A,$o,aTP_Lamp_sv(nat,fun(A,$o),Uu),Uua)
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_87
tff(fact_8260_ATP_Olambda__88,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_agi(A,fun(A,$o),Uu),Uua)
        <=> ( member(A,Uua,ring_1_Ints(A))
            & aa(A,$o,ord_less_eq(A,abs_abs(A,Uua)),Uu) ) ) ) ).

% ATP.lambda_88
tff(fact_8261_ATP_Olambda__89,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_bab(real,fun(nat,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Uu),aa(nat,real,semiring_1_of_nat(real),Uua)))),Uua) ).

% ATP.lambda_89
tff(fact_8262_ATP_Olambda__90,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_beo(real,fun(real,real),Uu),Uua) = powr(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,aa(real,fun(real,real),divide_divide(real),Uu),Uua)),Uua) ).

% ATP.lambda_90
tff(fact_8263_ATP_Olambda__91,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_mj(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))) ).

% ATP.lambda_91
tff(fact_8264_ATP_Olambda__92,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_adm(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,ord_less_eq(real,zero_zero(real)),Uua)
        & aa(real,$o,ord_less_eq(real,Uua),pi)
        & ( cos(real,Uua) = Uu ) ) ) ).

% ATP.lambda_92
tff(fact_8265_ATP_Olambda__93,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_yt(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_93
tff(fact_8266_ATP_Olambda__94,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_vi(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))),aa(nat,A,Uu,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)))) ) ).

% ATP.lambda_94
tff(fact_8267_ATP_Olambda__95,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_azg(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_95
tff(fact_8268_ATP_Olambda__96,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_vd(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,aa(nat,nat,suc,Uua))),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_96
tff(fact_8269_ATP_Olambda__97,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_afg(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),Uu,Uua),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_97
tff(fact_8270_ATP_Olambda__98,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_afe(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),Uu,Uua)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_98
tff(fact_8271_ATP_Olambda__99,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_nc(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_99
tff(fact_8272_ATP_Olambda__100,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_lu(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_100
tff(fact_8273_ATP_Olambda__101,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_tz(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_101
tff(fact_8274_ATP_Olambda__102,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_nb(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uua)) ) ).

% ATP.lambda_102
tff(fact_8275_ATP_Olambda__103,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_vh(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,Uua),one_one(nat)))) ) ).

% ATP.lambda_103
tff(fact_8276_ATP_Olambda__104,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_vg(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),one_one(nat)))) ) ).

% ATP.lambda_104
tff(fact_8277_ATP_Olambda__105,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_azh(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_105
tff(fact_8278_ATP_Olambda__106,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_wh(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uua)),aa(nat,A,Uu,aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_106
tff(fact_8279_ATP_Olambda__107,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_anj(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ( aa(A,$o,Uu,Uua)
            & ! [Y3: A] :
                ( aa(A,$o,Uu,Y3)
               => aa(A,$o,ord_less_eq(A,Uua),Y3) ) ) ) ) ).

% ATP.lambda_107
tff(fact_8280_ATP_Olambda__108,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_adt(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ( aa(A,$o,Uu,Uua)
            & ! [Y3: A] :
                ( aa(A,$o,Uu,Y3)
               => aa(A,$o,ord_less_eq(A,Y3),Uua) ) ) ) ) ).

% ATP.lambda_108
tff(fact_8281_ATP_Olambda__109,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_aag(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),Uu,Uua),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_109
tff(fact_8282_ATP_Olambda__110,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_aad(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),Uu,Uua)),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_110
tff(fact_8283_ATP_Olambda__111,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(B,A),aTP_Lamp_apg(fun(A,B),fun(A,product_prod(B,A)),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),aa(A,B,Uu,Uua)),Uua) ).

% ATP.lambda_111
tff(fact_8284_ATP_Olambda__112,axiom,
    ! [B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(B,real),Uua: B] : aa(B,real,aTP_Lamp_ati(fun(B,real),fun(B,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(B,real,Uu,Uua)),zero_zero(real)) ) ).

% ATP.lambda_112
tff(fact_8285_ATP_Olambda__113,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_acj(fun(A,real),fun(A,complex),Uu),Uua) = complex2(aa(A,real,Uu,Uua),zero_zero(real)) ).

% ATP.lambda_113
tff(fact_8286_ATP_Olambda__114,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,list(A),aTP_Lamp_alg(fun(B,A),fun(B,list(A)),Uu),Uua) = aa(list(A),list(A),cons(A,aa(B,A,Uu,Uua)),nil(A)) ).

% ATP.lambda_114
tff(fact_8287_ATP_Olambda__115,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_va(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_115
tff(fact_8288_ATP_Olambda__116,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_yl(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_116
tff(fact_8289_ATP_Olambda__117,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: list(A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_aka(list(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_ajz(list(A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,Uu,Uua)) ) ).

% ATP.lambda_117
tff(fact_8290_ATP_Olambda__118,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_bal(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),Uu)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat)))) ).

% ATP.lambda_118
tff(fact_8291_ATP_Olambda__119,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_bak(fun(nat,real),fun(nat,real),Uu),Uua) = aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_ayi(fun(nat,real),fun(nat,real),Uu)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ).

% ATP.lambda_119
tff(fact_8292_ATP_Olambda__120,axiom,
    ! [Uu: code_integer,Uua: $o] : aa($o,char,aa(code_integer,fun($o,char),aTP_Lamp_aos(code_integer,fun($o,char)),Uu),(Uua)) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aTP_Lamp_aor($o,fun(code_integer,fun($o,char)),(Uua))),code_bit_cut_integer(Uu)) ).

% ATP.lambda_120
tff(fact_8293_ATP_Olambda__121,axiom,
    ! [Uu: product_prod(vEBT_VEBTi,nat),Uua: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi))] : aa(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),heap_Time_Heap(vEBT_VEBTi),aTP_Lamp_pc(product_prod(vEBT_VEBTi,nat),fun(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),heap_Time_Heap(vEBT_VEBTi)),Uu),Uua) = aa(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi),product_case_prod(vEBT_VEBTi,nat,heap_Time_Heap(vEBT_VEBTi),aTP_Lamp_pb(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uua)),Uu) ).

% ATP.lambda_121
tff(fact_8294_ATP_Olambda__122,axiom,
    ! [Uu: product_prod(vEBT_VEBTi,nat),Uua: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o))] : aa(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),heap_Time_Heap($o),aTP_Lamp_ky(product_prod(vEBT_VEBTi,nat),fun(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),heap_Time_Heap($o)),Uu),Uua) = aa(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o),product_case_prod(vEBT_VEBTi,nat,heap_Time_Heap($o),aTP_Lamp_kx(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),Uua)),Uu) ).

% ATP.lambda_122
tff(fact_8295_ATP_Olambda__123,axiom,
    ! [Uu: product_prod(vEBT_VEBTi,nat),Uua: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat)))] : aa(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),heap_Time_Heap(option(nat)),aTP_Lamp_ko(product_prod(vEBT_VEBTi,nat),fun(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),heap_Time_Heap(option(nat))),Uu),Uua) = aa(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat)),product_case_prod(vEBT_VEBTi,nat,heap_Time_Heap(option(nat)),aTP_Lamp_kn(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uua)),Uu) ).

% ATP.lambda_123
tff(fact_8296_ATP_Olambda__124,axiom,
    ! [Uu: product_prod(vEBT_VEBTi,nat),Uua: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat)))] : aa(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),heap_Time_Heap(option(nat)),aTP_Lamp_kg(product_prod(vEBT_VEBTi,nat),fun(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),heap_Time_Heap(option(nat))),Uu),Uua) = aa(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat)),product_case_prod(vEBT_VEBTi,nat,heap_Time_Heap(option(nat)),aTP_Lamp_kf(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uua)),Uu) ).

% ATP.lambda_124
tff(fact_8297_ATP_Olambda__125,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_alk(list(A),fun(list(A),list(list(A))),Uu),Uua) = map(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_alj(list(A),fun(A,list(A))),Uua),Uu) ).

% ATP.lambda_125
tff(fact_8298_ATP_Olambda__126,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_bax(fun(A,B),fun(A,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua))),real_V7770717601297561774m_norm(A,Uua)) ) ).

% ATP.lambda_126
tff(fact_8299_ATP_Olambda__127,axiom,
    ! [Uu: product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),Uua: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi))] : aa(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),heap_Time_Heap(vEBT_VEBTi),aTP_Lamp_re(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),fun(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),heap_Time_Heap(vEBT_VEBTi)),Uu),Uua) = aa(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi),product_case_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(vEBT_VEBTi),product_case_prod(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_rd(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua))),Uu) ).

% ATP.lambda_127
tff(fact_8300_ATP_Olambda__128,axiom,
    ! [Uu: product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),Uua: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o))] : aa(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),heap_Time_Heap($o),aTP_Lamp_lo(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),fun(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),heap_Time_Heap($o)),Uu),Uua) = aa(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o),product_case_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap($o),product_case_prod(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aTP_Lamp_ln(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uua))),Uu) ).

% ATP.lambda_128
tff(fact_8301_ATP_Olambda__129,axiom,
    ! [Uu: product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),Uua: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat)))] : aa(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),heap_Time_Heap(option(nat)),aTP_Lamp_jo(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),fun(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),heap_Time_Heap(option(nat))),Uu),Uua) = aa(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat)),product_case_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(option(nat)),product_case_prod(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aTP_Lamp_jn(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua))),Uu) ).

% ATP.lambda_129
tff(fact_8302_ATP_Olambda__130,axiom,
    ! [Uu: product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),Uua: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat)))] : aa(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),heap_Time_Heap(option(nat)),aTP_Lamp_ix(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),fun(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),heap_Time_Heap(option(nat))),Uu),Uua) = aa(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat)),product_case_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(option(nat)),product_case_prod(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aTP_Lamp_iw(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua))),Uu) ).

% ATP.lambda_130
tff(fact_8303_ATP_Olambda__131,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_zj(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% ATP.lambda_131
tff(fact_8304_ATP_Olambda__132,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aby(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,suc,Uua)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,suc,Uua))) ) ).

% ATP.lambda_132
tff(fact_8305_ATP_Olambda__133,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aaj(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_133
tff(fact_8306_ATP_Olambda__134,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_zb(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_134
tff(fact_8307_ATP_Olambda__135,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_vp(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_135
tff(fact_8308_ATP_Olambda__136,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: num,Uua: num] : aa(num,word(A),aTP_Lamp_acs(num,fun(num,word(A)),Uu),Uua) = aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),Uu)),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,word(A),numeral_numeral(word(A)),Uua))) ) ).

% ATP.lambda_136
tff(fact_8309_ATP_Olambda__137,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_acu(num,fun(num,int),Uu),Uua) = aa(int,int,bit_se2584673776208193580ke_bit(int,aa(num,nat,numeral_numeral(nat),Uu)),aa(int,int,minus_minus(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(num,nat,numeral_numeral(nat),Uu))),aa(num,int,numeral_numeral(int),Uua))) ).

% ATP.lambda_137
tff(fact_8310_ATP_Olambda__138,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aaz(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uu)),Uua)) ) ).

% ATP.lambda_138
tff(fact_8311_ATP_Olambda__139,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_afq(nat,fun(real,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),abs_abs(real,Uua)),Uu)) ).

% ATP.lambda_139
tff(fact_8312_ATP_Olambda__140,axiom,
    ! [Uu: $o,Uua: nat] : aa(nat,nat,aa($o,fun(nat,nat),aTP_Lamp_ano($o,fun(nat,nat)),(Uu)),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa($o,nat,zero_neq_one_of_bool(nat),(Uu))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% ATP.lambda_140
tff(fact_8313_ATP_Olambda__141,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aak(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_141
tff(fact_8314_ATP_Olambda__142,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aal(A,fun(nat,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_142
tff(fact_8315_ATP_Olambda__143,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_bag(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_143
tff(fact_8316_ATP_Olambda__144,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_baf(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua)) ) ).

% ATP.lambda_144
tff(fact_8317_ATP_Olambda__145,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_xj(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),sin_coeff(Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_145
tff(fact_8318_ATP_Olambda__146,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_xh(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),cos_coeff(Uua)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_146
tff(fact_8319_ATP_Olambda__147,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_alh(list(list(A)),fun(A,list(list(A))),Uu),Uua) = map(list(A),list(A),cons(A,Uua),product_lists(A,Uu)) ).

% ATP.lambda_147
tff(fact_8320_ATP_Olambda__148,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_aoe(code_integer,fun(code_integer,num)),Uu),Uua) = $let(
        l2: num,
        l2:= code_num_of_integer(Uu),
        $let(
          l3: num,
          l3:= aa(num,num,aa(num,fun(num,num),plus_plus(num),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(num,num,aa(num,fun(num,num),plus_plus(num),l3),one2)) ) ) ).

% ATP.lambda_148
tff(fact_8321_ATP_Olambda__149,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_aod(code_integer,fun(code_integer,nat)),Uu),Uua) = $let(
        l2: nat,
        l2:= aa(code_integer,nat,code_nat_of_integer,Uu),
        $let(
          l3: nat,
          l3:= aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l2),l2),
          $ite(Uua = zero_zero(code_integer),l3,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),l3),one_one(nat))) ) ) ).

% ATP.lambda_149
tff(fact_8322_ATP_Olambda__150,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_zf(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_150
tff(fact_8323_ATP_Olambda__151,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_ali(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu) ).

% ATP.lambda_151
tff(fact_8324_ATP_Olambda__152,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_adi(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,real,exp(real),Uua) = Uu ) ) ).

% ATP.lambda_152
tff(fact_8325_ATP_Olambda__153,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_gh(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))) ) ).

% ATP.lambda_153
tff(fact_8326_ATP_Olambda__154,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: A,Uua: A] : aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_gk(A,fun(A,product_prod(A,A))),Uu),Uua) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)) ) ).

% ATP.lambda_154
tff(fact_8327_ATP_Olambda__155,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_xl(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),Uua)),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% ATP.lambda_155
tff(fact_8328_ATP_Olambda__156,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aku(list(A),fun(A,$o),Uu),Uua)
        <=> aa(A,$o,ord_less(A,Uua),aa(nat,A,nth(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ).

% ATP.lambda_156
tff(fact_8329_ATP_Olambda__157,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_akv(list(A),fun(A,$o),Uu),Uua)
        <=> ( Uua = aa(nat,A,nth(A,Uu),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uu)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% ATP.lambda_157
tff(fact_8330_ATP_Olambda__158,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aaa(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),aa(nat,nat,binomial(Uu),Uua)) ).

% ATP.lambda_158
tff(fact_8331_ATP_Olambda__159,axiom,
    ! [A: $tType,Uu: A,Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aTP_Lamp_akw(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Uu),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),zero_zero(nat))) ).

% ATP.lambda_159
tff(fact_8332_ATP_Olambda__160,axiom,
    ! [A: $tType,Uu: A,Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),aTP_Lamp_ajv(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Uu),Uua) = aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Uu),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),one_one(nat))) ).

% ATP.lambda_160
tff(fact_8333_ATP_Olambda__161,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aja(set(A),fun(A,$o),Uu),Uua)
    <=> ( Uu = aa(set(A),set(A),insert(A,Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_161
tff(fact_8334_ATP_Olambda__162,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_bad(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),times_times(real),Uu),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))))) ).

% ATP.lambda_162
tff(fact_8335_ATP_Olambda__163,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_azw(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or1337092689740270186AtMost(nat,zero_zero(nat),Uua)) ) ).

% ATP.lambda_163
tff(fact_8336_ATP_Olambda__164,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_ajj(fun(nat,A),fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),aa(nat,A,Uu,Uua)) ).

% ATP.lambda_164
tff(fact_8337_ATP_Olambda__165,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_apf(fun(A,B),fun(A,product_prod(A,B)),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),aa(A,B,Uu,Uua)) ).

% ATP.lambda_165
tff(fact_8338_ATP_Olambda__166,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aix(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Uua),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Uu),Uua))) ) ).

% ATP.lambda_166
tff(fact_8339_ATP_Olambda__167,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_azy(fun(nat,A),fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua))) ) ).

% ATP.lambda_167
tff(fact_8340_ATP_Olambda__168,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_acq(int,fun(int,int)),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uu),aa($o,int,zero_neq_one_of_bool(int),Uua != zero_zero(int))) ).

% ATP.lambda_168
tff(fact_8341_ATP_Olambda__169,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_azx(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu),aa(real,real,uminus_uminus(real),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua))))) ).

% ATP.lambda_169
tff(fact_8342_ATP_Olambda__170,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_azn(real,fun(nat,real),Uu),Uua) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Uu),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uua)))) ).

% ATP.lambda_170
tff(fact_8343_ATP_Olambda__171,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_azo(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_171
tff(fact_8344_ATP_Olambda__172,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_azz(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_172
tff(fact_8345_ATP_Olambda__173,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_azp(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_173
tff(fact_8346_ATP_Olambda__174,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_baa(fun(nat,A),fun(nat,A),Uu),Uua) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_174
tff(fact_8347_ATP_Olambda__175,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_azb(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_175
tff(fact_8348_ATP_Olambda__176,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_ald(nat,fun(list(A),$o),Uu),Uua)
    <=> aa(nat,$o,ord_less(nat,Uu),aa(list(A),nat,size_size(list(A)),Uua)) ) ).

% ATP.lambda_176
tff(fact_8349_ATP_Olambda__177,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_zg(A,fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_177
tff(fact_8350_ATP_Olambda__178,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_afn(A,fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_178
tff(fact_8351_ATP_Olambda__179,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_yr(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_179
tff(fact_8352_ATP_Olambda__180,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_bck(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> eventually(A,Uu,topolo7230453075368039082e_nhds(A,Uua)) ) ) ).

% ATP.lambda_180
tff(fact_8353_ATP_Olambda__181,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: nat,Uua: A] : aa(A,A,aTP_Lamp_axw(nat,fun(A,A),Uu),Uua) = comm_s3205402744901411588hammer(A,Uua,Uu) ) ).

% ATP.lambda_181
tff(fact_8354_ATP_Olambda__182,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: real] : aa(real,A,aTP_Lamp_bar(A,fun(real,A),Uu),Uua) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),Uu) ) ).

% ATP.lambda_182
tff(fact_8355_ATP_Olambda__183,axiom,
    ! [Uu: nat,Uua: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap($o),aTP_Lamp_hh(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)),Uu),Uua) = aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),vEBT_vebt_memberi,Uua),Uu) ).

% ATP.lambda_183
tff(fact_8356_ATP_Olambda__184,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_sy(set(A),fun(set(A),$o),Uu),Uua)
    <=> aa(set(A),$o,ord_less_eq(set(A),Uua),Uu) ) ).

% ATP.lambda_184
tff(fact_8357_ATP_Olambda__185,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_sw(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,ord_less_eq(nat,Uua),Uu) ) ).

% ATP.lambda_185
tff(fact_8358_ATP_Olambda__186,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_zn(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,ord_less_eq(A,Uua),Uu) ) ) ).

% ATP.lambda_186
tff(fact_8359_ATP_Olambda__187,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aim(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Uua),Uu) ).

% ATP.lambda_187
tff(fact_8360_ATP_Olambda__188,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_bap(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_188
tff(fact_8361_ATP_Olambda__189,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ahi(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_189
tff(fact_8362_ATP_Olambda__190,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_aot(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_190
tff(fact_8363_ATP_Olambda__191,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_sx(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,ord_less(nat,Uua),Uu) ) ).

% ATP.lambda_191
tff(fact_8364_ATP_Olambda__192,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_dm(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,ord_less(A,Uua),Uu) ) ) ).

% ATP.lambda_192
tff(fact_8365_ATP_Olambda__193,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_vv(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,ord_less(A,Uua),Uu) ) ) ).

% ATP.lambda_193
tff(fact_8366_ATP_Olambda__194,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ayx(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uua),Uu) ).

% ATP.lambda_194
tff(fact_8367_ATP_Olambda__195,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_bas(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_195
tff(fact_8368_ATP_Olambda__196,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_fy(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_196
tff(fact_8369_ATP_Olambda__197,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ahy(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_197
tff(fact_8370_ATP_Olambda__198,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aiq(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,minus_minus(nat,Uua),Uu) ).

% ATP.lambda_198
tff(fact_8371_ATP_Olambda__199,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ahg(A,fun(A,A),Uu),Uua) = aa(A,A,minus_minus(A,Uua),Uu) ) ).

% ATP.lambda_199
tff(fact_8372_ATP_Olambda__200,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ahe(A,fun(A,A),Uu),Uua) = aa(A,A,minus_minus(A,Uua),Uu) ) ).

% ATP.lambda_200
tff(fact_8373_ATP_Olambda__201,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_aqj(nat,fun(real,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu) ).

% ATP.lambda_201
tff(fact_8374_ATP_Olambda__202,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,code_integer,aTP_Lamp_aip(code_integer,fun(code_integer,code_integer),Uu),Uua) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),Uua),Uu) ).

% ATP.lambda_202
tff(fact_8375_ATP_Olambda__203,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ajg(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_203
tff(fact_8376_ATP_Olambda__204,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_ain(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_204
tff(fact_8377_ATP_Olambda__205,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_aow(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_205
tff(fact_8378_ATP_Olambda__206,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ahf(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_206
tff(fact_8379_ATP_Olambda__207,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: list(A),Uua: array(A)] : aa(array(A),assn,aTP_Lamp_aem(list(A),fun(array(A),assn),Uu),Uua) = aa(list(A),assn,snga_assn(A,Uua),Uu) ) ).

% ATP.lambda_207
tff(fact_8380_ATP_Olambda__208,axiom,
    ! [Uu: real,Uua: real] : aa(real,real,aTP_Lamp_aql(real,fun(real,real),Uu),Uua) = powr(real,Uua,Uu) ).

% ATP.lambda_208
tff(fact_8381_ATP_Olambda__209,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_su(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,dvd_dvd(nat,Uua),Uu) ) ).

% ATP.lambda_209
tff(fact_8382_ATP_Olambda__210,axiom,
    ! [Uu: int,Uua: int] :
      ( aa(int,$o,aTP_Lamp_tg(int,fun(int,$o),Uu),Uua)
    <=> aa(int,$o,dvd_dvd(int,Uua),Uu) ) ).

% ATP.lambda_210
tff(fact_8383_ATP_Olambda__211,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_gq(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,dvd_dvd(A,Uua),Uu) ) ) ).

% ATP.lambda_211
tff(fact_8384_ATP_Olambda__212,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_gp(B,fun(A,product_prod(A,B))),Uu),Uua) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uu) ).

% ATP.lambda_212
tff(fact_8385_ATP_Olambda__213,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_ajl(A,fun(nat,product_prod(nat,A)),Uu),Uua) = aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Uua),Uu) ).

% ATP.lambda_213
tff(fact_8386_ATP_Olambda__214,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_tq(A,fun(B,product_prod(B,A))),Uu),Uua) = aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uu) ).

% ATP.lambda_214
tff(fact_8387_ATP_Olambda__215,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_zl(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_215
tff(fact_8388_ATP_Olambda__216,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_alj(list(A),fun(A,list(A))),Uu),Uua) = aa(list(A),list(A),cons(A,Uua),Uu) ).

% ATP.lambda_216
tff(fact_8389_ATP_Olambda__217,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_azm(real,fun(nat,real),Uu),Uua) = aa(real,real,root(Uua),Uu) ).

% ATP.lambda_217
tff(fact_8390_ATP_Olambda__218,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_cc(set(A),fun(A,$o),Uu),Uua)
    <=> member(A,Uua,Uu) ) ).

% ATP.lambda_218
tff(fact_8391_ATP_Olambda__219,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_alc(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).

% ATP.lambda_219
tff(fact_8392_ATP_Olambda__220,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_gc(A,fun(A,$o),Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_220
tff(fact_8393_ATP_Olambda__221,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_alu(A,fun(list(A),list(A))),Uu),Uua) = aa(list(A),list(A),cons(A,Uu),nil(A)) ).

% ATP.lambda_221
tff(fact_8394_ATP_Olambda__222,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_alv(A,fun(list(A),list(list(A)))),Uu),Uua) = aa(list(list(A)),list(list(A)),cons(list(A),Uua),nil(list(A))) ).

% ATP.lambda_222
tff(fact_8395_ATP_Olambda__223,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_bdn(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,ord_less_eq(real,zero_zero(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_223
tff(fact_8396_ATP_Olambda__224,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] :
          ( aa(A,$o,aTP_Lamp_bdb(fun(A,real),fun(A,$o),Uu),Uua)
        <=> aa(real,$o,ord_less_eq(real,one_one(real)),aa(A,real,Uu,Uua)) ) ) ).

% ATP.lambda_224
tff(fact_8397_ATP_Olambda__225,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_bdg(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,ord_less_eq(real,one_one(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_225
tff(fact_8398_ATP_Olambda__226,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] :
      ( aa(A,$o,aTP_Lamp_bdt(fun(A,real),fun(A,$o),Uu),Uua)
    <=> aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_226
tff(fact_8399_ATP_Olambda__227,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ti(fun(A,nat),fun(A,$o),Uu),Uua)
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(A,nat,Uu,Uua)) ) ).

% ATP.lambda_227
tff(fact_8400_ATP_Olambda__228,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_wt(fun(nat,real),fun(nat,real),Uu),Uua) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))) ).

% ATP.lambda_228
tff(fact_8401_ATP_Olambda__229,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_ws(fun(nat,real),fun(nat,real),Uu),Uua) = aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)) ).

% ATP.lambda_229
tff(fact_8402_ATP_Olambda__230,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_beq(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),Uua)) ) ).

% ATP.lambda_230
tff(fact_8403_ATP_Olambda__231,axiom,
    ! [Uu: fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),$o),Uua: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi))] :
      ( aa(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),$o,aTP_Lamp_nk(fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),$o),fun(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),$o),Uu),Uua)
    <=> aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),$o,Uu,product_curry(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),product_curry(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(vEBT_VEBTi),Uua))) ) ).

% ATP.lambda_231
tff(fact_8404_ATP_Olambda__232,axiom,
    ! [Uu: fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o),Uua: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat)))] :
      ( aa(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),$o,aTP_Lamp_ju(fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o),fun(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),$o),Uu),Uua)
    <=> aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o,Uu,product_curry(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),product_curry(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(option(nat)),Uua))) ) ).

% ATP.lambda_232
tff(fact_8405_ATP_Olambda__233,axiom,
    ! [Uu: fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),$o),Uua: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o))] :
      ( aa(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),$o,aTP_Lamp_lv(fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),$o),fun(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),$o),Uu),Uua)
    <=> aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),$o,Uu,product_curry(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),product_curry(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap($o),Uua))) ) ).

% ATP.lambda_233
tff(fact_8406_ATP_Olambda__234,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(set(nat),A),Uua: nat] : aa(nat,A,aTP_Lamp_ayy(fun(set(nat),A),fun(nat,A),Uu),Uua) = aa(set(nat),A,Uu,set_ord_lessThan(nat,Uua)) ) ).

% ATP.lambda_234
tff(fact_8407_ATP_Olambda__235,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(set(nat),A),Uua: nat] : aa(nat,A,aTP_Lamp_ayz(fun(set(nat),A),fun(nat,A),Uu),Uua) = aa(set(nat),A,Uu,set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_235
tff(fact_8408_ATP_Olambda__236,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(real,A),Uua: nat] : aa(nat,A,aTP_Lamp_ben(fun(real,A),fun(nat,A),Uu),Uua) = aa(real,A,Uu,aa(nat,real,semiring_1_of_nat(real),Uua)) ) ).

% ATP.lambda_236
tff(fact_8409_ATP_Olambda__237,axiom,
    ! [A: $tType,Uu: fun(int,A),Uua: nat] : aa(nat,A,aTP_Lamp_bee(fun(int,A),fun(nat,A),Uu),Uua) = aa(int,A,Uu,aa(nat,int,semiring_1_of_nat(int),Uua)) ).

% ATP.lambda_237
tff(fact_8410_ATP_Olambda__238,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_apm(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,Uu,aa(real,real,uminus_uminus(real),Uua)) ).

% ATP.lambda_238
tff(fact_8411_ATP_Olambda__239,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_fk(fun(B,C),fun(product_prod(A,B),C),Uu),Uua) = aa(B,C,Uu,aa(product_prod(A,B),B,product_snd(A,B),Uua)) ).

% ATP.lambda_239
tff(fact_8412_ATP_Olambda__240,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_fi(fun(A,C),fun(product_prod(A,B),C),Uu),Uua) = aa(A,C,Uu,aa(product_prod(A,B),A,product_fst(A,B),Uua)) ).

% ATP.lambda_240
tff(fact_8413_ATP_Olambda__241,axiom,
    ! [Uu: fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),$o),Uua: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi))] :
      ( aa(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),$o,aTP_Lamp_qd(fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),$o),fun(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),$o),Uu),Uua)
    <=> aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),$o,Uu,product_curry(vEBT_VEBTi,nat,heap_Time_Heap(vEBT_VEBTi),Uua)) ) ).

% ATP.lambda_241
tff(fact_8414_ATP_Olambda__242,axiom,
    ! [Uu: fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o),Uua: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat)))] :
      ( aa(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),$o,aTP_Lamp_lp(fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o),fun(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),$o),Uu),Uua)
    <=> aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o,Uu,product_curry(vEBT_VEBTi,nat,heap_Time_Heap(option(nat)),Uua)) ) ).

% ATP.lambda_242
tff(fact_8415_ATP_Olambda__243,axiom,
    ! [Uu: fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),$o),Uua: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o))] :
      ( aa(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),$o,aTP_Lamp_ma(fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),$o),fun(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),$o),Uu),Uua)
    <=> aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),$o,Uu,product_curry(vEBT_VEBTi,nat,heap_Time_Heap($o),Uua)) ) ).

% ATP.lambda_243
tff(fact_8416_ATP_Olambda__244,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_anh(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_244
tff(fact_8417_ATP_Olambda__245,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_mw(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_245
tff(fact_8418_ATP_Olambda__246,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ayq(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_246
tff(fact_8419_ATP_Olambda__247,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_vz(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_247
tff(fact_8420_ATP_Olambda__248,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_yi(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_248
tff(fact_8421_ATP_Olambda__249,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_uw(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_249
tff(fact_8422_ATP_Olambda__250,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ajk(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_250
tff(fact_8423_ATP_Olambda__251,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_agb(A,fun(fun(A,B),B),Uu),Uua) = aa(A,B,Uua,Uu) ).

% ATP.lambda_251
tff(fact_8424_ATP_Olambda__252,axiom,
    ! [B: $tType,A: $tType,Uu: fun(set(A),A),Uua: B] : aa(B,A,aTP_Lamp_agg(fun(set(A),A),fun(B,A),Uu),Uua) = aa(set(A),A,Uu,bot_bot(set(A))) ).

% ATP.lambda_252
tff(fact_8425_ATP_Olambda__253,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_anv(A,fun(option(A),option(A))),Uu),Uua) = aa(A,option(A),some(A),case_option(A,A,Uu,ord_max(A,Uu),Uua)) ) ).

% ATP.lambda_253
tff(fact_8426_ATP_Olambda__254,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_adb(nat,fun(num,option(num)),Uu),Uua) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Uu,Uua))) ).

% ATP.lambda_254
tff(fact_8427_ATP_Olambda__255,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_acw(num,fun(nat,option(num)),Uu),Uua) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Uua,Uu))) ).

% ATP.lambda_255
tff(fact_8428_ATP_Olambda__256,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A)] : aa(list(A),fun(product_prod(list(A),list(A)),list(A)),aTP_Lamp_aln(fun(A,B),fun(list(A),fun(product_prod(list(A),list(A)),list(A))),Uu),Uua) = product_case_prod(list(A),list(A),list(A),aa(list(A),fun(list(A),fun(list(A),list(A))),aTP_Lamp_alm(fun(A,B),fun(list(A),fun(list(A),fun(list(A),list(A)))),Uu),Uua)) ) ).

% ATP.lambda_256
tff(fact_8429_ATP_Olambda__257,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,heap_Time_Heap(A)),Uua: B] : aa(B,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_akl(fun(B,heap_Time_Heap(A)),fun(B,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),Uu),Uua) = product_case_prod(heap_ext(product_unit),nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(B,fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),aTP_Lamp_akk(fun(B,heap_Time_Heap(A)),fun(B,fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))))),Uu),Uua)) ).

% ATP.lambda_257
tff(fact_8430_ATP_Olambda__258,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,assn)),Uua: A] : aa(A,assn,aTP_Lamp_aex(fun(B,fun(A,assn)),fun(A,assn),Uu),Uua) = ex_assn(B,aa(A,fun(B,assn),aTP_Lamp_aew(fun(B,fun(A,assn)),fun(A,fun(B,assn)),Uu),Uua)) ).

% ATP.lambda_258
tff(fact_8431_ATP_Olambda__259,axiom,
    ! [Uu: fun(nat,real),Uua: real] : aa(real,real,aTP_Lamp_arb(fun(nat,real),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),aTP_Lamp_ara(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua)) ).

% ATP.lambda_259
tff(fact_8432_ATP_Olambda__260,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_aqf(fun(nat,A),fun(A,A),Uu),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_xc(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua)) ) ).

% ATP.lambda_260
tff(fact_8433_ATP_Olambda__261,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_acm(nat,fun(nat,complex),Uu),Uua) = cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi)),aa(nat,real,semiring_1_of_nat(real),Uua))),aa(nat,real,semiring_1_of_nat(real),Uu))) ).

% ATP.lambda_261
tff(fact_8434_ATP_Olambda__262,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_asg(A,fun(A,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),Uu)),Uua)),aa(A,A,inverse_inverse(A),Uu))) ) ).

% ATP.lambda_262
tff(fact_8435_ATP_Olambda__263,axiom,
    ! [A: $tType,Uu: fun(heap_ext(product_unit),A),Uua: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_akc(fun(heap_ext(product_unit),A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Uu),Uua) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),aa(heap_ext(product_unit),A,Uu,Uua)),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uua),one_one(nat)))) ).

% ATP.lambda_263
tff(fact_8436_ATP_Olambda__264,axiom,
    ! [Uu: fun(real,real),Uua: real] :
      ( aa(real,$o,aTP_Lamp_bdz(fun(real,real),fun(real,$o),Uu),Uua)
    <=> ( aa(real,real,Uu,Uua) != zero_zero(real) ) ) ).

% ATP.lambda_264
tff(fact_8437_ATP_Olambda__265,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_ans(fun(B,option(A)),fun(B,$o),Uu),Uua)
    <=> ( aa(B,option(A),Uu,Uua) != none(A) ) ) ).

% ATP.lambda_265
tff(fact_8438_ATP_Olambda__266,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_aax(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_266
tff(fact_8439_ATP_Olambda__267,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_acp(nat,fun(nat,$o),Uu),Uua)
    <=> ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ) ).

% ATP.lambda_267
tff(fact_8440_ATP_Olambda__268,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aay(A,fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uu)),Uua))) ) ).

% ATP.lambda_268
tff(fact_8441_ATP_Olambda__269,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_aam(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,sin_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_269
tff(fact_8442_ATP_Olambda__270,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_aan(A,fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,cos_coeff(Uua)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uua))) ) ).

% ATP.lambda_270
tff(fact_8443_ATP_Olambda__271,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_zi(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_271
tff(fact_8444_ATP_Olambda__272,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_xm(A,fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),Uua))) ) ).

% ATP.lambda_272
tff(fact_8445_ATP_Olambda__273,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_azv(real,fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uua)) ).

% ATP.lambda_273
tff(fact_8446_ATP_Olambda__274,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),heap_Time_Heap(list(A)),aTP_Lamp_afw(A,fun(list(A),heap_Time_Heap(list(A))),Uu),Uua) = aa(list(A),heap_Time_Heap(list(A)),heap_Time_return(list(A)),aa(list(A),list(A),cons(A,Uu),Uua)) ).

% ATP.lambda_274
tff(fact_8447_ATP_Olambda__275,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_apk(A,fun(A,A),Uu),Uua) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)) ) ).

% ATP.lambda_275
tff(fact_8448_ATP_Olambda__276,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aml(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,ord_min(nat,Uu,Uua)) ).

% ATP.lambda_276
tff(fact_8449_ATP_Olambda__277,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_amk(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,ord_min(nat,Uua,Uu)) ).

% ATP.lambda_277
tff(fact_8450_ATP_Olambda__278,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_adl(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,Uu),Uua)) ).

% ATP.lambda_278
tff(fact_8451_ATP_Olambda__279,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_adk(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,Uua),Uu)) ).

% ATP.lambda_279
tff(fact_8452_ATP_Olambda__280,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_amf(A,fun(A,$o),Uu),Uua)
        <=> ~ aa(A,$o,ord_less_eq(A,Uu),Uua) ) ) ).

% ATP.lambda_280
tff(fact_8453_ATP_Olambda__281,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_vt(set(A),fun(A,$o),Uu),Uua)
    <=> ~ member(A,Uua,Uu) ) ).

% ATP.lambda_281
tff(fact_8454_ATP_Olambda__282,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_bdv(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_282
tff(fact_8455_ATP_Olambda__283,axiom,
    ! [A: $tType] :
      ( topological_t1_space(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_bcj(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_283
tff(fact_8456_ATP_Olambda__284,axiom,
    ! [A: $tType] :
      ( ( linorder(A)
        & no_top(A) )
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_bbx(A,fun(A,$o),Uu),Uua)
        <=> ( Uua != Uu ) ) ) ).

% ATP.lambda_284
tff(fact_8457_ATP_Olambda__285,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_ajs(A,fun(A,$o),Uu),Uua)
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_285
tff(fact_8458_ATP_Olambda__286,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_nh(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_286
tff(fact_8459_ATP_Olambda__287,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_abd(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_287
tff(fact_8460_ATP_Olambda__288,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_mq(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_288
tff(fact_8461_ATP_Olambda__289,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_ue(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_289
tff(fact_8462_ATP_Olambda__290,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_xu(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_290
tff(fact_8463_ATP_Olambda__291,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_auo(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_291
tff(fact_8464_ATP_Olambda__292,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_yb(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_292
tff(fact_8465_ATP_Olambda__293,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_avv(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_293
tff(fact_8466_ATP_Olambda__294,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_azj(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_294
tff(fact_8467_ATP_Olambda__295,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_ase(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_295
tff(fact_8468_ATP_Olambda__296,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_apn(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_296
tff(fact_8469_ATP_Olambda__297,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_atz(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_297
tff(fact_8470_ATP_Olambda__298,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_bed(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_298
tff(fact_8471_ATP_Olambda__299,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_avm(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,inverse_inverse(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_299
tff(fact_8472_ATP_Olambda__300,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_xp(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_300
tff(fact_8473_ATP_Olambda__301,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_tt(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_301
tff(fact_8474_ATP_Olambda__302,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,real,aTP_Lamp_go(fun(A,nat),fun(A,real),Uu),Uua) = aa(nat,real,semiring_1_of_nat(real),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_302
tff(fact_8475_ATP_Olambda__303,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,int,aTP_Lamp_vr(fun(A,nat),fun(A,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_303
tff(fact_8476_ATP_Olambda__304,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_asn(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_304
tff(fact_8477_ATP_Olambda__305,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ayw(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_305
tff(fact_8478_ATP_Olambda__306,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_zc(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_306
tff(fact_8479_ATP_Olambda__307,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,complex,aTP_Lamp_tr(fun(nat,real),fun(nat,complex),Uu),Uua) = aa(real,complex,real_Vector_of_real(complex),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_307
tff(fact_8480_ATP_Olambda__308,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_vq(fun(nat,real),fun(nat,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_308
tff(fact_8481_ATP_Olambda__309,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_uz(fun(nat,real),fun(nat,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_309
tff(fact_8482_ATP_Olambda__310,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_xa(fun(nat,real),fun(nat,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_310
tff(fact_8483_ATP_Olambda__311,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_tw(fun(B,real),fun(B,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(B,real,Uu,Uua)) ) ).

% ATP.lambda_311
tff(fact_8484_ATP_Olambda__312,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2191834092415804123ebra_1(A) )
     => ! [Uu: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_xr(fun(B,real),fun(B,A),Uu),Uua) = aa(real,A,real_Vector_of_real(A),aa(B,real,Uu,Uua)) ) ).

% ATP.lambda_312
tff(fact_8485_ATP_Olambda__313,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_art(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_313
tff(fact_8486_ATP_Olambda__314,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_auf(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_314
tff(fact_8487_ATP_Olambda__315,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_aws(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_315
tff(fact_8488_ATP_Olambda__316,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V2191834092415804123ebra_1(B)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: A] : aa(A,B,aTP_Lamp_awt(fun(A,real),fun(A,B),Uu),Uua) = aa(real,B,real_Vector_of_real(B),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_316
tff(fact_8489_ATP_Olambda__317,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_mx(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_317
tff(fact_8490_ATP_Olambda__318,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_wu(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_318
tff(fact_8491_ATP_Olambda__319,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_xb(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_319
tff(fact_8492_ATP_Olambda__320,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_um(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_320
tff(fact_8493_ATP_Olambda__321,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_arx(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_321
tff(fact_8494_ATP_Olambda__322,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_apj(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_322
tff(fact_8495_ATP_Olambda__323,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_atu(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_323
tff(fact_8496_ATP_Olambda__324,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_avc(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_324
tff(fact_8497_ATP_Olambda__325,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_avb(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_325
tff(fact_8498_ATP_Olambda__326,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_yg(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_326
tff(fact_8499_ATP_Olambda__327,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_xq(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_327
tff(fact_8500_ATP_Olambda__328,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_tu(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_328
tff(fact_8501_ATP_Olambda__329,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,heap_Time_Heap(A),aTP_Lamp_aog(fun(B,A),fun(B,heap_Time_Heap(A)),Uu),Uua) = aa(A,heap_Time_Heap(A),heap_Time_return(A),aa(B,A,Uu,Uua)) ).

% ATP.lambda_329
tff(fact_8502_ATP_Olambda__330,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_yu(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua) = aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_330
tff(fact_8503_ATP_Olambda__331,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_bdo(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_331
tff(fact_8504_ATP_Olambda__332,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_avf(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,artanh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_332
tff(fact_8505_ATP_Olambda__333,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ays(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arsinh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_333
tff(fact_8506_ATP_Olambda__334,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_awo(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arsinh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_334
tff(fact_8507_ATP_Olambda__335,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_asz(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_335
tff(fact_8508_ATP_Olambda__336,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ayo(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_336
tff(fact_8509_ATP_Olambda__337,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_awp(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arctan,aa(A,real,Uu,Uua)) ).

% ATP.lambda_337
tff(fact_8510_ATP_Olambda__338,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_arc(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcsin,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_338
tff(fact_8511_ATP_Olambda__339,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_bdc(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_339
tff(fact_8512_ATP_Olambda__340,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_avg(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arcosh(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_340
tff(fact_8513_ATP_Olambda__341,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_atf(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,arccos,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_341
tff(fact_8514_ATP_Olambda__342,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,product_prod(B,C)),Uua: A] : aa(A,C,aTP_Lamp_aus(fun(A,product_prod(B,C)),fun(A,C),Uu),Uua) = aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),Uu,Uua)) ) ).

% ATP.lambda_342
tff(fact_8515_ATP_Olambda__343,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,product_prod(B,C)),Uua: A] : aa(A,C,aTP_Lamp_awc(fun(A,product_prod(B,C)),fun(A,C),Uu),Uua) = aa(product_prod(B,C),C,product_snd(B,C),aa(A,product_prod(B,C),Uu,Uua)) ) ).

% ATP.lambda_343
tff(fact_8516_ATP_Olambda__344,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,product_prod(B,C)),Uua: A] : aa(A,B,aTP_Lamp_aur(fun(A,product_prod(B,C)),fun(A,B),Uu),Uua) = aa(product_prod(B,C),B,product_fst(B,C),aa(A,product_prod(B,C),Uu,Uua)) ) ).

% ATP.lambda_344
tff(fact_8517_ATP_Olambda__345,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,product_prod(B,C)),Uua: A] : aa(A,B,aTP_Lamp_awd(fun(A,product_prod(B,C)),fun(A,B),Uu),Uua) = aa(product_prod(B,C),B,product_fst(B,C),aa(A,product_prod(B,C),Uu,Uua)) ) ).

% ATP.lambda_345
tff(fact_8518_ATP_Olambda__346,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aua(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_346
tff(fact_8519_ATP_Olambda__347,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_avw(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,sgn_sgn(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_347
tff(fact_8520_ATP_Olambda__348,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_my(fun(nat,real),fun(nat,real),Uu),Uua) = abs_abs(real,aa(nat,real,Uu,Uua)) ).

% ATP.lambda_348
tff(fact_8521_ATP_Olambda__349,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_tv(fun(B,A),fun(B,A),Uu),Uua) = abs_abs(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_349
tff(fact_8522_ATP_Olambda__350,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_yf(fun(B,A),fun(B,A),Uu),Uua) = abs_abs(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_350
tff(fact_8523_ATP_Olambda__351,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aym(fun(A,real),fun(A,real),Uu),Uua) = abs_abs(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_351
tff(fact_8524_ATP_Olambda__352,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_avt(fun(A,real),fun(A,real),Uu),Uua) = abs_abs(real,aa(A,real,Uu,Uua)) ).

% ATP.lambda_352
tff(fact_8525_ATP_Olambda__353,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aya(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_353
tff(fact_8526_ATP_Olambda__354,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_aqp(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tanh(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_354
tff(fact_8527_ATP_Olambda__355,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_avp(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,tanh(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_355
tff(fact_8528_ATP_Olambda__356,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_awx(fun(A,B),fun(A,B),Uu),Uua) = sinh(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_356
tff(fact_8529_ATP_Olambda__357,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_aqe(fun(A,A),fun(A,A),Uu),Uua) = sinh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_357
tff(fact_8530_ATP_Olambda__358,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_awz(fun(A,B),fun(A,B),Uu),Uua) = sinh(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_358
tff(fact_8531_ATP_Olambda__359,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_awy(fun(A,B),fun(A,B),Uu),Uua) = cosh(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_359
tff(fact_8532_ATP_Olambda__360,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_aqd(fun(A,A),fun(A,A),Uu),Uua) = cosh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_360
tff(fact_8533_ATP_Olambda__361,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_axa(fun(A,B),fun(A,B),Uu),Uua) = cosh(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_361
tff(fact_8534_ATP_Olambda__362,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_atb(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,tan(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_362
tff(fact_8535_ATP_Olambda__363,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_avu(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,tan(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_363
tff(fact_8536_ATP_Olambda__364,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_asa(fun(A,real),fun(A,real),Uu),Uua) = sin(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_364
tff(fact_8537_ATP_Olambda__365,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_auy(fun(A,B),fun(A,B),Uu),Uua) = sin(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_365
tff(fact_8538_ATP_Olambda__366,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_apu(fun(A,A),fun(A,A),Uu),Uua) = sin(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_366
tff(fact_8539_ATP_Olambda__367,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ary(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,exp(real),aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_367
tff(fact_8540_ATP_Olambda__368,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_auz(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,exp(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_368
tff(fact_8541_ATP_Olambda__369,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_apr(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,exp(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_369
tff(fact_8542_ATP_Olambda__370,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_axb(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,exp(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_370
tff(fact_8543_ATP_Olambda__371,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_ym(fun(A,B),fun(A,B),Uu),Uua) = aa(B,B,exp(B),aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_371
tff(fact_8544_ATP_Olambda__372,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_avo(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,cot(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_372
tff(fact_8545_ATP_Olambda__373,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_baj(fun(nat,real),fun(nat,real),Uu),Uua) = cos(real,aa(nat,real,Uu,Uua)) ).

% ATP.lambda_373
tff(fact_8546_ATP_Olambda__374,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_asj(fun(A,real),fun(A,real),Uu),Uua) = cos(real,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_374
tff(fact_8547_ATP_Olambda__375,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,B,aTP_Lamp_aux(fun(A,B),fun(A,B),Uu),Uua) = cos(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_375
tff(fact_8548_ATP_Olambda__376,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_apl(fun(A,A),fun(A,A),Uu),Uua) = cos(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_376
tff(fact_8549_ATP_Olambda__377,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),Uua: A] : aa(A,assn,aTP_Lamp_aev(fun(A,fun(B,assn)),fun(A,assn),Uu),Uua) = ex_assn(B,aa(A,fun(B,assn),Uu,Uua)) ).

% ATP.lambda_377
tff(fact_8550_ATP_Olambda__378,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: real] : aa(real,real,aTP_Lamp_ate(fun(real,fun(nat,real)),fun(real,real),Uu),Uua) = suminf(real,aa(real,fun(nat,real),Uu,Uua)) ).

% ATP.lambda_378
tff(fact_8551_ATP_Olambda__379,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_Vector_banach(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_ayb(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_379
tff(fact_8552_ATP_Olambda__380,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: A] : aa(A,B,aTP_Lamp_uy(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_380
tff(fact_8553_ATP_Olambda__381,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_asx(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_381
tff(fact_8554_ATP_Olambda__382,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_ayr(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ) ).

% ATP.lambda_382
tff(fact_8555_ATP_Olambda__383,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_awq(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,sqrt,aa(A,real,Uu,Uua)) ).

% ATP.lambda_383
tff(fact_8556_ATP_Olambda__384,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_anu(fun(A,B),fun(A,fun(set(B),set(B))),Uu),Uua) = insert(B,aa(A,B,Uu,Uua)) ).

% ATP.lambda_384
tff(fact_8557_ATP_Olambda__385,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aug(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ~ aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_385
tff(fact_8558_ATP_Olambda__386,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_vu(fun(A,$o),fun(A,$o),Uu),Uua)
    <=> ~ aa(A,$o,Uu,Uua) ) ).

% ATP.lambda_386
tff(fact_8559_ATP_Olambda__387,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(A)
     => ! [Uu: fun(B,fun(A,$o)),Uua: B] :
          ( aa(B,$o,aTP_Lamp_bce(fun(B,fun(A,$o)),fun(B,$o),Uu),Uua)
        <=> ! [X_12: A] : aa(A,$o,aa(B,fun(A,$o),Uu,Uua),X_12) ) ) ).

% ATP.lambda_387
tff(fact_8560_ATP_Olambda__388,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,$o)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_aiw(fun(B,fun(A,$o)),fun(B,$o),Uu),Uua)
    <=> ! [X_12: A] : aa(A,$o,aa(B,fun(A,$o),Uu,Uua),X_12) ) ).

% ATP.lambda_388
tff(fact_8561_ATP_Olambda__389,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_amw(fun(A,fun(B,$o)),fun(A,$o),Uu),Uua)
    <=> ? [X_12: B] : aa(B,$o,aa(A,fun(B,$o),Uu,Uua),X_12) ) ).

% ATP.lambda_389
tff(fact_8562_ATP_Olambda__390,axiom,
    ! [A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu: fun(real,A),Uua: real] : aa(real,real,aTP_Lamp_aye(fun(real,A),fun(real,real),Uu),Uua) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(A,aa(real,A,Uu,Uua))) ) ).

% ATP.lambda_390
tff(fact_8563_ATP_Olambda__391,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B)
        & ring_1(C)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,C,aTP_Lamp_avk(fun(A,B),fun(A,C),Uu),Uua) = aa(int,C,ring_1_of_int(C),archim6421214686448440834_floor(B,aa(A,B,Uu,Uua))) ) ).

% ATP.lambda_391
tff(fact_8564_ATP_Olambda__392,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B)
        & ring_1(C)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,C,aTP_Lamp_avl(fun(A,B),fun(A,C),Uu),Uua) = aa(int,C,ring_1_of_int(C),archimedean_ceiling(B,aa(A,B,Uu,Uua))) ) ).

% ATP.lambda_392
tff(fact_8565_ATP_Olambda__393,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,fun(A,$o)),Uua: fun(B,A)] :
      ( aa(fun(B,A),$o,aTP_Lamp_agk(fun(B,fun(A,$o)),fun(fun(B,A),$o),Uu),Uua)
    <=> ! [X: B] : aa(A,$o,aa(B,fun(A,$o),Uu,X),aa(B,A,Uua,X)) ) ).

% ATP.lambda_393
tff(fact_8566_ATP_Olambda__394,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_amx(fun(A,fun(B,$o)),fun(B,$o),Uu),Uua)
    <=> ? [A7: A] : aa(B,$o,aa(A,fun(B,$o),Uu,A7),Uua) ) ).

% ATP.lambda_394
tff(fact_8567_ATP_Olambda__395,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_and(list(A),fun(A,$o),Uu),Uua)
    <=> ? [I4: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu),I4) )
          & aa(nat,$o,ord_less(nat,I4),aa(list(A),nat,size_size(list(A)),Uu)) ) ) ).

% ATP.lambda_395
tff(fact_8568_ATP_Olambda__396,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ane(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> ! [M6: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Uua),M6)
         => ( aa(nat,$o,Uu,M6)
          <=> aa(nat,$o,Uu,Uua) ) ) ) ).

% ATP.lambda_396
tff(fact_8569_ATP_Olambda__397,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_ang(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> ! [N12: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Uua),N12)
         => aa(nat,$o,Uu,N12) ) ) ).

% ATP.lambda_397
tff(fact_8570_ATP_Olambda__398,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_bbw(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ! [Y3: A] :
              ( aa(A,$o,ord_less_eq(A,Uua),Y3)
             => aa(A,$o,Uu,Y3) ) ) ) ).

% ATP.lambda_398
tff(fact_8571_ATP_Olambda__399,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_anf(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> ! [N12: nat] :
          ( aa(nat,$o,ord_less_eq(nat,Uua),N12)
         => ~ aa(nat,$o,Uu,N12) ) ) ).

% ATP.lambda_399
tff(fact_8572_ATP_Olambda__400,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,heap_Time_Heap(B)),Uua: fun(A,heap_Time_Heap(B))] :
      ( aa(fun(A,heap_Time_Heap(B)),$o,aTP_Lamp_ze(fun(A,heap_Time_Heap(B)),fun(fun(A,heap_Time_Heap(B)),$o),Uu),Uua)
    <=> ! [X: A] : refine_Imp_refines(B,aa(A,heap_Time_Heap(B),Uu,X),aa(A,heap_Time_Heap(B),Uua,X)) ) ).

% ATP.lambda_400
tff(fact_8573_ATP_Olambda__401,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,nat),Uua: fun(A,heap_Time_Heap(B))] :
      ( aa(fun(A,heap_Time_Heap(B)),$o,aTP_Lamp_zd(fun(A,nat),fun(fun(A,heap_Time_Heap(B)),$o),Uu),Uua)
    <=> ! [X: A] : time_TBOUND(B,aa(A,heap_Time_Heap(B),Uua,X),aa(A,nat,Uu,X)) ) ).

% ATP.lambda_401
tff(fact_8574_ATP_Olambda__402,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_amy(fun(B,option(A)),fun(A,$o),Uu),Uua)
    <=> ? [A7: B] : aa(B,option(A),Uu,A7) = aa(A,option(A),some(A),Uua) ) ).

% ATP.lambda_402
tff(fact_8575_ATP_Olambda__403,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ams(fun(B,A),fun(A,$o),Uu),Uua)
    <=> ? [X: B] : Uua = aa(B,A,Uu,X) ) ).

% ATP.lambda_403
tff(fact_8576_ATP_Olambda__404,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(A,fun(B,nat)),Uua: fun(product_prod(A,B),heap_Time_Heap(C))] :
      ( aa(fun(product_prod(A,B),heap_Time_Heap(C)),$o,aTP_Lamp_zh(fun(A,fun(B,nat)),fun(fun(product_prod(A,B),heap_Time_Heap(C)),$o),Uu),Uua)
    <=> ! [X: A,Xa3: B] : time_TBOUND(C,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),product_curry(A,B,heap_Time_Heap(C),Uua),X),Xa3),aa(B,nat,aa(A,fun(B,nat),Uu,X),Xa3)) ) ).

% ATP.lambda_404
tff(fact_8577_ATP_Olambda__405,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(A,option(B))] :
      ( aa(fun(A,option(B)),$o,aTP_Lamp_aik(fun(A,fun(B,$o)),fun(fun(A,option(B)),$o),Uu),Uua)
    <=> ! [X: A,Y3: B] :
          ( ( aa(A,option(B),Uua,X) = aa(B,option(B),some(B),Y3) )
         => aa(B,$o,aa(A,fun(B,$o),Uu,X),Y3) ) ) ).

% ATP.lambda_405
tff(fact_8578_ATP_Olambda__406,axiom,
    ! [D2: $tType,B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,fun(C,nat))),Uua: fun(product_prod(product_prod(A,B),C),heap_Time_Heap(D2))] :
      ( aa(fun(product_prod(product_prod(A,B),C),heap_Time_Heap(D2)),$o,aTP_Lamp_yw(fun(A,fun(B,fun(C,nat))),fun(fun(product_prod(product_prod(A,B),C),heap_Time_Heap(D2)),$o),Uu),Uua)
    <=> ! [X: A,Xa3: B,Xb4: C] : time_TBOUND(D2,aa(C,heap_Time_Heap(D2),aa(B,fun(C,heap_Time_Heap(D2)),aa(A,fun(B,fun(C,heap_Time_Heap(D2))),product_curry(A,B,fun(C,heap_Time_Heap(D2)),product_curry(product_prod(A,B),C,heap_Time_Heap(D2),Uua)),X),Xa3),Xb4),aa(C,nat,aa(B,fun(C,nat),aa(A,fun(B,fun(C,nat)),Uu,X),Xa3),Xb4)) ) ).

% ATP.lambda_406
tff(fact_8579_ATP_Olambda__407,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,$o))))),Uua: fun(A,heap_Time_Heap(B))] :
      ( aa(fun(A,heap_Time_Heap(B)),$o,aTP_Lamp_yz(fun(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,$o))))),fun(fun(A,heap_Time_Heap(B)),$o),Uu),Uua)
    <=> ! [X: A,H6: heap_ext(product_unit),H10: heap_ext(product_unit),R6: B,N8: nat] :
          ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),Uua,X),H6,H10,R6,N8)
         => aa(nat,$o,aa(B,fun(nat,$o),aa(heap_ext(product_unit),fun(B,fun(nat,$o)),aa(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,$o))),aa(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,$o)))),Uu,X),H6),H10),R6),N8) ) ) ).

% ATP.lambda_407
tff(fact_8580_ATP_Olambda__408,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: array(A)] : aa(array(A),heap_Time_Heap(array(A)),aTP_Lamp_akx(array(A),fun(array(A),heap_Time_Heap(array(A))),Uu),Uua) = aa(array(A),heap_Time_Heap(array(A)),heap_Time_ureturn(array(A)),Uu) ) ).

% ATP.lambda_408
tff(fact_8581_ATP_Olambda__409,axiom,
    ! [Uu: vEBT_VEBTi,Uua: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_qe(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uu),Uua) = heap_Time_Heap2(vEBT_VEBTi,aTP_Lamp_nl(heap_ext(product_unit),option(product_prod(vEBT_VEBTi,product_prod(heap_ext(product_unit),nat))))) ).

% ATP.lambda_409
tff(fact_8582_ATP_Olambda__410,axiom,
    ! [Uu: vEBT_VEBTi,Uua: nat] : aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aTP_Lamp_mb(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),Uu),Uua) = heap_Time_Heap2($o,aTP_Lamp_lw(heap_ext(product_unit),option(product_prod($o,product_prod(heap_ext(product_unit),nat))))) ).

% ATP.lambda_410
tff(fact_8583_ATP_Olambda__411,axiom,
    ! [Uu: vEBT_VEBTi,Uua: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aTP_Lamp_lq(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uu),Uua) = heap_Time_Heap2(option(nat),aTP_Lamp_jv(heap_ext(product_unit),option(product_prod(option(nat),product_prod(heap_ext(product_unit),nat))))) ).

% ATP.lambda_411
tff(fact_8584_ATP_Olambda__412,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uua: vEBT_VEBTi,Uub: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_qf(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uu),Uua),Uub) = vEBT_case_VEBTi(heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_pp(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),Uu),Uub),aTP_Lamp_om(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),Uub),Uua) ).

% ATP.lambda_412
tff(fact_8585_ATP_Olambda__413,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),Uua: vEBT_VEBTi,Uub: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_pb(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uu),Uua),Uub) = vEBT_case_VEBTi(heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_pa(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),Uu),Uub),aTP_Lamp_om(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),Uub),Uua) ).

% ATP.lambda_413
tff(fact_8586_ATP_Olambda__414,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),Uua: vEBT_VEBTi,Uub: nat] : aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aTP_Lamp_kx(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),Uu),Uua),Uub) = vEBT_case_VEBTi(heap_Time_Heap($o),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))),aTP_Lamp_kw(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))))),Uu),Uub),aTP_Lamp_hg(nat,fun($o,fun($o,heap_Time_Heap($o))),Uub),Uua) ).

% ATP.lambda_414
tff(fact_8587_ATP_Olambda__415,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),Uua: vEBT_VEBTi,Uub: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aTP_Lamp_kn(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uu),Uua),Uub) = vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_km(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Uu),Uub),aTP_Lamp_bk(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uub),Uua) ).

% ATP.lambda_415
tff(fact_8588_ATP_Olambda__416,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),Uua: vEBT_VEBTi,Uub: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aTP_Lamp_kf(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uu),Uua),Uub) = vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_ke(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Uu),Uub),aTP_Lamp_cl(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uub),Uua) ).

% ATP.lambda_416
tff(fact_8589_ATP_Olambda__417,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),Uua: vEBT_VEBTi,Uub: nat] : aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aTP_Lamp_mc(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),Uu),Uua),Uub) = vEBT_case_VEBTi(heap_Time_Heap($o),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))),aTP_Lamp_ig(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))))),Uu),Uub),aTP_Lamp_hg(nat,fun($o,fun($o,heap_Time_Heap($o))),Uub),Uua) ).

% ATP.lambda_417
tff(fact_8590_ATP_Olambda__418,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uua: vEBT_VEBTi,Uub: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aTP_Lamp_ls(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uu),Uua),Uub) = vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_ff(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Uu),Uub),aTP_Lamp_cl(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uub),Uua) ).

% ATP.lambda_418
tff(fact_8591_ATP_Olambda__419,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uua: vEBT_VEBTi,Uub: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aTP_Lamp_lr(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uu),Uua),Uub) = vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_ez(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Uu),Uub),aTP_Lamp_bk(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uub),Uua) ).

% ATP.lambda_419
tff(fact_8592_ATP_Olambda__420,axiom,
    ! [Uu: nat,Uua: $o,Uub: $o] :
      aa($o,heap_Time_Heap(option(nat)),aa($o,fun($o,heap_Time_Heap(option(nat))),aTP_Lamp_bk(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uu),(Uua)),(Uub)) = $ite(
        aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu),
        $ite(
          (Uub),
          aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),one_one(nat))),
          $ite((Uua),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),zero_zero(nat))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))) ),
        $ite(
          Uu = one_one(nat),
          $ite((Uua),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),zero_zero(nat))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))),
          aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)) ) ) ).

% ATP.lambda_420
tff(fact_8593_ATP_Olambda__421,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_wq(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(nat,real,Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,Uub),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% ATP.lambda_421
tff(fact_8594_ATP_Olambda__422,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_wr(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(nat,real,Uu,Uub),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_422
tff(fact_8595_ATP_Olambda__423,axiom,
    ! [Uu: num,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_gr(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(aa(code_integer,$o,ord_less_eq(code_integer,aa(num,code_integer,numeral_numeral(code_integer),Uu)),Uub),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,Uub),aa(num,code_integer,numeral_numeral(code_integer),Uu))),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_423
tff(fact_8596_ATP_Olambda__424,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_fu(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = $ite(aa(nat,$o,ord_less_eq(nat,aa(num,nat,numeral_numeral(nat),Uu)),Uub),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),one_one(nat))),aa(nat,nat,minus_minus(nat,Uub),aa(num,nat,numeral_numeral(nat),Uu))),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_424
tff(fact_8597_ATP_Olambda__425,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ft(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = $ite(aa(int,$o,ord_less_eq(int,aa(num,int,numeral_numeral(int),Uu)),Uub),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),one_one(int))),aa(int,int,minus_minus(int,Uub),aa(num,int,numeral_numeral(int),Uu))),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),Uua)),Uub)) ).

% ATP.lambda_425
tff(fact_8598_ATP_Olambda__426,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Uu: num,Uua: A,Uub: A] :
          aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),aTP_Lamp_fv(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = $ite(aa(A,$o,ord_less_eq(A,aa(num,A,numeral_numeral(A),Uu)),Uub),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),one_one(A))),aa(A,A,minus_minus(A,Uub),aa(num,A,numeral_numeral(A),Uu))),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),Uua)),Uub)) ) ).

% ATP.lambda_426
tff(fact_8599_ATP_Olambda__427,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: set(nat),Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_sn(set(nat),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(member(nat,Uub,Uu),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_427
tff(fact_8600_ATP_Olambda__428,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tx(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_428
tff(fact_8601_ATP_Olambda__429,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xt(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_429
tff(fact_8602_ATP_Olambda__430,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat,Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mm(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_430
tff(fact_8603_ATP_Olambda__431,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ty(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_431
tff(fact_8604_ATP_Olambda__432,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xs(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_432
tff(fact_8605_ATP_Olambda__433,axiom,
    ! [Uu: nat,Uua: $o,Uub: $o] :
      aa($o,heap_Time_Heap(option(nat)),aa($o,fun($o,heap_Time_Heap(option(nat))),aTP_Lamp_cl(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uu),(Uua)),(Uub)) = $ite(
        Uu = zero_zero(nat),
        $ite((Uub),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),one_one(nat))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)) ) ).

% ATP.lambda_433
tff(fact_8606_ATP_Olambda__434,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_aoj(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uu)),Uub))) ).

% ATP.lambda_434
tff(fact_8607_ATP_Olambda__435,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_aoh(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,abs_abs(code_integer,Uu)),Uub))) ).

% ATP.lambda_435
tff(fact_8608_ATP_Olambda__436,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),aTP_Lamp_aoi(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(Uub = zero_zero(code_integer),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),zero_zero(code_integer)),aa(code_integer,product_prod(code_integer,code_integer),aa(code_integer,fun(code_integer,product_prod(code_integer,code_integer)),product_Pair(code_integer,code_integer),aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),one_one(code_integer))),aa(code_integer,code_integer,minus_minus(code_integer,Uu),Uub))) ).

% ATP.lambda_436
tff(fact_8609_ATP_Olambda__437,axiom,
    ! [Uu: nat,Uua: $o,Uub: $o] :
      aa($o,heap_Time_Heap(vEBT_VEBTi),aa($o,fun($o,heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_om(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),Uu),(Uua)),(Uub)) = $ite(
        Uu = zero_zero(nat),
        aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi($true,(Uub))),
        $ite(Uu = one_one(nat),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi((Uua),$true)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi((Uua),(Uub)))) ) ).

% ATP.lambda_437
tff(fact_8610_ATP_Olambda__438,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: option(nat)] :
      aa(option(nat),heap_Time_Heap(option(nat)),aa(array(vEBT_VEBTi),fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_ce(nat,fun(array(vEBT_VEBTi),fun(option(nat),heap_Time_Heap(option(nat)))),Uu),Uua),Uub) = $ite(Uub = none(nat),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uua),aa(option(nat),nat,the2(nat),Uub)),aa(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aTP_Lamp_cd(nat,fun(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),Uu),Uub))) ).

% ATP.lambda_438
tff(fact_8611_ATP_Olambda__439,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: list(A)] :
      aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_ajr(fun(A,$o),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = $ite(aa(A,$o,Uu,Uua),aa(list(A),list(A),cons(A,Uua),Uub),Uub) ).

% ATP.lambda_439
tff(fact_8612_ATP_Olambda__440,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: set(A)] :
      aa(set(A),set(A),aa(A,fun(set(A),set(A)),aTP_Lamp_aoa(fun(A,$o),fun(A,fun(set(A),set(A))),Uu),Uua),Uub) = $ite(aa(A,$o,Uu,Uua),aa(set(A),set(A),insert(A,Uua),Uub),Uub) ).

% ATP.lambda_440
tff(fact_8613_ATP_Olambda__441,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,$o),Uua: fun(nat,A),Uub: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_so(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(aa(nat,$o,Uu,Uub),aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_441
tff(fact_8614_ATP_Olambda__442,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_uv(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_442
tff(fact_8615_ATP_Olambda__443,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: A] :
          aa(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_yh(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_443
tff(fact_8616_ATP_Olambda__444,axiom,
    ! [A: $tType,Uu: fun(heap_ext(product_unit),$o),Uua: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Uub: heap_ext(product_unit)] :
      aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_akh(fun(heap_ext(product_unit),$o),fun(fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub) = $ite(aa(heap_ext(product_unit),$o,Uu,Uub),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),Uua,Uub)),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))) ).

% ATP.lambda_444
tff(fact_8617_ATP_Olambda__445,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: B] :
      aa(B,option(A),aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_alr(fun(B,A),fun(fun(B,$o),fun(B,option(A))),Uu),Uua),Uub) = $ite(aa(B,$o,Uua,Uub),aa(A,option(A),some(A),aa(B,A,Uu,Uub)),none(A)) ).

% ATP.lambda_445
tff(fact_8618_ATP_Olambda__446,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] :
      ( aa(real,$o,aa(fun(real,real),fun(real,$o),aTP_Lamp_bdr(fun(real,real),fun(fun(real,real),fun(real,$o)),Uu),Uua),Uub)
    <=> has_field_derivative(real,Uu,aa(real,real,Uua,Uub),topolo174197925503356063within(real,Uub,top_top(set(real)))) ) ).

% ATP.lambda_446
tff(fact_8619_ATP_Olambda__447,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: filter(B),Uua: fun(B,A),Uub: A] :
          ( aa(A,$o,aa(fun(B,A),fun(A,$o),aTP_Lamp_bdd(filter(B),fun(fun(B,A),fun(A,$o)),Uu),Uua),Uub)
        <=> filterlim(B,A,Uua,topolo7230453075368039082e_nhds(A,Uub),Uu) ) ) ).

% ATP.lambda_447
tff(fact_8620_ATP_Olambda__448,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),Uua: nat,Uub: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap($o),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)),aTP_Lamp_kr(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o))),Uu),Uua),Uub) = aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),product_curry(vEBT_VEBTi,nat,heap_Time_Heap($o),Uu),Uub),Uua) ).

% ATP.lambda_448
tff(fact_8621_ATP_Olambda__449,axiom,
    ! [A: $tType,B: $tType,Uu: heap_Time_Heap(B),Uua: fun(B,heap_Time_Heap(A)),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(fun(B,heap_Time_Heap(A)),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aTP_Lamp_akm(heap_Time_Heap(B),fun(fun(B,heap_Time_Heap(A)),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),Uu),Uua),Uub) = case_option(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(B,product_prod(heap_ext(product_unit),nat)),none(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_case_prod(B,product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_akl(fun(B,heap_Time_Heap(A)),fun(B,fun(product_prod(heap_ext(product_unit),nat),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),Uua)),aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(B,Uu),Uub)) ).

% ATP.lambda_449
tff(fact_8622_ATP_Olambda__450,axiom,
    ! [D2: $tType,C: $tType,B: $tType,A: $tType,Uu: fun(D2,fun(B,C)),Uua: fun(A,D2),Uub: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,D2),fun(product_prod(A,B),C),aTP_Lamp_cr(fun(D2,fun(B,C)),fun(fun(A,D2),fun(product_prod(A,B),C)),Uu),Uua),Uub) = aa(B,C,aa(D2,fun(B,C),Uu,aa(A,D2,Uua,aa(product_prod(A,B),A,product_fst(A,B),Uub))),aa(product_prod(A,B),B,product_snd(A,B),Uub)) ).

% ATP.lambda_450
tff(fact_8623_ATP_Olambda__451,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aat(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,minus_minus(nat,Uua),Uub)) ) ).

% ATP.lambda_451
tff(fact_8624_ATP_Olambda__452,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aar(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),aa(nat,nat,minus_minus(nat,Uua),Uub)) ) ).

% ATP.lambda_452
tff(fact_8625_ATP_Olambda__453,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_bcf(fun(A,fun(B,$o)),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,aa(A,fun(B,$o),Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_453
tff(fact_8626_ATP_Olambda__454,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),Uua: nat,Uub: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap($o),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)),aTP_Lamp_ib(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o))),Uu),Uua),Uub) = aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),Uu,Uub),Uua) ).

% ATP.lambda_454
tff(fact_8627_ATP_Olambda__455,axiom,
    ! [Uu: fun(real,fun(nat,real)),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_atd(fun(real,fun(nat,real)),fun(nat,fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),Uu,Uub),Uua) ).

% ATP.lambda_455
tff(fact_8628_ATP_Olambda__456,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aah(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ) ).

% ATP.lambda_456
tff(fact_8629_ATP_Olambda__457,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aae(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(nat,fun(nat,A),Uu,Uub),Uua) ) ).

% ATP.lambda_457
tff(fact_8630_ATP_Olambda__458,axiom,
    ! [A: $tType,B: $tType,Uu: fun(fun(B,A),fun(B,A)),Uua: B,Uub: fun(B,A)] : aa(fun(B,A),A,aa(B,fun(fun(B,A),A),aTP_Lamp_agj(fun(fun(B,A),fun(B,A)),fun(B,fun(fun(B,A),A)),Uu),Uua),Uub) = aa(B,A,aa(fun(B,A),fun(B,A),Uu,Uub),Uua) ).

% ATP.lambda_458
tff(fact_8631_ATP_Olambda__459,axiom,
    ! [B: $tType,A: $tType,Uu: fun(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))),Uua: A,Uub: fun(A,heap_Time_Heap(B))] : aa(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B),aa(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B)),aTP_Lamp_mi(fun(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))),fun(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B))),Uu),Uua),Uub) = aa(A,heap_Time_Heap(B),aa(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B)),Uu,Uub),Uua) ).

% ATP.lambda_459
tff(fact_8632_ATP_Olambda__460,axiom,
    ! [B: $tType,A: $tType,Uu: fun(fun(A,option(B)),fun(A,option(B))),Uua: A,Uub: fun(A,option(B))] : aa(fun(A,option(B)),option(B),aa(A,fun(fun(A,option(B)),option(B)),aTP_Lamp_aig(fun(fun(A,option(B)),fun(A,option(B))),fun(A,fun(fun(A,option(B)),option(B))),Uu),Uua),Uub) = aa(A,option(B),aa(fun(A,option(B)),fun(A,option(B)),Uu,Uub),Uua) ).

% ATP.lambda_460
tff(fact_8633_ATP_Olambda__461,axiom,
    ! [B: $tType,A: $tType,Uu: fun(fun(A,B),fun(A,B)),Uua: A,Uub: fun(A,B)] : aa(fun(A,B),B,aa(A,fun(fun(A,B),B),aTP_Lamp_ais(fun(fun(A,B),fun(A,B)),fun(A,fun(fun(A,B),B)),Uu),Uua),Uub) = aa(A,B,aa(fun(A,B),fun(A,B),Uu,Uub),Uua) ).

% ATP.lambda_461
tff(fact_8634_ATP_Olambda__462,axiom,
    ! [C: $tType,D2: $tType,B: $tType,A: $tType,Uu: fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2))),Uua: fun(A,heap_Time_Heap(B)),Uub: C] : aa(C,heap_Time_Heap(D2),aa(fun(A,heap_Time_Heap(B)),fun(C,heap_Time_Heap(D2)),aTP_Lamp_jq(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2))),fun(fun(A,heap_Time_Heap(B)),fun(C,heap_Time_Heap(D2))),Uu),Uua),Uub) = aa(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2),aa(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2)),Uu,Uub),Uua) ).

% ATP.lambda_462
tff(fact_8635_ATP_Olambda__463,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(B,fun(A,B)),Uua: A,Uub: B] : aa(B,B,aa(A,fun(B,B),aTP_Lamp_apy(fun(B,fun(A,B)),fun(A,fun(B,B)),Uu),Uua),Uub) = aa(A,B,aa(B,fun(A,B),Uu,Uub),Uua) ) ).

% ATP.lambda_463
tff(fact_8636_ATP_Olambda__464,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_yd(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_464
tff(fact_8637_ATP_Olambda__465,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: C,Uub: B] : aa(B,A,aa(C,fun(B,A),aTP_Lamp_uc(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uua),Uub) = aa(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_465
tff(fact_8638_ATP_Olambda__466,axiom,
    ! [B: $tType,A: $tType] :
      ( finite_finite(A)
     => ! [Uu: fun(B,fun(A,$o)),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_bcd(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,aa(B,fun(A,$o),Uu,Uub),Uua) ) ) ).

% ATP.lambda_466
tff(fact_8639_ATP_Olambda__467,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,assn)),Uua: A,Uub: B] : aa(B,assn,aa(A,fun(B,assn),aTP_Lamp_aew(fun(B,fun(A,assn)),fun(A,fun(B,assn)),Uu),Uua),Uub) = aa(A,assn,aa(B,fun(A,assn),Uu,Uub),Uua) ).

% ATP.lambda_467
tff(fact_8640_ATP_Olambda__468,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,$o)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_aiv(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,aa(B,fun(A,$o),Uu,Uub),Uua) ) ).

% ATP.lambda_468
tff(fact_8641_ATP_Olambda__469,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_avi(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu,Uub),Uua) ) ).

% ATP.lambda_469
tff(fact_8642_ATP_Olambda__470,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_avq(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu,Uub),Uua) ) ).

% ATP.lambda_470
tff(fact_8643_ATP_Olambda__471,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(B,fun(A,C)),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_agd(fun(B,fun(A,C)),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,aa(B,fun(A,C),Uu,Uub),Uua) ).

% ATP.lambda_471
tff(fact_8644_ATP_Olambda__472,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu: fun(A,fun(nat,B)),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_us(fun(A,fun(nat,B)),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(A,fun(nat,B),Uu,Uub),Uua) ) ).

% ATP.lambda_472
tff(fact_8645_ATP_Olambda__473,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_arp(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_473
tff(fact_8646_ATP_Olambda__474,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V3459762299906320749_field(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_asr(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_474
tff(fact_8647_ATP_Olambda__475,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_awh(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_475
tff(fact_8648_ATP_Olambda__476,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo4987421752381908075d_mult(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_awf(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_476
tff(fact_8649_ATP_Olambda__477,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo5987344860129210374id_add(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_auc(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_477
tff(fact_8650_ATP_Olambda__478,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,assn)),Uua: B,Uub: A] : aa(A,assn,aa(B,fun(A,assn),aTP_Lamp_aeu(fun(A,fun(B,assn)),fun(B,fun(A,assn)),Uu),Uua),Uub) = aa(B,assn,aa(A,fun(B,assn),Uu,Uub),Uua) ).

% ATP.lambda_478
tff(fact_8651_ATP_Olambda__479,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_awl(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_479
tff(fact_8652_ATP_Olambda__480,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_awj(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_480
tff(fact_8653_ATP_Olambda__481,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_awv(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_481
tff(fact_8654_ATP_Olambda__482,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_xz(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_482
tff(fact_8655_ATP_Olambda__483,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,fun(B,C)),Uua: B,Uub: A] : aa(A,C,aa(B,fun(A,C),aTP_Lamp_up(fun(A,fun(B,C)),fun(B,fun(A,C)),Uu),Uua),Uub) = aa(B,C,aa(A,fun(B,C),Uu,Uub),Uua) ) ).

% ATP.lambda_483
tff(fact_8656_ATP_Olambda__484,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_abl(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_abk(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_484
tff(fact_8657_ATP_Olambda__485,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_abj(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_abi(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_485
tff(fact_8658_ATP_Olambda__486,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_abh(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_abg(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_486
tff(fact_8659_ATP_Olambda__487,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_abf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_abe(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_487
tff(fact_8660_ATP_Olambda__488,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat,Uua: A,Uub: nat] :
          aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_we(nat,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(Uub = Uu,one_one(A),zero_zero(A))),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_488
tff(fact_8661_ATP_Olambda__489,axiom,
    ! [Uu: code_integer,Uua: code_integer,Uub: code_integer] :
      aa(code_integer,product_prod(code_integer,$o),aa(code_integer,fun(code_integer,product_prod(code_integer,$o)),aTP_Lamp_aok(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),Uu),Uua),Uub) = aa($o,product_prod(code_integer,$o),
        aa(code_integer,fun($o,product_prod(code_integer,$o)),product_Pair(code_integer,$o),
          $ite(aa(code_integer,$o,ord_less(code_integer,zero_zero(code_integer)),Uu),Uua,aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),Uua)),Uub))),
        Uub = one_one(code_integer)) ).

% ATP.lambda_489
tff(fact_8662_ATP_Olambda__490,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aai(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aah(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_490
tff(fact_8663_ATP_Olambda__491,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aaf(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aae(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_491
tff(fact_8664_ATP_Olambda__492,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_ayd(fun(nat,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_mz(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_492
tff(fact_8665_ATP_Olambda__493,axiom,
    ! [Uu: $o,Uua: code_integer,Uub: $o] : aa($o,char,aa(code_integer,fun($o,char),aTP_Lamp_aor($o,fun(code_integer,fun($o,char)),(Uu)),Uua),(Uub)) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aa($o,fun(code_integer,fun($o,char)),aTP_Lamp_aoq($o,fun($o,fun(code_integer,fun($o,char))),(Uu)),(Uub))),code_bit_cut_integer(Uua)) ).

% ATP.lambda_493
tff(fact_8666_ATP_Olambda__494,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ahc(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_ahb(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_494
tff(fact_8667_ATP_Olambda__495,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_aha(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_agz(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_495
tff(fact_8668_ATP_Olambda__496,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_agy(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_agx(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_496
tff(fact_8669_ATP_Olambda__497,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_agw(rat,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = aa(product_prod(int,int),product_prod(int,int),product_case_prod(int,int,product_prod(int,int),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_agv(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_497
tff(fact_8670_ATP_Olambda__498,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_agt(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aa(int,fun(int,fun(int,$o)),aTP_Lamp_ags(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_498
tff(fact_8671_ATP_Olambda__499,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_agq(rat,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aa(int,fun(int,fun(int,$o)),aTP_Lamp_agp(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_499
tff(fact_8672_ATP_Olambda__500,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_awm(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_awl(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_500
tff(fact_8673_ATP_Olambda__501,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_awk(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_awj(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_501
tff(fact_8674_ATP_Olambda__502,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V4412858255891104859lgebra(C)
        & comm_ring_1(C)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_awi(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_awh(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_502
tff(fact_8675_ATP_Olambda__503,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4987421752381908075d_mult(C)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_awg(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_awf(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_503
tff(fact_8676_ATP_Olambda__504,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_ass(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_asr(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub),Uu) ) ).

% ATP.lambda_504
tff(fact_8677_ATP_Olambda__505,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_ye(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(B,A,aa(C,fun(B,A),aTP_Lamp_yd(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub),Uua) ) ).

% ATP.lambda_505
tff(fact_8678_ATP_Olambda__506,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aww(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_awv(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_506
tff(fact_8679_ATP_Olambda__507,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(C)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_aud(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_auc(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_507
tff(fact_8680_ATP_Olambda__508,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: B] : aa(B,C,aa(fun(A,fun(B,C)),fun(B,C),aTP_Lamp_arq(set(A),fun(fun(A,fun(B,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_arp(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uub)),Uu) ) ).

% ATP.lambda_508
tff(fact_8681_ATP_Olambda__509,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(nat,B)),Uub: nat] : aa(nat,B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_ut(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),Uu),Uua),Uub) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(nat,fun(A,B),aTP_Lamp_us(fun(A,fun(nat,B)),fun(nat,fun(A,B)),Uua),Uub)),Uu) ) ).

% ATP.lambda_509
tff(fact_8682_ATP_Olambda__510,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(B),Uub: C] : aa(C,A,aa(set(B),fun(C,A),aTP_Lamp_ud(fun(B,fun(C,A)),fun(set(B),fun(C,A)),Uu),Uua),Uub) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(C,fun(B,A),aTP_Lamp_uc(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua) ) ).

% ATP.lambda_510
tff(fact_8683_ATP_Olambda__511,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_aqq(fun(nat,fun(real,real)),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,Uub),zero_zero(real))),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_511
tff(fact_8684_ATP_Olambda__512,axiom,
    ! [Uu: real,Uua: fun(nat,fun(real,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(real,real)),fun(nat,real),aTP_Lamp_aqr(real,fun(fun(nat,fun(real,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uua,Uub),zero_zero(real))),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_512
tff(fact_8685_ATP_Olambda__513,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Uu: real,Uua: fun(nat,fun(A,real)),Uub: nat] : aa(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_xe(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(A,real,aa(nat,fun(A,real),Uua,Uub),zero_zero(A))),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ) ).

% ATP.lambda_513
tff(fact_8686_ATP_Olambda__514,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_aba(fun(nat,A),fun(nat,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_aac(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_514
tff(fact_8687_ATP_Olambda__515,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_axx(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu,Uua))),Uub) ) ).

% ATP.lambda_515
tff(fact_8688_ATP_Olambda__516,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_atr(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub))),aa(A,A,Uu,Uua))),Uub) ) ).

% ATP.lambda_516
tff(fact_8689_ATP_Olambda__517,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ua(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Uub))),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_517
tff(fact_8690_ATP_Olambda__518,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_att(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua))),aa(A,A,minus_minus(A,Uub),Uua)) ) ).

% ATP.lambda_518
tff(fact_8691_ATP_Olambda__519,axiom,
    ! [A: $tType] :
      ( ( inverse(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ats(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu,Uub)),aa(A,A,Uu,Uua))),aa(A,A,minus_minus(A,Uub),Uua)) ) ).

% ATP.lambda_519
tff(fact_8692_ATP_Olambda__520,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_aqz(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Uub)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_520
tff(fact_8693_ATP_Olambda__521,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_xf(real,fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,Uua,Uub)),semiring_char_0_fact(real,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_521
tff(fact_8694_ATP_Olambda__522,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_wy(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),aa(nat,A,Uu,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uub),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% ATP.lambda_522
tff(fact_8695_ATP_Olambda__523,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_aob(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,ord_less(nat,aa(list(A),nat,size_size(list(A)),Uua)),aa(list(A),nat,size_size(list(A)),Uub))
        | ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
          & member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Uua),Uub),lex(A,Uu)) ) ) ) ).

% ATP.lambda_523
tff(fact_8696_ATP_Olambda__524,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_ann(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uua) = aa(list(A),nat,size_size(list(A)),Uub) )
        & ? [Xys: list(A),X: A,Y3: A,Xs6: list(A),Ys6: list(A)] :
            ( ( Uua = append(A,Xys,aa(list(A),list(A),cons(A,X),Xs6)) )
            & ( Uub = append(A,Xys,aa(list(A),list(A),cons(A,Y3),Ys6)) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3),Uu) ) ) ) ).

% ATP.lambda_524
tff(fact_8697_ATP_Olambda__525,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_all(nat,fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & aa(set(A),$o,ord_less_eq(set(A),set2(A,Uub)),set2(A,Uua)) ) ) ).

% ATP.lambda_525
tff(fact_8698_ATP_Olambda__526,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_st(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,ord_less_eq(set(A),set2(A,Uub)),Uu)
        & aa(nat,$o,ord_less_eq(nat,aa(list(A),nat,size_size(list(A)),Uub)),Uua) ) ) ).

% ATP.lambda_526
tff(fact_8699_ATP_Olambda__527,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_ss(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,ord_less_eq(set(A),set2(A,Uub)),Uu)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_527
tff(fact_8700_ATP_Olambda__528,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_aqm(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,diffs(A,Uu)),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_528
tff(fact_8701_ATP_Olambda__529,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_ajq(fun(A,$o),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,set2(A,Uua))
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_529
tff(fact_8702_ATP_Olambda__530,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afm(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))) ) ).

% ATP.lambda_530
tff(fact_8703_ATP_Olambda__531,axiom,
    ! [Uu: list(vEBT_VEBT),Uua: array(vEBT_VEBTi),Uub: list(vEBT_VEBTi)] : aa(list(vEBT_VEBTi),assn,aa(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn),aTP_Lamp_aer(list(vEBT_VEBT),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,Uua),Uub)),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Uu),Uub)) ).

% ATP.lambda_531
tff(fact_8704_ATP_Olambda__532,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_wx(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_532
tff(fact_8705_ATP_Olambda__533,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_xd(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uu),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_533
tff(fact_8706_ATP_Olambda__534,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_xg(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,diffs(A,Uua),Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uub)) ) ).

% ATP.lambda_534
tff(fact_8707_ATP_Olambda__535,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_zq(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Uua),Uub)),aa(nat,nat,minus_minus(nat,Uu),Uub)) ).

% ATP.lambda_535
tff(fact_8708_ATP_Olambda__536,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_gd(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member(A,Uub,Uua) ) ) ).

% ATP.lambda_536
tff(fact_8709_ATP_Olambda__537,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ack(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,ord_less_eq(int,Uu),Uua)
        & aa(int,$o,ord_less(int,Uua),Uub) ) ) ).

% ATP.lambda_537
tff(fact_8710_ATP_Olambda__538,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_tb(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,ord_less_eq(int,Uu),Uub)
        & aa(int,$o,ord_less_eq(int,Uub),Uua) ) ) ).

% ATP.lambda_538
tff(fact_8711_ATP_Olambda__539,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_acl(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,ord_less_eq(int,Uu),Uub)
        & aa(int,$o,ord_less(int,Uua),Uub) ) ) ).

% ATP.lambda_539
tff(fact_8712_ATP_Olambda__540,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_tf(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,ord_less_eq(int,Uu),Uub)
        & aa(int,$o,ord_less(int,Uub),Uua) ) ) ).

% ATP.lambda_540
tff(fact_8713_ATP_Olambda__541,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_te(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,ord_less(int,Uu),Uub)
        & aa(int,$o,ord_less_eq(int,Uub),Uua) ) ) ).

% ATP.lambda_541
tff(fact_8714_ATP_Olambda__542,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_tc(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,ord_less(int,Uu),Uub)
        & aa(int,$o,ord_less(int,Uub),Uua) ) ) ).

% ATP.lambda_542
tff(fact_8715_ATP_Olambda__543,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_fx(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,vEBT_vebt_member(Uu),Uub)
        & aa(nat,$o,ord_less(nat,Uua),Uub) ) ) ).

% ATP.lambda_543
tff(fact_8716_ATP_Olambda__544,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_fw(vEBT_VEBT,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,vEBT_vebt_member(Uu),Uub)
        & aa(nat,$o,ord_less(nat,Uub),Uua) ) ) ).

% ATP.lambda_544
tff(fact_8717_ATP_Olambda__545,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ahx(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,dvd_dvd(nat,Uub),Uua)
        & aa(nat,$o,dvd_dvd(nat,Uub),Uu) ) ) ).

% ATP.lambda_545
tff(fact_8718_ATP_Olambda__546,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_ail(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,dvd_dvd(int,Uub),Uua)
        & aa(int,$o,dvd_dvd(int,Uub),Uu) ) ) ).

% ATP.lambda_546
tff(fact_8719_ATP_Olambda__547,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_sp(set(A),fun(set(A),fun(set(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),set(A),minus_minus(set(A),Uub),Uu) = aa(set(A),set(A),minus_minus(set(A),Uua),Uu) ) ) ).

% ATP.lambda_547
tff(fact_8720_ATP_Olambda__548,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: set(A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ani(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_548
tff(fact_8721_ATP_Olambda__549,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_uu(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_549
tff(fact_8722_ATP_Olambda__550,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_aoc(fun(A,$o),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uua)
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_550
tff(fact_8723_ATP_Olambda__551,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_gf(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_551
tff(fact_8724_ATP_Olambda__552,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_gg(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_552
tff(fact_8725_ATP_Olambda__553,axiom,
    ! [A: $tType,Uu: nat,Uua: heap_Time_Heap(A),Uub: A] : aa(A,heap_Time_Heap(list(A)),aa(heap_Time_Heap(A),fun(A,heap_Time_Heap(list(A))),aTP_Lamp_afx(nat,fun(heap_Time_Heap(A),fun(A,heap_Time_Heap(list(A)))),Uu),Uua),Uub) = heap_Time_bind(list(A),list(A),vEBT_VEBT_replicatei(A,Uu,Uua),aTP_Lamp_afw(A,fun(list(A),heap_Time_Heap(list(A))),Uub)) ).

% ATP.lambda_553
tff(fact_8726_ATP_Olambda__554,axiom,
    ! [A: $tType,Uu: heap_Time_Heap(A),Uua: nat,Uub: A] : aa(A,heap_Time_Heap(list(A)),aa(nat,fun(A,heap_Time_Heap(list(A))),aTP_Lamp_afz(heap_Time_Heap(A),fun(nat,fun(A,heap_Time_Heap(list(A)))),Uu),Uua),Uub) = heap_Time_bind(list(A),list(A),vEBT_VEBT_replicatei(A,Uua,Uu),aTP_Lamp_afw(A,fun(list(A),heap_Time_Heap(list(A))),Uub)) ).

% ATP.lambda_554
tff(fact_8727_ATP_Olambda__555,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_to(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_555
tff(fact_8728_ATP_Olambda__556,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajb(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_556
tff(fact_8729_ATP_Olambda__557,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_tm(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_557
tff(fact_8730_ATP_Olambda__558,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: set(B),Uub: B] :
          ( aa(B,$o,aa(set(B),fun(B,$o),aTP_Lamp_ajc(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_558
tff(fact_8731_ATP_Olambda__559,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: $o] : aa($o,assn,aa(vEBT_VEBTi,fun($o,assn),aTP_Lamp_adz(vEBT_VEBT,fun(vEBT_VEBTi,fun($o,assn)),Uu),Uua),(Uub)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu),Uua)),pure_assn((Uub) = vEBT_VEBT_minNull(Uu))) ).

% ATP.lambda_559
tff(fact_8732_ATP_Olambda__560,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option(nat)] : aa(option(nat),assn,aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_adx(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu),Uua)),pure_assn(Uub = vEBT_vebt_mint(Uu))) ).

% ATP.lambda_560
tff(fact_8733_ATP_Olambda__561,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option(nat)] : aa(option(nat),assn,aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_ady(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu),Uua)),pure_assn(Uub = vEBT_vebt_maxt(Uu))) ).

% ATP.lambda_561
tff(fact_8734_ATP_Olambda__562,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: list(A),Uub: nat] : aa(nat,assn,aa(list(A),fun(nat,assn),aTP_Lamp_aen(array(A),fun(list(A),fun(nat,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(A),assn,snga_assn(A,Uu),Uua)),pure_assn(Uub = aa(list(A),nat,size_size(list(A)),Uua))) ) ).

% ATP.lambda_562
tff(fact_8735_ATP_Olambda__563,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: list(A),Uub: list(A)] : aa(list(A),assn,aa(list(A),fun(list(A),assn),aTP_Lamp_aes(array(A),fun(list(A),fun(list(A),assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(A),assn,snga_assn(A,Uu),Uua)),pure_assn(Uub = Uua)) ) ).

% ATP.lambda_563
tff(fact_8736_ATP_Olambda__564,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_fz(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & ~ member(A,Uub,Uua) ) ) ).

% ATP.lambda_564
tff(fact_8737_ATP_Olambda__565,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ahw(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uu) ) ).

% ATP.lambda_565
tff(fact_8738_ATP_Olambda__566,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aaq(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_566
tff(fact_8739_ATP_Olambda__567,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_abc(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_567
tff(fact_8740_ATP_Olambda__568,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aic(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_568
tff(fact_8741_ATP_Olambda__569,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aia(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_569
tff(fact_8742_ATP_Olambda__570,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aib(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_570
tff(fact_8743_ATP_Olambda__571,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_ahz(A,fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_571
tff(fact_8744_ATP_Olambda__572,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aet(set(product_prod(B,A)),fun(B,fun(A,$o)),Uu),Uua),Uub)
    <=> member(product_prod(B,A),aa(A,product_prod(B,A),aa(B,fun(A,product_prod(B,A)),product_Pair(B,A),Uua),Uub),Uu) ) ).

% ATP.lambda_572
tff(fact_8745_ATP_Olambda__573,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_kp(set(product_prod(A,B)),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub),Uu) ) ).

% ATP.lambda_573
tff(fact_8746_ATP_Olambda__574,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ale(list(list(A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,nth(A,aa(nat,list(A),nth(list(A),Uu),Uub)),Uua) ).

% ATP.lambda_574
tff(fact_8747_ATP_Olambda__575,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_td(nat,fun(complex,fun(complex,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uu) = Uua ) ) ).

% ATP.lambda_575
tff(fact_8748_ATP_Olambda__576,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa(complex,$o,aa(nat,fun(complex,$o),aTP_Lamp_acr(complex,fun(nat,fun(complex,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uub),Uua) = Uu ) ) ).

% ATP.lambda_576
tff(fact_8749_ATP_Olambda__577,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_agh(A,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,ring_1_Ints(A))
            & aa(A,$o,ord_less_eq(A,Uu),Uub)
            & aa(A,$o,ord_less_eq(A,Uub),Uua) ) ) ) ).

% ATP.lambda_577
tff(fact_8750_ATP_Olambda__578,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_akr(fun(A,B),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(A,B,Uu,aa(nat,A,nth(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_578
tff(fact_8751_ATP_Olambda__579,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_nf(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_579
tff(fact_8752_ATP_Olambda__580,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_ne(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,suc,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_580
tff(fact_8753_ATP_Olambda__581,axiom,
    ! [C: $tType,D2: $tType,B: $tType,A: $tType,Uu: fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(C)),Uua: fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2))),Uub: fun(A,heap_Time_Heap(B))] : aa(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2),aa(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2))),fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2)),aTP_Lamp_jr(fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(C)),fun(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2))),fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2))),Uu),Uua),Uub) = heap_Time_bind(C,D2,aa(fun(A,heap_Time_Heap(B)),heap_Time_Heap(C),Uu,Uub),aa(fun(A,heap_Time_Heap(B)),fun(C,heap_Time_Heap(D2)),aTP_Lamp_jq(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D2))),fun(fun(A,heap_Time_Heap(B)),fun(C,heap_Time_Heap(D2))),Uua),Uub)) ).

% ATP.lambda_581
tff(fact_8754_ATP_Olambda__582,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,int),Uub: nat] : aa(nat,real,aa(fun(nat,int),fun(nat,real),aTP_Lamp_bai(fun(nat,real),fun(fun(nat,int),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,minus_minus(real,aa(nat,real,Uu,Uub)),aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),aa(nat,int,Uua,Uub))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))) ).

% ATP.lambda_582
tff(fact_8755_ATP_Olambda__583,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_ara(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),aa(nat,nat,suc,Uub))) ).

% ATP.lambda_583
tff(fact_8756_ATP_Olambda__584,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_ni(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,Uu,Uub)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uub)) ).

% ATP.lambda_584
tff(fact_8757_ATP_Olambda__585,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_abp(fun(nat,nat),fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uub)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uua),Uub)) ).

% ATP.lambda_585
tff(fact_8758_ATP_Olambda__586,axiom,
    ! [B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(nat,B),Uua: B,Uub: nat] : aa(nat,B,aa(B,fun(nat,B),aTP_Lamp_ayf(fun(nat,B),fun(B,fun(nat,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uu,Uub)),aa(nat,B,aa(B,fun(nat,B),power_power(B),Uua),Uub)) ) ).

% ATP.lambda_586
tff(fact_8759_ATP_Olambda__587,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_aac(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_587
tff(fact_8760_ATP_Olambda__588,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_mz(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_588
tff(fact_8761_ATP_Olambda__589,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_nd(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_589
tff(fact_8762_ATP_Olambda__590,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_xc(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_590
tff(fact_8763_ATP_Olambda__591,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_aao(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_591
tff(fact_8764_ATP_Olambda__592,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_abb(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_592
tff(fact_8765_ATP_Olambda__593,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_sq(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,Uu,Uub)
        & aa(nat,$o,ord_less(nat,Uub),Uua) ) ) ).

% ATP.lambda_593
tff(fact_8766_ATP_Olambda__594,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_akp(fun(A,B),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(A,B,Uu,Uub)),aa(A,B,Uu,aa(nat,A,nth(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% ATP.lambda_594
tff(fact_8767_ATP_Olambda__595,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_akq(fun(A,B),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uu,aa(nat,A,nth(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ).

% ATP.lambda_595
tff(fact_8768_ATP_Olambda__596,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_arg(fun(A,real),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_596
tff(fact_8769_ATP_Olambda__597,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aue(fun(A,real),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_597
tff(fact_8770_ATP_Olambda__598,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,real),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_awu(fun(A,real),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_598
tff(fact_8771_ATP_Olambda__599,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] :
      ( aa(A,$o,aa(fun(A,real),fun(A,$o),aTP_Lamp_bbu(fun(A,real),fun(fun(A,real),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(real,$o,ord_less_eq(real,aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_599
tff(fact_8772_ATP_Olambda__600,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_bbv(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_600
tff(fact_8773_ATP_Olambda__601,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_bbs(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_601
tff(fact_8774_ATP_Olambda__602,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_bcw(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_602
tff(fact_8775_ATP_Olambda__603,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_bds(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,Uu,Uub)),aa(real,real,Uua,Uub)) ).

% ATP.lambda_603
tff(fact_8776_ATP_Olambda__604,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xw(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_604
tff(fact_8777_ATP_Olambda__605,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_asp(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_605
tff(fact_8778_ATP_Olambda__606,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_asc(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_606
tff(fact_8779_ATP_Olambda__607,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_aph(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_607
tff(fact_8780_ATP_Olambda__608,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_atv(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_608
tff(fact_8781_ATP_Olambda__609,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_bec(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_609
tff(fact_8782_ATP_Olambda__610,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_bej(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_610
tff(fact_8783_ATP_Olambda__611,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ave(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_611
tff(fact_8784_ATP_Olambda__612,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_bea(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,Uua,Uub)),aa(real,real,Uu,Uub)) ).

% ATP.lambda_612
tff(fact_8785_ATP_Olambda__613,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_afp(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uub)),aa(nat,nat,Uua,Uub)) ).

% ATP.lambda_613
tff(fact_8786_ATP_Olambda__614,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_afo(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_614
tff(fact_8787_ATP_Olambda__615,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xv(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_615
tff(fact_8788_ATP_Olambda__616,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ari(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_616
tff(fact_8789_ATP_Olambda__617,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_aqa(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_617
tff(fact_8790_ATP_Olambda__618,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aub(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_618
tff(fact_8791_ATP_Olambda__619,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4211221413907600880p_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axd(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_619
tff(fact_8792_ATP_Olambda__620,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_bdw(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_620
tff(fact_8793_ATP_Olambda__621,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_avz(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_621
tff(fact_8794_ATP_Olambda__622,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_beh(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_622
tff(fact_8795_ATP_Olambda__623,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axg(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_623
tff(fact_8796_ATP_Olambda__624,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_avh(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_624
tff(fact_8797_ATP_Olambda__625,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_bef(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),aa(A,real,Uu,Uub)) ).

% ATP.lambda_625
tff(fact_8798_ATP_Olambda__626,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_vf(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_626
tff(fact_8799_ATP_Olambda__627,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_azi(fun(nat,real),fun(fun(nat,real),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,minus_minus(real,aa(nat,real,Uu,Uub)),aa(nat,real,Uua,Uub)) ).

% ATP.lambda_627
tff(fact_8800_ATP_Olambda__628,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mv(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_628
tff(fact_8801_ATP_Olambda__629,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_uk(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_629
tff(fact_8802_ATP_Olambda__630,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_arm(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_630
tff(fact_8803_ATP_Olambda__631,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_aqh(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_631
tff(fact_8804_ATP_Olambda__632,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_auu(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_632
tff(fact_8805_ATP_Olambda__633,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1633459387980952147up_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axr(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_633
tff(fact_8806_ATP_Olambda__634,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axv(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_634
tff(fact_8807_ATP_Olambda__635,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axs(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_635
tff(fact_8808_ATP_Olambda__636,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wf(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uua,Uub)),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_636
tff(fact_8809_ATP_Olambda__637,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_vs(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,minus_minus(nat,aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_637
tff(fact_8810_ATP_Olambda__638,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] : aa(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_vy(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Uu),Uua),Uub) = aa(nat,nat,minus_minus(nat,aa(A,nat,Uua,Uub)),aa(A,nat,Uu,Uub)) ).

% ATP.lambda_638
tff(fact_8811_ATP_Olambda__639,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axu(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uua,Uub)),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_639
tff(fact_8812_ATP_Olambda__640,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1898628316856586783d_mult(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_axl(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_640
tff(fact_8813_ATP_Olambda__641,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1898628316856586783d_mult(B)
     => ! [Uu: fun(A,B),Uua: fun(A,nat),Uub: A] : aa(A,B,aa(fun(A,nat),fun(A,B),aTP_Lamp_axk(fun(A,B),fun(fun(A,nat),fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(A,nat,Uua,Uub)) ) ).

% ATP.lambda_641
tff(fact_8814_ATP_Olambda__642,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axn(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = ord_min(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_642
tff(fact_8815_ATP_Olambda__643,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axm(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = ord_min(B,aa(A,B,Uu,Uub),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_643
tff(fact_8816_ATP_Olambda__644,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo1944317154257567458pology(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axq(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,ord_max(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_644
tff(fact_8817_ATP_Olambda__645,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_axp(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,ord_max(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_645
tff(fact_8818_ATP_Olambda__646,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,complex,aa(fun(A,real),fun(A,complex),aTP_Lamp_awn(fun(A,real),fun(fun(A,real),fun(A,complex)),Uu),Uua),Uub) = complex2(aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_646
tff(fact_8819_ATP_Olambda__647,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ms(fun(nat,A),fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,Uu,Uub)),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_647
tff(fact_8820_ATP_Olambda__648,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,A),Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_uf(fun(B,A),fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_648
tff(fact_8821_ATP_Olambda__649,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_arv(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_649
tff(fact_8822_ATP_Olambda__650,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_apw(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_650
tff(fact_8823_ATP_Olambda__651,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo6943815403480290642id_add(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aup(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_651
tff(fact_8824_ATP_Olambda__652,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_bdx(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_652
tff(fact_8825_ATP_Olambda__653,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_beb(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_653
tff(fact_8826_ATP_Olambda__654,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo6943815403480290642id_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_avn(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_654
tff(fact_8827_ATP_Olambda__655,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_ajd(fun(A,B),fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_655
tff(fact_8828_ATP_Olambda__656,axiom,
    ! [Uu: fun(real,real),Uua: fun(real,real),Uub: real] : aa(real,real,aa(fun(real,real),fun(real,real),aTP_Lamp_aqo(fun(real,real),fun(fun(real,real),fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),aa(real,real,Uua,Uub)) ).

% ATP.lambda_656
tff(fact_8829_ATP_Olambda__657,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_asv(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_657
tff(fact_8830_ATP_Olambda__658,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ayv(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_658
tff(fact_8831_ATP_Olambda__659,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_avs(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uu,Uub),aa(A,real,Uua,Uub)) ).

% ATP.lambda_659
tff(fact_8832_ATP_Olambda__660,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_azq(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_660
tff(fact_8833_ATP_Olambda__661,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_awb(fun(A,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,log(aa(A,real,Uu,Uub)),aa(A,real,Uua,Uub)) ).

% ATP.lambda_661
tff(fact_8834_ATP_Olambda__662,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_atq(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_662
tff(fact_8835_ATP_Olambda__663,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: A] : aa(A,product_prod(B,C),aa(fun(A,C),fun(A,product_prod(B,C)),aTP_Lamp_ava(fun(A,B),fun(fun(A,C),fun(A,product_prod(B,C))),Uu),Uua),Uub) = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(A,B,Uu,Uub)),aa(A,C,Uua,Uub)) ) ).

% ATP.lambda_663
tff(fact_8836_ATP_Olambda__664,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bcb(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_664
tff(fact_8837_ATP_Olambda__665,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_sz(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_665
tff(fact_8838_ATP_Olambda__666,axiom,
    ! [B: $tType,Uu: fun(B,$o),Uua: fun(B,$o),Uub: B] :
      ( aa(B,$o,aa(fun(B,$o),fun(B,$o),aTP_Lamp_anx(fun(B,$o),fun(fun(B,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,$o,Uu,Uub)
        & aa(B,$o,Uua,Uub) ) ) ).

% ATP.lambda_666
tff(fact_8839_ATP_Olambda__667,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ta(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_667
tff(fact_8840_ATP_Olambda__668,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ajp(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uua,Uub)
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_668
tff(fact_8841_ATP_Olambda__669,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_apa(fun(A,B),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).

% ATP.lambda_669
tff(fact_8842_ATP_Olambda__670,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_bci(fun(nat,A),fun(fun(nat,A),fun(nat,$o)),Uu),Uua),Uub)
        <=> ( aa(nat,A,Uu,Uub) = aa(nat,A,Uua,Uub) ) ) ) ).

% ATP.lambda_670
tff(fact_8843_ATP_Olambda__671,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_bbl(fun(B,A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> ( aa(B,A,Uu,Uub) = aa(B,A,Uua,Uub) ) ) ).

% ATP.lambda_671
tff(fact_8844_ATP_Olambda__672,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_bcg(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_672
tff(fact_8845_ATP_Olambda__673,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] :
          ( aa(A,$o,aa(fun(A,A),fun(A,$o),aTP_Lamp_bch(fun(A,A),fun(fun(A,A),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,A,Uu,Uub) = aa(A,A,Uua,Uub) ) ) ) ).

% ATP.lambda_673
tff(fact_8846_ATP_Olambda__674,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_bcv(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_674
tff(fact_8847_ATP_Olambda__675,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bcc(fun(A,$o),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,Uu,Uub)
      <=> aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_675
tff(fact_8848_ATP_Olambda__676,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_bbn(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_676
tff(fact_8849_ATP_Olambda__677,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: A] :
      ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_bbt(fun(A,B),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uub) = aa(A,B,Uua,Uub) ) ) ).

% ATP.lambda_677
tff(fact_8850_ATP_Olambda__678,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bdm(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(A,B,Uu,Uub)),aa(int,B,ring_1_of_int(B),archimedean_ceiling(B,Uua))) ) ) ).

% ATP.lambda_678
tff(fact_8851_ATP_Olambda__679,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_apo(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Uu,Uub)),aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_679
tff(fact_8852_ATP_Olambda__680,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,real),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_zu(fun(nat,real),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_680
tff(fact_8853_ATP_Olambda__681,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: fun(B,real),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_zt(fun(B,real),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(B,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_681
tff(fact_8854_ATP_Olambda__682,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,real),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ars(fun(A,real),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),Uua) ) ).

% ATP.lambda_682
tff(fact_8855_ATP_Olambda__683,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_yc(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(C,A,aa(B,fun(C,A),Uu,Uub),Uua) ) ).

% ATP.lambda_683
tff(fact_8856_ATP_Olambda__684,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo4987421752381908075d_mult(C)
     => ! [Uu: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_avj(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu),Uua),Uub) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),Uua,Uub),Uu) ) ).

% ATP.lambda_684
tff(fact_8857_ATP_Olambda__685,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(B,fun(C,A)),Uua: set(C),Uub: B] : aa(B,A,aa(set(C),fun(B,A),aTP_Lamp_ub(fun(B,fun(C,A)),fun(set(C),fun(B,A)),Uu),Uua),Uub) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(B,fun(C,A),Uu,Uub)),Uua) ) ).

% ATP.lambda_685
tff(fact_8858_ATP_Olambda__686,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: set(A),Uua: fun(B,fun(A,B)),Uub: B] : aa(B,B,aa(fun(B,fun(A,B)),fun(B,B),aTP_Lamp_apz(set(A),fun(fun(B,fun(A,B)),fun(B,B)),Uu),Uua),Uub) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(B,fun(A,B),Uua,Uub)),Uu) ) ).

% ATP.lambda_686
tff(fact_8859_ATP_Olambda__687,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( topolo5987344860129210374id_add(C)
     => ! [Uu: set(A),Uua: fun(B,fun(A,C)),Uub: B] : aa(B,C,aa(fun(B,fun(A,C)),fun(B,C),aTP_Lamp_avr(set(A),fun(fun(B,fun(A,C)),fun(B,C)),Uu),Uua),Uub) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),Uua,Uub)),Uu) ) ).

% ATP.lambda_687
tff(fact_8860_ATP_Olambda__688,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_sr(fun(nat,nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less_eq(nat,aa(nat,nat,Uu,Uub)),Uua) ) ).

% ATP.lambda_688
tff(fact_8861_ATP_Olambda__689,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bcy(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_689
tff(fact_8862_ATP_Olambda__690,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bcp(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_690
tff(fact_8863_ATP_Olambda__691,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ur(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_691
tff(fact_8864_ATP_Olambda__692,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_mp(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_692
tff(fact_8865_ATP_Olambda__693,axiom,
    ! [B: $tType,A: $tType] :
      ( field(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ul(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_693
tff(fact_8866_ATP_Olambda__694,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_api(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_694
tff(fact_8867_ATP_Olambda__695,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_avd(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_695
tff(fact_8868_ATP_Olambda__696,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ayl(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_696
tff(fact_8869_ATP_Olambda__697,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_bbj(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,ord_less(A,aa(B,A,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_697
tff(fact_8870_ATP_Olambda__698,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bbh(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_698
tff(fact_8871_ATP_Olambda__699,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aly(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_699
tff(fact_8872_ATP_Olambda__700,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_mt(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_700
tff(fact_8873_ATP_Olambda__701,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_ui(fun(B,A),fun(A,fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_701
tff(fact_8874_ATP_Olambda__702,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_aro(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_702
tff(fact_8875_ATP_Olambda__703,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aqb(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_703
tff(fact_8876_ATP_Olambda__704,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_axf(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_704
tff(fact_8877_ATP_Olambda__705,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_aei(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(A,assn,Uu,Uub)),Uua) ).

% ATP.lambda_705
tff(fact_8878_ATP_Olambda__706,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_avy(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_706
tff(fact_8879_ATP_Olambda__707,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_axi(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_707
tff(fact_8880_ATP_Olambda__708,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_asi(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uub)),Uu) ) ).

% ATP.lambda_708
tff(fact_8881_ATP_Olambda__709,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vx(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_709
tff(fact_8882_ATP_Olambda__710,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ayj(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_710
tff(fact_8883_ATP_Olambda__711,axiom,
    ! [B: $tType,A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_atn(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uua,Uub)),Uu) ) ).

% ATP.lambda_711
tff(fact_8884_ATP_Olambda__712,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_wi(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,minus_minus(A,aa(nat,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_712
tff(fact_8885_ATP_Olambda__713,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_axt(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_713
tff(fact_8886_ATP_Olambda__714,axiom,
    ! [Uu: fun(real,real),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_aqk(fun(real,real),fun(nat,fun(real,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,Uu,Uub)),Uua) ).

% ATP.lambda_714
tff(fact_8887_ATP_Olambda__715,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,A),Uua: nat,Uub: B] : aa(B,A,aa(nat,fun(B,A),aTP_Lamp_xx(fun(B,A),fun(nat,fun(B,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(B,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_715
tff(fact_8888_ATP_Olambda__716,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_asl(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_716
tff(fact_8889_ATP_Olambda__717,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: nat,Uub: A] : aa(A,A,aa(nat,fun(A,A),aTP_Lamp_aqi(fun(A,A),fun(nat,fun(A,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,Uu,Uub)),Uua) ) ).

% ATP.lambda_717
tff(fact_8890_ATP_Olambda__718,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_auq(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_718
tff(fact_8891_ATP_Olambda__719,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V8999393235501362500lgebra(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_bei(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_719
tff(fact_8892_ATP_Olambda__720,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_awa(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_720
tff(fact_8893_ATP_Olambda__721,axiom,
    ! [A: $tType,B: $tType] :
      ( ( power(B)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: nat,Uub: A] : aa(A,B,aa(nat,fun(A,B),aTP_Lamp_axj(fun(A,B),fun(nat,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_721
tff(fact_8894_ATP_Olambda__722,axiom,
    ! [A: $tType,Uu: nat,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_atp(nat,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uua,Uub)),Uu) ).

% ATP.lambda_722
tff(fact_8895_ATP_Olambda__723,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_ahn(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),Uua) ) ).

% ATP.lambda_723
tff(fact_8896_ATP_Olambda__724,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(C,heap_Time_Heap(B)),Uua: fun(B,heap_Time_Heap(A)),Uub: C] : aa(C,heap_Time_Heap(A),aa(fun(B,heap_Time_Heap(A)),fun(C,heap_Time_Heap(A)),aTP_Lamp_jp(fun(C,heap_Time_Heap(B)),fun(fun(B,heap_Time_Heap(A)),fun(C,heap_Time_Heap(A))),Uu),Uua),Uub) = heap_Time_bind(B,A,aa(C,heap_Time_Heap(B),Uu,Uub),Uua) ).

% ATP.lambda_724
tff(fact_8897_ATP_Olambda__725,axiom,
    ! [Uu: fun(real,real),Uua: real,Uub: real] : aa(real,real,aa(real,fun(real,real),aTP_Lamp_aqn(fun(real,real),fun(real,fun(real,real)),Uu),Uua),Uub) = powr(real,aa(real,real,Uu,Uub),Uua) ).

% ATP.lambda_725
tff(fact_8898_ATP_Olambda__726,axiom,
    ! [A: $tType,Uu: real,Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_beg(real,fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = powr(real,aa(A,real,Uua,Uub),Uu) ).

% ATP.lambda_726
tff(fact_8899_ATP_Olambda__727,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(B,A),Uua: set(A),Uub: B] :
          ( aa(B,$o,aa(set(A),fun(B,$o),aTP_Lamp_ber(fun(B,A),fun(set(A),fun(B,$o)),Uu),Uua),Uub)
        <=> member(A,aa(B,A,Uu,Uub),Uua) ) ) ).

% ATP.lambda_727
tff(fact_8900_ATP_Olambda__728,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: set(B),Uub: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_bes(fun(A,B),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
        <=> member(B,aa(A,B,Uu,Uub),Uua) ) ) ).

% ATP.lambda_728
tff(fact_8901_ATP_Olambda__729,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_aoy(set(A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> member(A,aa(B,A,Uua,Uub),Uu) ) ).

% ATP.lambda_729
tff(fact_8902_ATP_Olambda__730,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: $o,Uub: A] :
      ( aa(A,$o,aa($o,fun(A,$o),aTP_Lamp_bca(fun(A,$o),fun($o,fun(A,$o)),Uu),(Uua)),Uub)
    <=> ( aa(A,$o,Uu,Uub)
        | (Uua) ) ) ).

% ATP.lambda_730
tff(fact_8903_ATP_Olambda__731,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [Uu: fun(A,word(B)),Uua: word(B),Uub: A] :
          ( aa(A,$o,aa(word(B),fun(A,$o),aTP_Lamp_akz(fun(A,word(B)),fun(word(B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,word(B),Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_731
tff(fact_8904_ATP_Olambda__732,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bbo(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_732
tff(fact_8905_ATP_Olambda__733,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo8865339358273720382pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bbq(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_733
tff(fact_8906_ATP_Olambda__734,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_aks(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_734
tff(fact_8907_ATP_Olambda__735,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_aju(nat,fun(fun(nat,A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_ajt(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,map(nat,A,Uua,upt(zero_zero(nat),Uu)),Uub)) ) ).

% ATP.lambda_735
tff(fact_8908_ATP_Olambda__736,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: A,Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_akb(nat,fun(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(array(A),heap_ext(product_unit)),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),product_case_prod(array(A),heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aTP_Lamp_ajt(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu)),array_alloc(A,replicate(A,Uu,Uua),Uub)) ) ).

% ATP.lambda_736
tff(fact_8909_ATP_Olambda__737,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(vEBT_VEBTi)] : aa(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_si(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Uu),Uua),Uub) = heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert(aa(list(vEBT_VEBTi),nat,size_size(list(vEBT_VEBTi)),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,Uua))),aa(list(vEBT_VEBTi),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(list(vEBT_VEBTi),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aTP_Lamp_sh(nat,fun(nat,fun(list(vEBT_VEBTi),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),Uu),Uua),Uub)) ).

% ATP.lambda_737
tff(fact_8910_ATP_Olambda__738,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(vEBT_VEBTi)] : aa(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sf(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Uu),Uua),Uub) = heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert(aa(list(vEBT_VEBTi),nat,size_size(list(vEBT_VEBTi)),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua)),aa(list(vEBT_VEBTi),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(list(vEBT_VEBTi),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aTP_Lamp_se(nat,fun(nat,fun(list(vEBT_VEBTi),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),Uu),Uua),Uub)) ).

% ATP.lambda_738
tff(fact_8911_ATP_Olambda__739,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_ge(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_739
tff(fact_8912_ATP_Olambda__740,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afl(nat,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uu),Uub))) ) ).

% ATP.lambda_740
tff(fact_8913_ATP_Olambda__741,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: real,Uua: fun(nat,A),Uub: nat] : aa(nat,real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_nj(real,fun(fun(nat,A),fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,Uua,Uub))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ) ).

% ATP.lambda_741
tff(fact_8914_ATP_Olambda__742,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_bdf(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uub))),aa(nat,real,Uua,Uub)) ) ) ).

% ATP.lambda_742
tff(fact_8915_ATP_Olambda__743,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bcz(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( archim6421214686448440834_floor(B,aa(A,B,Uu,Uub)) = archim6421214686448440834_floor(B,Uua) ) ) ) ).

% ATP.lambda_743
tff(fact_8916_ATP_Olambda__744,axiom,
    ! [A: $tType,B: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bda(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( archimedean_ceiling(B,aa(A,B,Uu,Uub)) = archimedean_ceiling(B,Uua) ) ) ) ).

% ATP.lambda_744
tff(fact_8917_ATP_Olambda__745,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ace(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua))) ) ).

% ATP.lambda_745
tff(fact_8918_ATP_Olambda__746,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi)] : aa(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sg(nat,fun(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Uu),Uua),Uub) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,vEBT_V739175172307565963ildupi(aa(nat,nat,suc,Uua)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sc(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),Uu),Uub)) ).

% ATP.lambda_746
tff(fact_8919_ATP_Olambda__747,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi)] : aa(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sl(nat,fun(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Uu),Uua),Uub) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,vEBT_vebt_buildupi(aa(nat,nat,suc,Uua)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sc(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),Uu),Uub)) ).

% ATP.lambda_747
tff(fact_8920_ATP_Olambda__748,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(B)
        & topolo2564578578187576103pology(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bdl(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(B,Uua))),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_748
tff(fact_8921_ATP_Olambda__749,axiom,
    ! [Uu: array(vEBT_VEBTi),Uua: nat,Uub: nat] : aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aTP_Lamp_hj(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),Uu),Uua),Uub) = heap_Time_bind(nat,$o,array_len(vEBT_VEBTi,Uu),aa(nat,fun(nat,heap_Time_Heap($o)),aa(nat,fun(nat,fun(nat,heap_Time_Heap($o))),aTP_Lamp_hi(array(vEBT_VEBTi),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))),Uu),Uua),Uub)) ).

% ATP.lambda_749
tff(fact_8922_ATP_Olambda__750,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi)] : aa(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sd(nat,fun(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Uu),Uua),Uub) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,vEBT_V739175172307565963ildupi(Uua),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sc(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),Uu),Uub)) ).

% ATP.lambda_750
tff(fact_8923_ATP_Olambda__751,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi)] : aa(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sj(nat,fun(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Uu),Uua),Uub) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,vEBT_vebt_buildupi(Uua),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sc(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),Uu),Uub)) ).

% ATP.lambda_751
tff(fact_8924_ATP_Olambda__752,axiom,
    ! [Uu: nat,Uua: option(nat),Uub: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aTP_Lamp_cd(nat,fun(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),Uu),Uua),Uub) = heap_Time_bind(option(nat),option(nat),vEBT_vebt_minti(Uub),aa(option(nat),fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_aa(nat,fun(option(nat),fun(option(nat),heap_Time_Heap(option(nat)))),Uu),Uua)) ).

% ATP.lambda_752
tff(fact_8925_ATP_Olambda__753,axiom,
    ! [Uu: nat,Uua: option(nat),Uub: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aTP_Lamp_ab(nat,fun(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),Uu),Uua),Uub) = heap_Time_bind(option(nat),option(nat),vEBT_vebt_maxti(Uub),aa(option(nat),fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_aa(nat,fun(option(nat),fun(option(nat),heap_Time_Heap(option(nat)))),Uu),Uua)) ).

% ATP.lambda_753
tff(fact_8926_ATP_Olambda__754,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(vEBT_VEBTi)] : aa(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sm(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Uu),Uua),Uub) = heap_Time_bind(array(vEBT_VEBTi),vEBT_VEBTi,array_of_list(vEBT_VEBTi,Uub),aa(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sl(nat,fun(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Uu),Uua)) ).

% ATP.lambda_754
tff(fact_8927_ATP_Olambda__755,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(vEBT_VEBTi)] : aa(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sk(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Uu),Uua),Uub) = heap_Time_bind(array(vEBT_VEBTi),vEBT_VEBTi,array_of_list(vEBT_VEBTi,Uub),aa(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sj(nat,fun(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Uu),Uua)) ).

% ATP.lambda_755
tff(fact_8928_ATP_Olambda__756,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] : aa(A,fun(list(A),list(A)),aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_alq(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),Uu),Uua),Uub) = case_list(list(A),A,Uua,aa(A,fun(A,fun(list(A),list(A))),aa(list(A),fun(A,fun(A,fun(list(A),list(A)))),aTP_Lamp_alp(fun(A,B),fun(list(A),fun(A,fun(A,fun(list(A),list(A))))),Uu),Uua),Uub)) ) ).

% ATP.lambda_756
tff(fact_8929_ATP_Olambda__757,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: nat,Uua: nat,Uub: word(A)] :
          ( aa(word(A),$o,aa(nat,fun(word(A),$o),aTP_Lamp_ami(nat,fun(nat,fun(word(A),$o)),Uu),Uua),Uub)
        <=> aa(word(A),$o,ord_less(word(A),Uub),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Uu)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Uua))) ) ) ).

% ATP.lambda_757
tff(fact_8930_ATP_Olambda__758,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_amj(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less(nat,Uub),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uua))) ) ).

% ATP.lambda_758
tff(fact_8931_ATP_Olambda__759,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afi(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(nat,A,Uu,set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_759
tff(fact_8932_ATP_Olambda__760,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afj(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_760
tff(fact_8933_ATP_Olambda__761,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_afh(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uua))) ) ).

% ATP.lambda_761
tff(fact_8934_ATP_Olambda__762,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_vm(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_762
tff(fact_8935_ATP_Olambda__763,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_vk(A,fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uub)),Uua)) ) ).

% ATP.lambda_763
tff(fact_8936_ATP_Olambda__764,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: array(A),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ajt(nat,fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uua),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uub),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),one_one(nat)))) ) ).

% ATP.lambda_764
tff(fact_8937_ATP_Olambda__765,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ahd(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> member(A,Uub,aa(set(A),set(A),minus_minus(set(A),Uua),aa(set(A),set(A),insert(A,Uu),bot_bot(set(A))))) ) ).

% ATP.lambda_765
tff(fact_8938_ATP_Olambda__766,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: list(A),Uua: array(A),Uub: heap_ext(product_unit)] : aa(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_ajz(list(A),fun(array(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Uu),Uua),Uub) = aa(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat)),aa(array(A),fun(product_prod(heap_ext(product_unit),nat),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),product_Pair(array(A),product_prod(heap_ext(product_unit),nat)),Uua),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uub),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(list(A),nat,size_size(list(A)),Uu)))) ) ).

% ATP.lambda_766
tff(fact_8939_ATP_Olambda__767,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: array(A)] : aa(array(A),assn,aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_ajm(nat,fun(fun(nat,A),fun(array(A),assn)),Uu),Uua),Uub) = aa(list(A),assn,snga_assn(A,Uub),map(nat,A,Uua,upt(zero_zero(nat),Uu))) ) ).

% ATP.lambda_767
tff(fact_8940_ATP_Olambda__768,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wo(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_768
tff(fact_8941_ATP_Olambda__769,axiom,
    ! [Uu: real,Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_azu(real,fun(real,fun(nat,real)),Uu),Uua),Uub) = aa(real,real,aa(real,fun(real,real),divide_divide(real),Uua),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uu),Uub)) ).

% ATP.lambda_769
tff(fact_8942_ATP_Olambda__770,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_vb(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)) ) ).

% ATP.lambda_770
tff(fact_8943_ATP_Olambda__771,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_vl(nat,fun(nat,fun(nat,nat)),Uu),Uua),Uub) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uu),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ).

% ATP.lambda_771
tff(fact_8944_ATP_Olambda__772,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: A,Uub: array(A)] : aa(array(A),assn,aa(A,fun(array(A),assn),aTP_Lamp_aft(nat,fun(A,fun(array(A),assn)),Uu),Uua),Uub) = aa(list(A),assn,snga_assn(A,Uub),replicate(A,Uu,Uua)) ) ).

% ATP.lambda_772
tff(fact_8945_ATP_Olambda__773,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,option(A)),Uua: list(B),Uub: A] : aa(A,list(A),aa(list(B),fun(A,list(A)),aTP_Lamp_alt(fun(B,option(A)),fun(list(B),fun(A,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),cons(A,Uub),map_filter(B,A,Uu,Uua)) ).

% ATP.lambda_773
tff(fact_8946_ATP_Olambda__774,axiom,
    ! [Uu: real,Uua: real,Uub: real] :
      ( aa(real,$o,aa(real,fun(real,$o),aTP_Lamp_bdy(real,fun(real,fun(real,$o)),Uu),Uua),Uub)
    <=> member(real,Uub,set_or5935395276787703475ssThan(real,Uu,Uua)) ) ).

% ATP.lambda_774
tff(fact_8947_ATP_Olambda__775,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Uu: real,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_zk(real,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_775
tff(fact_8948_ATP_Olambda__776,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: real,Uub: nat] : aa(nat,A,aa(real,fun(nat,A),aTP_Lamp_zm(fun(nat,A),fun(real,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_776
tff(fact_8949_ATP_Olambda__777,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: real,Uub: A] : aa(A,B,aa(real,fun(A,B),aTP_Lamp_arr(fun(A,B),fun(real,fun(A,B)),Uu),Uua),Uub) = aa(B,B,real_V8093663219630862766scaleR(B,Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_777
tff(fact_8950_ATP_Olambda__778,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: real,Uub: A] : aa(A,A,aa(real,fun(A,A),aTP_Lamp_apx(fun(A,A),fun(real,fun(A,A)),Uu),Uua),Uub) = aa(A,A,real_V8093663219630862766scaleR(A,Uua),aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_778
tff(fact_8951_ATP_Olambda__779,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aTP_Lamp_bco(B,fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,Uu),aa(A,B,Uua,Uub)) ) ) ).

% ATP.lambda_779
tff(fact_8952_ATP_Olambda__780,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1944317154257567458pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bcx(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_780
tff(fact_8953_ATP_Olambda__781,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bcq(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_781
tff(fact_8954_ATP_Olambda__782,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bbr(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less_eq(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_782
tff(fact_8955_ATP_Olambda__783,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_anp(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_783
tff(fact_8956_ATP_Olambda__784,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B] :
          ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_bbi(fun(B,A),fun(A,fun(B,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,ord_less(A,Uua),aa(B,A,Uu,Uub)) ) ) ).

% ATP.lambda_784
tff(fact_8957_ATP_Olambda__785,axiom,
    ! [B: $tType,A: $tType] :
      ( unboun7993243217541854897norder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bbf(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_785
tff(fact_8958_ATP_Olambda__786,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo2564578578187576103pology(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bbg(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_786
tff(fact_8959_ATP_Olambda__787,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_alz(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_787
tff(fact_8960_ATP_Olambda__788,axiom,
    ! [B: $tType,Uu: assn,Uua: fun(B,assn),Uub: B] : aa(B,assn,aa(fun(B,assn),fun(B,assn),aTP_Lamp_aej(assn,fun(fun(B,assn),fun(B,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu),aa(B,assn,Uua,Uub)) ).

% ATP.lambda_788
tff(fact_8961_ATP_Olambda__789,axiom,
    ! [A: $tType,Uu: assn,Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_aey(assn,fun(fun(A,assn),fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu),aa(A,assn,Uua,Uub)) ).

% ATP.lambda_789
tff(fact_8962_ATP_Olambda__790,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_mo(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_790
tff(fact_8963_ATP_Olambda__791,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_uj(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_791
tff(fact_8964_ATP_Olambda__792,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vw(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_792
tff(fact_8965_ATP_Olambda__793,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_ayk(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_793
tff(fact_8966_ATP_Olambda__794,axiom,
    ! [A: $tType,B: $tType] :
      ( ( field(A)
        & topolo4211221413907600880p_mult(A) )
     => ! [Uu: A,Uua: fun(B,A),Uub: B] : aa(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ato(A,fun(fun(B,A),fun(B,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uu),aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_794
tff(fact_8967_ATP_Olambda__795,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_mu(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(nat,A,Uu,Uub)) ) ).

% ATP.lambda_795
tff(fact_8968_ATP_Olambda__796,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_arn(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_796
tff(fact_8969_ATP_Olambda__797,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aqc(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_797
tff(fact_8970_ATP_Olambda__798,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_axe(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_798
tff(fact_8971_ATP_Olambda__799,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_afa(fun(A,assn),fun(assn,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uua),aa(A,assn,Uu,Uub)) ).

% ATP.lambda_799
tff(fact_8972_ATP_Olambda__800,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V4412858255891104859lgebra(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_avx(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_800
tff(fact_8973_ATP_Olambda__801,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4211221413907600880p_mult(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_axh(fun(A,B),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),times_times(B),Uua),aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_801
tff(fact_8974_ATP_Olambda__802,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: fun(B,nat),Uub: B] : aa(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_yj(A,fun(fun(B,nat),fun(B,A)),Uu),Uua),Uub) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(B,nat,Uua,Uub)) ) ).

% ATP.lambda_802
tff(fact_8975_ATP_Olambda__803,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V2822296259951069270ebra_1(B)
     => ! [Uu: fun(A,nat),Uua: B,Uub: A] : aa(A,B,aa(B,fun(A,B),aTP_Lamp_bac(fun(A,nat),fun(B,fun(A,B)),Uu),Uua),Uub) = aa(nat,B,aa(B,fun(nat,B),power_power(B),Uua),aa(A,nat,Uu,Uub)) ) ).

% ATP.lambda_803
tff(fact_8976_ATP_Olambda__804,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo1633459387980952147up_add(B)
     => ! [Uu: B,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_awe(B,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(B,B,aa(B,fun(B,B),plus_plus(B),Uu),aa(A,B,Uua,Uub)) ) ).

% ATP.lambda_804
tff(fact_8977_ATP_Olambda__805,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_ayt(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ) ).

% ATP.lambda_805
tff(fact_8978_ATP_Olambda__806,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: nat,Uub: A] : aa(A,real,aa(nat,fun(A,real),aTP_Lamp_axc(fun(A,real),fun(nat,fun(A,real)),Uu),Uua),Uub) = aa(real,real,root(Uua),aa(A,real,Uu,Uub)) ).

% ATP.lambda_806
tff(fact_8979_ATP_Olambda__807,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bby($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_807
tff(fact_8980_ATP_Olambda__808,axiom,
    ! [A: $tType,Uu: $o,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_bbz($o,fun(fun(A,$o),fun(A,$o)),(Uu)),Uua),Uub)
    <=> ( (Uu)
        | aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_808
tff(fact_8981_ATP_Olambda__809,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: $o] :
      ( aa($o,$o,aa(A,fun($o,$o),aTP_Lamp_ant(fun(A,$o),fun(A,fun($o,$o)),Uu),Uua),(Uub))
    <=> ( (Uub)
        & aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_809
tff(fact_8982_ATP_Olambda__810,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_apb(fun(A,B),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( Uub = aa(A,B,Uu,Uua) ) ) ).

% ATP.lambda_810
tff(fact_8983_ATP_Olambda__811,axiom,
    ! [A: $tType,Uu: assn,Uua: A,Uub: A] : aa(A,assn,aa(A,fun(A,assn),aTP_Lamp_aeh(assn,fun(A,fun(A,assn)),Uu),Uua),Uub) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu),pure_assn(Uub = Uua)) ).

% ATP.lambda_811
tff(fact_8984_ATP_Olambda__812,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ys(A,fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,Uua),Uub))) ) ).

% ATP.lambda_812
tff(fact_8985_ATP_Olambda__813,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: real,Uub: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aTP_Lamp_bep(fun(A,B),fun(real,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(real,$o,ord_less_eq(real,Uua),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uub))) ) ) ).

% ATP.lambda_813
tff(fact_8986_ATP_Olambda__814,axiom,
    ! [A: $tType,Uu: list(A),Uua: A,Uub: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_ame(list(A),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = append(A,Uub,Uu) ).

% ATP.lambda_814
tff(fact_8987_ATP_Olambda__815,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_tk(fun(A,nat),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),
          $ite(Uub = Uua,aa(nat,nat,suc,aa(A,nat,Uu,Uub)),aa(A,nat,Uu,Uub))) ) ).

% ATP.lambda_815
tff(fact_8988_ATP_Olambda__816,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_tl(fun(A,nat),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),
          $ite(aa(A,$o,Uua,Uub),aa(A,nat,Uu,Uub),zero_zero(nat))) ) ).

% ATP.lambda_816
tff(fact_8989_ATP_Olambda__817,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] :
      ( aa(A,$o,aa(fun(A,nat),fun(A,$o),aTP_Lamp_tj(fun(A,nat),fun(fun(A,nat),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,minus_minus(nat,aa(A,nat,Uu,Uub)),aa(A,nat,Uua,Uub))) ) ).

% ATP.lambda_817
tff(fact_8990_ATP_Olambda__818,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_yn(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_818
tff(fact_8991_ATP_Olambda__819,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wb(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uub))) ) ).

% ATP.lambda_819
tff(fact_8992_ATP_Olambda__820,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_azc(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)) ) ).

% ATP.lambda_820
tff(fact_8993_ATP_Olambda__821,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ayu(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,Uub),Uua)) ) ).

% ATP.lambda_821
tff(fact_8994_ATP_Olambda__822,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_atx(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_822
tff(fact_8995_ATP_Olambda__823,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aqg(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)) ) ).

% ATP.lambda_823
tff(fact_8996_ATP_Olambda__824,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_bbb(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_824
tff(fact_8997_ATP_Olambda__825,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_mn(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_825
tff(fact_8998_ATP_Olambda__826,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ayn(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_826
tff(fact_8999_ATP_Olambda__827,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_yk(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_827
tff(fact_9000_ATP_Olambda__828,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ux(fun(nat,A),fun(nat,fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)) ) ).

% ATP.lambda_828
tff(fact_9001_ATP_Olambda__829,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,$o),Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_bcn(fun(A,$o),fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ) ).

% ATP.lambda_829
tff(fact_9002_ATP_Olambda__830,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_auv(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_830
tff(fact_9003_ATP_Olambda__831,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,B),Uua: A,Uub: A] : aa(A,B,aa(A,fun(A,B),aTP_Lamp_axz(fun(A,B),fun(A,fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_831
tff(fact_9004_ATP_Olambda__832,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_apv(fun(A,A),fun(A,fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uua)) ) ).

% ATP.lambda_832
tff(fact_9005_ATP_Olambda__833,axiom,
    ! [A: $tType,B: $tType,Uu: fun(product_prod(A,B),$o),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_aep(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> aa(product_prod(A,B),$o,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ) ).

% ATP.lambda_833
tff(fact_9006_ATP_Olambda__834,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(product_prod(A,B),C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_co(fun(product_prod(A,B),C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(product_prod(A,B),C,Uu,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua),Uub)) ).

% ATP.lambda_834
tff(fact_9007_ATP_Olambda__835,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_ajo(fun(A,$o),fun(list(A),fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,Uu,aa(nat,A,nth(A,Uua),Uub)) ) ).

% ATP.lambda_835
tff(fact_9008_ATP_Olambda__836,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: A,Uua: fun(A,B),Uub: A] : aa(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aty(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = aa(A,B,Uua,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uub)) ) ).

% ATP.lambda_836
tff(fact_9009_ATP_Olambda__837,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wa(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(nat,A,Uua,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uu)) ) ).

% ATP.lambda_837
tff(fact_9010_ATP_Olambda__838,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(real,real),Uua: fun(A,real),Uub: A] : aa(A,real,aa(fun(A,real),fun(A,real),aTP_Lamp_ash(fun(real,real),fun(fun(A,real),fun(A,real)),Uu),Uua),Uub) = aa(real,real,Uu,aa(A,real,Uua,Uub)) ) ).

% ATP.lambda_838
tff(fact_9011_ATP_Olambda__839,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(real,A),Uua: fun(nat,real),Uub: nat] : aa(nat,A,aa(fun(nat,real),fun(nat,A),aTP_Lamp_bel(fun(real,A),fun(fun(nat,real),fun(nat,A)),Uu),Uua),Uub) = aa(real,A,Uu,aa(nat,real,Uua,Uub)) ) ).

% ATP.lambda_839
tff(fact_9012_ATP_Olambda__840,axiom,
    ! [B: $tType,C: $tType,D2: $tType,A: $tType,Uu: fun(D2,fun(B,C)),Uua: fun(A,D2),Uub: A] : aa(A,fun(B,C),aa(fun(A,D2),fun(A,fun(B,C)),aTP_Lamp_cs(fun(D2,fun(B,C)),fun(fun(A,D2),fun(A,fun(B,C))),Uu),Uua),Uub) = aa(D2,fun(B,C),Uu,aa(A,D2,Uua,Uub)) ).

% ATP.lambda_840
tff(fact_9013_ATP_Olambda__841,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,A),Uua: fun(B,C),Uub: B] : aa(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_ahj(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_841
tff(fact_9014_ATP_Olambda__842,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(num,B),Uub: num] : aa(num,A,aa(fun(num,B),fun(num,A),aTP_Lamp_adf(fun(B,A),fun(fun(num,B),fun(num,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(num,B,Uua,Uub)) ).

% ATP.lambda_842
tff(fact_9015_ATP_Olambda__843,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(nat,B),Uub: nat] : aa(nat,A,aa(fun(nat,B),fun(nat,A),aTP_Lamp_acx(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(nat,B,Uua,Uub)) ).

% ATP.lambda_843
tff(fact_9016_ATP_Olambda__844,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,B),Uub: C] : aa(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_cn(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_844
tff(fact_9017_ATP_Olambda__845,axiom,
    ! [A: $tType] :
      ( topolo3112930676232923870pology(A)
     => ! [Uu: fun(A,$o),Uua: fun(nat,A),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,A),fun(nat,$o),aTP_Lamp_bbc(fun(A,$o),fun(fun(nat,A),fun(nat,$o)),Uu),Uua),Uub)
        <=> aa(A,$o,Uu,aa(nat,A,Uua,Uub)) ) ) ).

% ATP.lambda_845
tff(fact_9018_ATP_Olambda__846,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo3112930676232923870pology(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_aze(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_846
tff(fact_9019_ATP_Olambda__847,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_baq(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_847
tff(fact_9020_ATP_Olambda__848,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ark(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_848
tff(fact_9021_ATP_Olambda__849,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_bao(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_849
tff(fact_9022_ATP_Olambda__850,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_ban(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_850
tff(fact_9023_ATP_Olambda__851,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,A),Uub: nat] : aa(nat,B,aa(fun(nat,A),fun(nat,B),aTP_Lamp_azf(fun(A,B),fun(fun(nat,A),fun(nat,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(nat,A,Uua,Uub)) ) ).

% ATP.lambda_851
tff(fact_9024_ATP_Olambda__852,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_auk(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_852
tff(fact_9025_ATP_Olambda__853,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_apt(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_853
tff(fact_9026_ATP_Olambda__854,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_aum(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ) ).

% ATP.lambda_854
tff(fact_9027_ATP_Olambda__855,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_bbm(fun(A,$o),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> aa(A,$o,Uu,aa(B,A,Uua,Uub)) ) ).

% ATP.lambda_855
tff(fact_9028_ATP_Olambda__856,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,B),Uua: fun(C,A),Uub: C] : aa(C,B,aa(fun(C,A),fun(C,B),aTP_Lamp_bat(fun(A,B),fun(fun(C,A),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,aa(C,A,Uua,Uub)) ).

% ATP.lambda_856
tff(fact_9029_ATP_Olambda__857,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( topological_t2_space(C)
        & topological_t2_space(A)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(C,A),Uua: fun(A,B),Uub: C] : aa(C,B,aa(fun(A,B),fun(C,B),aTP_Lamp_aun(fun(C,A),fun(fun(A,B),fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uua,aa(C,A,Uu,Uub)) ) ).

% ATP.lambda_857
tff(fact_9030_ATP_Olambda__858,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(A,C),Uub: B] : aa(B,C,aa(fun(A,C),fun(B,C),aTP_Lamp_aho(fun(B,A),fun(fun(A,C),fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu,Uub)) ).

% ATP.lambda_858
tff(fact_9031_ATP_Olambda__859,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_arf(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_859
tff(fact_9032_ATP_Olambda__860,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_auw(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_860
tff(fact_9033_ATP_Olambda__861,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topolo4958980785337419405_space(A)
        & topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_bct(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_861
tff(fact_9034_ATP_Olambda__862,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: fun(A,A),Uub: A] : aa(A,A,aa(fun(A,A),fun(A,A),aTP_Lamp_aps(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uua,aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_862
tff(fact_9035_ATP_Olambda__863,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topological_t2_space(B)
        & topolo4958980785337419405_space(C) )
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aui(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_863
tff(fact_9036_ATP_Olambda__864,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_bbk(fun(A,B),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> aa(B,$o,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_864
tff(fact_9037_ATP_Olambda__865,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_acn(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_865
tff(fact_9038_ATP_Olambda__866,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: fun(A,B),Uua: fun(B,C),Uub: A] : aa(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_aco(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_866
tff(fact_9039_ATP_Olambda__867,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(A,C),Uua: A,Uub: B] : aa(B,C,aa(A,fun(B,C),aTP_Lamp_fj(fun(A,C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uu,Uua) ).

% ATP.lambda_867
tff(fact_9040_ATP_Olambda__868,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: C] : aa(C,B,aa(A,fun(C,B),aTP_Lamp_aoz(fun(A,B),fun(A,fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,Uua) ).

% ATP.lambda_868
tff(fact_9041_ATP_Olambda__869,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: fun(B,heap_Time_Heap(A)),Uub: product_unit] : aa(product_unit,heap_Time_Heap(A),aa(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A)),aTP_Lamp_aiu(B,fun(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A))),Uu),Uua),Uub) = aa(B,heap_Time_Heap(A),Uua,Uu) ).

% ATP.lambda_869
tff(fact_9042_ATP_Olambda__870,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sc(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),Uu),Uua),Uub) = aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Uu)),Uua,Uub)) ).

% ATP.lambda_870
tff(fact_9043_ATP_Olambda__871,axiom,
    ! [A: $tType,Uu: heap_Time_Heap(A),Uua: heap_ext(product_unit),Uub: A] : aa(A,fun(product_prod(heap_ext(product_unit),nat),$o),aa(heap_ext(product_unit),fun(A,fun(product_prod(heap_ext(product_unit),nat),$o)),aTP_Lamp_akg(heap_Time_Heap(A),fun(heap_ext(product_unit),fun(A,fun(product_prod(heap_ext(product_unit),nat),$o))),Uu),Uua),Uub) = product_case_prod(heap_ext(product_unit),nat,$o,aa(A,fun(heap_ext(product_unit),fun(nat,$o)),aa(heap_ext(product_unit),fun(A,fun(heap_ext(product_unit),fun(nat,$o))),aTP_Lamp_akf(heap_Time_Heap(A),fun(heap_ext(product_unit),fun(A,fun(heap_ext(product_unit),fun(nat,$o)))),Uu),Uua),Uub)) ).

% ATP.lambda_871
tff(fact_9044_ATP_Olambda__872,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & real_Vector_banach(B)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(nat,B),Uub: A] : aa(A,B,aa(fun(nat,B),fun(A,B),aTP_Lamp_ayh(fun(A,B),fun(fun(nat,B),fun(A,B)),Uu),Uua),Uub) = suminf(B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_ayg(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub)) ) ).

% ATP.lambda_872
tff(fact_9045_ATP_Olambda__873,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_atk(fun(nat,A),fun(A,fun(A,A)),Uu),Uua),Uub) = suminf(A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_atj(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub)) ) ).

% ATP.lambda_873
tff(fact_9046_ATP_Olambda__874,axiom,
    ! [Uu: nat,Uua: $o,Uub: $o] :
      aa($o,heap_Time_Heap($o),aa($o,fun($o,heap_Time_Heap($o)),aTP_Lamp_hg(nat,fun($o,fun($o,heap_Time_Heap($o))),Uu),(Uua)),(Uub)) = aa($o,heap_Time_Heap($o),heap_Time_return($o),
        $ite(
          Uu = zero_zero(nat),
          (Uua),
          $ite(Uu = one_one(nat),(Uub),$false) )) ).

% ATP.lambda_874
tff(fact_9047_ATP_Olambda__875,axiom,
    ! [Uu: nat,Uua: nat,Uub: option(nat)] : aa(option(nat),heap_Time_Heap(option(nat)),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_a(nat,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uu),Uua),Uub) = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),Uua))),Uub)) ).

% ATP.lambda_875
tff(fact_9048_ATP_Olambda__876,axiom,
    ! [Uu: nat,Uua: option(nat),Uub: option(nat)] : aa(option(nat),heap_Time_Heap(option(nat)),aa(option(nat),fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_aa(nat,fun(option(nat),fun(option(nat),heap_Time_Heap(option(nat)))),Uu),Uua),Uub) = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),Uua)),Uub)) ).

% ATP.lambda_876
tff(fact_9049_ATP_Olambda__877,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_na(fun(nat,A),fun(A,fun(nat,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub))) ) ).

% ATP.lambda_877
tff(fact_9050_ATP_Olambda__878,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A] : aa(A,real,aa(fun(A,B),fun(A,real),aTP_Lamp_yq(fun(A,B),fun(fun(A,B),fun(A,real)),Uu),Uua),Uub) = real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uub)),aa(A,B,Uua,Uub))) ) ).

% ATP.lambda_878
tff(fact_9051_ATP_Olambda__879,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_bah(fun(nat,real),fun(real,fun(nat,real)),Uu),Uua),Uub) = cos(real,aa(real,real,minus_minus(real,aa(nat,real,Uu,Uub)),Uua)) ).

% ATP.lambda_879
tff(fact_9052_ATP_Olambda__880,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bcs(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_880
tff(fact_9053_ATP_Olambda__881,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bdu(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_881
tff(fact_9054_ATP_Olambda__882,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bcu(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_882
tff(fact_9055_ATP_Olambda__883,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t1_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_bbp(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) != Uua ) ) ) ).

% ATP.lambda_883
tff(fact_9056_ATP_Olambda__884,axiom,
    ! [C: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A,Uua: fun(C,A),Uub: C] :
          ( aa(C,$o,aa(fun(C,A),fun(C,$o),aTP_Lamp_bcr(A,fun(fun(C,A),fun(C,$o)),Uu),Uua),Uub)
        <=> ( aa(C,A,Uua,Uub) != Uu ) ) ) ).

% ATP.lambda_884
tff(fact_9057_ATP_Olambda__885,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,assn,aa(nat,fun(nat,assn),aTP_Lamp_adv(nat,fun(nat,fun(nat,assn)),Uu),Uua),Uub) = pure_assn(Uub = vEBT_VEBT_high(Uu,Uua)) ).

% ATP.lambda_885
tff(fact_9058_ATP_Olambda__886,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,assn,aa(nat,fun(nat,assn),aTP_Lamp_adw(nat,fun(nat,fun(nat,assn)),Uu),Uua),Uub) = pure_assn(Uub = vEBT_VEBT_low(Uu,Uua)) ).

% ATP.lambda_886
tff(fact_9059_ATP_Olambda__887,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & archim2362893244070406136eiling(A)
        & topolo2564578578187576103pology(A) )
     => ! [Uu: fun(B,real),Uua: fun(real,A),Uub: B] : aa(B,real,aa(fun(real,A),fun(B,real),aTP_Lamp_ath(fun(B,real),fun(fun(real,A),fun(B,real)),Uu),Uua),Uub) = aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(A,aa(real,A,Uua,aa(B,real,Uu,Uub)))) ) ).

% ATP.lambda_887
tff(fact_9060_ATP_Olambda__888,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,C)),Uua: set(C),Uub: A] :
      ( aa(A,$o,aa(set(C),fun(A,$o),aTP_Lamp_amu(fun(A,fun(B,C)),fun(set(C),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [B7: B] : member(C,aa(B,C,aa(A,fun(B,C),Uu,Uub),B7),Uua) ) ).

% ATP.lambda_888
tff(fact_9061_ATP_Olambda__889,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_amq(fun(B,A),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [L4: B] :
          ( ( Uub = aa(B,A,Uu,L4) )
          & member(B,L4,Uua) ) ) ).

% ATP.lambda_889
tff(fact_9062_ATP_Olambda__890,axiom,
    ! [B: $tType,C: $tType,Uu: fun(B,set(C)),Uua: fun(C,$o),Uub: B] :
      ( aa(B,$o,aa(fun(C,$o),fun(B,$o),aTP_Lamp_anw(fun(B,set(C)),fun(fun(C,$o),fun(B,$o)),Uu),Uua),Uub)
    <=> ! [X: C] :
          ( member(C,X,aa(B,set(C),Uu,Uub))
         => aa(C,$o,Uua,X) ) ) ).

% ATP.lambda_890
tff(fact_9063_ATP_Olambda__891,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_amr(fun(B,A),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X: B] :
          ( ( Uub = aa(B,A,Uu,X) )
          & aa(B,$o,Uua,X) ) ) ).

% ATP.lambda_891
tff(fact_9064_ATP_Olambda__892,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(A,B),Uub: B] :
      ( aa(B,$o,aa(fun(A,B),fun(B,$o),aTP_Lamp_amz(fun(A,$o),fun(fun(A,B),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [X: A] :
          ( ( Uub = aa(A,B,Uua,X) )
          & aa(A,$o,Uu,X) ) ) ).

% ATP.lambda_892
tff(fact_9065_ATP_Olambda__893,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: list(B),Uub: A] :
      ( aa(A,$o,aa(list(B),fun(A,$o),aTP_Lamp_anc(fun(B,option(A)),fun(list(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X: B] :
          ( member(B,X,set2(B,Uua))
          & ( aa(B,option(A),Uu,X) = aa(A,option(A),some(A),Uub) ) ) ) ).

% ATP.lambda_893
tff(fact_9066_ATP_Olambda__894,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_anb(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ! [X: A] :
          ( member(A,X,Uu)
         => aa(A,$o,aa(B,fun(A,$o),Uua,Uub),X) ) ) ).

% ATP.lambda_894
tff(fact_9067_ATP_Olambda__895,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(A,fun(B,C)),Uua: set(C),Uub: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aa(set(C),fun(product_prod(A,B),$o),aTP_Lamp_amt(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),$o)),Uu),Uua),Uub)
    <=> ? [P7: product_prod(A,B)] :
          ( ( Uub = P7 )
          & member(C,aa(B,C,aa(A,fun(B,C),Uu,aa(product_prod(A,B),A,product_fst(A,B),P7)),aa(product_prod(A,B),B,product_snd(A,B),P7)),Uua) ) ) ).

% ATP.lambda_895
tff(fact_9068_ATP_Olambda__896,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_bba(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> ! [N8: nat] :
              ( aa(nat,$o,ord_less_eq(nat,Uub),N8)
             => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or7035219750837199246ssThan(nat,Uub,N8)))),aa(nat,real,Uua,Uub)) ) ) ) ).

% ATP.lambda_896
tff(fact_9069_ATP_Olambda__897,axiom,
    ! [A: $tType] :
      ( ( real_V8037385150606011577_space(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,real),Uub: nat] :
          ( aa(nat,$o,aa(fun(nat,real),fun(nat,$o),aTP_Lamp_bdq(fun(nat,A),fun(fun(nat,real),fun(nat,$o)),Uu),Uua),Uub)
        <=> ! [A7: nat] :
              ( aa(nat,$o,ord_less_eq(nat,Uub),A7)
             => ! [B7: nat] :
                  ( aa(nat,$o,ord_less(nat,A7),B7)
                 => aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,Uu),set_or3652927894154168847AtMost(nat,A7,B7)))),aa(nat,real,Uua,A7)) ) ) ) ) ).

% ATP.lambda_897
tff(fact_9070_ATP_Olambda__898,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,$o),Uua: fun(B,fun(A,$o)),Uub: B] :
      ( aa(B,$o,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_amo(fun(A,$o),fun(fun(B,fun(A,$o)),fun(B,$o)),Uu),Uua),Uub)
    <=> ? [Y3: A] :
          ( aa(A,$o,Uu,Y3)
          & aa(A,$o,aa(B,fun(A,$o),Uua,Uub),Y3) ) ) ).

% ATP.lambda_898
tff(fact_9071_ATP_Olambda__899,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: fun(A,fun(B,$o)),Uub: fun(A,B)] :
      ( aa(fun(A,B),$o,aa(fun(A,fun(B,$o)),fun(fun(A,B),$o),aTP_Lamp_air(B,fun(fun(A,fun(B,$o)),fun(fun(A,B),$o)),Uu),Uua),Uub)
    <=> ! [X: A] :
          ( ( aa(A,B,Uub,X) != Uu )
         => aa(B,$o,aa(A,fun(B,$o),Uua,X),aa(A,B,Uub,X)) ) ) ).

% ATP.lambda_899
tff(fact_9072_ATP_Olambda__900,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,fun(B,$o)),Uub: product_prod(A,A)] :
      ( aa(product_prod(A,A),$o,aa(fun(B,fun(B,$o)),fun(product_prod(A,A),$o),aTP_Lamp_anm(fun(B,A),fun(fun(B,fun(B,$o)),fun(product_prod(A,A),$o)),Uu),Uua),Uub)
    <=> ? [A7: B,B7: B] :
          ( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uu,A7)),aa(B,A,Uu,B7)) )
          & aa(B,$o,aa(B,fun(B,$o),Uua,A7),B7) ) ) ).

% ATP.lambda_900
tff(fact_9073_ATP_Olambda__901,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(list(A)),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(list(A)),fun(list(A),$o),aTP_Lamp_amv(set(A),fun(set(list(A)),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [X: A,Xs3: list(A)] :
          ( ( Uub = aa(list(A),list(A),cons(A,X),Xs3) )
          & member(A,X,Uu)
          & member(list(A),Xs3,Uua) ) ) ).

% ATP.lambda_901
tff(fact_9074_ATP_Olambda__902,axiom,
    ! [B: $tType,A: $tType,E: $tType,C: $tType,D2: $tType,Uu: fun(E,fun(A,fun(C,fun(B,fun(D2,$o))))),Uua: E,Uub: fun(A,option(product_prod(B,product_prod(C,D2))))] :
      ( aa(fun(A,option(product_prod(B,product_prod(C,D2)))),$o,aa(E,fun(fun(A,option(product_prod(B,product_prod(C,D2)))),$o),aTP_Lamp_aio(fun(E,fun(A,fun(C,fun(B,fun(D2,$o))))),fun(E,fun(fun(A,option(product_prod(B,product_prod(C,D2)))),$o)),Uu),Uua),Uub)
    <=> ! [H6: A,H10: C,R6: B,N8: D2] :
          ( ( aa(A,option(product_prod(B,product_prod(C,D2))),Uub,H6) = aa(product_prod(B,product_prod(C,D2)),option(product_prod(B,product_prod(C,D2))),some(product_prod(B,product_prod(C,D2))),aa(product_prod(C,D2),product_prod(B,product_prod(C,D2)),aa(B,fun(product_prod(C,D2),product_prod(B,product_prod(C,D2))),product_Pair(B,product_prod(C,D2)),R6),aa(D2,product_prod(C,D2),aa(C,fun(D2,product_prod(C,D2)),product_Pair(C,D2),H10),N8))) )
         => aa(D2,$o,aa(B,fun(D2,$o),aa(C,fun(B,fun(D2,$o)),aa(A,fun(C,fun(B,fun(D2,$o))),aa(E,fun(A,fun(C,fun(B,fun(D2,$o)))),Uu,Uua),H6),H10),R6),N8) ) ) ).

% ATP.lambda_902
tff(fact_9075_ATP_Olambda__903,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aTP_Lamp_nm(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uu),Uua),Uub) = heap_Time_Heap2(vEBT_VEBTi,aTP_Lamp_nl(heap_ext(product_unit),option(product_prod(vEBT_VEBTi,product_prod(heap_ext(product_unit),nat))))) ).

% ATP.lambda_903
tff(fact_9076_ATP_Olambda__904,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat] : aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),aTP_Lamp_lx(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uu),Uua),Uub) = heap_Time_Heap2($o,aTP_Lamp_lw(heap_ext(product_unit),option(product_prod($o,product_prod(heap_ext(product_unit),nat))))) ).

% ATP.lambda_904
tff(fact_9077_ATP_Olambda__905,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),aTP_Lamp_jw(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uu),Uua),Uub) = heap_Time_Heap2(option(nat),aTP_Lamp_jv(heap_ext(product_unit),option(product_prod(option(nat),product_prod(heap_ext(product_unit),nat))))) ).

% ATP.lambda_905
tff(fact_9078_ATP_Olambda__906,axiom,
    ! [C: $tType,A: $tType,B: $tType,D2: $tType,Uu: fun(B,fun(C,fun(D2,A))),Uua: D2,Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(D2,fun(B,fun(C,A)),aTP_Lamp_fm(fun(B,fun(C,fun(D2,A))),fun(D2,fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(D2,A,aa(C,fun(D2,A),aa(B,fun(C,fun(D2,A)),Uu,Uub),Uuc),Uua) ).

% ATP.lambda_906
tff(fact_9079_ATP_Olambda__907,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aTP_Lamp_rd(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uu),Uua),Uub),Uuc) = vEBT_case_VEBTi(heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_rc(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uuc),aTP_Lamp_om(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),Uuc),Uub) ).

% ATP.lambda_907
tff(fact_9080_ATP_Olambda__908,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aTP_Lamp_on(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uu),Uua),Uub),Uuc) = vEBT_case_VEBTi(heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_ol(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uuc),aTP_Lamp_om(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),Uuc),Uub) ).

% ATP.lambda_908
tff(fact_9081_ATP_Olambda__909,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),aTP_Lamp_ln(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uu),Uua),Uub),Uuc) = vEBT_case_VEBTi(heap_Time_Heap($o),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))))),aTP_Lamp_lm(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))))),Uu),Uua),Uuc),aTP_Lamp_hg(nat,fun($o,fun($o,heap_Time_Heap($o))),Uuc),Uub) ).

% ATP.lambda_909
tff(fact_9082_ATP_Olambda__910,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),aTP_Lamp_jn(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uu),Uua),Uub),Uuc) = vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aTP_Lamp_jm(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uuc),aTP_Lamp_bk(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uuc),Uub) ).

% ATP.lambda_910
tff(fact_9083_ATP_Olambda__911,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),aTP_Lamp_iw(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uu),Uua),Uub),Uuc) = vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aTP_Lamp_iv(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uuc),aTP_Lamp_cl(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uuc),Uub) ).

% ATP.lambda_911
tff(fact_9084_ATP_Olambda__912,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),aTP_Lamp_ly(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uu),Uua),Uub),Uuc) = vEBT_case_VEBTi(heap_Time_Heap($o),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))))),aTP_Lamp_ia(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))))),Uu),Uua),Uuc),aTP_Lamp_hg(nat,fun($o,fun($o,heap_Time_Heap($o))),Uuc),Uub) ).

% ATP.lambda_912
tff(fact_9085_ATP_Olambda__913,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),aTP_Lamp_jy(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uu),Uua),Uub),Uuc) = vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aTP_Lamp_et(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uuc),aTP_Lamp_cl(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uuc),Uub) ).

% ATP.lambda_913
tff(fact_9086_ATP_Olambda__914,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: vEBT_VEBT,Uub: vEBT_VEBTi,Uuc: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),aTP_Lamp_jx(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uu),Uua),Uub),Uuc) = vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aTP_Lamp_ee(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uuc),aTP_Lamp_bk(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uuc),Uub) ).

% ATP.lambda_914
tff(fact_9087_ATP_Olambda__915,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_abg(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)
            & aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uuc) ),
            aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_915
tff(fact_9088_ATP_Olambda__916,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_abk(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub)
            & ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uuc) ),
            aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),
            zero_zero(A) ) ) ).

% ATP.lambda_916
tff(fact_9089_ATP_Olambda__917,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] :
          aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_abi(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Uub),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),one_one(int))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,binomial(Uub),Uuc))))),semiring_char_0_fact(real,Uub))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))),zero_zero(A)) ) ).

% ATP.lambda_917
tff(fact_9090_ATP_Olambda__918,axiom,
    ! [Uu: array(vEBT_VEBTi),Uua: nat,Uub: nat,Uuc: nat] :
      aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(nat,fun(nat,fun(nat,heap_Time_Heap($o))),aTP_Lamp_hi(array(vEBT_VEBTi),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,ord_less(nat,Uua),Uuc),heap_Time_bind(vEBT_VEBTi,$o,aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uu),Uua),aTP_Lamp_hh(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)),Uub)),aa($o,heap_Time_Heap($o),heap_Time_return($o),$false)) ).

% ATP.lambda_918
tff(fact_9091_ATP_Olambda__919,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_abs(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,ord_less(nat,Uuc),Uu),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu,zero_zero(A),aa(nat,A,Uub,aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_919
tff(fact_9092_ATP_Olambda__920,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_abu(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(
            aa(nat,$o,ord_less(nat,Uuc),Uu),
            aa(nat,A,Uua,Uuc),
            $ite(Uuc = Uu,one_one(A),aa(nat,A,Uub,aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))) ) ) ).

% ATP.lambda_920
tff(fact_9093_ATP_Olambda__921,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_abv(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,ord_less(nat,Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_921
tff(fact_9094_ATP_Olambda__922,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(fun(nat,A),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_abt(nat,fun(fun(nat,A),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,ord_less(nat,Uuc),Uu),aa(nat,A,Uua,Uuc),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_922
tff(fact_9095_ATP_Olambda__923,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: set(nat),Uub: fun(nat,A),Uuc: nat] :
          aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_wg(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = $ite(member(nat,Uuc,Uua),aa(nat,A,Uub,Uuc),aa(nat,A,Uu,Uuc)) ) ).

% ATP.lambda_923
tff(fact_9096_ATP_Olambda__924,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_yp(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_924
tff(fact_9097_ATP_Olambda__925,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: A,Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ve(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uu,aa(A,B,Uua,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_925
tff(fact_9098_ATP_Olambda__926,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,option(A)),Uua: B,Uub: A,Uuc: B] :
      aa(B,option(A),aa(A,fun(B,option(A)),aa(B,fun(A,fun(B,option(A))),aTP_Lamp_agn(fun(B,option(A)),fun(B,fun(A,fun(B,option(A)))),Uu),Uua),Uub),Uuc) = $ite(Uuc = Uua,aa(A,option(A),some(A),Uub),aa(B,option(A),Uu,Uuc)) ).

% ATP.lambda_926
tff(fact_9099_ATP_Olambda__927,axiom,
    ! [Uu: list(vEBT_VEBT),Uua: nat,Uub: nat,Uuc: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap($o),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o))),aTP_Lamp_gs(list(vEBT_VEBT),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)))),Uu),Uua),Uub),Uuc) = aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),vEBT_V854960066525838166emberi,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uu),Uua)),Uuc),Uub) ).

% ATP.lambda_927
tff(fact_9100_ATP_Olambda__928,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: product_prod(nat,nat),Uuc: option(nat)] :
      aa(option(nat),heap_Time_Heap(option(nat)),aa(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat)))),aTP_Lamp_ac(nat,fun(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat))))),Uu),Uua),Uub),Uuc) = $ite(
        Uuc = none(nat),
        $ite(aa(nat,$o,ord_less(nat,aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uub)),xa),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uub))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))),
        heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uua),aa(option(nat),nat,the2(nat),Uuc)),aa(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aTP_Lamp_ab(nat,fun(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),Uu),Uuc)) ) ).

% ATP.lambda_928
tff(fact_9101_ATP_Olambda__929,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aTP_Lamp_ah(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),Uu),Uua),Uub),Uuc) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uu),one_one(nat)),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),
        $ite(aa(nat,$o,ord_less(nat,aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uuc)),xa),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uuc))),heap_Time_bind(nat,option(nat),vEBT_VEBT_lowi(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))),aTP_Lamp_ag(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc))) ) ).

% ATP.lambda_929
tff(fact_9102_ATP_Olambda__930,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: option(nat),Uuc: product_unit] :
      aa(product_unit,heap_Time_Heap(option(nat)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aTP_Lamp_cu(nat,fun(array(vEBT_VEBTi),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),Uu),Uua),Uub),Uuc) = $ite(Uub = none(nat),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uua),aa(option(nat),nat,the2(nat),Uub)),aa(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aTP_Lamp_cd(nat,fun(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),Uu),Uub))) ).

% ATP.lambda_930
tff(fact_9103_ATP_Olambda__931,axiom,
    ! [B: $tType,A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,B),Uua: fun(A,$o),Uub: fun(A,B),Uuc: A] :
          aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,$o),fun(fun(A,B),fun(A,B)),aTP_Lamp_auh(fun(A,B),fun(fun(A,$o),fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = $ite(aa(A,$o,Uua,Uuc),aa(A,B,Uu,Uuc),aa(A,B,Uub,Uuc)) ) ).

% ATP.lambda_931
tff(fact_9104_ATP_Olambda__932,axiom,
    ! [Uu: option(product_prod(nat,nat)),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aTP_Lamp_ai(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc) = case_option(heap_Time_Heap(option(nat)),product_prod(nat,nat),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aTP_Lamp_ah(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),Uua),Uub),Uuc),Uu) ).

% ATP.lambda_932
tff(fact_9105_ATP_Olambda__933,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: fun(A,B),Uub: $o,Uuc: A] :
      aa(A,B,aa($o,fun(A,B),aa(fun(A,B),fun($o,fun(A,B)),aTP_Lamp_agc(fun(A,B),fun(fun(A,B),fun($o,fun(A,B))),Uu),Uua),(Uub)),Uuc) = $ite((Uub),aa(A,B,Uu,Uuc),aa(A,B,Uua,Uuc)) ).

% ATP.lambda_933
tff(fact_9106_ATP_Olambda__934,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_vo(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_vn(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uub),aa(nat,nat,suc,zero_zero(nat)))),Uuc))) ) ).

% ATP.lambda_934
tff(fact_9107_ATP_Olambda__935,axiom,
    ! [Uu: $o,Uua: $o,Uub: code_integer,Uuc: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aTP_Lamp_aoq($o,fun($o,fun(code_integer,fun($o,char))),(Uu)),(Uua)),Uub),(Uuc)) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aTP_Lamp_aop($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),(Uu)),(Uua)),(Uuc))),code_bit_cut_integer(Uub)) ).

% ATP.lambda_935
tff(fact_9108_ATP_Olambda__936,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: set(C),Uuc: B] : aa(B,A,aa(set(C),fun(B,A),aa(fun(C,A),fun(set(C),fun(B,A)),aTP_Lamp_uh(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_ug(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_936
tff(fact_9109_ATP_Olambda__937,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A,Uuc: A] :
          aa(A,fun(list(A),list(A)),aa(A,fun(A,fun(list(A),list(A))),aa(list(A),fun(A,fun(A,fun(list(A),list(A)))),aTP_Lamp_alp(fun(A,B),fun(list(A),fun(A,fun(A,fun(list(A),list(A))))),Uu),Uua),Uub),Uuc) = case_list(list(A),A,
            $ite(aa(B,$o,ord_less_eq(B,aa(A,B,Uu,Uub)),aa(A,B,Uu,Uuc)),Uua,aa(list(A),list(A),cons(A,Uuc),aa(list(A),list(A),cons(A,Uub),nil(A)))),
            aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_alo(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),Uu),Uua)) ) ).

% ATP.lambda_937
tff(fact_9110_ATP_Olambda__938,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: nat] :
          aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_abz(nat,fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              $ite(
                Uuc = zero_zero(nat),
                aa(A,A,uminus_uminus(A),Uub),
                $ite(Uuc = Uu,one_one(A),zero_zero(A)) )),
            aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_938
tff(fact_9111_ATP_Olambda__939,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_ya(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_xz(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc),collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_th(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_939
tff(fact_9112_ATP_Olambda__940,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: B] : aa(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_uq(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_up(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_th(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_940
tff(fact_9113_ATP_Olambda__941,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_abr(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_abq(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uub),Uuc)) ).

% ATP.lambda_941
tff(fact_9114_ATP_Olambda__942,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_abn(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_abm(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uuc)),set_ord_atMost(nat,Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uuc)) ) ).

% ATP.lambda_942
tff(fact_9115_ATP_Olambda__943,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: nat,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_aqw(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,Uua)),Uuc)),zero_zero(real))),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uub),Uuc)) ).

% ATP.lambda_943
tff(fact_9116_ATP_Olambda__944,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: nat,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_aqu(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uuc)),zero_zero(real))),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uub),Uuc)) ).

% ATP.lambda_944
tff(fact_9117_ATP_Olambda__945,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_aqs(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,Uuc),Uua)),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,Uub),Uua)),Uuc)) ).

% ATP.lambda_945
tff(fact_9118_ATP_Olambda__946,axiom,
    ! [Uu: fun(nat,fun(real,real)),Uua: real,Uub: real,Uuc: nat] : aa(nat,real,aa(real,fun(nat,real),aa(real,fun(real,fun(nat,real)),aTP_Lamp_aqt(fun(nat,fun(real,real)),fun(real,fun(real,fun(nat,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(nat,fun(real,real),Uu,Uuc),Uub)),semiring_char_0_fact(real,Uuc))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,minus_minus(real,Uua),Uub)),Uuc)) ).

% ATP.lambda_946
tff(fact_9119_ATP_Olambda__947,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_wj(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uua)),aa(nat,nat,minus_minus(nat,Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uub)) ) ).

% ATP.lambda_947
tff(fact_9120_ATP_Olambda__948,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_vn(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),Uuc))) ) ).

% ATP.lambda_948
tff(fact_9121_ATP_Olambda__949,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A,Uub: fun(nat,A),Uuc: nat] : aa(nat,A,aa(fun(nat,A),fun(nat,A),aa(A,fun(fun(nat,A),fun(nat,A)),aTP_Lamp_bae(fun(A,A),fun(A,fun(fun(nat,A),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),aa(nat,A,Uub,Uuc)))),aa(A,A,Uu,Uua))),aa(nat,A,Uub,Uuc)) ) ).

% ATP.lambda_949
tff(fact_9122_ATP_Olambda__950,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,option(B)),Uua: fun(product_prod(A,B),$o),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),aTP_Lamp_gl(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( aa(A,option(B),Uu,Uub) = aa(B,option(B),some(B),Uuc) )
        & aa(product_prod(A,B),$o,Uua,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uub),Uuc)) ) ) ).

% ATP.lambda_950
tff(fact_9123_ATP_Olambda__951,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Uu: fun(A,B),Uua: set(B),Uub: B,Uuc: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aa(set(B),fun(B,fun(A,$o)),aTP_Lamp_bbd(fun(A,B),fun(set(B),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(B,aa(A,B,Uu,Uuc),Uua)
            & ( aa(A,B,Uu,Uuc) != Uub ) ) ) ) ).

% ATP.lambda_951
tff(fact_9124_ATP_Olambda__952,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: set(B),Uuc: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_bbe(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(B,aa(A,B,Uu,Uuc),Uub)
            & ( aa(A,B,Uu,Uuc) != Uua ) ) ) ) ).

% ATP.lambda_952
tff(fact_9125_ATP_Olambda__953,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_zw(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),comm_s3205402744901411588hammer(A,Uu,Uuc))),comm_s3205402744901411588hammer(A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_953
tff(fact_9126_ATP_Olambda__954,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_zs(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,semiring_1_of_nat(nat),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uu),Uuc))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_954
tff(fact_9127_ATP_Olambda__955,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_zv(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_955
tff(fact_9128_ATP_Olambda__956,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_zy(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc))),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Uub),Uuc)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc)))) ) ).

% ATP.lambda_956
tff(fact_9129_ATP_Olambda__957,axiom,
    ! [A: $tType,Uu: set(product_prod(A,A)),Uua: nat,Uub: list(A),Uuc: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_ank(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uua )
        & ( aa(list(A),nat,size_size(list(A)),Uuc) = Uua )
        & ? [Xys: list(A),X: A,Y3: A,Xs6: list(A),Ys6: list(A)] :
            ( ( Uub = append(A,Xys,aa(list(A),list(A),cons(A,X),Xs6)) )
            & ( Uuc = append(A,Xys,aa(list(A),list(A),cons(A,Y3),Ys6)) )
            & member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),X),Y3),Uu) ) ) ) ).

% ATP.lambda_957
tff(fact_9130_ATP_Olambda__958,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: nat] : aa(nat,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o))),aTP_Lamp_hk(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o)))),Uu),Uua),Uub),Uuc) = heap_Time_bind(nat,$o,vEBT_VEBT_lowi(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap($o)),aTP_Lamp_hj(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),Uub),Uuc)) ).

% ATP.lambda_958
tff(fact_9131_ATP_Olambda__959,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_wm(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),aa(nat,nat,minus_minus(nat,Uua),aa(nat,nat,suc,Uuc)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)) ) ).

% ATP.lambda_959
tff(fact_9132_ATP_Olambda__960,axiom,
    ! [A: $tType,Uu: $o,Uua: A,Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(A,fun(A,fun(A,$o)),aTP_Lamp_adr($o,fun(A,fun(A,fun(A,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( ( (Uu)
         => ( Uuc = Uua ) )
        & ( ~ (Uu)
         => ( Uuc = Uub ) ) ) ) ).

% ATP.lambda_960
tff(fact_9133_ATP_Olambda__961,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_ahk(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,aa(set(B),set(A),image(B,A,Uu),Uua))
        & aa(A,$o,Uub,Uuc) ) ) ).

% ATP.lambda_961
tff(fact_9134_ATP_Olambda__962,axiom,
    ! [A: $tType,B: $tType,Uu: set(B),Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_un(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_962
tff(fact_9135_ATP_Olambda__963,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,fun(B,$o)),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_th(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uuc),Uub) ) ) ).

% ATP.lambda_963
tff(fact_9136_ATP_Olambda__964,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_wk(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),Uua)),aa(nat,nat,minus_minus(nat,Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),aa(nat,nat,minus_minus(nat,Uub),Uuc)))) ) ).

% ATP.lambda_964
tff(fact_9137_ATP_Olambda__965,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: set(A),Uua: fun(A,A),Uub: fun(A,A),Uuc: A] :
          ( aa(A,$o,aa(fun(A,A),fun(A,$o),aa(fun(A,A),fun(fun(A,A),fun(A,$o)),aTP_Lamp_bcm(set(A),fun(fun(A,A),fun(fun(A,A),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
           => ( aa(A,A,Uua,Uuc) = aa(A,A,Uub,Uuc) ) ) ) ) ).

% ATP.lambda_965
tff(fact_9138_ATP_Olambda__966,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(A,B),Uub: A,Uuc: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_ahm(set(A),fun(fun(A,B),fun(A,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,B,Uua,Uuc) = aa(A,B,Uua,Uub) ) ) ) ).

% ATP.lambda_966
tff(fact_9139_ATP_Olambda__967,axiom,
    ! [B: $tType,A: $tType,Uu: set(B),Uua: fun(B,A),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_afs(set(B),fun(fun(B,A),fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uu)
        & ( aa(B,A,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_967
tff(fact_9140_ATP_Olambda__968,axiom,
    ! [A: $tType,C: $tType,Uu: set(A),Uua: fun(A,C),Uub: C,Uuc: A] :
      ( aa(A,$o,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_ahq(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,C,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_968
tff(fact_9141_ATP_Olambda__969,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: fun(A,B),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_aht(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uu)
        & ( aa(A,B,Uua,Uuc) = Uub ) ) ) ).

% ATP.lambda_969
tff(fact_9142_ATP_Olambda__970,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: set(B),Uuc: B] :
      ( aa(B,$o,aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_aox(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uub)
        & member(A,aa(B,A,Uua,Uuc),Uu) ) ) ).

% ATP.lambda_970
tff(fact_9143_ATP_Olambda__971,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Uu: A,Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_wl(A,fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),aa(nat,nat,minus_minus(nat,Uua),Uuc))) ) ).

% ATP.lambda_971
tff(fact_9144_ATP_Olambda__972,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_zr(nat,fun(nat,fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Uu),Uuc)),aa(nat,nat,binomial(Uua),aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_972
tff(fact_9145_ATP_Olambda__973,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: list(A),Uuc: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aa(list(A),fun(list(A),fun(list(A),list(A))),aTP_Lamp_alm(fun(A,B),fun(list(A),fun(list(A),fun(list(A),list(A)))),Uu),Uua),Uub),Uuc) = append(A,aa(list(A),list(A),linorder_sort_key(A,B,Uu),Uua),append(A,Uub,aa(list(A),list(A),linorder_sort_key(A,B,Uu),Uuc))) ) ).

% ATP.lambda_973
tff(fact_9146_ATP_Olambda__974,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_ags(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,ord_less_eq(int,aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_974
tff(fact_9147_ATP_Olambda__975,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_agp(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub)) ) ).

% ATP.lambda_975
tff(fact_9148_ATP_Olambda__976,axiom,
    ! [A: $tType,B: $tType,Uu: A,Uua: B,Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(B,fun(A,fun(B,$o)),aTP_Lamp_adp(A,fun(B,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_976
tff(fact_9149_ATP_Olambda__977,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: fun(A,$o),Uuc: B] :
      ( aa(B,$o,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_ahl(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uua)
        & aa(A,$o,Uub,aa(B,A,Uu,Uuc)) ) ) ).

% ATP.lambda_977
tff(fact_9150_ATP_Olambda__978,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_tn(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
            & ( aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != one_one(B) ) ) ) ) ).

% ATP.lambda_978
tff(fact_9151_ATP_Olambda__979,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,B),Uuc: A] :
          ( aa(A,$o,aa(fun(A,B),fun(A,$o),aa(fun(A,B),fun(fun(A,B),fun(A,$o)),aTP_Lamp_tp(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( member(A,Uuc,Uu)
            & ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_979
tff(fact_9152_ATP_Olambda__980,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: $o] : aa($o,assn,aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_aea(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),Uu),Uua),Uub),(Uuc)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu),Uua)),pure_assn((Uuc) = aa(nat,$o,vEBT_vebt_member(Uu),Uub))) ).

% ATP.lambda_980
tff(fact_9153_ATP_Olambda__981,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option(nat)] : aa(option(nat),assn,aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_aeb(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu),Uua)),pure_assn(Uuc = vEBT_vebt_succ(Uu,Uub))) ).

% ATP.lambda_981
tff(fact_9154_ATP_Olambda__982,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: nat,Uuc: option(nat)] : aa(option(nat),assn,aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_aec(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Uu),Uua)),pure_assn(Uuc = vEBT_vebt_pred(Uu,Uub))) ).

% ATP.lambda_982
tff(fact_9155_ATP_Olambda__983,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: list(A),Uub: array(A),Uuc: A] : aa(A,assn,aa(array(A),fun(A,assn),aa(list(A),fun(array(A),fun(A,assn)),aTP_Lamp_aek(nat,fun(list(A),fun(array(A),fun(A,assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(A),assn,snga_assn(A,Uub),Uua)),pure_assn(Uuc = aa(nat,A,nth(A,Uua),Uu))) ) ).

% ATP.lambda_983
tff(fact_9156_ATP_Olambda__984,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_bam(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub)))),aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,Uu,Uub)),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub))))) ) ).

% ATP.lambda_984
tff(fact_9157_ATP_Olambda__985,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_ng(fun(nat,A),fun(nat,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uua))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uuc)) ) ).

% ATP.lambda_985
tff(fact_9158_ATP_Olambda__986,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: list(B),Uuc: nat] : aa(nat,A,aa(list(B),fun(nat,A),aa(A,fun(list(B),fun(nat,A)),aTP_Lamp_aje(fun(B,A),fun(A,fun(list(B),fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,aa(nat,B,nth(B,Uub),Uuc))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_986
tff(fact_9159_ATP_Olambda__987,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A,Uub: A,Uuc: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_atj(fun(nat,A),fun(A,fun(A,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uub)),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc))),Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,Uuc),aa(nat,nat,suc,zero_zero(nat))))))) ) ).

% ATP.lambda_987
tff(fact_9160_ATP_Olambda__988,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_Vector_banach(B)
        & real_V3459762299906320749_field(B)
        & topological_t2_space(A) )
     => ! [Uu: fun(A,B),Uua: fun(nat,B),Uub: A,Uuc: nat] : aa(nat,B,aa(A,fun(nat,B),aa(fun(nat,B),fun(A,fun(nat,B)),aTP_Lamp_ayg(fun(A,B),fun(fun(nat,B),fun(A,fun(nat,B))),Uu),Uua),Uub),Uuc) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,Uua,Uuc)),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),Uuc)) ) ).

% ATP.lambda_988
tff(fact_9161_ATP_Olambda__989,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_asy(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(A,real,Uu,Uua)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% ATP.lambda_989
tff(fact_9162_ATP_Olambda__990,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_acf(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,Uuc),Uub)),one_one(nat)))) ) ).

% ATP.lambda_990
tff(fact_9163_ATP_Olambda__991,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: set(B),Uuc: A] :
          ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_bdi(fun(A,B),fun(B,fun(set(B),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> member(B,aa(A,B,Uu,Uuc),aa(set(B),set(B),minus_minus(set(B),Uub),aa(set(B),set(B),insert(B,Uua),bot_bot(set(B))))) ) ) ).

% ATP.lambda_991
tff(fact_9164_ATP_Olambda__992,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [Uu: fun(B,A),Uua: A,Uub: B,Uuc: A] : aa(A,A,aa(B,fun(A,A),aa(A,fun(B,fun(A,A)),aTP_Lamp_ajf(fun(B,A),fun(A,fun(B,fun(A,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,Uu,Uub)),aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uuc)) ) ).

% ATP.lambda_992
tff(fact_9165_ATP_Olambda__993,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat,Uuc: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_abq(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),Uu),Uua),Uub),Uuc) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,Uu,Uuc)),aa(nat,nat,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ).

% ATP.lambda_993
tff(fact_9166_ATP_Olambda__994,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_abe(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_994
tff(fact_9167_ATP_Olambda__995,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: fun(nat,A),Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(fun(nat,A),fun(nat,fun(nat,A)),aTP_Lamp_abm(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,Uuc)),aa(nat,A,Uua,aa(nat,nat,minus_minus(nat,Uub),Uuc))) ) ).

% ATP.lambda_995
tff(fact_9168_ATP_Olambda__996,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( semiring_0(A)
     => ! [Uu: fun(B,A),Uua: fun(C,A),Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(fun(C,A),fun(B,fun(C,A)),aTP_Lamp_ug(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(B,A,Uu,Uub)),aa(C,A,Uua,Uuc)) ) ).

% ATP.lambda_996
tff(fact_9169_ATP_Olambda__997,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_xy(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(B,C,aa(A,fun(B,C),Uua,Uuc),collect(B,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_un(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_997
tff(fact_9170_ATP_Olambda__998,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(B),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,$o)),Uuc: A] : aa(A,C,aa(fun(A,fun(B,$o)),fun(A,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C)),aTP_Lamp_uo(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),Uu),Uua),Uub),Uuc) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(A,fun(B,C),Uua,Uuc)),collect(B,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_un(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_998
tff(fact_9171_ATP_Olambda__999,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_atc(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),cos(real,aa(A,real,Uu,Uua))),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% ATP.lambda_999
tff(fact_9172_ATP_Olambda__1000,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ata(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uub)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% ATP.lambda_1000
tff(fact_9173_ATP_Olambda__1001,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_arz(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,exp(real),aa(A,real,Uu,Uub))) ) ).

% ATP.lambda_1001
tff(fact_9174_ATP_Olambda__1002,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_asb(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),cos(real,aa(A,real,Uu,Uub))) ) ).

% ATP.lambda_1002
tff(fact_9175_ATP_Olambda__1003,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_aso(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(A,real,Uu,Uua))) ) ).

% ATP.lambda_1003
tff(fact_9176_ATP_Olambda__1004,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_ard(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% ATP.lambda_1004
tff(fact_9177_ATP_Olambda__1005,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,real),fun(A,fun(A,real)),aTP_Lamp_ask(fun(A,real),fun(fun(A,real),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uuc)),aa(real,real,uminus_uminus(real),sin(real,aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_1005
tff(fact_9178_ATP_Olambda__1006,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: A,Uub: fun(A,real),Uuc: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(A,fun(fun(A,real),fun(A,real)),aTP_Lamp_atg(fun(A,real),fun(A,fun(fun(A,real),fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uub,Uuc)),aa(real,real,inverse_inverse(real),aa(real,real,uminus_uminus(real),aa(real,real,sqrt,aa(real,real,minus_minus(real,one_one(real)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(A,real,Uu,Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))) ) ).

% ATP.lambda_1006
tff(fact_9179_ATP_Olambda__1007,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_alo(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(list(A),list(A))),list(A),product_case_prod(list(A),product_prod(list(A),list(A)),list(A),aTP_Lamp_aln(fun(A,B),fun(list(A),fun(product_prod(list(A),list(A)),list(A))),Uu)),linorder_part(A,B,Uu,aa(A,B,Uu,aa(nat,A,nth(A,Uua),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(list(A),nat,size_size(list(A)),Uua)),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Uua)) ) ).

% ATP.lambda_1007
tff(fact_9180_ATP_Olambda__1008,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_bay(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc))),aa(A,B,Uu,Uub))),aa(A,B,Uua,Uuc)))),real_V7770717601297561774m_norm(A,Uuc)) ) ).

% ATP.lambda_1008
tff(fact_9181_ATP_Olambda__1009,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,real,aa(A,fun(A,real),aa(fun(A,B),fun(A,fun(A,real)),aTP_Lamp_bau(fun(A,B),fun(fun(A,B),fun(A,fun(A,real))),Uu),Uua),Uub),Uuc) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(B,aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub))))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub))) ) ).

% ATP.lambda_1009
tff(fact_9182_ATP_Olambda__1010,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: list(vEBT_VEBTi),Uuc: option(nat)] : aa(option(nat),assn,aa(list(vEBT_VEBTi),fun(option(nat),assn),aa(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn)),aTP_Lamp_aee(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn(Uuc = vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),Uu)))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_1010
tff(fact_9183_ATP_Olambda__1011,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: list(vEBT_VEBTi),Uuc: option(nat)] : aa(option(nat),assn,aa(list(vEBT_VEBTi),fun(option(nat),assn),aa(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn)),aTP_Lamp_aed(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn(Uuc = vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),Uu)))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,Uua),Uub)) ).

% ATP.lambda_1011
tff(fact_9184_ATP_Olambda__1012,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: fun(A,$o),Uua: A,Uub: set(A),Uuc: A] :
          ( aa(A,$o,aa(set(A),fun(A,$o),aa(A,fun(set(A),fun(A,$o)),aTP_Lamp_bcl(fun(A,$o),fun(A,fun(set(A),fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> ( ( Uuc != Uua )
           => ( member(A,Uuc,Uub)
             => aa(A,$o,Uu,Uuc) ) ) ) ) ).

% ATP.lambda_1012
tff(fact_9185_ATP_Olambda__1013,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(C) )
     => ! [Uu: fun(A,B),Uua: fun(A,C),Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(fun(A,C),fun(real,fun(A,$o)),aTP_Lamp_bdj(fun(A,B),fun(fun(A,C),fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,ord_less_eq(real,real_V7770717601297561774m_norm(C,aa(A,C,Uua,Uuc))),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uuc))),Uub)) ) ) ).

% ATP.lambda_1013
tff(fact_9186_ATP_Olambda__1014,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: filter(A),Uuc: A] : aa(A,B,aa(filter(A),fun(A,B),aa(fun(A,B),fun(filter(A),fun(A,B)),aTP_Lamp_baz(fun(A,B),fun(fun(A,B),fun(filter(A),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_aru(A,A)))))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(A,B,Uu,topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_aru(A,A))))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),topolo3827282254853284352ce_Lim(A,A,Uub,aTP_Lamp_aru(A,A)))))) ) ).

% ATP.lambda_1014
tff(fact_9187_ATP_Olambda__1015,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: A,Uub: fun(A,B),Uuc: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(A,fun(fun(A,B),fun(A,B)),aTP_Lamp_bav(fun(A,B),fun(A,fun(fun(A,B),fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uua)))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uub,Uuc)),aa(A,B,Uub,Uua))),aa(A,B,Uu,aa(A,A,minus_minus(A,Uuc),Uua)))) ) ).

% ATP.lambda_1015
tff(fact_9188_ATP_Olambda__1016,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: A] : aa(A,B,aa(A,fun(A,B),aa(fun(A,B),fun(A,fun(A,B)),aTP_Lamp_baw(fun(A,B),fun(fun(A,B),fun(A,fun(A,B))),Uu),Uua),Uub),Uuc) = aa(B,B,real_V8093663219630862766scaleR(B,aa(real,real,inverse_inverse(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Uuc),Uub)))),aa(B,B,minus_minus(B,aa(B,B,minus_minus(B,aa(A,B,Uu,Uuc)),aa(A,B,Uu,Uub))),aa(A,B,Uua,aa(A,A,minus_minus(A,Uuc),Uub)))) ) ).

% ATP.lambda_1016
tff(fact_9189_ATP_Olambda__1017,axiom,
    ! [Uu: option(product_prod(nat,nat)),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aTP_Lamp_bc(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(vEBT_is_Node(ta)),aa(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))),aTP_Lamp_bb(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1017
tff(fact_9190_ATP_Olambda__1018,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),Uua: array(vEBT_VEBTi),Uub: nat,Uuc: nat] : aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),aTP_Lamp_kt(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))),Uu),Uua),Uub),Uuc) = heap_Time_bind(nat,$o,array_len(vEBT_VEBTi,Uua),aa(nat,fun(nat,heap_Time_Heap($o)),aa(nat,fun(nat,fun(nat,heap_Time_Heap($o))),aa(array(vEBT_VEBTi),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))),aTP_Lamp_ks(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(array(vEBT_VEBTi),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1018
tff(fact_9191_ATP_Olambda__1019,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),Uua: array(vEBT_VEBTi),Uub: nat,Uuc: nat] : aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),aTP_Lamp_id(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))),Uu),Uua),Uub),Uuc) = heap_Time_bind(nat,$o,array_len(vEBT_VEBTi,Uua),aa(nat,fun(nat,heap_Time_Heap($o)),aa(nat,fun(nat,fun(nat,heap_Time_Heap($o))),aa(array(vEBT_VEBTi),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))),aTP_Lamp_ic(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(array(vEBT_VEBTi),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1019
tff(fact_9192_ATP_Olambda__1020,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(vEBT_VEBTi),Uuc: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(list(vEBT_VEBTi),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(list(vEBT_VEBTi),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aTP_Lamp_sh(nat,fun(nat,fun(list(vEBT_VEBTi),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),Uu),Uua),Uub),Uuc) = heap_Time_bind(array(vEBT_VEBTi),vEBT_VEBTi,array_of_list(vEBT_VEBTi,Uub),aa(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sg(nat,fun(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Uu),Uua)) ).

% ATP.lambda_1020
tff(fact_9193_ATP_Olambda__1021,axiom,
    ! [Uu: nat,Uua: nat,Uub: list(vEBT_VEBTi),Uuc: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(list(vEBT_VEBTi),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(list(vEBT_VEBTi),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aTP_Lamp_se(nat,fun(nat,fun(list(vEBT_VEBTi),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),Uu),Uua),Uub),Uuc) = heap_Time_bind(array(vEBT_VEBTi),vEBT_VEBTi,array_of_list(vEBT_VEBTi,Uub),aa(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_sd(nat,fun(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Uu),Uua)) ).

% ATP.lambda_1021
tff(fact_9194_ATP_Olambda__1022,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: nat] : aa(nat,list(A),aa(A,fun(nat,list(A)),aa(list(A),fun(A,fun(nat,list(A))),aTP_Lamp_afv(A,fun(list(A),fun(A,fun(nat,list(A)))),Uu),Uua),Uub),Uuc) = aa(list(A),list(A),cons(A,Uu),list_update(A,Uua,Uuc,Uub)) ).

% ATP.lambda_1022
tff(fact_9195_ATP_Olambda__1023,axiom,
    ! [B: $tType,A: $tType,Uu: assn,Uua: fun(B,fun(A,assn)),Uub: B,Uuc: A] : aa(A,assn,aa(B,fun(A,assn),aa(fun(B,fun(A,assn)),fun(B,fun(A,assn)),aTP_Lamp_aef(assn,fun(fun(B,fun(A,assn)),fun(B,fun(A,assn))),Uu),Uua),Uub),Uuc) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu),aa(A,assn,aa(B,fun(A,assn),Uua,Uub),Uuc)) ).

% ATP.lambda_1023
tff(fact_9196_ATP_Olambda__1024,axiom,
    ! [A: $tType,B: $tType,Uu: $o,Uua: fun(A,fun(B,$o)),Uub: A,Uuc: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aa(fun(A,fun(B,$o)),fun(A,fun(B,$o)),aTP_Lamp_fg($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_1024
tff(fact_9197_ATP_Olambda__1025,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,heap_Time_Heap(A)),Uua: B,Uub: heap_ext(product_unit),Uuc: nat] : aa(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(B,fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),aTP_Lamp_akk(fun(B,heap_Time_Heap(A)),fun(B,fun(heap_ext(product_unit),fun(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))))),Uu),Uua),Uub),Uuc) = heap_Time_timeFrame(A,Uuc,aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,aa(B,heap_Time_Heap(A),Uu,Uua)),Uub)) ).

% ATP.lambda_1025
tff(fact_9198_ATP_Olambda__1026,axiom,
    ! [Uu: option(product_prod(nat,nat)),Uua: nat,Uub: list(vEBT_VEBT),Uuc: vEBT_VEBT] : aa(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)))))),aTP_Lamp_az(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),aa(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)))),product_Pair(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),Uu),aa(product_prod(list(vEBT_VEBT),vEBT_VEBT),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),aa(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),product_Pair(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),Uua),aa(vEBT_VEBT,product_prod(list(vEBT_VEBT),vEBT_VEBT),aa(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(list(vEBT_VEBT),vEBT_VEBT)),product_Pair(list(vEBT_VEBT),vEBT_VEBT),Uub),Uuc))) ).

% ATP.lambda_1026
tff(fact_9199_ATP_Olambda__1027,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ahs(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,B,Uua,collect(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_ahq(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1027
tff(fact_9200_ATP_Olambda__1028,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_ahv(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,C,Uub,collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_aht(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_1028
tff(fact_9201_ATP_Olambda__1029,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ahr(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,Uua),collect(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_ahq(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_1029
tff(fact_9202_ATP_Olambda__1030,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_ahu(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,Uub),collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_aht(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_1030
tff(fact_9203_ATP_Olambda__1031,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: real,Uuc: A] :
          ( aa(A,$o,aa(real,fun(A,$o),aa(nat,fun(real,fun(A,$o)),aTP_Lamp_bdp(fun(nat,A),fun(nat,fun(real,fun(A,$o))),Uu),Uua),Uub),Uuc)
        <=> aa(real,$o,ord_less_eq(real,Uub),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_mz(fun(nat,A),fun(A,fun(nat,A)),Uu),Uuc)),set_ord_atMost(nat,Uua)))) ) ) ).

% ATP.lambda_1031
tff(fact_9204_ATP_Olambda__1032,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aa(list(A),fun(A,fun(list(A),A)),aTP_Lamp_amm(A,fun(list(A),fun(A,fun(list(A),A))),Uu),Uua),Uub),Uuc) = ord_min(A,Uu,min_list(A,Uua)) ) ).

% ATP.lambda_1032
tff(fact_9205_ATP_Olambda__1033,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun($o,fun($o,B)),Uub: $o,Uuc: $o] : aa($o,A,aa($o,fun($o,A),aa(fun($o,fun($o,B)),fun($o,fun($o,A)),aTP_Lamp_fo(fun(B,A),fun(fun($o,fun($o,B)),fun($o,fun($o,A))),Uu),Uua),(Uub)),(Uuc)) = aa(B,A,Uu,aa($o,B,aa($o,fun($o,B),Uua,(Uub)),(Uuc))) ).

% ATP.lambda_1033
tff(fact_9206_ATP_Olambda__1034,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: fun(B,A),Uua: fun(C,fun(list(C),B)),Uub: C,Uuc: list(C)] : aa(list(C),A,aa(C,fun(list(C),A),aa(fun(C,fun(list(C),B)),fun(C,fun(list(C),A)),aTP_Lamp_als(fun(B,A),fun(fun(C,fun(list(C),B)),fun(C,fun(list(C),A))),Uu),Uua),Uub),Uuc) = aa(B,A,Uu,aa(list(C),B,aa(C,fun(list(C),B),Uua,Uub),Uuc)) ).

% ATP.lambda_1034
tff(fact_9207_ATP_Olambda__1035,axiom,
    ! [A: $tType,C: $tType,B: $tType,D2: $tType,Uu: fun(B,A),Uua: fun(C,fun(D2,B)),Uub: C,Uuc: D2] : aa(D2,A,aa(C,fun(D2,A),aa(fun(C,fun(D2,B)),fun(C,fun(D2,A)),aTP_Lamp_cm(fun(B,A),fun(fun(C,fun(D2,B)),fun(C,fun(D2,A))),Uu),Uua),Uub),Uuc) = aa(B,A,Uu,aa(D2,B,aa(C,fun(D2,B),Uua,Uub),Uuc)) ).

% ATP.lambda_1035
tff(fact_9208_ATP_Olambda__1036,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_aff(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_1036
tff(fact_9209_ATP_Olambda__1037,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_afd(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),aa(nat,nat,suc,Uuc))) ) ).

% ATP.lambda_1037
tff(fact_9210_ATP_Olambda__1038,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_yo(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_1038
tff(fact_9211_ATP_Olambda__1039,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat,Uuc: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_vc(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),Uu),Uua),Uub),Uuc) = aa(nat,A,Uu,aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uub),Uua)),Uuc)) ) ).

% ATP.lambda_1039
tff(fact_9212_ATP_Olambda__1040,axiom,
    ! [C: $tType,A: $tType,B: $tType,Uu: fun(C,fun(B,A)),Uua: fun(C,C),Uub: fun(fun(B,A),C),Uuc: fun(B,A)] : aa(fun(B,A),fun(B,A),aa(fun(fun(B,A),C),fun(fun(B,A),fun(B,A)),aa(fun(C,C),fun(fun(fun(B,A),C),fun(fun(B,A),fun(B,A))),aTP_Lamp_agf(fun(C,fun(B,A)),fun(fun(C,C),fun(fun(fun(B,A),C),fun(fun(B,A),fun(B,A)))),Uu),Uua),Uub),Uuc) = aa(C,fun(B,A),Uu,aa(C,C,Uua,aa(fun(B,A),C,Uub,Uuc))) ).

% ATP.lambda_1040
tff(fact_9213_ATP_Olambda__1041,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(C,fun(A,heap_Time_Heap(B))),Uua: fun(C,C),Uub: fun(fun(A,heap_Time_Heap(B)),C),Uuc: fun(A,heap_Time_Heap(B))] : aa(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B)),aa(fun(fun(A,heap_Time_Heap(B)),C),fun(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))),aa(fun(C,C),fun(fun(fun(A,heap_Time_Heap(B)),C),fun(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B)))),aTP_Lamp_mf(fun(C,fun(A,heap_Time_Heap(B))),fun(fun(C,C),fun(fun(fun(A,heap_Time_Heap(B)),C),fun(fun(A,heap_Time_Heap(B)),fun(A,heap_Time_Heap(B))))),Uu),Uua),Uub),Uuc) = aa(C,fun(A,heap_Time_Heap(B)),Uu,aa(C,C,Uua,aa(fun(A,heap_Time_Heap(B)),C,Uub,Uuc))) ).

% ATP.lambda_1041
tff(fact_9214_ATP_Olambda__1042,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(C,fun(A,option(B))),Uua: fun(C,C),Uub: fun(fun(A,option(B)),C),Uuc: fun(A,option(B))] : aa(fun(A,option(B)),fun(A,option(B)),aa(fun(fun(A,option(B)),C),fun(fun(A,option(B)),fun(A,option(B))),aa(fun(C,C),fun(fun(fun(A,option(B)),C),fun(fun(A,option(B)),fun(A,option(B)))),aTP_Lamp_aii(fun(C,fun(A,option(B))),fun(fun(C,C),fun(fun(fun(A,option(B)),C),fun(fun(A,option(B)),fun(A,option(B))))),Uu),Uua),Uub),Uuc) = aa(C,fun(A,option(B)),Uu,aa(C,C,Uua,aa(fun(A,option(B)),C,Uub,Uuc))) ).

% ATP.lambda_1042
tff(fact_9215_ATP_Olambda__1043,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(C,fun(A,B)),Uua: fun(C,C),Uub: fun(fun(A,B),C),Uuc: fun(A,B)] : aa(fun(A,B),fun(A,B),aa(fun(fun(A,B),C),fun(fun(A,B),fun(A,B)),aa(fun(C,C),fun(fun(fun(A,B),C),fun(fun(A,B),fun(A,B))),aTP_Lamp_aie(fun(C,fun(A,B)),fun(fun(C,C),fun(fun(fun(A,B),C),fun(fun(A,B),fun(A,B)))),Uu),Uua),Uub),Uuc) = aa(C,fun(A,B),Uu,aa(C,C,Uua,aa(fun(A,B),C,Uub,Uuc))) ).

% ATP.lambda_1043
tff(fact_9216_ATP_Olambda__1044,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A,Uuc: list(A)] : aa(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),aa(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))),aa(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))))),aTP_Lamp_alx(fun(A,B),fun(B,fun(A,fun(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))))))),Uu),Uua),Uub),Uuc) = product_case_prod(list(A),list(A),product_prod(list(A),product_prod(list(A),list(A))),aa(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A))))),aa(A,fun(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A)))))),aa(B,fun(A,fun(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A))))))),aTP_Lamp_alw(fun(A,B),fun(B,fun(A,fun(list(A),fun(list(A),fun(list(A),product_prod(list(A),product_prod(list(A),list(A)))))))),Uu),Uua),Uub),Uuc)) ) ).

% ATP.lambda_1044
tff(fact_9217_ATP_Olambda__1045,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_agx(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_1045
tff(fact_9218_ATP_Olambda__1046,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_agv(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uub),Uua))),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_1046
tff(fact_9219_ATP_Olambda__1047,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V8999393235501362500lgebra(A) )
     => ! [Uu: fun(B,A),Uua: B,Uub: fun(B,A),Uuc: B] : aa(B,A,aa(fun(B,A),fun(B,A),aa(B,fun(fun(B,A),fun(B,A)),aTP_Lamp_asf(fun(B,A),fun(B,fun(fun(B,A),fun(B,A))),Uu),Uua),Uub),Uuc) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua))),aa(B,A,Uub,Uuc))),aa(A,A,inverse_inverse(A),aa(B,A,Uu,Uua)))) ) ).

% ATP.lambda_1047
tff(fact_9220_ATP_Olambda__1048,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_ahb(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uub)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uuc))) ).

% ATP.lambda_1048
tff(fact_9221_ATP_Olambda__1049,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aa(int,fun(int,fun(int,product_prod(int,int))),aTP_Lamp_agz(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uu),Uua),Uub),Uuc) = normalize(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,aa(int,fun(int,int),times_times(int),Uu),Uuc)),aa(int,int,aa(int,fun(int,int),times_times(int),Uua),Uub))) ).

% ATP.lambda_1049
tff(fact_9222_ATP_Olambda__1050,axiom,
    ! [A: $tType,B: $tType,Uu: set(A),Uua: set(B),Uub: B,Uuc: fun(A,B)] :
      ( aa(fun(A,B),$o,aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_yy(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),Uu),Uua),Uub),Uuc)
    <=> ! [X: A] :
          ( ( member(A,X,Uu)
           => member(B,aa(A,B,Uuc,X),Uua) )
          & ( ~ member(A,X,Uu)
           => ( aa(A,B,Uuc,X) = Uub ) ) ) ) ).

% ATP.lambda_1050
tff(fact_9223_ATP_Olambda__1051,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B,Uub: set(B),Uuc: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_amp(fun(B,A),fun(B,fun(set(B),fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [L4: B] :
          ( ( Uuc = aa(B,A,Uu,L4) )
          & ( ( L4 = Uua )
            | member(B,L4,Uub) ) ) ) ).

% ATP.lambda_1051
tff(fact_9224_ATP_Olambda__1052,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(A,$o),Uua: fun(B,$o),Uub: fun(A,fun(B,C)),Uuc: C] :
      ( aa(C,$o,aa(fun(A,fun(B,C)),fun(C,$o),aa(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o)),aTP_Lamp_ana(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),Uu),Uua),Uub),Uuc)
    <=> ? [X: A,Y3: B] :
          ( ( Uuc = aa(B,C,aa(A,fun(B,C),Uub,X),Y3) )
          & aa(A,$o,Uu,X)
          & aa(B,$o,Uua,Y3) ) ) ).

% ATP.lambda_1052
tff(fact_9225_ATP_Olambda__1053,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),Uua: list(vEBT_VEBT),Uub: nat,Uuc: nat,Uud: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap($o),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)))),aTP_Lamp_kz(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc),Uud) = aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),product_curry(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),product_curry(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap($o),Uu)),aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),Uub)),Uud),Uuc) ).

% ATP.lambda_1053
tff(fact_9226_ATP_Olambda__1054,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uua: list(vEBT_VEBT),Uub: nat,Uuc: nat,Uud: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap($o),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)))),aTP_Lamp_hn(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc),Uud) = aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),Uu,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),Uub)),Uud),Uuc) ).

% ATP.lambda_1054
tff(fact_9227_ATP_Olambda__1055,axiom,
    ! [Uu: nat,Uua: option(product_prod(nat,nat)),Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi] :
      aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aTP_Lamp_qc(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),Uu),Uua),Uub),Uuc),Uud) = case_option(heap_Time_Heap(vEBT_VEBTi),product_prod(nat,nat),
        $ite(aa(nat,$o,ord_less_eq(nat,Uub),one_one(nat)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(Uua,Uub,Uuc,Uud)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uu),Uu)),Uub,Uuc,Uud))),
        aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_qb(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))))),Uu),Uua),Uub),Uuc),Uud),Uua) ).

% ATP.lambda_1055
tff(fact_9228_ATP_Olambda__1056,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),Uua: array(vEBT_VEBTi),Uub: nat,Uuc: nat,Uud: nat] :
      aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(nat,fun(nat,fun(nat,heap_Time_Heap($o))),aa(array(vEBT_VEBTi),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))),aTP_Lamp_ks(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(array(vEBT_VEBTi),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc),Uud) = $ite(aa(nat,$o,ord_less(nat,Uub),Uud),heap_Time_bind(vEBT_VEBTi,$o,aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uua),Uub),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)),aTP_Lamp_kr(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o))),Uu),Uuc)),aa($o,heap_Time_Heap($o),heap_Time_return($o),$false)) ).

% ATP.lambda_1056
tff(fact_9229_ATP_Olambda__1057,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),Uua: array(vEBT_VEBTi),Uub: nat,Uuc: nat,Uud: nat] :
      aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(nat,fun(nat,fun(nat,heap_Time_Heap($o))),aa(array(vEBT_VEBTi),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))),aTP_Lamp_ic(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(array(vEBT_VEBTi),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc),Uud) = $ite(aa(nat,$o,ord_less(nat,Uub),Uud),heap_Time_bind(vEBT_VEBTi,$o,aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uua),Uub),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)),aTP_Lamp_ib(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o))),Uu),Uuc)),aa($o,heap_Time_Heap($o),heap_Time_return($o),$false)) ).

% ATP.lambda_1057
tff(fact_9230_ATP_Olambda__1058,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: product_prod(nat,nat),Uud: option(nat)] :
      aa(option(nat),heap_Time_Heap(option(nat)),aa(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat))))),aTP_Lamp_bd(nat,fun(nat,fun(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(
        Uud = none(nat),
        $ite(aa(nat,$o,ord_less(nat,aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uuc)),Uu),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uuc))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))),
        heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uub),aa(option(nat),nat,the2(nat),Uud)),aa(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aTP_Lamp_ab(nat,fun(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),Uua),Uud)) ) ).

% ATP.lambda_1058
tff(fact_9231_ATP_Olambda__1059,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: product_prod(nat,nat),Uuc: option(nat),Uud: product_unit] :
      aa(product_unit,heap_Time_Heap(option(nat)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aTP_Lamp_ak(nat,fun(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(
        Uuc = none(nat),
        $ite(aa(nat,$o,ord_less(nat,aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uub)),xa),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uub))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))),
        heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uua),aa(option(nat),nat,the2(nat),Uuc)),aa(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aTP_Lamp_ab(nat,fun(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),Uu),Uuc)) ) ).

% ATP.lambda_1059
tff(fact_9232_ATP_Olambda__1060,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aTP_Lamp_bi(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uua),one_one(nat)),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),
        $ite(aa(nat,$o,ord_less(nat,aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uud)),Uu),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uud))),heap_Time_bind(nat,option(nat),vEBT_VEBT_lowi(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))),aTP_Lamp_bh(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud))) ) ).

% ATP.lambda_1060
tff(fact_9233_ATP_Olambda__1061,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aTP_Lamp_cj(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc),Uud) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uua),one_one(nat)),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),
        $ite(
          aa(nat,$o,ord_less(nat,Uu),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uud)),
          aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uud))),
          $ite(aa(nat,$o,ord_less_eq(nat,aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uud)),Uu),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),heap_Time_bind(nat,option(nat),vEBT_VEBT_lowi(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),aTP_Lamp_ci(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc))) ) ) ).

% ATP.lambda_1061
tff(fact_9234_ATP_Olambda__1062,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: nat,Uud: nat] :
      aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))),aTP_Lamp_hl(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc),Uud) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uua),one_one(nat)),
        aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),
        $ite(
          Uu = Uuc,
          aa($o,heap_Time_Heap($o),heap_Time_return($o),$true),
          $ite(
            Uu = Uud,
            aa($o,heap_Time_Heap($o),heap_Time_return($o),$true),
            $ite(
              aa(nat,$o,ord_less(nat,Uu),Uuc),
              aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),
              $ite(aa(nat,$o,ord_less(nat,Uud),Uu),aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),heap_Time_bind(nat,$o,vEBT_VEBT_highi(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o))),aTP_Lamp_hk(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o)))),Uu),Uua),Uub))) ) ) ) ) ).

% ATP.lambda_1062
tff(fact_9235_ATP_Olambda__1063,axiom,
    ! [Uu: nat,Uua: option(product_prod(nat,nat)),Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aTP_Lamp_ck(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc),Uud) = case_option(heap_Time_Heap(option(nat)),product_prod(nat,nat),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aTP_Lamp_cj(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),Uu),Uub),Uuc),Uud),Uua) ).

% ATP.lambda_1063
tff(fact_9236_ATP_Olambda__1064,axiom,
    ! [Uu: nat,Uua: option(product_prod(nat,nat)),Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aTP_Lamp_bj(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc),Uud) = case_option(heap_Time_Heap(option(nat)),product_prod(nat,nat),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aTP_Lamp_bi(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),Uu),Uub),Uuc),Uud),Uua) ).

% ATP.lambda_1064
tff(fact_9237_ATP_Olambda__1065,axiom,
    ! [Uu: nat,Uua: option(product_prod(nat,nat)),Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))),aTP_Lamp_hm(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc),Uud) = case_option(heap_Time_Heap($o),product_prod(nat,nat),aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),product_case_prod(nat,nat,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))),aTP_Lamp_hl(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))),Uu),Uub),Uuc)),Uua) ).

% ATP.lambda_1065
tff(fact_9238_ATP_Olambda__1066,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(C,fun(B,A)),Uua: fun(C,C),Uub: fun(fun(B,A),C),Uuc: B,Uud: fun(B,A)] : aa(fun(B,A),A,aa(B,fun(fun(B,A),A),aa(fun(fun(B,A),C),fun(B,fun(fun(B,A),A)),aa(fun(C,C),fun(fun(fun(B,A),C),fun(B,fun(fun(B,A),A))),aTP_Lamp_age(fun(C,fun(B,A)),fun(fun(C,C),fun(fun(fun(B,A),C),fun(B,fun(fun(B,A),A)))),Uu),Uua),Uub),Uuc),Uud) = aa(B,A,aa(C,fun(B,A),Uu,aa(C,C,Uua,aa(fun(B,A),C,Uub,Uud))),Uuc) ).

% ATP.lambda_1066
tff(fact_9239_ATP_Olambda__1067,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(C,fun(A,heap_Time_Heap(B))),Uua: fun(C,C),Uub: fun(fun(A,heap_Time_Heap(B)),C),Uuc: A,Uud: fun(A,heap_Time_Heap(B))] : aa(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B),aa(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B)),aa(fun(fun(A,heap_Time_Heap(B)),C),fun(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B))),aa(fun(C,C),fun(fun(fun(A,heap_Time_Heap(B)),C),fun(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B)))),aTP_Lamp_me(fun(C,fun(A,heap_Time_Heap(B))),fun(fun(C,C),fun(fun(fun(A,heap_Time_Heap(B)),C),fun(A,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(B))))),Uu),Uua),Uub),Uuc),Uud) = aa(A,heap_Time_Heap(B),aa(C,fun(A,heap_Time_Heap(B)),Uu,aa(C,C,Uua,aa(fun(A,heap_Time_Heap(B)),C,Uub,Uud))),Uuc) ).

% ATP.lambda_1067
tff(fact_9240_ATP_Olambda__1068,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(C,fun(A,option(B))),Uua: fun(C,C),Uub: fun(fun(A,option(B)),C),Uuc: A,Uud: fun(A,option(B))] : aa(fun(A,option(B)),option(B),aa(A,fun(fun(A,option(B)),option(B)),aa(fun(fun(A,option(B)),C),fun(A,fun(fun(A,option(B)),option(B))),aa(fun(C,C),fun(fun(fun(A,option(B)),C),fun(A,fun(fun(A,option(B)),option(B)))),aTP_Lamp_aih(fun(C,fun(A,option(B))),fun(fun(C,C),fun(fun(fun(A,option(B)),C),fun(A,fun(fun(A,option(B)),option(B))))),Uu),Uua),Uub),Uuc),Uud) = aa(A,option(B),aa(C,fun(A,option(B)),Uu,aa(C,C,Uua,aa(fun(A,option(B)),C,Uub,Uud))),Uuc) ).

% ATP.lambda_1068
tff(fact_9241_ATP_Olambda__1069,axiom,
    ! [B: $tType,C: $tType,A: $tType,Uu: fun(C,fun(A,B)),Uua: fun(C,C),Uub: fun(fun(A,B),C),Uuc: A,Uud: fun(A,B)] : aa(fun(A,B),B,aa(A,fun(fun(A,B),B),aa(fun(fun(A,B),C),fun(A,fun(fun(A,B),B)),aa(fun(C,C),fun(fun(fun(A,B),C),fun(A,fun(fun(A,B),B))),aTP_Lamp_aid(fun(C,fun(A,B)),fun(fun(C,C),fun(fun(fun(A,B),C),fun(A,fun(fun(A,B),B)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,B,aa(C,fun(A,B),Uu,aa(C,C,Uua,aa(fun(A,B),C,Uub,Uud))),Uuc) ).

% ATP.lambda_1069
tff(fact_9242_ATP_Olambda__1070,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( real_V822414075346904944vector(C)
        & real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,fun(A,B)),Uua: fun(C,A),Uub: C,Uuc: fun(C,A),Uud: C] : aa(C,B,aa(fun(C,A),fun(C,B),aa(C,fun(fun(C,A),fun(C,B)),aa(fun(C,A),fun(C,fun(fun(C,A),fun(C,B))),aTP_Lamp_arl(fun(A,fun(A,B)),fun(fun(C,A),fun(C,fun(fun(C,A),fun(C,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,B,aa(A,fun(A,B),Uu,aa(C,A,Uua,Uub)),aa(C,A,Uuc,Uud)) ) ).

% ATP.lambda_1070
tff(fact_9243_ATP_Olambda__1071,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B] : aa(B,C,aa(B,fun(B,C),aa(fun(A,fun(B,C)),fun(B,fun(B,C)),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C))),aTP_Lamp_asu(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,C)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_ast(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu),Uua),Uub),Uuc),Uud)),Uu) ) ).

% ATP.lambda_1071
tff(fact_9244_ATP_Olambda__1072,axiom,
    ! [Uu: option(product_prod(nat,nat)),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))),aTP_Lamp_bb(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc),Uud) = aa(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),heap_Time_Heap(option(nat)),product_case_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))),aTP_Lamp_ay(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc)),vEBT_case_VEBT(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),aTP_Lamp_az(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))))),aTP_Lamp_ba($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),ta)) ).

% ATP.lambda_1072
tff(fact_9245_ATP_Olambda__1073,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_acd(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aa(A,fun(A,fun(nat,fun(nat,A))),aTP_Lamp_acc(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uua),Uub),Uuc),Uud)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_1073
tff(fact_9246_ATP_Olambda__1074,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: code_integer,Uud: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aTP_Lamp_aop($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),(Uu)),(Uua)),(Uub)),Uuc),(Uud)) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aTP_Lamp_aoo($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),(Uu)),(Uua)),(Uub)),(Uud))),code_bit_cut_integer(Uuc)) ).

% ATP.lambda_1074
tff(fact_9247_ATP_Olambda__1075,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [Uu: fun(code_integer,product_prod(A,A)),Uua: A,Uub: code_integer,Uuc: A,Uud: A] : aa(A,list(A),aa(A,fun(A,list(A)),aa(code_integer,fun(A,fun(A,list(A))),aa(A,fun(code_integer,fun(A,fun(A,list(A)))),aTP_Lamp_aga(fun(code_integer,product_prod(A,A)),fun(A,fun(code_integer,fun(A,fun(A,list(A))))),Uu),Uua),Uub),Uuc),Uud) = append(A,code_T4080844693773952564amples(A,Uu,Uua,aa(code_integer,code_integer,minus_minus(code_integer,Uub),one_one(code_integer))),aa(list(A),list(A),cons(A,Uuc),aa(list(A),list(A),cons(A,Uud),nil(A)))) ) ).

% ATP.lambda_1075
tff(fact_9248_ATP_Olambda__1076,axiom,
    ! [B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,assn)),Uua: list(A),Uub: list(B),Uuc: fun(C,assn),Uud: C] : aa(C,assn,aa(fun(C,assn),fun(C,assn),aa(list(B),fun(fun(C,assn),fun(C,assn)),aa(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn))),aTP_Lamp_afk(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn)))),Uu),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,Uu,Uua),Uub)),aa(C,assn,Uuc,Uud)) ).

% ATP.lambda_1076
tff(fact_9249_ATP_Olambda__1077,axiom,
    ! [Uu: nat,Uua: fun(nat,fun(real,real)),Uub: real,Uuc: nat,Uud: real] : aa(real,real,aa(nat,fun(real,real),aa(real,fun(nat,fun(real,real)),aa(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real))),aTP_Lamp_aqv(nat,fun(fun(nat,fun(real,real)),fun(real,fun(nat,fun(real,real)))),Uu),Uua),Uub),Uuc),Uud) = aa(real,real,minus_minus(real,aa(real,real,aa(nat,fun(real,real),Uua,Uuc),Uud)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(real,fun(nat,real),aa(nat,fun(real,fun(nat,real)),aTP_Lamp_aqu(fun(nat,fun(real,real)),fun(nat,fun(real,fun(nat,real))),Uua),Uuc),Uud)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,Uu),Uuc)))),aa(real,real,aa(real,fun(real,real),times_times(real),Uub),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),Uud),aa(nat,nat,minus_minus(nat,Uu),Uuc))),semiring_char_0_fact(real,aa(nat,nat,minus_minus(nat,Uu),Uuc)))))) ).

% ATP.lambda_1077
tff(fact_9250_ATP_Olambda__1078,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: nat,Uua: fun(nat,A),Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(fun(nat,A),fun(A,fun(A,fun(nat,A))),aTP_Lamp_acg(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_acf(fun(nat,A),fun(A,fun(nat,fun(nat,A))),Uua),Uuc),Uud)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uud),Uu))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud)) ) ).

% ATP.lambda_1078
tff(fact_9251_ATP_Olambda__1079,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_acb(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uud)),Uua)),one_one(A))),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_1079
tff(fact_9252_ATP_Olambda__1080,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_abw(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Uu)),Uua)),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uuc),aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_1080
tff(fact_9253_ATP_Olambda__1081,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: A,Uub: A,Uuc: A,Uud: nat] : aa(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aa(A,fun(A,fun(A,fun(nat,A))),aTP_Lamp_abx(nat,fun(A,fun(A,fun(A,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),Uua)),Uud)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),Uub)),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Uub),Uuc)),aa(nat,nat,minus_minus(nat,Uu),Uud))) ) ).

% ATP.lambda_1081
tff(fact_9254_ATP_Olambda__1082,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: A,Uuc: nat,Uud: nat] : aa(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aa(A,fun(A,fun(nat,fun(nat,A))),aTP_Lamp_acc(fun(nat,A),fun(A,fun(A,fun(nat,fun(nat,A)))),Uu),Uua),Uub),Uuc),Uud) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,Uu,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uuc),Uud)),one_one(nat)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uub),Uud))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uuc)) ) ).

% ATP.lambda_1082
tff(fact_9255_ATP_Olambda__1083,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( topolo4958980785337419405_space(C)
        & topolo4958980785337419405_space(B) )
     => ! [Uu: fun(A,B),Uua: B,Uub: fun(B,C),Uuc: C,Uud: A] :
          ( aa(A,$o,aa(C,fun(A,$o),aa(fun(B,C),fun(C,fun(A,$o)),aa(B,fun(fun(B,C),fun(C,fun(A,$o))),aTP_Lamp_bde(fun(A,B),fun(B,fun(fun(B,C),fun(C,fun(A,$o)))),Uu),Uua),Uub),Uuc),Uud)
        <=> ( ( aa(A,B,Uu,Uud) = Uua )
           => ( aa(B,C,Uub,Uua) = Uuc ) ) ) ) ).

% ATP.lambda_1083
tff(fact_9256_ATP_Olambda__1084,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V3459762299906320749_field(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: nat,Uud: A] : aa(A,B,aa(nat,fun(A,B),aa(A,fun(nat,fun(A,B)),aa(fun(A,B),fun(A,fun(nat,fun(A,B))),aTP_Lamp_asm(fun(A,B),fun(fun(A,B),fun(A,fun(nat,fun(A,B)))),Uu),Uua),Uub),Uuc),Uud) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,semiring_1_of_nat(B),Uuc)),aa(A,B,Uua,Uud))),aa(nat,B,aa(B,fun(nat,B),power_power(B),aa(A,B,Uu,Uub)),aa(nat,nat,minus_minus(nat,Uuc),one_one(nat)))) ) ).

% ATP.lambda_1084
tff(fact_9257_ATP_Olambda__1085,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: list(A),Uub: array(A),Uuc: A,Uud: array(A)] : aa(array(A),assn,aa(A,fun(array(A),assn),aa(array(A),fun(A,fun(array(A),assn)),aa(list(A),fun(array(A),fun(A,fun(array(A),assn))),aTP_Lamp_aeo(nat,fun(list(A),fun(array(A),fun(A,fun(array(A),assn)))),Uu),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(A),assn,snga_assn(A,Uub),list_update(A,Uua,Uu,Uuc))),pure_assn(Uud = Uub)) ) ).

% ATP.lambda_1085
tff(fact_9258_ATP_Olambda__1086,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: list(vEBT_VEBT),Uud: nat] : aa(nat,heap_Time_Heap($o),aa(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o)))),aTP_Lamp_gy(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind(nat,$o,vEBT_VEBT_lowi(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap($o)),aa(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o)))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o))))),aTP_Lamp_gx(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_1086
tff(fact_9259_ATP_Olambda__1087,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: list(vEBT_VEBT),Uud: product_unit] : aa(product_unit,heap_Time_Heap($o),aa(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o)))),aTP_Lamp_gz(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind(nat,$o,vEBT_VEBT_highi(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o)))),aTP_Lamp_gy(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_1087
tff(fact_9260_ATP_Olambda__1088,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),aTP_Lamp_ci(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind(nat,option(nat),vEBT_VEBT_highi(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aTP_Lamp_ch(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_1088
tff(fact_9261_ATP_Olambda__1089,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: nat] : aa(nat,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o)))),aTP_Lamp_ku(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind(nat,$o,vEBT_VEBT_lowi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),aTP_Lamp_kt(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))),Uu),Uuc),Uud)) ).

% ATP.lambda_1089
tff(fact_9262_ATP_Olambda__1090,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: nat] : aa(nat,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o)))),aTP_Lamp_ie(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind(nat,$o,vEBT_VEBT_lowi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),aTP_Lamp_id(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))),Uu),Uuc),Uud)) ).

% ATP.lambda_1090
tff(fact_9263_ATP_Olambda__1091,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: nat,Uud: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aTP_Lamp_ch(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uua),Uud),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_cg(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_1091
tff(fact_9264_ATP_Olambda__1092,axiom,
    ! [Uu: array(vEBT_VEBTi),Uua: list(vEBT_VEBT),Uub: nat,Uuc: nat,Uud: product_unit] : aa(product_unit,heap_Time_Heap($o),aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aTP_Lamp_gt(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind(vEBT_VEBTi,$o,aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uu),Uub),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o))),aTP_Lamp_gs(list(vEBT_VEBT),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)))),Uua),Uub),Uuc)) ).

% ATP.lambda_1092
tff(fact_9265_ATP_Olambda__1093,axiom,
    ! [A: $tType,Uu: heap_Time_Heap(A),Uua: heap_ext(product_unit),Uub: A,Uuc: heap_ext(product_unit),Uud: nat] :
      ( aa(nat,$o,aa(heap_ext(product_unit),fun(nat,$o),aa(A,fun(heap_ext(product_unit),fun(nat,$o)),aa(heap_ext(product_unit),fun(A,fun(heap_ext(product_unit),fun(nat,$o))),aTP_Lamp_akf(heap_Time_Heap(A),fun(heap_ext(product_unit),fun(A,fun(heap_ext(product_unit),fun(nat,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),heap_Time_execute(A,Uu),Uua) = aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),Uub),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),Uuc),Uud))) ) ) ).

% ATP.lambda_1093
tff(fact_9266_ATP_Olambda__1094,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: product_prod(nat,nat),Uud: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))),aTP_Lamp_ag(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind(nat,option(nat),vEBT_VEBT_highi(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aTP_Lamp_af(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_1094
tff(fact_9267_ATP_Olambda__1095,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBT,Uuc: nat,Uud: option(nat)] :
      aa(option(nat),heap_Time_Heap(option(nat)),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(nat,fun(option(nat),heap_Time_Heap(option(nat))))),aTP_Lamp_cv(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc),Uud) = heap_Time_bind(product_unit,option(nat),
        refine_Imp_assert(( ( Uud = none(nat) )
          = ( vEBT_vebt_succ(Uub,Uuc) = none(nat) ) )),
        aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aTP_Lamp_cu(nat,fun(array(vEBT_VEBTi),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),Uu),Uua),Uud)) ).

% ATP.lambda_1095
tff(fact_9268_ATP_Olambda__1096,axiom,
    ! [B: $tType,A: $tType,Uu: assn,Uua: fun(B,fun(A,assn)),Uub: B,Uuc: nat,Uud: list(A)] : aa(list(A),assn,aa(nat,fun(list(A),assn),aa(B,fun(nat,fun(list(A),assn)),aa(fun(B,fun(A,assn)),fun(B,fun(nat,fun(list(A),assn))),aTP_Lamp_aeg(assn,fun(fun(B,fun(A,assn)),fun(B,fun(nat,fun(list(A),assn)))),Uu),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uu),aa(list(A),assn,vEBT_List_list_assn(B,A,Uua,replicate(B,Uuc,Uub)),Uud)) ).

% ATP.lambda_1096
tff(fact_9269_ATP_Olambda__1097,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,assn)),Uua: list(A),Uub: list(B),Uuc: nat,Uud: assn] : aa(assn,assn,aa(nat,fun(assn,assn),aa(list(B),fun(nat,fun(assn,assn)),aa(list(A),fun(list(B),fun(nat,fun(assn,assn))),aTP_Lamp_afr(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),Uu),Uua),Uub),Uuc),Uud) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uud),aa(B,assn,aa(A,fun(B,assn),Uu,aa(nat,A,nth(A,Uua),Uuc)),aa(nat,B,nth(B,Uub),Uuc))) ).

% ATP.lambda_1097
tff(fact_9270_ATP_Olambda__1098,axiom,
    ! [Uu: nat,Uua: option(product_prod(nat,nat)),Uub: nat,Uuc: array(vEBT_VEBTi),Uud: option(product_prod(nat,nat))] : aa(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o))))),aTP_Lamp_hc(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud) = product_case_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)))))),aTP_Lamp_hb(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_1098
tff(fact_9271_ATP_Olambda__1099,axiom,
    ! [Uu: option(product_prod(nat,nat)),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: option(product_prod(nat,nat))] : aa(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))),aTP_Lamp_ay(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud) = product_case_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))),aTP_Lamp_ax(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_1099
tff(fact_9272_ATP_Olambda__1100,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aTP_Lamp_nn(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),Uu),Uua),Uub),Uuc),Uud) = aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uua),Uud)),Uu,Uub,Uuc)) ).

% ATP.lambda_1100
tff(fact_9273_ATP_Olambda__1101,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] :
      aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_sb(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = case_option(heap_Time_Heap(vEBT_VEBTi),product_prod(nat,nat),
        $ite(aa(nat,$o,ord_less_eq(nat,Uuc),one_one(nat)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(Uub,Uuc,Uud,Uue)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uua),Uua)),Uuc,Uud,Uue))),
        aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_sa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_1101
tff(fact_9274_ATP_Olambda__1102,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] :
      aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_pp(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = case_option(heap_Time_Heap(vEBT_VEBTi),product_prod(nat,nat),
        $ite(aa(nat,$o,ord_less_eq(nat,Uuc),one_one(nat)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(Uub,Uuc,Uud,Uue)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uua),Uua)),Uuc,Uud,Uue))),
        aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_po(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_1102
tff(fact_9275_ATP_Olambda__1103,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] :
      aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_pa(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = case_option(heap_Time_Heap(vEBT_VEBTi),product_prod(nat,nat),
        $ite(aa(nat,$o,ord_less_eq(nat,Uuc),one_one(nat)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(Uub,Uuc,Uud,Uue)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uua),Uua)),Uuc,Uud,Uue))),
        aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_oz(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_1103
tff(fact_9276_ATP_Olambda__1104,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: product_prod(nat,nat),Uud: option(nat),Uue: product_unit] :
      aa(product_unit,heap_Time_Heap(option(nat)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),aTP_Lamp_bl(nat,fun(nat,fun(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        Uud = none(nat),
        $ite(aa(nat,$o,ord_less(nat,aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uuc)),Uu),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uuc))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))),
        heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uub),aa(option(nat),nat,the2(nat),Uud)),aa(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aTP_Lamp_ab(nat,fun(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),Uua),Uud)) ) ).

% ATP.lambda_1104
tff(fact_9277_ATP_Olambda__1105,axiom,
    ! [Uu: nat,Uua: option(product_prod(nat,nat)),Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_qb(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(aa(nat,$o,ord_less_eq(nat,Uub),one_one(nat)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(Uua,Uub,Uuc,Uud)),heap_Time_bind(nat,vEBT_VEBTi,aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uue)),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_qa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),Uu),Uub),Uuc),Uud),Uue))) ).

% ATP.lambda_1105
tff(fact_9278_ATP_Olambda__1106,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aTP_Lamp_kl(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uub),one_one(nat)),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),
        $ite(aa(nat,$o,ord_less(nat,aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uue)),Uua),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uue))),heap_Time_bind(nat,option(nat),vEBT_VEBT_lowi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))),aTP_Lamp_kk(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue))) ) ).

% ATP.lambda_1106
tff(fact_9279_ATP_Olambda__1107,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aTP_Lamp_ey(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uub),one_one(nat)),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),
        $ite(aa(nat,$o,ord_less(nat,aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uue)),Uua),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uue))),heap_Time_bind(nat,option(nat),vEBT_VEBT_lowi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))),aTP_Lamp_ex(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue))) ) ).

% ATP.lambda_1107
tff(fact_9280_ATP_Olambda__1108,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: list(vEBT_VEBT),Uud: vEBT_VEBT,Uue: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aTP_Lamp_au(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uu),one_one(nat)),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),
        $ite(aa(nat,$o,ord_less(nat,aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uue)),xa),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uue))),heap_Time_bind(nat,option(nat),vEBT_VEBT_lowi(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))),aTP_Lamp_at(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue))) ) ).

% ATP.lambda_1108
tff(fact_9281_ATP_Olambda__1109,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aTP_Lamp_kd(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uub),one_one(nat)),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),
        $ite(
          aa(nat,$o,ord_less(nat,Uua),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uue)),
          aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uue))),
          $ite(aa(nat,$o,ord_less_eq(nat,aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uue)),Uua),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),heap_Time_bind(nat,option(nat),vEBT_VEBT_lowi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))),aTP_Lamp_kc(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud))) ) ) ).

% ATP.lambda_1109
tff(fact_9282_ATP_Olambda__1110,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aTP_Lamp_fe(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uub),one_one(nat)),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),
        $ite(
          aa(nat,$o,ord_less(nat,Uua),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uue)),
          aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uue))),
          $ite(aa(nat,$o,ord_less_eq(nat,aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uue)),Uua),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),heap_Time_bind(nat,option(nat),vEBT_VEBT_lowi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))),aTP_Lamp_fd(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud))) ) ) ).

% ATP.lambda_1110
tff(fact_9283_ATP_Olambda__1111,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: nat,Uue: nat] :
      aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))),aTP_Lamp_kv(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uub),one_one(nat)),
        aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),
        $ite(
          Uua = Uud,
          aa($o,heap_Time_Heap($o),heap_Time_return($o),$true),
          $ite(
            Uua = Uue,
            aa($o,heap_Time_Heap($o),heap_Time_return($o),$true),
            $ite(
              aa(nat,$o,ord_less(nat,Uua),Uud),
              aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),
              $ite(aa(nat,$o,ord_less(nat,Uue),Uua),aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),heap_Time_bind(nat,$o,vEBT_VEBT_highi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o)))),aTP_Lamp_ku(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc))) ) ) ) ) ).

% ATP.lambda_1111
tff(fact_9284_ATP_Olambda__1112,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: nat,Uue: nat] :
      aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))),aTP_Lamp_if(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uub),one_one(nat)),
        aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),
        $ite(
          Uua = Uud,
          aa($o,heap_Time_Heap($o),heap_Time_return($o),$true),
          $ite(
            Uua = Uue,
            aa($o,heap_Time_Heap($o),heap_Time_return($o),$true),
            $ite(
              aa(nat,$o,ord_less(nat,Uua),Uud),
              aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),
              $ite(aa(nat,$o,ord_less(nat,Uue),Uua),aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),heap_Time_bind(nat,$o,vEBT_VEBT_highi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o)))),aTP_Lamp_ie(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc))) ) ) ) ) ).

% ATP.lambda_1112
tff(fact_9285_ATP_Olambda__1113,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_km(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = case_option(heap_Time_Heap(option(nat)),product_prod(nat,nat),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aTP_Lamp_kl(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),Uu),Uua),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_1113
tff(fact_9286_ATP_Olambda__1114,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_ke(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = case_option(heap_Time_Heap(option(nat)),product_prod(nat,nat),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aTP_Lamp_kd(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),Uu),Uua),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_1114
tff(fact_9287_ATP_Olambda__1115,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_ff(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = case_option(heap_Time_Heap(option(nat)),product_prod(nat,nat),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aTP_Lamp_fe(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),Uu),Uua),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_1115
tff(fact_9288_ATP_Olambda__1116,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_ez(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = case_option(heap_Time_Heap(option(nat)),product_prod(nat,nat),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aTP_Lamp_ey(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),Uu),Uua),Uuc),Uud),Uue),Uub) ).

% ATP.lambda_1116
tff(fact_9289_ATP_Olambda__1117,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: product_unit] : aa(product_unit,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o))))),aTP_Lamp_he(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = case_option(heap_Time_Heap($o),product_prod(nat,nat),aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),product_case_prod(nat,nat,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))))),aTP_Lamp_hd(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud)),Uub) ).

% ATP.lambda_1117
tff(fact_9290_ATP_Olambda__1118,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))),aTP_Lamp_kw(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = case_option(heap_Time_Heap($o),product_prod(nat,nat),aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),product_case_prod(nat,nat,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))),aTP_Lamp_kv(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))))),Uu),Uua),Uuc),Uud)),Uub) ).

% ATP.lambda_1118
tff(fact_9291_ATP_Olambda__1119,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))),aTP_Lamp_ig(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = case_option(heap_Time_Heap($o),product_prod(nat,nat),aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),product_case_prod(nat,nat,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))),aTP_Lamp_if(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))))),Uu),Uua),Uuc),Uud)),Uub) ).

% ATP.lambda_1119
tff(fact_9292_ATP_Olambda__1120,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: code_integer,Uue: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aTP_Lamp_aoo($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),(Uu)),(Uua)),(Uub)),(Uuc)),Uud),(Uue)) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aTP_Lamp_aon($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uue))),code_bit_cut_integer(Uud)) ).

% ATP.lambda_1120
tff(fact_9293_ATP_Olambda__1121,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_pz(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(nat,vEBT_VEBTi,
        $ite(aa(nat,$o,ord_less(nat,Uu),Uud),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uud),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uu)),
        aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_py(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1121
tff(fact_9294_ATP_Olambda__1122,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat,Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] :
      aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_no(nat,fun(nat,fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(nat,vEBT_VEBTi,
        $ite(aa(nat,$o,ord_less(nat,Uua),Uub),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uub),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uua)),
        aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aTP_Lamp_nn(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),Uu),Uuc),Uud),Uue)) ).

% ATP.lambda_1122
tff(fact_9295_ATP_Olambda__1123,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( real_V3459762299906320749_field(C)
        & real_V822414075346904944vector(B) )
     => ! [Uu: set(A),Uua: fun(A,fun(B,C)),Uub: fun(A,fun(B,C)),Uuc: B,Uud: B,Uue: A] : aa(A,C,aa(B,fun(A,C),aa(B,fun(B,fun(A,C)),aa(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C)))),aTP_Lamp_ast(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,C)),fun(B,fun(B,fun(A,C))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(C,C,aa(C,fun(C,C),times_times(C),aa(B,C,aa(A,fun(B,C),Uub,Uue),Uud)),groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_asr(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc),aa(set(A),set(A),minus_minus(set(A),Uu),aa(set(A),set(A),insert(A,Uue),bot_bot(set(A)))))) ) ).

% ATP.lambda_1123
tff(fact_9296_ATP_Olambda__1124,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V3459762299906320749_field(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_asd(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(B,B,minus_minus(B,aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uud,Uue)))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uuc,Uub)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_1124
tff(fact_9297_ATP_Olambda__1125,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,real),Uud: fun(A,real),Uue: A] : aa(A,real,aa(fun(A,real),fun(A,real),aa(fun(A,real),fun(fun(A,real),fun(A,real)),aa(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))),aa(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real)))),aTP_Lamp_asw(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,real),fun(fun(A,real),fun(A,real))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,aa(A,real,Uu,Uub),aa(A,real,Uuc,Uub))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uud,Uue)),aa(real,real,ln_ln(real),aa(A,real,Uu,Uub)))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),aa(A,real,Uua,Uue)),aa(A,real,Uuc,Uub))),aa(A,real,Uu,Uub)))) ) ).

% ATP.lambda_1125
tff(fact_9298_ATP_Olambda__1126,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V822414075346904944vector(B) )
     => ! [Uu: fun(A,real),Uua: fun(A,real),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_arh(fun(A,real),fun(fun(A,real),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,real_V8093663219630862766scaleR(B,aa(A,real,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_1126
tff(fact_9299_ATP_Olambda__1127,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V4412858255891104859lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_arj(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uu,Uub)),aa(A,B,Uud,Uue))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_1127
tff(fact_9300_ATP_Olambda__1128,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & real_V8999393235501362500lgebra(B) )
     => ! [Uu: fun(A,B),Uua: fun(A,B),Uub: A,Uuc: fun(A,B),Uud: fun(A,B),Uue: A] : aa(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aa(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),aa(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B)))),aTP_Lamp_asq(fun(A,B),fun(fun(A,B),fun(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,uminus_uminus(B),aa(A,B,Uu,Uub))),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))),aa(A,B,Uud,Uue))),aa(B,B,inverse_inverse(B),aa(A,B,Uuc,Uub))))),aa(B,B,aa(B,fun(B,B),divide_divide(B),aa(A,B,Uua,Uue)),aa(A,B,Uuc,Uub))) ) ).

% ATP.lambda_1128
tff(fact_9301_ATP_Olambda__1129,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: product_prod(nat,nat),Uue: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))),aTP_Lamp_bh(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(nat,option(nat),vEBT_VEBT_highi(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aTP_Lamp_bg(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1129
tff(fact_9302_ATP_Olambda__1130,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: list(vEBT_VEBT),Uue: nat] : aa(nat,heap_Time_Heap($o),aa(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o)))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o))))),aTP_Lamp_lf(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(nat,$o,vEBT_VEBT_lowi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap($o)),aa(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o)))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o)))))),aTP_Lamp_le(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1130
tff(fact_9303_ATP_Olambda__1131,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: list(vEBT_VEBT),Uue: nat] : aa(nat,heap_Time_Heap($o),aa(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o)))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o))))),aTP_Lamp_ht(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(nat,$o,vEBT_VEBT_lowi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap($o)),aa(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o)))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o)))))),aTP_Lamp_hs(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1131
tff(fact_9304_ATP_Olambda__1132,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: list(vEBT_VEBT),Uue: product_unit] : aa(product_unit,heap_Time_Heap($o),aa(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o)))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o))))),aTP_Lamp_lg(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(nat,$o,vEBT_VEBT_highi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o)))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o))))),aTP_Lamp_lf(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_1132
tff(fact_9305_ATP_Olambda__1133,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))),aTP_Lamp_kc(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(nat,option(nat),vEBT_VEBT_highi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aTP_Lamp_kb(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),Uu),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1133
tff(fact_9306_ATP_Olambda__1134,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: list(vEBT_VEBT),Uue: product_unit] : aa(product_unit,heap_Time_Heap($o),aa(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o)))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o))))),aTP_Lamp_hu(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(nat,$o,vEBT_VEBT_highi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o)))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o))))),aTP_Lamp_ht(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_1134
tff(fact_9307_ATP_Olambda__1135,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))),aTP_Lamp_fd(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(nat,option(nat),vEBT_VEBT_highi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aTP_Lamp_fc(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),Uu),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1135
tff(fact_9308_ATP_Olambda__1136,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: product_prod(nat,nat),Uud: nat,Uue: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aTP_Lamp_af(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uua),Uue),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aTP_Lamp_ae(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1136
tff(fact_9309_ATP_Olambda__1137,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aTP_Lamp_kb(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uub),Uue),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aTP_Lamp_ka(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1137
tff(fact_9310_ATP_Olambda__1138,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aTP_Lamp_fc(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uub),Uue),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aTP_Lamp_fb(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1138
tff(fact_9311_ATP_Olambda__1139,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),Uua: array(vEBT_VEBTi),Uub: list(vEBT_VEBT),Uuc: nat,Uud: nat,Uue: product_unit] : aa(product_unit,heap_Time_Heap($o),aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),aTP_Lamp_la(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(vEBT_VEBTi,$o,aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uua),Uuc),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)))),aTP_Lamp_kz(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o))))),Uu),Uub),Uuc),Uud)) ).

% ATP.lambda_1139
tff(fact_9312_ATP_Olambda__1140,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uua: array(vEBT_VEBTi),Uub: list(vEBT_VEBT),Uuc: nat,Uud: nat,Uue: product_unit] : aa(product_unit,heap_Time_Heap($o),aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),aTP_Lamp_ho(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(vEBT_VEBTi,$o,aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uua),Uuc),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)))),aTP_Lamp_hn(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o))))),Uu),Uub),Uuc),Uud)) ).

% ATP.lambda_1140
tff(fact_9313_ATP_Olambda__1141,axiom,
    ! [Uu: fun(a,b),Uua: b,Uub: a,Uuc: list(a),Uud: list(a),Uue: list(a)] :
      aa(list(a),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a)))),aa(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a))))),aa(a,fun(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a)))))),aa(b,fun(a,fun(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a))))))),aTP_Lamp_alw(fun(a,b),fun(b,fun(a,fun(list(a),fun(list(a),fun(list(a),product_prod(list(a),product_prod(list(a),list(a)))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = $let(
        x: b,
        x:= aa(a,b,Uu,Uub),
        $ite(
          aa(b,$o,ord_less(b,x),Uua),
          aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),aa(list(a),list(a),cons(a,Uub),Uuc)),aa(list(a),product_prod(list(a),list(a)),aa(list(a),fun(list(a),product_prod(list(a),list(a))),product_Pair(list(a),list(a)),Uud),Uue)),
          $ite(aa(b,$o,ord_less(b,Uua),x),aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),Uuc),aa(list(a),product_prod(list(a),list(a)),aa(list(a),fun(list(a),product_prod(list(a),list(a))),product_Pair(list(a),list(a)),Uud),aa(list(a),list(a),cons(a,Uub),Uue))),aa(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a))),aa(list(a),fun(product_prod(list(a),list(a)),product_prod(list(a),product_prod(list(a),list(a)))),product_Pair(list(a),product_prod(list(a),list(a))),Uuc),aa(list(a),product_prod(list(a),list(a)),aa(list(a),fun(list(a),product_prod(list(a),list(a))),product_Pair(list(a),list(a)),aa(list(a),list(a),cons(a,Uub),Uud)),Uue))) ) ) ).

% ATP.lambda_1141
tff(fact_9314_ATP_Olambda__1142,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: list(vEBT_VEBT),Uud: nat,Uue: nat] :
      aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o)))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o))))),aTP_Lamp_gx(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(product_unit,$o,
        refine_Imp_assert(( ( Uue = vEBT_VEBT_low(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
          & ( Uud = vEBT_VEBT_high(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) )),
        aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))),aTP_Lamp_gw(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1142
tff(fact_9315_ATP_Olambda__1143,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBT,Uuc: product_prod(nat,nat),Uud: nat,Uue: option(nat)] :
      aa(option(nat),heap_Time_Heap(option(nat)),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))),aTP_Lamp_al(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(product_unit,option(nat),
        refine_Imp_assert(( ( Uue = none(nat) )
          = ( vEBT_vebt_pred(Uub,Uud) = none(nat) ) )),
        aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aTP_Lamp_ak(nat,fun(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),Uu),Uua),Uuc),Uue)) ).

% ATP.lambda_1143
tff(fact_9316_ATP_Olambda__1144,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_dk(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(vEBT_is_Node(Uu)),aa(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))),aTP_Lamp_dj(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1144
tff(fact_9317_ATP_Olambda__1145,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_cb(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(vEBT_is_Node(Uu)),aa(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))),aTP_Lamp_ca(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1145
tff(fact_9318_ATP_Olambda__1146,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))),aTP_Lamp_hf(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(product_unit,$o,refine_Imp_assert(vEBT_is_Node(Uu)),aa(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o))))),aTP_Lamp_he(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_1146
tff(fact_9319_ATP_Olambda__1147,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: product_prod(nat,nat),Uue: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_qa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(nat,vEBT_VEBTi,aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uud)),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_pz(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),Uu),Uua),Uub),Uuc),Uue)) ).

% ATP.lambda_1147
tff(fact_9320_ATP_Olambda__1148,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_cg(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = heap_Time_bind(option(nat),option(nat),vEBT_vebt_maxti(Uue),aa(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))),aa(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))),aTP_Lamp_cf(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1148
tff(fact_9321_ATP_Olambda__1149,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi] : aa(vEBT_VEBTi,A,aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,A),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A))),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),aTP_Lamp_ga(fun(B,A),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,A,Uu,aa(vEBT_VEBTi,B,aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,B),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B))),Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1149
tff(fact_9322_ATP_Olambda__1150,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT] : aa(vEBT_VEBT,A,aa(list(vEBT_VEBT),fun(vEBT_VEBT,A),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A))),aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),aTP_Lamp_fn(fun(B,A),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A))))),Uu),Uua),Uub),Uuc),Uud),Uue) = aa(B,A,Uu,aa(vEBT_VEBT,B,aa(list(vEBT_VEBT),fun(vEBT_VEBT,B),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B))),Uua,Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1150
tff(fact_9323_ATP_Olambda__1151,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: option(product_prod(nat,nat))] : aa(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o)))))),aTP_Lamp_lj(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = product_case_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))))))),aTP_Lamp_li(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1151
tff(fact_9324_ATP_Olambda__1152,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: option(product_prod(nat,nat))] : aa(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o)))))),aTP_Lamp_hx(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = product_case_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))))))),aTP_Lamp_hw(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1152
tff(fact_9325_ATP_Olambda__1153,axiom,
    ! [Uu: nat,Uua: option(product_prod(nat,nat)),Uub: nat,Uuc: array(vEBT_VEBTi),Uud: option(product_prod(nat,nat)),Uue: nat] : aa(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)))))),aTP_Lamp_hb(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = product_case_prod(list(vEBT_VEBT),vEBT_VEBT,heap_Time_Heap($o),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))))))),aTP_Lamp_ha(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1153
tff(fact_9326_ATP_Olambda__1154,axiom,
    ! [Uu: option(product_prod(nat,nat)),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: option(product_prod(nat,nat)),Uue: nat] : aa(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))),aTP_Lamp_ax(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue) = product_case_prod(list(vEBT_VEBT),vEBT_VEBT,heap_Time_Heap(option(nat)),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))),aTP_Lamp_aw(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1154
tff(fact_9327_ATP_Olambda__1155,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi] :
      aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_rc(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = case_option(heap_Time_Heap(vEBT_VEBTi),product_prod(nat,nat),
        $ite(aa(nat,$o,ord_less_eq(nat,Uud),one_one(nat)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(Uuc,Uud,Uue,Uuf)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uub),Uub)),Uud,Uue,Uuf))),
        aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_rb(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uuc) ).

% ATP.lambda_1155
tff(fact_9328_ATP_Olambda__1156,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi] :
      aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_ol(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = case_option(heap_Time_Heap(vEBT_VEBTi),product_prod(nat,nat),
        $ite(aa(nat,$o,ord_less_eq(nat,Uud),one_one(nat)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(Uuc,Uud,Uue,Uuf)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uub),Uub)),Uud,Uue,Uuf))),
        aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_ok(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uuc) ).

% ATP.lambda_1156
tff(fact_9329_ATP_Olambda__1157,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: vEBT_VEBTi,Uuf: option(nat)] :
      aa(option(nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))),aa(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))),aTP_Lamp_cf(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(
        ( ( Uuf != none(nat) )
        & vEBT_VEBT_less(aa(nat,option(nat),some(nat),Uuc),Uuf) ),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_vebt_succi,Uue),Uuc),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_a(nat,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uu),Uud)),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_vebt_succi,Uub),Uud),aa(array(vEBT_VEBTi),fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_ce(nat,fun(array(vEBT_VEBTi),fun(option(nat),heap_Time_Heap(option(nat)))),Uu),Uua)) ) ).

% ATP.lambda_1157
tff(fact_9330_ATP_Olambda__1158,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_sa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(aa(nat,$o,ord_less_eq(nat,Uuc),one_one(nat)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(Uub,Uuc,Uud,Uue)),heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert(vEBT_is_Node(Uu)),aa(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_rz(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf))) ).

% ATP.lambda_1158
tff(fact_9331_ATP_Olambda__1159,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_po(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(aa(nat,$o,ord_less_eq(nat,Uuc),one_one(nat)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(Uub,Uuc,Uud,Uue)),heap_Time_bind(nat,vEBT_VEBTi,aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uuf)),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_pn(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uuc),Uud),Uue),Uuf))) ).

% ATP.lambda_1159
tff(fact_9332_ATP_Olambda__1160,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_oz(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(aa(nat,$o,ord_less_eq(nat,Uuc),one_one(nat)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(Uub,Uuc,Uud,Uue)),heap_Time_bind(nat,vEBT_VEBTi,aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uuf)),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_oy(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uuc),Uud),Uue),Uuf))) ).

% ATP.lambda_1160
tff(fact_9333_ATP_Olambda__1161,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),aTP_Lamp_bv(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uua),one_one(nat)),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),
        $ite(aa(nat,$o,ord_less(nat,aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uuf)),Uu),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uuf))),heap_Time_bind(nat,option(nat),vEBT_VEBT_lowi(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))))),aTP_Lamp_bu(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf))) ) ).

% ATP.lambda_1161
tff(fact_9334_ATP_Olambda__1162,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),aTP_Lamp_de(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uua),one_one(nat)),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),
        $ite(
          aa(nat,$o,ord_less(nat,Uu),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uuf)),
          aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uuf))),
          $ite(aa(nat,$o,ord_less_eq(nat,aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uuf)),Uu),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),heap_Time_bind(nat,option(nat),vEBT_VEBT_lowi(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))))))),aTP_Lamp_dd(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue))) ) ) ).

% ATP.lambda_1162
tff(fact_9335_ATP_Olambda__1163,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: nat,Uuf: nat] :
      aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))))),aTP_Lamp_hd(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uuc),one_one(nat)),
        aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),
        $ite(
          Uua = Uue,
          aa($o,heap_Time_Heap($o),heap_Time_return($o),$true),
          $ite(
            Uua = Uuf,
            aa($o,heap_Time_Heap($o),heap_Time_return($o),$true),
            $ite(
              aa(nat,$o,ord_less(nat,Uua),Uue),
              aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),
              $ite(aa(nat,$o,ord_less(nat,Uuf),Uua),aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),aa(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),heap_Time_Heap($o),product_case_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o))))),aTP_Lamp_hc(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o)))))),Uua),Uub),Uuc),Uud)),vEBT_case_VEBT(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),aTP_Lamp_az(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))))),aTP_Lamp_ba($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uu))) ) ) ) ) ).

% ATP.lambda_1163
tff(fact_9336_ATP_Olambda__1164,axiom,
    ! [Uu: option(product_prod(nat,nat)),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))),aTP_Lamp_av(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = case_option(heap_Time_Heap(option(nat)),product_prod(nat,nat),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),aa(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aTP_Lamp_au(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),Uua),Uub),Uuc),Uud),Uue),Uu) ).

% ATP.lambda_1164
tff(fact_9337_ATP_Olambda__1165,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: product_unit] : aa(product_unit,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)))))),aTP_Lamp_ll(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = case_option(heap_Time_Heap($o),product_prod(nat,nat),aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),product_case_prod(nat,nat,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))))),aTP_Lamp_lk(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue)),Uuc) ).

% ATP.lambda_1165
tff(fact_9338_ATP_Olambda__1166,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: product_unit] : aa(product_unit,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)))))),aTP_Lamp_hz(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = case_option(heap_Time_Heap($o),product_prod(nat,nat),aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),product_case_prod(nat,nat,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))))),aTP_Lamp_hy(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue)),Uuc) ).

% ATP.lambda_1166
tff(fact_9339_ATP_Olambda__1167,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))),aTP_Lamp_dj(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),heap_Time_Heap(option(nat)),product_case_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))),aTP_Lamp_di(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue)),vEBT_case_VEBT(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),aTP_Lamp_az(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))))),aTP_Lamp_ba($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uu)) ).

% ATP.lambda_1167
tff(fact_9340_ATP_Olambda__1168,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))),aTP_Lamp_ca(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),heap_Time_Heap(option(nat)),product_case_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))),aTP_Lamp_bz(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue)),vEBT_case_VEBT(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),aTP_Lamp_az(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))))),aTP_Lamp_ba($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uu)) ).

% ATP.lambda_1168
tff(fact_9341_ATP_Olambda__1169,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: code_integer,Uuf: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aTP_Lamp_aon($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),Uue),(Uuf)) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aTP_Lamp_aom($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uuf))),code_bit_cut_integer(Uue)) ).

% ATP.lambda_1169
tff(fact_9342_ATP_Olambda__1170,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_py(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(nat,vEBT_VEBTi,
        $ite(aa(nat,$o,ord_less(nat,Uu),Uud),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uu),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uud)),
        aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_px(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1170
tff(fact_9343_ATP_Olambda__1171,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_pm(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(nat,vEBT_VEBTi,
        $ite(aa(nat,$o,ord_less(nat,Uua),Uue),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uue),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uua)),
        aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_pl(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1171
tff(fact_9344_ATP_Olambda__1172,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_ox(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(nat,vEBT_VEBTi,
        $ite(aa(nat,$o,ord_less(nat,Uua),Uue),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uue),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uua)),
        aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_ow(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1172
tff(fact_9345_ATP_Olambda__1173,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))))))),aTP_Lamp_dd(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(nat,option(nat),vEBT_VEBT_highi(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aTP_Lamp_dc(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1173
tff(fact_9346_ATP_Olambda__1174,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))),aTP_Lamp_kk(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(nat,option(nat),vEBT_VEBT_highi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aTP_Lamp_kj(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1174
tff(fact_9347_ATP_Olambda__1175,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))),aTP_Lamp_ex(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(nat,option(nat),vEBT_VEBT_highi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aTP_Lamp_ew(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1175
tff(fact_9348_ATP_Olambda__1176,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: product_prod(nat,nat),Uue: nat,Uuf: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aTP_Lamp_bg(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uub),Uuf),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aTP_Lamp_bf(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1176
tff(fact_9349_ATP_Olambda__1177,axiom,
    ! [Uu: nat,Uua: list(vEBT_VEBT),Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aa(nat,fun(nat,fun(nat,$o)),aa(nat,fun(nat,fun(nat,fun(nat,$o))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o))))),aTP_Lamp_amn(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)
    <=> ( aa(nat,$o,ord_less_eq(nat,Uue),Uuf)
        & aa(nat,$o,ord_less(nat,Uuf),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu))
        & ! [I4: nat] :
            ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uud))
           => ( ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),I4)),X_12)
            <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Uub),I4) ) )
        & $ite(
            Uue = Uuf,
            ! [X: vEBT_VEBT] :
              ( member(vEBT_VEBT,X,set2(vEBT_VEBT,Uua))
             => ~ ? [X6: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X6) ),
            ( vEBT_V5917875025757280293ildren(Uuc,Uua,Uuf)
            & ! [X: nat] :
                ( aa(nat,$o,ord_less(nat,X),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uu))
               => ( vEBT_V5917875025757280293ildren(Uuc,Uua,X)
                 => ( aa(nat,$o,ord_less(nat,Uue),X)
                    & aa(nat,$o,ord_less_eq(nat,X),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_1177
tff(fact_9350_ATP_Olambda__1178,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: list(vEBT_VEBT),Uud: vEBT_VEBT,Uue: product_prod(nat,nat),Uuf: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))),aTP_Lamp_at(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(nat,option(nat),vEBT_VEBT_highi(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aTP_Lamp_as(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1178
tff(fact_9351_ATP_Olambda__1179,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: list(vEBT_VEBT),Uue: nat,Uuf: nat] :
      aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o)))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o)))))),aTP_Lamp_le(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(product_unit,$o,
        refine_Imp_assert(( ( Uuf = vEBT_VEBT_low(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
          & ( Uue = vEBT_VEBT_high(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) )),
        aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))))),aTP_Lamp_ld(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1179
tff(fact_9352_ATP_Olambda__1180,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: list(vEBT_VEBT),Uue: nat,Uuf: nat] :
      aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o)))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o)))))),aTP_Lamp_hs(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(product_unit,$o,
        refine_Imp_assert(( ( Uuf = vEBT_VEBT_low(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
          & ( Uue = vEBT_VEBT_high(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) )),
        aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))))),aTP_Lamp_hr(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1180
tff(fact_9353_ATP_Olambda__1181,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBT,Uud: product_prod(nat,nat),Uue: nat,Uuf: option(nat)] :
      aa(option(nat),heap_Time_Heap(option(nat)),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat))))))),aTP_Lamp_bm(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(product_unit,option(nat),
        refine_Imp_assert(( ( Uuf = none(nat) )
          = ( vEBT_vebt_pred(Uuc,Uue) = none(nat) ) )),
        aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),aTP_Lamp_bl(nat,fun(nat,fun(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))),Uu),Uua),Uub),Uud),Uuf)) ).

% ATP.lambda_1181
tff(fact_9354_ATP_Olambda__1182,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: list(vEBT_VEBT),Uud: nat,Uue: nat,Uuf: nat] : aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(nat,fun(nat,fun(nat,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))))),aTP_Lamp_gv(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(product_unit,$o,refine_Imp_assert(Uuf = aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uuc)),aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))))),aTP_Lamp_gu(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1182
tff(fact_9355_ATP_Olambda__1183,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aTP_Lamp_jm(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(vEBT_is_Node(Uua)),aa(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))))),aTP_Lamp_jl(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1183
tff(fact_9356_ATP_Olambda__1184,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aTP_Lamp_iv(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(vEBT_is_Node(Uua)),aa(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))))),aTP_Lamp_iu(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1184
tff(fact_9357_ATP_Olambda__1185,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aTP_Lamp_et(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(vEBT_is_Node(Uua)),aa(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))))),aTP_Lamp_es(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1185
tff(fact_9358_ATP_Olambda__1186,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aTP_Lamp_ee(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(vEBT_is_Node(Uua)),aa(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))))),aTP_Lamp_ed(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1186
tff(fact_9359_ATP_Olambda__1187,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))))),aTP_Lamp_lm(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(product_unit,$o,refine_Imp_assert(vEBT_is_Node(Uua)),aa(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)))))),aTP_Lamp_ll(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1187
tff(fact_9360_ATP_Olambda__1188,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))))),aTP_Lamp_ia(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(product_unit,$o,refine_Imp_assert(vEBT_is_Node(Uua)),aa(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)))))),aTP_Lamp_hz(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1188
tff(fact_9361_ATP_Olambda__1189,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_pn(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(nat,vEBT_VEBTi,aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uue)),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_pm(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uub),Uuc),Uud),Uuf)) ).

% ATP.lambda_1189
tff(fact_9362_ATP_Olambda__1190,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_oy(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(nat,vEBT_VEBTi,aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uue)),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aTP_Lamp_ox(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),Uu),Uua),Uub),Uuc),Uud),Uuf)) ).

% ATP.lambda_1190
tff(fact_9363_ATP_Olambda__1191,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: product_prod(nat,nat),Uud: nat,Uue: nat,Uuf: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aTP_Lamp_ae(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(option(nat),option(nat),vEBT_vebt_minti(Uuf),aa(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))),aa(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))),aTP_Lamp_ad(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1191
tff(fact_9364_ATP_Olambda__1192,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aTP_Lamp_ka(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(option(nat),option(nat),vEBT_vebt_maxti(Uuf),aa(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))),aa(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))),aTP_Lamp_jz(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1192
tff(fact_9365_ATP_Olambda__1193,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aTP_Lamp_fb(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(option(nat),option(nat),vEBT_vebt_maxti(Uuf),aa(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))),aa(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))),aTP_Lamp_fa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1193
tff(fact_9366_ATP_Olambda__1194,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: list(vEBT_VEBT),Uud: nat,Uue: nat,Uuf: product_unit] : aa(product_unit,heap_Time_Heap($o),aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))),aTP_Lamp_gw(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = heap_Time_bind(nat,$o,array_len(vEBT_VEBTi,Uub),aa(nat,fun(nat,heap_Time_Heap($o)),aa(nat,fun(nat,fun(nat,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))))),aTP_Lamp_gv(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_1194
tff(fact_9367_ATP_Olambda__1195,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,assn)),Uua: A,Uub: list(A),Uuc: B,Uud: list(B),Uue: nat,Uuf: assn] : aa(assn,assn,aa(nat,fun(assn,assn),aa(list(B),fun(nat,fun(assn,assn)),aa(B,fun(list(B),fun(nat,fun(assn,assn))),aa(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))),aa(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn))))),aTP_Lamp_afu(fun(A,fun(B,assn)),fun(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Uuf),aa(B,assn,aa(A,fun(B,assn),Uu,aa(nat,A,nth(A,aa(list(A),list(A),cons(A,Uua),Uub)),Uue)),aa(nat,B,nth(B,aa(list(B),list(B),cons(B,Uuc),Uud)),Uue))) ).

% ATP.lambda_1195
tff(fact_9368_ATP_Olambda__1196,axiom,
    ! [Uu: nat,Uua: option(product_prod(nat,nat)),Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: option(product_prod(nat,nat))] : aa(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_ry(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = product_case_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_rx(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1196
tff(fact_9369_ATP_Olambda__1197,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: option(product_prod(nat,nat)),Uuf: nat] : aa(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))))))),aTP_Lamp_li(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = product_case_prod(list(vEBT_VEBT),vEBT_VEBT,heap_Time_Heap($o),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))))))),aTP_Lamp_lh(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1197
tff(fact_9370_ATP_Olambda__1198,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: option(product_prod(nat,nat)),Uuf: nat] : aa(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o))))))),aTP_Lamp_hw(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = product_case_prod(list(vEBT_VEBT),vEBT_VEBT,heap_Time_Heap($o),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))))))),aTP_Lamp_hv(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1198
tff(fact_9371_ATP_Olambda__1199,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: option(product_prod(nat,nat))] : aa(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))),aTP_Lamp_di(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = product_case_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))),aTP_Lamp_dh(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1199
tff(fact_9372_ATP_Olambda__1200,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: option(product_prod(nat,nat))] : aa(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))),aTP_Lamp_bz(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf) = product_case_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))),aTP_Lamp_by(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1200
tff(fact_9373_ATP_Olambda__1201,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: $o,Uug: code_integer] : aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),aTP_Lamp_aol($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uuf)),Uug) = char2((Uu),(Uua),(Uub),(Uuc),(Uud),(Uue),(Uuf)) ).

% ATP.lambda_1201
tff(fact_9374_ATP_Olambda__1202,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi,Uug: option(nat)] :
      aa(option(nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))),aa(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))),aTP_Lamp_jz(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        ( ( Uug != none(nat) )
        & vEBT_VEBT_less(aa(nat,option(nat),some(nat),Uud),Uug) ),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),product_curry(vEBT_VEBTi,nat,heap_Time_Heap(option(nat)),Uu),Uuf),Uud),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_a(nat,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uua),Uue)),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),product_curry(vEBT_VEBTi,nat,heap_Time_Heap(option(nat)),Uu),Uuc),Uue),aa(array(vEBT_VEBTi),fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_ce(nat,fun(array(vEBT_VEBTi),fun(option(nat),heap_Time_Heap(option(nat)))),Uua),Uub)) ) ).

% ATP.lambda_1202
tff(fact_9375_ATP_Olambda__1203,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi,Uug: option(nat)] :
      aa(option(nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))),aa(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))),aTP_Lamp_fa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        ( ( Uug != none(nat) )
        & vEBT_VEBT_less(aa(nat,option(nat),some(nat),Uud),Uug) ),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),Uu,Uuf),Uud),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_a(nat,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uua),Uue)),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),Uu,Uuc),Uue),aa(array(vEBT_VEBTi),fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_ce(nat,fun(array(vEBT_VEBTi),fun(option(nat),heap_Time_Heap(option(nat)))),Uua),Uub)) ) ).

% ATP.lambda_1203
tff(fact_9376_ATP_Olambda__1204,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: product_prod(nat,nat),Uud: nat,Uue: nat,Uuf: vEBT_VEBTi,Uug: option(nat)] :
      aa(option(nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))),aa(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))),aTP_Lamp_ad(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        ( ( Uug != none(nat) )
        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),Uud),Uug) ),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_vebt_predi,Uuf),Uud),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_a(nat,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uu),Uue)),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_vebt_predi,Uub),Uue),aa(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat)))),aTP_Lamp_ac(nat,fun(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat))))),Uu),Uua),Uuc)) ) ).

% ATP.lambda_1204
tff(fact_9377_ATP_Olambda__1205,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: list(vEBT_VEBT),Uud: nat,Uue: nat,Uuf: nat,Uug: product_unit] :
      aa(product_unit,heap_Time_Heap($o),aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))))),aTP_Lamp_gu(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa(nat,$o,ord_less(nat,Uud),Uuf),
        heap_Time_bind(product_unit,$o,
          refine_Imp_assert(( ( Uud = vEBT_VEBT_high(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
            & aa(nat,$o,ord_less(nat,Uud),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uuc)) )),
          aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aTP_Lamp_gt(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),Uub),Uuc),Uud),Uue)),
        aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) ) ).

% ATP.lambda_1205
tff(fact_9378_ATP_Olambda__1206,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_rb(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(aa(nat,$o,ord_less_eq(nat,Uud),one_one(nat)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(Uuc,Uud,Uue,Uuf)),heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert(vEBT_is_Node(Uua)),aa(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_ra(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))) ).

% ATP.lambda_1206
tff(fact_9379_ATP_Olambda__1207,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_ok(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(aa(nat,$o,ord_less_eq(nat,Uud),one_one(nat)),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(Uuc,Uud,Uue,Uuf)),heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert(vEBT_is_Node(Uua)),aa(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_oj(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))) ).

% ATP.lambda_1207
tff(fact_9380_ATP_Olambda__1208,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))))),aTP_Lamp_jg(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uub),one_one(nat)),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),
        $ite(aa(nat,$o,ord_less(nat,aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uug)),Uua),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uug))),heap_Time_bind(nat,option(nat),vEBT_VEBT_lowi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))))),aTP_Lamp_jf(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))) ) ).

% ATP.lambda_1208
tff(fact_9381_ATP_Olambda__1209,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))))),aTP_Lamp_dy(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uub),one_one(nat)),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),
        $ite(aa(nat,$o,ord_less(nat,aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uug)),Uua),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uug))),heap_Time_bind(nat,option(nat),vEBT_VEBT_lowi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))))),aTP_Lamp_dx(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug))) ) ).

% ATP.lambda_1209
tff(fact_9382_ATP_Olambda__1210,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))))),aTP_Lamp_ip(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uub),one_one(nat)),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),
        $ite(
          aa(nat,$o,ord_less(nat,Uua),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uug)),
          aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uug))),
          $ite(aa(nat,$o,ord_less_eq(nat,aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uug)),Uua),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),heap_Time_bind(nat,option(nat),vEBT_VEBT_lowi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))))))),aTP_Lamp_io(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf))) ) ) ).

% ATP.lambda_1210
tff(fact_9383_ATP_Olambda__1211,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))))),aTP_Lamp_en(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uub),one_one(nat)),
        aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),
        $ite(
          aa(nat,$o,ord_less(nat,Uua),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uug)),
          aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uug))),
          $ite(aa(nat,$o,ord_less_eq(nat,aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uug)),Uua),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),heap_Time_bind(nat,option(nat),vEBT_VEBT_lowi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))))))),aTP_Lamp_em(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf))) ) ) ).

% ATP.lambda_1211
tff(fact_9384_ATP_Olambda__1212,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: nat,Uug: nat] :
      aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))))),aTP_Lamp_lk(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uud),one_one(nat)),
        aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),
        $ite(
          Uub = Uuf,
          aa($o,heap_Time_Heap($o),heap_Time_return($o),$true),
          $ite(
            Uub = Uug,
            aa($o,heap_Time_Heap($o),heap_Time_return($o),$true),
            $ite(
              aa(nat,$o,ord_less(nat,Uub),Uuf),
              aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),
              $ite(aa(nat,$o,ord_less(nat,Uug),Uub),aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),aa(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),heap_Time_Heap($o),product_case_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o)))))),aTP_Lamp_lj(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o))))))),Uu),Uub),Uuc),Uud),Uue)),vEBT_case_VEBT(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),aTP_Lamp_az(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))))),aTP_Lamp_ba($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uua))) ) ) ) ) ).

% ATP.lambda_1212
tff(fact_9385_ATP_Olambda__1213,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: nat,Uug: nat] :
      aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))))),aTP_Lamp_hy(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $ite(
        aa(nat,$o,ord_less_eq(nat,Uud),one_one(nat)),
        aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),
        $ite(
          Uub = Uuf,
          aa($o,heap_Time_Heap($o),heap_Time_return($o),$true),
          $ite(
            Uub = Uug,
            aa($o,heap_Time_Heap($o),heap_Time_return($o),$true),
            $ite(
              aa(nat,$o,ord_less(nat,Uub),Uuf),
              aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),
              $ite(aa(nat,$o,ord_less(nat,Uug),Uub),aa($o,heap_Time_Heap($o),heap_Time_return($o),$false),aa(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),heap_Time_Heap($o),product_case_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o)))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o)))))),aTP_Lamp_hx(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($o))))))),Uu),Uub),Uuc),Uud),Uue)),vEBT_case_VEBT(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),aTP_Lamp_az(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))))),aTP_Lamp_ba($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uua))) ) ) ) ) ).

% ATP.lambda_1213
tff(fact_9386_ATP_Olambda__1214,axiom,
    ! [Uu: nat,Uua: option(product_prod(nat,nat)),Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))),aTP_Lamp_df(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = case_option(heap_Time_Heap(option(nat)),product_prod(nat,nat),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),aa(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),aTP_Lamp_de(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))))),Uu),Uub),Uuc),Uud),Uue),Uuf),Uua) ).

% ATP.lambda_1214
tff(fact_9387_ATP_Olambda__1215,axiom,
    ! [Uu: nat,Uua: option(product_prod(nat,nat)),Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))),aTP_Lamp_bw(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = case_option(heap_Time_Heap(option(nat)),product_prod(nat,nat),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),aa(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),aTP_Lamp_bv(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))))),Uu),Uub),Uuc),Uud),Uue),Uuf),Uua) ).

% ATP.lambda_1215
tff(fact_9388_ATP_Olambda__1216,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))))),aTP_Lamp_jl(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = aa(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),heap_Time_Heap(option(nat)),product_case_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))))),aTP_Lamp_jk(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)),vEBT_case_VEBT(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),aTP_Lamp_az(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))))),aTP_Lamp_ba($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uua)) ).

% ATP.lambda_1216
tff(fact_9389_ATP_Olambda__1217,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))))),aTP_Lamp_iu(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = aa(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),heap_Time_Heap(option(nat)),product_case_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))))),aTP_Lamp_it(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)),vEBT_case_VEBT(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),aTP_Lamp_az(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))))),aTP_Lamp_ba($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uua)) ).

% ATP.lambda_1217
tff(fact_9390_ATP_Olambda__1218,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))))),aTP_Lamp_es(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = aa(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),heap_Time_Heap(option(nat)),product_case_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))))),aTP_Lamp_er(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)),vEBT_case_VEBT(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),aTP_Lamp_az(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))))),aTP_Lamp_ba($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uua)) ).

% ATP.lambda_1218
tff(fact_9391_ATP_Olambda__1219,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat)))))))),aTP_Lamp_ed(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_unit,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = aa(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),heap_Time_Heap(option(nat)),product_case_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))))),aTP_Lamp_ec(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)),vEBT_case_VEBT(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),aTP_Lamp_az(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))))),aTP_Lamp_ba($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uua)) ).

% ATP.lambda_1219
tff(fact_9392_ATP_Olambda__1220,axiom,
    ! [Uu: $o,Uua: $o,Uub: $o,Uuc: $o,Uud: $o,Uue: $o,Uuf: code_integer,Uug: $o] : aa($o,char,aa(code_integer,fun($o,char),aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aTP_Lamp_aom($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),Uuf),(Uug)) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aa($o,fun(code_integer,fun($o,char)),aa($o,fun($o,fun(code_integer,fun($o,char))),aa($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),aa($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))),aa($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))))),aTP_Lamp_aol($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char)))))))),(Uu)),(Uua)),(Uub)),(Uuc)),(Uud)),(Uue)),(Uug))),code_bit_cut_integer(Uuf)) ).

% ATP.lambda_1220
tff(fact_9393_ATP_Olambda__1221,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: product_prod(nat,nat),Uug: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_rz(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = aa(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),heap_Time_Heap(vEBT_VEBTi),product_case_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_ry(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))))))),Uua),Uub),Uuc),Uud),Uue),Uuf)),vEBT_case_VEBT(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),aTP_Lamp_az(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))))),aTP_Lamp_ba($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uu)) ).

% ATP.lambda_1221
tff(fact_9394_ATP_Olambda__1222,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_pl(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(nat,vEBT_VEBTi,
        $ite(aa(nat,$o,ord_less(nat,Uua),Uue),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uua),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uue)),
        aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_pk(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1222
tff(fact_9395_ATP_Olambda__1223,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_ow(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(nat,vEBT_VEBTi,
        $ite(aa(nat,$o,ord_less(nat,Uua),Uue),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uua),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uue)),
        aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_ov(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1223
tff(fact_9396_ATP_Olambda__1224,axiom,
    ! [Uu: nat,Uua: vEBT_VEBTi,Uub: nat,Uuc: nat,Uud: nat,Uue: nat,Uuf: $o,Uug: array(vEBT_VEBTi)] :
      aa(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),aa($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_pq(nat,fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),(Uuf)),Uug) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,
        $ite((Uuf),aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),vEBT_vebt_inserti,Uua),Uue),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),Uua)),
        aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_no(nat,fun(nat,fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),Uu),Uub),Uuc),Uud),Uug)) ).

% ATP.lambda_1224
tff(fact_9397_ATP_Olambda__1225,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: product_prod(nat,nat),Uug: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))))),aTP_Lamp_bu(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(nat,option(nat),vEBT_VEBT_highi(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),aTP_Lamp_bt(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1225
tff(fact_9398_ATP_Olambda__1226,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))))))),aTP_Lamp_io(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(nat,option(nat),vEBT_VEBT_highi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),aTP_Lamp_in(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1226
tff(fact_9399_ATP_Olambda__1227,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat)))))))),aTP_Lamp_em(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(nat,option(nat),vEBT_VEBT_highi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),aTP_Lamp_el(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1227
tff(fact_9400_ATP_Olambda__1228,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_px(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(nat,vEBT_VEBTi,vEBT_VEBT_lowi(Uuf,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_pw(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1228
tff(fact_9401_ATP_Olambda__1229,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: nat,Uug: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aTP_Lamp_kj(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uuc),Uug),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ki(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1229
tff(fact_9402_ATP_Olambda__1230,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: nat,Uug: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aTP_Lamp_ew(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uuc),Uug),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ev(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1230
tff(fact_9403_ATP_Olambda__1231,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: list(vEBT_VEBT),Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aTP_Lamp_cz(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uua),Uuf),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aTP_Lamp_cy(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1231
tff(fact_9404_ATP_Olambda__1232,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: list(vEBT_VEBT),Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi] :
      aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aTP_Lamp_cy(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uuc),Uuf),
        heap_Time_bind(option(nat),option(nat),vEBT_vebt_maxti(Uug),aa(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))),aTP_Lamp_cx(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uud),Uue),Uuf),Uug),aktnode)) ) ).

% ATP.lambda_1232
tff(fact_9405_ATP_Olambda__1233,axiom,
    ! [Uu: option(product_prod(nat,nat)),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: option(product_prod(nat,nat)),Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT] :
      aa(vEBT_VEBT,heap_Time_Heap(option(nat)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))),aTP_Lamp_aw(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(product_unit,option(nat),
        refine_Imp_assert(( ( Uud = Uu )
          & ( Uue = Uua )
          & vEBT_is_Node(ta) )),
        aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))),aTP_Lamp_av(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uuc),Uuf),Uug)) ).

% ATP.lambda_1233
tff(fact_9406_ATP_Olambda__1234,axiom,
    ! [Uu: nat,Uua: option(product_prod(nat,nat)),Uub: nat,Uuc: array(vEBT_VEBTi),Uud: option(product_prod(nat,nat)),Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT] :
      aa(vEBT_VEBT,heap_Time_Heap($o),aa(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))))))),aTP_Lamp_ha(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(product_unit,$o,
        refine_Imp_assert(( ( Uua = Uud )
          & ( Uub = Uue ) )),
        aa(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o)))),aTP_Lamp_gz(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o))))),Uu),Uub),Uuc),Uuf)) ).

% ATP.lambda_1234
tff(fact_9407_ATP_Olambda__1235,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aTP_Lamp_dc(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uuf = vEBT_VEBT_low(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_db(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1235
tff(fact_9408_ATP_Olambda__1236,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: list(vEBT_VEBT),Uud: vEBT_VEBT,Uue: product_prod(nat,nat),Uuf: nat,Uug: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aTP_Lamp_as(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uuf = vEBT_VEBT_low(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ar(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1236
tff(fact_9409_ATP_Olambda__1237,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: list(vEBT_VEBT),Uue: nat,Uuf: nat,Uug: nat] : aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(nat,fun(nat,fun(nat,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))))),aTP_Lamp_lc(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(product_unit,$o,refine_Imp_assert(Uug = aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uud)),aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))))),aTP_Lamp_lb(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1237
tff(fact_9410_ATP_Olambda__1238,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: list(vEBT_VEBT),Uue: nat,Uuf: nat,Uug: nat] : aa(nat,heap_Time_Heap($o),aa(nat,fun(nat,heap_Time_Heap($o)),aa(nat,fun(nat,fun(nat,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))))),aTP_Lamp_hq(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(product_unit,$o,refine_Imp_assert(Uug = aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uud)),aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))))),aTP_Lamp_hp(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1238
tff(fact_9411_ATP_Olambda__1239,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: list(vEBT_VEBT),Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aTP_Lamp_da(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(aa(nat,$o,ord_less(nat,Uuf),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uuc))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aTP_Lamp_cz(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1239
tff(fact_9412_ATP_Olambda__1240,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: product_prod(nat,nat),Uue: nat,Uuf: nat,Uug: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aTP_Lamp_bf(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(option(nat),option(nat),vEBT_vebt_minti(Uug),aa(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))),aa(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))))),aTP_Lamp_be(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1240
tff(fact_9413_ATP_Olambda__1241,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: list(vEBT_VEBT),Uue: nat,Uuf: nat,Uug: product_unit] : aa(product_unit,heap_Time_Heap($o),aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))))),aTP_Lamp_ld(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(nat,$o,array_len(vEBT_VEBTi,Uuc),aa(nat,fun(nat,heap_Time_Heap($o)),aa(nat,fun(nat,fun(nat,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))))),aTP_Lamp_lc(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1241
tff(fact_9414_ATP_Olambda__1242,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: list(vEBT_VEBT),Uue: nat,Uuf: nat,Uug: product_unit] : aa(product_unit,heap_Time_Heap($o),aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))))),aTP_Lamp_hr(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = heap_Time_bind(nat,$o,array_len(vEBT_VEBTi,Uuc),aa(nat,fun(nat,heap_Time_Heap($o)),aa(nat,fun(nat,fun(nat,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))))),aTP_Lamp_hq(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf)) ).

% ATP.lambda_1242
tff(fact_9415_ATP_Olambda__1243,axiom,
    ! [Uu: nat,Uua: option(product_prod(nat,nat)),Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: option(product_prod(nat,nat)),Uug: nat] : aa(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_rx(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = product_case_prod(list(vEBT_VEBT),vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_rw(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1243
tff(fact_9416_ATP_Olambda__1244,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: product_prod(nat,nat),Uug: option(product_prod(nat,nat))] : aa(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_qz(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = product_case_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_qy(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1244
tff(fact_9417_ATP_Olambda__1245,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: product_prod(nat,nat),Uug: option(product_prod(nat,nat))] : aa(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_oi(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = product_case_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_oh(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1245
tff(fact_9418_ATP_Olambda__1246,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: option(product_prod(nat,nat))] : aa(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))))),aTP_Lamp_jk(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = product_case_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_jj(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1246
tff(fact_9419_ATP_Olambda__1247,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: option(product_prod(nat,nat))] : aa(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))))),aTP_Lamp_it(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = product_case_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_is(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1247
tff(fact_9420_ATP_Olambda__1248,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: option(product_prod(nat,nat))] : aa(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))))),aTP_Lamp_er(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = product_case_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_eq(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1248
tff(fact_9421_ATP_Olambda__1249,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: option(product_prod(nat,nat))] : aa(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat))))))))),aTP_Lamp_ec(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = product_case_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_eb(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1249
tff(fact_9422_ATP_Olambda__1250,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: option(product_prod(nat,nat)),Uug: nat] : aa(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))),aTP_Lamp_dh(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = product_case_prod(list(vEBT_VEBT),vEBT_VEBT,heap_Time_Heap(option(nat)),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_dg(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1250
tff(fact_9423_ATP_Olambda__1251,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: option(product_prod(nat,nat)),Uug: nat] : aa(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))),aTP_Lamp_by(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug) = product_case_prod(list(vEBT_VEBT),vEBT_VEBT,heap_Time_Heap(option(nat)),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_bx(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1251
tff(fact_9424_ATP_Olambda__1252,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: product_prod(nat,nat),Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: option(nat)] :
      aa(option(nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))),aa(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))))),aTP_Lamp_be(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $ite(
        ( ( Uuh != none(nat) )
        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),Uue),Uuh) ),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_vebt_predi,Uug),Uue),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_a(nat,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uua),Uuf)),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_vebt_predi,Uuc),Uuf),aa(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat))))),aTP_Lamp_bd(nat,fun(nat,fun(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uud)) ) ).

% ATP.lambda_1252
tff(fact_9425_ATP_Olambda__1253,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: list(vEBT_VEBT),Uue: nat,Uuf: nat,Uug: nat,Uuh: product_unit] :
      aa(product_unit,heap_Time_Heap($o),aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))))),aTP_Lamp_lb(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $ite(
        aa(nat,$o,ord_less(nat,Uue),Uug),
        heap_Time_bind(product_unit,$o,
          refine_Imp_assert(( ( Uue = vEBT_VEBT_high(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
            & aa(nat,$o,ord_less(nat,Uue),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uud)) )),
          aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),aTP_Lamp_la(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))),Uu),Uuc),Uud),Uue),Uuf)),
        aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) ) ).

% ATP.lambda_1253
tff(fact_9426_ATP_Olambda__1254,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: list(vEBT_VEBT),Uue: nat,Uuf: nat,Uug: nat,Uuh: product_unit] :
      aa(product_unit,heap_Time_Heap($o),aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))))),aTP_Lamp_hp(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $ite(
        aa(nat,$o,ord_less(nat,Uue),Uug),
        heap_Time_bind(product_unit,$o,
          refine_Imp_assert(( ( Uue = vEBT_VEBT_high(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
            & aa(nat,$o,ord_less(nat,Uue),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uud)) )),
          aa(nat,fun(product_unit,heap_Time_Heap($o)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))),aa(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),aTP_Lamp_ho(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o)))))),Uu),Uuc),Uud),Uue),Uuf)),
        aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) ) ).

% ATP.lambda_1254
tff(fact_9427_ATP_Olambda__1255,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_jh(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = case_option(heap_Time_Heap(option(nat)),product_prod(nat,nat),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),aa(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))))),aTP_Lamp_jg(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uub) ).

% ATP.lambda_1255
tff(fact_9428_ATP_Olambda__1256,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_iq(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = case_option(heap_Time_Heap(option(nat)),product_prod(nat,nat),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),aa(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))))),aTP_Lamp_ip(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uub) ).

% ATP.lambda_1256
tff(fact_9429_ATP_Olambda__1257,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_eo(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = case_option(heap_Time_Heap(option(nat)),product_prod(nat,nat),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),aa(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))))),aTP_Lamp_en(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uub) ).

% ATP.lambda_1257
tff(fact_9430_ATP_Olambda__1258,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_dz(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = case_option(heap_Time_Heap(option(nat)),product_prod(nat,nat),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)),aa(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat)))))))),aTP_Lamp_dy(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),heap_Time_Heap(option(nat))))))))),Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uub) ).

% ATP.lambda_1258
tff(fact_9431_ATP_Olambda__1259,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: product_prod(nat,nat),Uuh: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_ra(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = aa(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),heap_Time_Heap(vEBT_VEBTi),product_case_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_qz(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uub),Uuc),Uud),Uue),Uuf),Uug)),vEBT_case_VEBT(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),aTP_Lamp_az(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))))),aTP_Lamp_ba($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uua)) ).

% ATP.lambda_1259
tff(fact_9432_ATP_Olambda__1260,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: product_prod(nat,nat),Uuh: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_oj(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = aa(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),heap_Time_Heap(vEBT_VEBTi),product_case_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_oi(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uub),Uuc),Uud),Uue),Uuf),Uug)),vEBT_case_VEBT(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),aTP_Lamp_az(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))))),aTP_Lamp_ba($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uua)) ).

% ATP.lambda_1260
tff(fact_9433_ATP_Olambda__1261,axiom,
    ! [Uu: nat,Uua: vEBT_VEBTi,Uub: vEBT_VEBT,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: $o,Uuh: array(vEBT_VEBTi)] :
      aa(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),aa($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_rf(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),(Uug)),Uuh) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,
        $ite((Uug),aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),vEBT_V3964819847710782039nserti,Uub),Uua),Uuf),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),Uua)),
        aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_no(nat,fun(nat,fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),Uu),Uuc),Uud),Uue),Uuh)) ).

% ATP.lambda_1261
tff(fact_9434_ATP_Olambda__1262,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: $o,Uuh: array(vEBT_VEBTi)] :
      aa(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),aa($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_oo(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),(Uug)),Uuh) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,
        $ite((Uug),aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),product_curry(vEBT_VEBTi,nat,heap_Time_Heap(vEBT_VEBTi),Uu),Uub),Uuf),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),Uub)),
        aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_no(nat,fun(nat,fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),Uua),Uuc),Uud),Uue),Uuh)) ).

% ATP.lambda_1262
tff(fact_9435_ATP_Olambda__1263,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uua: nat,Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: $o,Uuh: array(vEBT_VEBTi)] :
      aa(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),aa($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_pd(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),(Uug)),Uuh) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,
        $ite((Uug),aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),Uu,Uub),Uuf),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),Uub)),
        aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_no(nat,fun(nat,fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),Uua),Uuc),Uud),Uue),Uuh)) ).

% ATP.lambda_1263
tff(fact_9436_ATP_Olambda__1264,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: $o,Uuh: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_pr(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),(Uug)),Uuh) = heap_Time_bind(array(vEBT_VEBTi),vEBT_VEBTi,array_upd(vEBT_VEBTi,Uuf,Uuh,Uua),aa($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_pq(nat,fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))),Uu),Uub),Uuc),Uud),Uue),Uuf),(Uug))) ).

% ATP.lambda_1264
tff(fact_9437_ATP_Olambda__1265,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_prod(nat,nat),Uuh: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))))),aTP_Lamp_jf(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(nat,option(nat),vEBT_VEBT_highi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_je(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1265
tff(fact_9438_ATP_Olambda__1266,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_prod(nat,nat),Uuh: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))))))))),aTP_Lamp_dx(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(nat,option(nat),vEBT_VEBT_highi(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_dw(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1266
tff(fact_9439_ATP_Olambda__1267,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_pw(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(nat,vEBT_VEBTi,vEBT_VEBT_highi(Uuf,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_pv(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1267
tff(fact_9440_ATP_Olambda__1268,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_pk(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(nat,vEBT_VEBTi,vEBT_VEBT_lowi(Uug,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_pj(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1268
tff(fact_9441_ATP_Olambda__1269,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_ov(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(nat,vEBT_VEBTi,vEBT_VEBT_lowi(Uug,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_ou(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1269
tff(fact_9442_ATP_Olambda__1270,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: list(vEBT_VEBT),Uud: vEBT_VEBT,Uue: product_prod(nat,nat),Uuf: nat,Uug: nat,Uuh: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ap(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uua),Uug),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ao(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1270
tff(fact_9443_ATP_Olambda__1271,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ik(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uub),Uug),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ij(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1271
tff(fact_9444_ATP_Olambda__1272,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ei(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uub),Uug),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),aTP_Lamp_eh(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1272
tff(fact_9445_ATP_Olambda__1273,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: list(vEBT_VEBT),Uud: vEBT_VEBT,Uue: product_prod(nat,nat),Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] :
      aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ao(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uuc),Uug),
        heap_Time_bind(option(nat),option(nat),vEBT_vebt_minti(Uuh),aa(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_an(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uud),Uue),Uuf),Uug),Uuh),aktnode)) ) ).

% ATP.lambda_1273
tff(fact_9446_ATP_Olambda__1274,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] :
      aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ij(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uud),Uug),
        heap_Time_bind(option(nat),option(nat),vEBT_vebt_maxti(Uuh),aa(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_ii(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),aktnode)) ) ).

% ATP.lambda_1274
tff(fact_9447_ATP_Olambda__1275,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] :
      aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),aTP_Lamp_eh(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uud),Uug),
        heap_Time_bind(option(nat),option(nat),vEBT_vebt_maxti(Uuh),aa(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_eg(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),aktnode)) ) ).

% ATP.lambda_1275
tff(fact_9448_ATP_Olambda__1276,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: option(product_prod(nat,nat)),Uuf: nat,Uug: list(vEBT_VEBT),Uuh: vEBT_VEBT] :
      aa(vEBT_VEBT,heap_Time_Heap($o),aa(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))))))),aTP_Lamp_lh(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(product_unit,$o,
        refine_Imp_assert(( ( Uub = Uue )
          & ( Uuc = Uuf ) )),
        aa(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o)))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o))))),aTP_Lamp_lg(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o)))))),Uu),Uua),Uuc),Uud),Uug)) ).

% ATP.lambda_1276
tff(fact_9449_ATP_Olambda__1277,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: option(product_prod(nat,nat)),Uuf: nat,Uug: list(vEBT_VEBT),Uuh: vEBT_VEBT] :
      aa(vEBT_VEBT,heap_Time_Heap($o),aa(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))),aa(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))))),aa(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o)))))))),aTP_Lamp_hv(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($o))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(product_unit,$o,
        refine_Imp_assert(( ( Uub = Uue )
          & ( Uuc = Uuf ) )),
        aa(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o)),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o))),aa(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o)))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o))))),aTP_Lamp_hu(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o)))))),Uu),Uua),Uuc),Uud),Uug)) ).

% ATP.lambda_1277
tff(fact_9450_ATP_Olambda__1278,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: product_prod(nat,nat),Uug: nat,Uuh: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),aTP_Lamp_bt(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uug = vEBT_VEBT_low(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_bs(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1278
tff(fact_9451_ATP_Olambda__1279,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: nat,Uuh: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),aTP_Lamp_in(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uug = vEBT_VEBT_low(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_im(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1279
tff(fact_9452_ATP_Olambda__1280,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: nat,Uuh: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),aTP_Lamp_el(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uug = vEBT_VEBT_low(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_ek(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1280
tff(fact_9453_ATP_Olambda__1281,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_db(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uug = vEBT_VEBT_high(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aTP_Lamp_da(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1281
tff(fact_9454_ATP_Olambda__1282,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: list(vEBT_VEBT),Uud: vEBT_VEBT,Uue: product_prod(nat,nat),Uuf: nat,Uug: nat,Uuh: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ar(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uug = vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uu),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_aq(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1282
tff(fact_9455_ATP_Olambda__1283,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi,Uug: vEBT_VEBT,Uuh: option(nat)] : aa(option(nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))),aTP_Lamp_cx(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uuh = vEBT_vebt_maxt(Uug)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_cw(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1283
tff(fact_9456_ATP_Olambda__1284,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: list(vEBT_VEBT),Uud: vEBT_VEBT,Uue: product_prod(nat,nat),Uuf: nat,Uug: nat,Uuh: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_aq(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(aa(nat,$o,ord_less(nat,Uug),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uuc))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ap(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1284
tff(fact_9457_ATP_Olambda__1285,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_il(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(aa(nat,$o,ord_less(nat,Uug),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uud))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ik(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1285
tff(fact_9458_ATP_Olambda__1286,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ej(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(aa(nat,$o,ord_less(nat,Uug),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uud))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ei(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug)) ).

% ATP.lambda_1286
tff(fact_9459_ATP_Olambda__1287,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_pt(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind($o,vEBT_VEBTi,vEBT_VEBT_minNulli(Uuh),aa(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_ps(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1287
tff(fact_9460_ATP_Olambda__1288,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ki(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(option(nat),option(nat),vEBT_vebt_minti(Uuh),aa(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))),aa(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_kh(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1288
tff(fact_9461_ATP_Olambda__1289,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ev(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = heap_Time_bind(option(nat),option(nat),vEBT_vebt_minti(Uuh),aa(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))),aa(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_eu(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1289
tff(fact_9462_ATP_Olambda__1290,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: product_prod(nat,nat),Uug: option(product_prod(nat,nat)),Uuh: nat] : aa(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_qy(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = product_case_prod(list(vEBT_VEBT),vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_qx(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1290
tff(fact_9463_ATP_Olambda__1291,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: product_prod(nat,nat),Uug: option(product_prod(nat,nat)),Uuh: nat] : aa(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_oh(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = product_case_prod(list(vEBT_VEBT),vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_og(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1291
tff(fact_9464_ATP_Olambda__1292,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: option(product_prod(nat,nat)),Uuh: nat] : aa(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_jj(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = product_case_prod(list(vEBT_VEBT),vEBT_VEBT,heap_Time_Heap(option(nat)),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_ji(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1292
tff(fact_9465_ATP_Olambda__1293,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: option(product_prod(nat,nat)),Uuh: nat] : aa(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_is(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = product_case_prod(list(vEBT_VEBT),vEBT_VEBT,heap_Time_Heap(option(nat)),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_ir(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1293
tff(fact_9466_ATP_Olambda__1294,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: option(product_prod(nat,nat)),Uuh: nat] : aa(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_eq(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = product_case_prod(list(vEBT_VEBT),vEBT_VEBT,heap_Time_Heap(option(nat)),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_ep(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1294
tff(fact_9467_ATP_Olambda__1295,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: option(product_prod(nat,nat)),Uuh: nat] : aa(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))),aa(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_eb(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh) = product_case_prod(list(vEBT_VEBT),vEBT_VEBT,heap_Time_Heap(option(nat)),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_ea(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1295
tff(fact_9468_ATP_Olambda__1296,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: nat,Uue: nat,Uuf: vEBT_VEBTi,Uug: vEBT_VEBT,Uuh: option(nat),Uui: product_unit] :
      aa(product_unit,heap_Time_Heap(option(nat)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_cw(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = $ite(
        ( ( Uuh != none(nat) )
        & vEBT_VEBT_less(aa(nat,option(nat),some(nat),Uud),Uuh) ),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),vEBT_VEBT_vebt_succi,Uug),Uuf),Uud),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_a(nat,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uu),Uue)),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),vEBT_VEBT_vebt_succi,Uuc),Uub),Uue),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(nat,fun(option(nat),heap_Time_Heap(option(nat))))),aTP_Lamp_cv(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))),Uu),Uua),Uuc),Uue)) ) ).

% ATP.lambda_1296
tff(fact_9469_ATP_Olambda__1297,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi,Uui: option(nat)] :
      aa(option(nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))),aa(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_kh(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = $ite(
        ( ( Uui != none(nat) )
        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),Uuf),Uui) ),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),product_curry(vEBT_VEBTi,nat,heap_Time_Heap(option(nat)),Uu),Uuh),Uuf),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_a(nat,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uub),Uug)),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),product_curry(vEBT_VEBTi,nat,heap_Time_Heap(option(nat)),Uu),Uud),Uug),aa(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat))))),aTP_Lamp_bd(nat,fun(nat,fun(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat)))))),Uua),Uub),Uuc),Uue)) ) ).

% ATP.lambda_1297
tff(fact_9470_ATP_Olambda__1298,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi,Uui: option(nat)] :
      aa(option(nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))),aa(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_eu(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = $ite(
        ( ( Uui != none(nat) )
        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),Uuf),Uui) ),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),Uu,Uuh),Uuf),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_a(nat,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uub),Uug)),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),Uu,Uud),Uug),aa(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat))),aa(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat))))),aTP_Lamp_bd(nat,fun(nat,fun(array(vEBT_VEBTi),fun(product_prod(nat,nat),fun(option(nat),heap_Time_Heap(option(nat)))))),Uua),Uub),Uuc),Uue)) ) ).

% ATP.lambda_1298
tff(fact_9471_ATP_Olambda__1299,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: product_prod(nat,nat),Uue: option(product_prod(nat,nat)),Uuf: nat,Uug: list(vEBT_VEBT),Uuh: vEBT_VEBT,Uui: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_rv(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = aa(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi),product_case_prod(nat,nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_ru(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uuf),Uug),Uuh)),aa(option(product_prod(nat,nat)),product_prod(nat,nat),the2(product_prod(nat,nat)),Uue)) ).

% ATP.lambda_1299
tff(fact_9472_ATP_Olambda__1300,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_rs(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(nat,vEBT_VEBTi,
        $ite(aa(nat,$o,ord_less(nat,Uu),Uuh),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uuh),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uu)),
        aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_rr(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1300
tff(fact_9473_ATP_Olambda__1301,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: $o,Uui: array(vEBT_VEBTi)] :
      aa(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),aa($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_qg(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh)),Uui) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,
        $ite((Uuh),aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),product_curry(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),product_curry(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(vEBT_VEBTi),Uu)),Uuc),Uub),Uug),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),Uub)),
        aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_no(nat,fun(nat,fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),Uua),Uud),Uue),Uuf),Uui)) ).

% ATP.lambda_1301
tff(fact_9474_ATP_Olambda__1302,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: $o,Uui: array(vEBT_VEBTi)] :
      aa(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),aa($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_np(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh)),Uui) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,
        $ite((Uuh),aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uu,Uuc),Uub),Uug),aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),Uub)),
        aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_no(nat,fun(nat,fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),Uua),Uud),Uue),Uuf),Uui)) ).

% ATP.lambda_1302
tff(fact_9475_ATP_Olambda__1303,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: $o,Uui: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_rg(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh)),Uui) = heap_Time_bind(array(vEBT_VEBTi),vEBT_VEBTi,array_upd(vEBT_VEBTi,Uug,Uui,Uua),aa($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_rf(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh))) ).

% ATP.lambda_1303
tff(fact_9476_ATP_Olambda__1304,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: $o,Uui: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_pe(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh)),Uui) = heap_Time_bind(array(vEBT_VEBTi),vEBT_VEBTi,array_upd(vEBT_VEBTi,Uug,Uui,Uub),aa($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_pd(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),(Uuh))) ).

% ATP.lambda_1304
tff(fact_9477_ATP_Olambda__1305,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: $o,Uui: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_op(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh)),Uui) = heap_Time_bind(array(vEBT_VEBTi),vEBT_VEBTi,array_upd(vEBT_VEBTi,Uug,Uui,Uub),aa($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_oo(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),(Uuh))) ).

% ATP.lambda_1305
tff(fact_9478_ATP_Olambda__1306,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_pj(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(nat,vEBT_VEBTi,vEBT_VEBT_highi(Uug,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_pi(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1306
tff(fact_9479_ATP_Olambda__1307,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_ou(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(nat,vEBT_VEBTi,vEBT_VEBT_highi(Uug,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_ot(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1307
tff(fact_9480_ATP_Olambda__1308,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: product_prod(nat,nat),Uug: nat,Uuh: nat,Uui: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_bq(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uub),Uuh),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_bp(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1308
tff(fact_9481_ATP_Olambda__1309,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: nat,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi,Uui: $o] : aa($o,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_ps(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui)) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),vEBT_vebt_inserti,Uuh),Uuf),aa($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_pr(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uug),(Uui))) ).

% ATP.lambda_1309
tff(fact_9482_ATP_Olambda__1310,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: product_prod(nat,nat),Uug: nat,Uuh: nat,Uui: vEBT_VEBTi] :
      aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_bp(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uud),Uuh),
        heap_Time_bind(option(nat),option(nat),vEBT_vebt_minti(Uui),aa(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))))),aTP_Lamp_bo(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uui),aktnode)) ) ).

% ATP.lambda_1310
tff(fact_9483_ATP_Olambda__1311,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: option(product_prod(nat,nat)),Uug: nat,Uuh: list(vEBT_VEBT),Uui: vEBT_VEBT] :
      aa(vEBT_VEBT,heap_Time_Heap(option(nat)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_dg(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(product_unit,option(nat),
        refine_Imp_assert(( ( Uuf = Uub )
          & ( Uug = Uuc )
          & vEBT_is_Node(Uu) )),
        aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))),aTP_Lamp_df(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))))),Uua),Uub),Uuc),Uud),Uue),Uuh),Uui)) ).

% ATP.lambda_1311
tff(fact_9484_ATP_Olambda__1312,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: option(product_prod(nat,nat)),Uug: nat,Uuh: list(vEBT_VEBT),Uui: vEBT_VEBT] :
      aa(vEBT_VEBT,heap_Time_Heap(option(nat)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_bx(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(product_unit,option(nat),
        refine_Imp_assert(( ( Uuf = Uub )
          & ( Uug = Uuc )
          & vEBT_is_Node(Uu) )),
        aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))),aTP_Lamp_bw(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))))),Uua),Uub),Uuc),Uud),Uue),Uuh),Uui)) ).

% ATP.lambda_1312
tff(fact_9485_ATP_Olambda__1313,axiom,
    ! [Uu: nat,Uua: option(product_prod(nat,nat)),Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: option(product_prod(nat,nat)),Uug: nat,Uuh: list(vEBT_VEBT),Uui: vEBT_VEBT] :
      aa(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi),aa(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_rw(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(product_unit,vEBT_VEBTi,
        refine_Imp_assert(( ( Uua = Uuf )
          & ( Uub = Uug ) )),
        aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_rv(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1313
tff(fact_9486_ATP_Olambda__1314,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_prod(nat,nat),Uuh: nat,Uui: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_je(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uuh = vEBT_VEBT_low(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_jd(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1314
tff(fact_9487_ATP_Olambda__1315,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_prod(nat,nat),Uuh: nat,Uui: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_dw(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uuh = vEBT_VEBT_low(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_dv(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1315
tff(fact_9488_ATP_Olambda__1316,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: product_prod(nat,nat),Uug: nat,Uuh: nat,Uui: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_bs(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uuh = vEBT_VEBT_high(Uu,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_br(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1316
tff(fact_9489_ATP_Olambda__1317,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_im(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uuh = vEBT_VEBT_high(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_il(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1317
tff(fact_9490_ATP_Olambda__1318,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_ek(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uuh = vEBT_VEBT_high(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_ej(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1318
tff(fact_9491_ATP_Olambda__1319,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: product_prod(nat,nat),Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: vEBT_VEBT,Uui: option(nat)] : aa(option(nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_an(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uui = vEBT_vebt_mint(Uuh)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_am(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1319
tff(fact_9492_ATP_Olambda__1320,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: vEBT_VEBT,Uui: option(nat)] : aa(option(nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_ii(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uui = vEBT_vebt_maxt(Uuh)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_ih(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1320
tff(fact_9493_ATP_Olambda__1321,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: vEBT_VEBT,Uui: option(nat)] : aa(option(nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))),aTP_Lamp_eg(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uui = vEBT_vebt_maxt(Uuh)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_ef(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1321
tff(fact_9494_ATP_Olambda__1322,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: product_prod(nat,nat),Uug: nat,Uuh: nat,Uui: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_br(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(aa(nat,$o,ord_less(nat,Uuh),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uud))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aTP_Lamp_bq(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1322
tff(fact_9495_ATP_Olambda__1323,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: product_prod(nat,nat),Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: nat,Uui: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_rt(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(nat,vEBT_VEBTi,aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uud)),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_rs(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1323
tff(fact_9496_ATP_Olambda__1324,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: product_prod(nat,nat),Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: nat,Uui: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_ru(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(nat,vEBT_VEBTi,aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uud)),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_rt(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_1324
tff(fact_9497_ATP_Olambda__1325,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_pg(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind($o,vEBT_VEBTi,vEBT_VEBT_minNulli(Uui),aa(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_pf(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1325
tff(fact_9498_ATP_Olambda__1326,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_or(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind($o,vEBT_VEBTi,vEBT_VEBT_minNulli(Uui),aa(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_oq(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1326
tff(fact_9499_ATP_Olambda__1327,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_pv(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui) = heap_Time_bind(nat,vEBT_VEBTi,array_len(vEBT_VEBTi,Uub),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_pu(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1327
tff(fact_9500_ATP_Olambda__1328,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_pu(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $ite(
        ( aa(nat,$o,ord_less(nat,Uui),Uuj)
        & ~ ( ( Uu = Uud )
            | ( Uu = Uue ) ) ),
        heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uub),Uui),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_pt(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uui)),
        aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uud),Uue)),Uua,Uub,Uuc)) ) ).

% ATP.lambda_1328
tff(fact_9501_ATP_Olambda__1329,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: vEBT_VEBT,Uui: option(nat),Uuj: product_unit] :
      aa(product_unit,heap_Time_Heap(option(nat)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_ih(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $ite(
        ( ( Uui != none(nat) )
        & vEBT_VEBT_less(aa(nat,option(nat),some(nat),Uue),Uui) ),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),product_curry(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),product_curry(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(option(nat)),Uu)),Uuh),Uug),Uue),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_a(nat,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uua),Uuf)),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),product_curry(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),product_curry(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(option(nat)),Uu)),Uud),Uuc),Uuf),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(nat,fun(option(nat),heap_Time_Heap(option(nat))))),aTP_Lamp_cv(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))),Uua),Uub),Uud),Uuf)) ) ).

% ATP.lambda_1329
tff(fact_9502_ATP_Olambda__1330,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: vEBT_VEBT,Uui: option(nat),Uuj: product_unit] :
      aa(product_unit,heap_Time_Heap(option(nat)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_ef(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $ite(
        ( ( Uui != none(nat) )
        & vEBT_VEBT_less(aa(nat,option(nat),some(nat),Uue),Uui) ),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uu,Uuh),Uug),Uue),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_a(nat,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uua),Uuf)),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uu,Uud),Uuc),Uuf),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),aa(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(nat,fun(option(nat),heap_Time_Heap(option(nat))))),aTP_Lamp_cv(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))),Uua),Uub),Uud),Uuf)) ) ).

% ATP.lambda_1330
tff(fact_9503_ATP_Olambda__1331,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: vEBT_VEBT,Uud: product_prod(nat,nat),Uue: nat,Uuf: nat,Uug: vEBT_VEBTi,Uuh: vEBT_VEBT,Uui: option(nat),Uuj: product_unit] :
      aa(product_unit,heap_Time_Heap(option(nat)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_am(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $ite(
        ( ( Uui != none(nat) )
        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),Uue),Uui) ),
        heap_Time_bind(option(nat),option(nat),f(Uuh,Uug,Uue),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_a(nat,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uu),Uuf)),
        heap_Time_bind(option(nat),option(nat),f(Uuc,Uub,Uuf),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))),aTP_Lamp_al(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat))))))),Uu),Uua),Uuc),Uud),Uuf)) ) ).

% ATP.lambda_1331
tff(fact_9504_ATP_Olambda__1332,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: option(product_prod(nat,nat)),Uug: nat,Uuh: list(vEBT_VEBT),Uui: vEBT_VEBT,Uuj: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_qw(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = aa(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi),product_case_prod(nat,nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_qv(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uug),Uuh),Uui)),aa(option(product_prod(nat,nat)),product_prod(nat,nat),the2(product_prod(nat,nat)),Uuf)) ).

% ATP.lambda_1332
tff(fact_9505_ATP_Olambda__1333,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: option(product_prod(nat,nat)),Uug: nat,Uuh: list(vEBT_VEBT),Uui: vEBT_VEBT,Uuj: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_of(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = aa(product_prod(nat,nat),heap_Time_Heap(vEBT_VEBTi),product_case_prod(nat,nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_oe(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uug),Uuh),Uui)),aa(option(product_prod(nat,nat)),product_prod(nat,nat),the2(product_prod(nat,nat)),Uuf)) ).

% ATP.lambda_1333
tff(fact_9506_ATP_Olambda__1334,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_qt(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(nat,vEBT_VEBTi,
        $ite(aa(nat,$o,ord_less(nat,Uua),Uui),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uui),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uua)),
        aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_qs(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1334
tff(fact_9507_ATP_Olambda__1335,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_oc(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(nat,vEBT_VEBTi,
        $ite(aa(nat,$o,ord_less(nat,Uua),Uui),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uui),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uua)),
        aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_ob(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1335
tff(fact_9508_ATP_Olambda__1336,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_rr(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $let(
        xn2: nat,
        xn2:= 
          $ite(aa(nat,$o,ord_less(nat,Uu),Uug),Uug,Uu),
        heap_Time_bind(nat,vEBT_VEBTi,
          $ite(aa(nat,$o,ord_less(nat,Uu),Uuh),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uu),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uuh)),
          aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_rq(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uuh),Uui),Uuj),xn2)) ) ).

% ATP.lambda_1336
tff(fact_9509_ATP_Olambda__1337,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: $o] : aa($o,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_oq(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),(Uuj)) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),product_curry(vEBT_VEBTi,nat,heap_Time_Heap(vEBT_VEBTi),Uu),Uui),Uug),aa($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_op(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uuh),(Uuj))) ).

% ATP.lambda_1337
tff(fact_9510_ATP_Olambda__1338,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: $o,Uuj: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_qh(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui)),Uuj) = heap_Time_bind(array(vEBT_VEBTi),vEBT_VEBTi,array_upd(vEBT_VEBTi,Uuh,Uuj,Uub),aa($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_qg(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui))) ).

% ATP.lambda_1338
tff(fact_9511_ATP_Olambda__1339,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: $o,Uuj: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_nq(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui)),Uuj) = heap_Time_bind(array(vEBT_VEBTi),vEBT_VEBTi,array_upd(vEBT_VEBTi,Uuh,Uuj,Uub),aa($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_np(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui))) ).

% ATP.lambda_1339
tff(fact_9512_ATP_Olambda__1340,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: $o] : aa($o,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_pf(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),(Uuj)) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),Uu,Uui),Uug),aa($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_pe(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uuh),(Uuj))) ).

% ATP.lambda_1340
tff(fact_9513_ATP_Olambda__1341,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: list(vEBT_VEBT),Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_rk(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uua),Uui),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_rj(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1341
tff(fact_9514_ATP_Olambda__1342,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_prod(nat,nat),Uuh: nat,Uui: nat,Uuj: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_jb(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uuc),Uui),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_ja(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1342
tff(fact_9515_ATP_Olambda__1343,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_prod(nat,nat),Uuh: nat,Uui: nat,Uuj: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_dt(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uuc),Uui),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_ds(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1343
tff(fact_9516_ATP_Olambda__1344,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_prod(nat,nat),Uuh: nat,Uui: nat,Uuj: vEBT_VEBTi] :
      aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_ja(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uue),Uui),
        heap_Time_bind(option(nat),option(nat),vEBT_vebt_minti(Uuj),aa(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))))),aTP_Lamp_iz(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj),aktnode)) ) ).

% ATP.lambda_1344
tff(fact_9517_ATP_Olambda__1345,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_prod(nat,nat),Uuh: nat,Uui: nat,Uuj: vEBT_VEBTi] :
      aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_ds(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uue),Uui),
        heap_Time_bind(option(nat),option(nat),vEBT_vebt_minti(Uuj),aa(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))))),aTP_Lamp_dr(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj),aktnode)) ) ).

% ATP.lambda_1345
tff(fact_9518_ATP_Olambda__1346,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: option(product_prod(nat,nat)),Uuh: nat,Uui: list(vEBT_VEBT),Uuj: vEBT_VEBT] :
      aa(vEBT_VEBT,heap_Time_Heap(option(nat)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_ji(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(product_unit,option(nat),
        refine_Imp_assert(( ( Uug = Uuc )
          & ( Uuh = Uud )
          & vEBT_is_Node(Uua) )),
        aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_jh(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uub),Uuc),Uud),Uue),Uuf),Uui),Uuj)) ).

% ATP.lambda_1346
tff(fact_9519_ATP_Olambda__1347,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: option(product_prod(nat,nat)),Uuh: nat,Uui: list(vEBT_VEBT),Uuj: vEBT_VEBT] :
      aa(vEBT_VEBT,heap_Time_Heap(option(nat)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_ir(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(product_unit,option(nat),
        refine_Imp_assert(( ( Uug = Uuc )
          & ( Uuh = Uud )
          & vEBT_is_Node(Uua) )),
        aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_iq(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uub),Uuc),Uud),Uue),Uuf),Uui),Uuj)) ).

% ATP.lambda_1347
tff(fact_9520_ATP_Olambda__1348,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: option(product_prod(nat,nat)),Uuh: nat,Uui: list(vEBT_VEBT),Uuj: vEBT_VEBT] :
      aa(vEBT_VEBT,heap_Time_Heap(option(nat)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_ep(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(product_unit,option(nat),
        refine_Imp_assert(( ( Uug = Uuc )
          & ( Uuh = Uud )
          & vEBT_is_Node(Uua) )),
        aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_eo(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uub),Uuc),Uud),Uue),Uuf),Uui),Uuj)) ).

% ATP.lambda_1348
tff(fact_9521_ATP_Olambda__1349,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: vEBT_VEBT,Uub: nat,Uuc: option(product_prod(nat,nat)),Uud: nat,Uue: array(vEBT_VEBTi),Uuf: vEBT_VEBTi,Uug: option(product_prod(nat,nat)),Uuh: nat,Uui: list(vEBT_VEBT),Uuj: vEBT_VEBT] :
      aa(vEBT_VEBT,heap_Time_Heap(option(nat)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))),aa(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))),aa(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_ea(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(product_unit,option(nat),
        refine_Imp_assert(( ( Uug = Uuc )
          & ( Uuh = Uud )
          & vEBT_is_Node(Uua) )),
        aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat))))))))),aTP_Lamp_dz(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(option(nat)))))))))),Uu),Uub),Uuc),Uud),Uue),Uuf),Uui),Uuj)) ).

% ATP.lambda_1349
tff(fact_9522_ATP_Olambda__1350,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: product_prod(nat,nat),Uug: option(product_prod(nat,nat)),Uuh: nat,Uui: list(vEBT_VEBT),Uuj: vEBT_VEBT] :
      aa(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi),aa(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_qx(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(product_unit,vEBT_VEBTi,
        refine_Imp_assert(( ( Uub = Uug )
          & ( Uuc = Uuh ) )),
        aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_qw(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1350
tff(fact_9523_ATP_Olambda__1351,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: option(product_prod(nat,nat)),Uuc: nat,Uud: array(vEBT_VEBTi),Uue: vEBT_VEBTi,Uuf: product_prod(nat,nat),Uug: option(product_prod(nat,nat)),Uuh: nat,Uui: list(vEBT_VEBT),Uuj: vEBT_VEBT] :
      aa(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi),aa(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))))),aa(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_og(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(product_unit,vEBT_VEBTi,
        refine_Imp_assert(( ( Uub = Uug )
          & ( Uuc = Uuh ) )),
        aa(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_of(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1351
tff(fact_9524_ATP_Olambda__1352,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_prod(nat,nat),Uuh: nat,Uui: nat,Uuj: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_jd(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uui = vEBT_VEBT_high(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_jc(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1352
tff(fact_9525_ATP_Olambda__1353,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_prod(nat,nat),Uuh: nat,Uui: nat,Uuj: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_dv(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uui = vEBT_VEBT_high(Uua,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_du(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1353
tff(fact_9526_ATP_Olambda__1354,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: product_prod(nat,nat),Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi,Uui: vEBT_VEBT,Uuj: option(nat)] : aa(option(nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))))),aTP_Lamp_bo(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uuj = vEBT_vebt_mint(Uui)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))))),aTP_Lamp_bn(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1354
tff(fact_9527_ATP_Olambda__1355,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: list(vEBT_VEBT),Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_rl(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert(aa(nat,$o,ord_less(nat,Uui),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uuc))),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_rk(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1355
tff(fact_9528_ATP_Olambda__1356,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_prod(nat,nat),Uuh: nat,Uui: nat,Uuj: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_jc(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(aa(nat,$o,ord_less(nat,Uui),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uue))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_jb(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1356
tff(fact_9529_ATP_Olambda__1357,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: product_prod(nat,nat),Uuh: nat,Uui: nat,Uuj: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_du(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(aa(nat,$o,ord_less(nat,Uui),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uue))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))))))))),aTP_Lamp_dt(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1357
tff(fact_9530_ATP_Olambda__1358,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: nat,Uug: list(vEBT_VEBT),Uuh: vEBT_VEBT,Uui: nat,Uuj: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_qu(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(nat,vEBT_VEBTi,aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uue)),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_qt(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1358
tff(fact_9531_ATP_Olambda__1359,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: nat,Uug: list(vEBT_VEBT),Uuh: vEBT_VEBT,Uui: nat,Uuj: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_od(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(nat,vEBT_VEBTi,aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),aa(product_prod(nat,nat),nat,product_snd(nat,nat),Uue)),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_oc(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1359
tff(fact_9532_ATP_Olambda__1360,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: nat,Uug: list(vEBT_VEBT),Uuh: vEBT_VEBT,Uui: nat,Uuj: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_qv(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(nat,vEBT_VEBTi,aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uue)),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_qu(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1360
tff(fact_9533_ATP_Olambda__1361,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: product_prod(nat,nat),Uuf: nat,Uug: list(vEBT_VEBT),Uuh: vEBT_VEBT,Uui: nat,Uuj: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_oe(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(nat,vEBT_VEBTi,aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uue)),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_od(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(product_prod(nat,nat),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui)) ).

% ATP.lambda_1361
tff(fact_9534_ATP_Olambda__1362,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: list(vEBT_VEBT),Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_rj(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind($o,vEBT_VEBTi,vEBT_VEBT_minNulli(Uuj),aa(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_ri(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1362
tff(fact_9535_ATP_Olambda__1363,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_pi(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(nat,vEBT_VEBTi,array_len(vEBT_VEBTi,Uuc),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_ph(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1363
tff(fact_9536_ATP_Olambda__1364,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_ot(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj) = heap_Time_bind(nat,vEBT_VEBTi,array_len(vEBT_VEBTi,Uuc),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_os(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1364
tff(fact_9537_ATP_Olambda__1365,axiom,
    ! [Uu: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_ph(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = $ite(
        ( aa(nat,$o,ord_less(nat,Uuj),Uuk)
        & ~ ( ( Uua = Uue )
            | ( Uua = Uuf ) ) ),
        heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uuc),Uuj),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_pg(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj)),
        aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uue),Uuf)),Uub,Uuc,Uud)) ) ).

% ATP.lambda_1365
tff(fact_9538_ATP_Olambda__1366,axiom,
    ! [Uu: fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_os(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = $ite(
        ( aa(nat,$o,ord_less(nat,Uuj),Uuk)
        & ~ ( ( Uua = Uue )
            | ( Uua = Uuf ) ) ),
        heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uuc),Uuj),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_or(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uub),Uuc),Uud),Uuf),Uug),Uuh),Uui),Uuj)),
        aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uue),Uuf)),Uub,Uuc,Uud)) ) ).

% ATP.lambda_1366
tff(fact_9539_ATP_Olambda__1367,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: product_prod(nat,nat),Uuf: nat,Uug: nat,Uuh: vEBT_VEBTi,Uui: vEBT_VEBT,Uuj: option(nat),Uuk: product_unit] :
      aa(product_unit,heap_Time_Heap(option(nat)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))))),aTP_Lamp_bn(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = $ite(
        ( ( Uuj != none(nat) )
        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),Uuf),Uuj) ),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),vEBT_VEBT_vebt_predi,Uui),Uuh),Uuf),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_a(nat,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uua),Uug)),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),vEBT_VEBT_vebt_predi,Uud),Uuc),Uug),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat))))))),aTP_Lamp_bm(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))))),Uu),Uua),Uub),Uud),Uue),Uug)) ) ).

% ATP.lambda_1367
tff(fact_9540_ATP_Olambda__1368,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_qs(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = $let(
        xn2: nat,
        xn2:= 
          $ite(aa(nat,$o,ord_less(nat,Uua),Uuh),Uuh,Uua),
        heap_Time_bind(nat,vEBT_VEBTi,
          $ite(aa(nat,$o,ord_less(nat,Uua),Uui),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uua),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uui)),
          aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aTP_Lamp_qr(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uui),Uuj),Uuk),xn2)) ) ).

% ATP.lambda_1368
tff(fact_9541_ATP_Olambda__1369,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_ob(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = $let(
        xn2: nat,
        xn2:= 
          $ite(aa(nat,$o,ord_less(nat,Uua),Uuh),Uuh,Uua),
        heap_Time_bind(nat,vEBT_VEBTi,
          $ite(aa(nat,$o,ord_less(nat,Uua),Uui),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uua),aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),Uui)),
          aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aTP_Lamp_oa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uui),Uuj),Uuk),xn2)) ) ).

% ATP.lambda_1369
tff(fact_9542_ATP_Olambda__1370,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_rq(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_Time_bind(nat,vEBT_VEBTi,vEBT_VEBT_lowi(Uui,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aTP_Lamp_rp(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk)) ).

% ATP.lambda_1370
tff(fact_9543_ATP_Olambda__1371,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_ql(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uub),Uuj),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_qk(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1371
tff(fact_9544_ATP_Olambda__1372,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_nu(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uub),Uuj),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_nt(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1372
tff(fact_9545_ATP_Olambda__1373,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: list(vEBT_VEBT),Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: vEBT_VEBTi,Uuk: $o] : aa($o,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_ri(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),(Uuk)) = heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert((Uuk) = vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uuc),Uui))),aa($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),aTP_Lamp_rh(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),(Uuk))) ).

% ATP.lambda_1373
tff(fact_9546_ATP_Olambda__1374,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: vEBT_VEBT,Uuf: product_prod(nat,nat),Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: vEBT_VEBT,Uuk: option(nat)] : aa(option(nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))))),aTP_Lamp_iz(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uuk = vEBT_vebt_mint(Uuj)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))))),aTP_Lamp_iy(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk)) ).

% ATP.lambda_1374
tff(fact_9547_ATP_Olambda__1375,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: vEBT_VEBT,Uuf: product_prod(nat,nat),Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: vEBT_VEBT,Uuk: option(nat)] : aa(option(nat),heap_Time_Heap(option(nat)),aa(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat)))))))))))),aTP_Lamp_dr(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),heap_Time_Heap(option(nat))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uuk = vEBT_vebt_mint(Uuj)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))))),aTP_Lamp_dq(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk)) ).

% ATP.lambda_1375
tff(fact_9548_ATP_Olambda__1376,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_qm(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert(aa(nat,$o,ord_less(nat,Uuj),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uud))),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_ql(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1376
tff(fact_9549_ATP_Olambda__1377,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_nv(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert(aa(nat,$o,ord_less(nat,Uuj),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),Uud))),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_nu(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1377
tff(fact_9550_ATP_Olambda__1378,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_qk(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_Time_bind($o,vEBT_VEBTi,vEBT_VEBT_minNulli(Uuk),aa(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))))),aTP_Lamp_qj(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk)) ).

% ATP.lambda_1378
tff(fact_9551_ATP_Olambda__1379,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_nt(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk) = heap_Time_bind($o,vEBT_VEBTi,vEBT_VEBT_minNulli(Uuk),aa(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))))),aTP_Lamp_ns(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk)) ).

% ATP.lambda_1379
tff(fact_9552_ATP_Olambda__1380,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: vEBT_VEBT,Uuf: product_prod(nat,nat),Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: vEBT_VEBT,Uuk: option(nat),Uul: product_unit] :
      aa(product_unit,heap_Time_Heap(option(nat)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))))),aTP_Lamp_iy(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul) = $ite(
        ( ( Uuk != none(nat) )
        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),Uug),Uuk) ),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),product_curry(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),product_curry(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(option(nat)),Uu)),Uuj),Uui),Uug),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_a(nat,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uub),Uuh)),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),product_curry(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),product_curry(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(option(nat)),Uu)),Uue),Uud),Uuh),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat))))))),aTP_Lamp_bm(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))))),Uua),Uub),Uuc),Uue),Uuf),Uuh)) ) ).

% ATP.lambda_1380
tff(fact_9553_ATP_Olambda__1381,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: vEBT_VEBT,Uuf: product_prod(nat,nat),Uug: nat,Uuh: nat,Uui: vEBT_VEBTi,Uuj: vEBT_VEBT,Uuk: option(nat),Uul: product_unit] :
      aa(product_unit,heap_Time_Heap(option(nat)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),aa(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))),aa(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))))))))))),aTP_Lamp_dq(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul) = $ite(
        ( ( Uuk != none(nat) )
        & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),Uug),Uuk) ),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uu,Uuj),Uui),Uug),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_a(nat,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uub),Uuh)),
        heap_Time_bind(option(nat),option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uu,Uue),Uud),Uuh),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aa(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),aa(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat))))))),aTP_Lamp_bm(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))))),Uua),Uub),Uuc),Uue),Uuf),Uuh)) ) ).

% ATP.lambda_1381
tff(fact_9554_ATP_Olambda__1382,axiom,
    ! [Uu: nat,Uua: array(vEBT_VEBTi),Uub: vEBT_VEBTi,Uuc: list(vEBT_VEBT),Uud: vEBT_VEBT,Uue: nat,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: vEBT_VEBTi,Uuk: $o,Uul: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),aTP_Lamp_rh(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),(Uuk)),Uul) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),vEBT_V3964819847710782039nserti,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uuc),Uui)),Uuj),Uuh),aa($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_rg(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uud),Uue),Uuf),Uug),Uui),(Uuk))) ).

% ATP.lambda_1382
tff(fact_9555_ATP_Olambda__1383,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aTP_Lamp_qr(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul) = heap_Time_bind(nat,vEBT_VEBTi,vEBT_VEBT_lowi(Uuj,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),aTP_Lamp_qq(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul)) ).

% ATP.lambda_1383
tff(fact_9556_ATP_Olambda__1384,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aTP_Lamp_oa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul) = heap_Time_bind(nat,vEBT_VEBTi,vEBT_VEBT_lowi(Uuj,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),aTP_Lamp_nz(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul)) ).

% ATP.lambda_1384
tff(fact_9557_ATP_Olambda__1385,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aTP_Lamp_rp(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul) = heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert(Uul = vEBT_VEBT_low(Uuj,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uud),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))),aTP_Lamp_ro(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul)) ).

% ATP.lambda_1385
tff(fact_9558_ATP_Olambda__1386,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi,Uul: $o] : aa($o,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))))),aTP_Lamp_qj(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul)) = heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert((Uul) = vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uud),Uuj))),aa($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))),aTP_Lamp_qi(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul))) ).

% ATP.lambda_1386
tff(fact_9559_ATP_Olambda__1387,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi,Uul: $o] : aa($o,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))))),aTP_Lamp_ns(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul)) = heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert((Uul) = vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uud),Uuj))),aa($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))),aTP_Lamp_nr(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul))) ).

% ATP.lambda_1387
tff(fact_9560_ATP_Olambda__1388,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi,Uul: $o,Uum: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))),aTP_Lamp_qi(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul)),Uum) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),product_curry(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),product_curry(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(vEBT_VEBTi),Uu)),aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uud),Uuj)),Uuk),Uui),aa($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_qh(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uuj),(Uul))) ).

% ATP.lambda_1388
tff(fact_9561_ATP_Olambda__1389,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: list(vEBT_VEBT),Uue: vEBT_VEBT,Uuf: nat,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: vEBT_VEBTi,Uul: $o,Uum: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))),aTP_Lamp_nr(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),(Uul)),Uum) = heap_Time_bind(vEBT_VEBTi,vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uu,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uud),Uuj)),Uuk),Uui),aa($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_nq(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uuj),(Uul))) ).

% ATP.lambda_1389
tff(fact_9562_ATP_Olambda__1390,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))),aTP_Lamp_ro(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum) = heap_Time_bind(nat,vEBT_VEBTi,vEBT_VEBT_highi(Uui,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uua),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),aTP_Lamp_rn(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul)) ).

% ATP.lambda_1390
tff(fact_9563_ATP_Olambda__1391,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),aTP_Lamp_qq(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum) = heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert(Uum = vEBT_VEBT_low(Uuk,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uue),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))))),aTP_Lamp_qp(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum)) ).

% ATP.lambda_1391
tff(fact_9564_ATP_Olambda__1392,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),aTP_Lamp_nz(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum) = heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert(Uum = vEBT_VEBT_low(Uuk,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uue),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))))),aTP_Lamp_ny(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum)) ).

% ATP.lambda_1392
tff(fact_9565_ATP_Olambda__1393,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),aTP_Lamp_rn(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum) = heap_Time_bind(nat,vEBT_VEBTi,array_len(vEBT_VEBTi,Uub),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))),aTP_Lamp_rm(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum)) ).

% ATP.lambda_1393
tff(fact_9566_ATP_Olambda__1394,axiom,
    ! [Uu: nat,Uua: nat,Uub: array(vEBT_VEBTi),Uuc: vEBT_VEBTi,Uud: nat,Uue: list(vEBT_VEBT),Uuf: vEBT_VEBT,Uug: nat,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))),aTP_Lamp_rm(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun) = $ite(
        ( aa(nat,$o,ord_less(nat,Uum),Uun)
        & ~ ( ( Uu = Uug )
            | ( Uu = Uuh ) ) ),
        heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert(Uum = vEBT_VEBT_high(Uuj,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uud),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_rl(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),Uua),Uub),Uuc),Uue),Uuf),Uuh),Uui),Uuk),Uul),Uum)),
        aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uug),Uuh)),Uua,Uub,Uuc)) ) ).

% ATP.lambda_1394
tff(fact_9567_ATP_Olambda__1395,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))))),aTP_Lamp_qp(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun) = heap_Time_bind(nat,vEBT_VEBTi,vEBT_VEBT_highi(Uuj,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))),aTP_Lamp_qo(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum)) ).

% ATP.lambda_1395
tff(fact_9568_ATP_Olambda__1396,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: product_unit] : aa(product_unit,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))))),aTP_Lamp_ny(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun) = heap_Time_bind(nat,vEBT_VEBTi,vEBT_VEBT_highi(Uuj,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uub),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))),aTP_Lamp_nx(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum)) ).

% ATP.lambda_1396
tff(fact_9569_ATP_Olambda__1397,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))),aTP_Lamp_qo(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun) = heap_Time_bind(nat,vEBT_VEBTi,array_len(vEBT_VEBTi,Uuc),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))))),aTP_Lamp_qn(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun)) ).

% ATP.lambda_1397
tff(fact_9570_ATP_Olambda__1398,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))),aTP_Lamp_nx(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun) = heap_Time_bind(nat,vEBT_VEBTi,array_len(vEBT_VEBTi,Uuc),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))))),aTP_Lamp_nw(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun)) ).

% ATP.lambda_1398
tff(fact_9571_ATP_Olambda__1399,axiom,
    ! [Uu: fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: nat,Uuo: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))))),aTP_Lamp_qn(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun),Uuo) = $ite(
        ( aa(nat,$o,ord_less(nat,Uun),Uuo)
        & ~ ( ( Uua = Uuh )
            | ( Uua = Uui ) ) ),
        heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert(Uun = vEBT_VEBT_high(Uuk,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uue),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_qm(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uub),Uuc),Uud),Uuf),Uug),Uui),Uuj),Uul),Uum),Uun)),
        aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uuh),Uui)),Uub,Uuc,Uud)) ) ).

% ATP.lambda_1399
tff(fact_9572_ATP_Olambda__1400,axiom,
    ! [Uu: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uua: nat,Uub: nat,Uuc: array(vEBT_VEBTi),Uud: vEBT_VEBTi,Uue: nat,Uuf: list(vEBT_VEBT),Uug: vEBT_VEBT,Uuh: nat,Uui: nat,Uuj: nat,Uuk: nat,Uul: nat,Uum: nat,Uun: nat,Uuo: nat] :
      aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))),aa(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))),aa(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi))))))))))))))),aTP_Lamp_nw(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj),Uuk),Uul),Uum),Uun),Uuo) = $ite(
        ( aa(nat,$o,ord_less(nat,Uun),Uuo)
        & ~ ( ( Uua = Uuh )
            | ( Uua = Uui ) ) ),
        heap_Time_bind(product_unit,vEBT_VEBTi,refine_Imp_assert(Uun = vEBT_VEBT_high(Uuk,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Uue),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))),aa(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),aa(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))),aa(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))),aa(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))),aa(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))),aa(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))),aa(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))),aa(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_nv(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uub),Uuc),Uud),Uuf),Uug),Uui),Uuj),Uul),Uum),Uun)),
        aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Uuh),Uui)),Uub,Uuc,Uud)) ) ).

% ATP.lambda_1400
tff(fact_9573_ATP_Olambda__1401,axiom,
    ! [Uu: option(product_prod(nat,nat)),Uua: nat,Uub: list(vEBT_VEBT),Uuc: vEBT_VEBT] :
      ( aa(vEBT_VEBT,$o,aa(list(vEBT_VEBT),fun(vEBT_VEBT,$o),aa(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,$o)),aa(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,$o))),aTP_Lamp_fp(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,$o)))),Uu),Uua),Uub),Uuc)
    <=> $true ) ).

% ATP.lambda_1401
tff(fact_9574_ATP_Olambda__1402,axiom,
    ! [B: $tType,A: $tType,Uu: $o,Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_afc($o,fun(A,fun(B,$o)),(Uu)),Uua),Uub)
    <=> (Uu) ) ).

% ATP.lambda_1402
tff(fact_9575_ATP_Olambda__1403,axiom,
    ! [A: $tType,Uu: A,Uua: heap_ext(product_unit),Uub: nat] : aa(nat,A,aa(heap_ext(product_unit),fun(nat,A),aTP_Lamp_aki(A,fun(heap_ext(product_unit),fun(nat,A)),Uu),Uua),Uub) = Uu ).

% ATP.lambda_1403
tff(fact_9576_ATP_Olambda__1404,axiom,
    ! [A: $tType,B: $tType,Uu: heap_Time_Heap(B),Uua: A] : aa(A,heap_Time_Heap(B),aTP_Lamp_js(heap_Time_Heap(B),fun(A,heap_Time_Heap(B)),Uu),Uua) = Uu ).

% ATP.lambda_1404
tff(fact_9577_ATP_Olambda__1405,axiom,
    ! [A: $tType,Uu: heap_Time_Heap(A),Uua: product_unit] : aa(product_unit,heap_Time_Heap(A),aTP_Lamp_gb(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),Uu),Uua) = Uu ).

% ATP.lambda_1405
tff(fact_9578_ATP_Olambda__1406,axiom,
    ! [Uu: heap_ext(product_unit),Uua: nat] : aa(nat,heap_ext(product_unit),aa(heap_ext(product_unit),fun(nat,heap_ext(product_unit)),aTP_Lamp_akn(heap_ext(product_unit),fun(nat,heap_ext(product_unit))),Uu),Uua) = Uu ).

% ATP.lambda_1406
tff(fact_9579_ATP_Olambda__1407,axiom,
    ! [Uu: assn,Uua: product_unit] : aa(product_unit,assn,aTP_Lamp_aeq(assn,fun(product_unit,assn),Uu),Uua) = Uu ).

% ATP.lambda_1407
tff(fact_9580_ATP_Olambda__1408,axiom,
    ! [A: $tType,Uu: assn,Uua: A] : aa(A,assn,aTP_Lamp_aez(assn,fun(A,assn),Uu),Uua) = Uu ).

% ATP.lambda_1408
tff(fact_9581_ATP_Olambda__1409,axiom,
    ! [A: $tType,B: $tType,Uu: option(B),Uua: A] : aa(A,option(B),aTP_Lamp_aij(option(B),fun(A,option(B)),Uu),Uua) = Uu ).

% ATP.lambda_1409
tff(fact_9582_ATP_Olambda__1410,axiom,
    ! [Uu: $o,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_anl($o,fun(nat,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_1410
tff(fact_9583_ATP_Olambda__1411,axiom,
    ! [A: $tType,Uu: $o,Uua: A] :
      ( aa(A,$o,aTP_Lamp_afb($o,fun(A,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_1411
tff(fact_9584_ATP_Olambda__1412,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(B,C),Uua: A] : aa(A,fun(B,C),aTP_Lamp_fl(fun(B,C),fun(A,fun(B,C)),Uu),Uua) = Uu ).

% ATP.lambda_1412
tff(fact_9585_ATP_Olambda__1413,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_jt(C,fun(product_prod(A,B),C),Uu),Uua) = Uu ).

% ATP.lambda_1413
tff(fact_9586_ATP_Olambda__1414,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_arw(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1414
tff(fact_9587_ATP_Olambda__1415,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topolo4958980785337419405_space(B)
        & topological_t2_space(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_awr(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1415
tff(fact_9588_ATP_Olambda__1416,axiom,
    ! [A: $tType,B: $tType] :
      ( topolo4958980785337419405_space(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_atl(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1416
tff(fact_9589_ATP_Olambda__1417,axiom,
    ! [A: $tType,B: $tType] :
      ( ( topological_t2_space(B)
        & topolo8386298272705272623_space(A) )
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_auj(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1417
tff(fact_9590_ATP_Olambda__1418,axiom,
    ! [A: $tType,B: $tType] :
      ( topological_t2_space(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_axo(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1418
tff(fact_9591_ATP_Olambda__1419,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ahh(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1419
tff(fact_9592_ATP_Olambda__1420,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ahp(B,fun(A,B),Uu),Uua) = Uu ).

% ATP.lambda_1420
tff(fact_9593_ATP_Olambda__1421,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_mr(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1421
tff(fact_9594_ATP_Olambda__1422,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_apq(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1422
tff(fact_9595_ATP_Olambda__1423,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ayp(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1423
tff(fact_9596_ATP_Olambda__1424,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_aul(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1424
tff(fact_9597_ATP_Olambda__1425,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_ama(A,fun(list(A),A)),Uu),Uua) = Uu ) ).

% ATP.lambda_1425
tff(fact_9598_ATP_Olambda__1426,axiom,
    ! [B: $tType,A: $tType] :
      ( ( zero(A)
        & topological_t2_space(A)
        & topolo8386298272705272623_space(B) )
     => ! [Uu: A,Uua: B] : aa(B,A,aTP_Lamp_atw(A,fun(B,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1426
tff(fact_9599_ATP_Olambda__1427,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ajn(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1427
tff(fact_9600_ATP_Olambda__1428,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_amg(A,fun(list(A),A)),Uu),Uua) = Uu ).

% ATP.lambda_1428
tff(fact_9601_ATP_Olambda__1429,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aji(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_1429
tff(fact_9602_ATP_Olambda__1430,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_cp(A,fun(B,A)),Uu),Uua) = Uu ).

% ATP.lambda_1430
tff(fact_9603_ATP_Olambda__1431,axiom,
    ! [Uu: heap_ext(product_unit),Uua: nat] : aa(nat,nat,aa(heap_ext(product_unit),fun(nat,nat),aTP_Lamp_akd(heap_ext(product_unit),fun(nat,nat)),Uu),Uua) = Uua ).

% ATP.lambda_1431
tff(fact_9604_ATP_Olambda__1432,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,A,aa(B,fun(A,A),aTP_Lamp_cq(B,fun(A,A)),Uu),Uua) = Uua ).

% ATP.lambda_1432
tff(fact_9605_ATP_Olambda__1433,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_amd(A,fun(list(A),list(A))),Uu),Uua) = Uua ).

% ATP.lambda_1433
tff(fact_9606_ATP_Olambda__1434,axiom,
    ! [Uu: $o,Uua: $o] : aa($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))),aa($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)))),aTP_Lamp_ba($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),(Uu)),(Uua)) = undefined(product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)))) ).

% ATP.lambda_1434
tff(fact_9607_ATP_Olambda__1435,axiom,
    ! [Uu: $o,Uua: $o] :
      ( aa($o,$o,aa($o,fun($o,$o),aTP_Lamp_fq($o,fun($o,$o)),(Uu)),(Uua))
    <=> $false ) ).

% ATP.lambda_1435
tff(fact_9608_ATP_Olambda__1436,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_amc(A,fun(list(A),$o)),Uu),Uua)
    <=> $false ) ).

% ATP.lambda_1436
tff(fact_9609_ATP_Olambda__1437,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_amb(A,fun(list(A),$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1437
tff(fact_9610_ATP_Olambda__1438,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_fh(A,fun(B,$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1438
tff(fact_9611_ATP_Olambda__1439,axiom,
    ! [Uu: complex] : aa(complex,complex,aTP_Lamp_wc(complex,complex),Uu) = Uu ).

% ATP.lambda_1439
tff(fact_9612_ATP_Olambda__1440,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: word(A)] : aa(word(A),word(A),aTP_Lamp_ape(word(A),word(A)),Uu) = Uu ) ).

% ATP.lambda_1440
tff(fact_9613_ATP_Olambda__1441,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_vj(nat,nat),Uu) = Uu ).

% ATP.lambda_1441
tff(fact_9614_ATP_Olambda__1442,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_wv(int,int),Uu) = Uu ).

% ATP.lambda_1442
tff(fact_9615_ATP_Olambda__1443,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_aou(B,B),Uu) = Uu ) ).

% ATP.lambda_1443
tff(fact_9616_ATP_Olambda__1444,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_aov(B,B),Uu) = Uu ) ).

% ATP.lambda_1444
tff(fact_9617_ATP_Olambda__1445,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_aru(A,A),Uu) = Uu ) ).

% ATP.lambda_1445
tff(fact_9618_ATP_Olambda__1446,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_atm(A,A),Uu) = Uu ) ).

% ATP.lambda_1446
tff(fact_9619_ATP_Olambda__1447,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_app(A,A),Uu) = Uu ) ).

% ATP.lambda_1447
tff(fact_9620_ATP_Olambda__1448,axiom,
    ! [A: $tType] :
      ( topological_t2_space(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_aut(A,A),Uu) = Uu ) ).

% ATP.lambda_1448
tff(fact_9621_ATP_Olambda__1449,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A] : aa(A,A,aTP_Lamp_bdk(A,A),Uu) = Uu ) ).

% ATP.lambda_1449
tff(fact_9622_ATP_Olambda__1450,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_akt(A,A),Uu) = Uu ) ).

% ATP.lambda_1450
tff(fact_9623_ATP_Olambda__1451,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_fs(A,A),Uu) = Uu ) ).

% ATP.lambda_1451
tff(fact_9624_ATP_Olambda__1452,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_ct(A,A),Uu) = Uu ).

% ATP.lambda_1452
tff(fact_9625_ATP_Olambda__1453,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_any(C,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_1453
tff(fact_9626_ATP_Olambda__1454,axiom,
    ! [A: $tType,D2: $tType,Uu: A] : aa(A,set(D2),aTP_Lamp_anz(A,set(D2)),Uu) = bot_bot(set(D2)) ).

% ATP.lambda_1454
tff(fact_9627_ATP_Olambda__1455,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_ml(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1455
tff(fact_9628_ATP_Olambda__1456,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_lz(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1456
tff(fact_9629_ATP_Olambda__1457,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_ts(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1457
tff(fact_9630_ATP_Olambda__1458,axiom,
    ! [A: $tType,B: $tType] :
      ( ( real_V822414075346904944vector(B)
        & real_V822414075346904944vector(A) )
     => ! [Uu: A] : aa(A,B,aTP_Lamp_are(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_1458
tff(fact_9631_ATP_Olambda__1459,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_fr(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1459
tff(fact_9632_ATP_Olambda__1460,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_aiy(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_1460
tff(fact_9633_ATP_Olambda__1461,axiom,
    ! [Uu: product_unit] : aa(product_unit,assn,aTP_Lamp_ait(product_unit,assn),Uu) = one_one(assn) ).

% ATP.lambda_1461
tff(fact_9634_ATP_Olambda__1462,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_xo(B,A),Uu) = one_one(A) ) ).

% ATP.lambda_1462
tff(fact_9635_ATP_Olambda__1463,axiom,
    ! [Uu: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(vEBT_VEBTi,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_nl(heap_ext(product_unit),option(product_prod(vEBT_VEBTi,product_prod(heap_ext(product_unit),nat)))),Uu) = none(product_prod(vEBT_VEBTi,product_prod(heap_ext(product_unit),nat))) ).

% ATP.lambda_1463
tff(fact_9636_ATP_Olambda__1464,axiom,
    ! [Uu: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(option(nat),product_prod(heap_ext(product_unit),nat))),aTP_Lamp_jv(heap_ext(product_unit),option(product_prod(option(nat),product_prod(heap_ext(product_unit),nat)))),Uu) = none(product_prod(option(nat),product_prod(heap_ext(product_unit),nat))) ).

% ATP.lambda_1464
tff(fact_9637_ATP_Olambda__1465,axiom,
    ! [Uu: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod($o,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_lw(heap_ext(product_unit),option(product_prod($o,product_prod(heap_ext(product_unit),nat)))),Uu) = none(product_prod($o,product_prod(heap_ext(product_unit),nat))) ).

% ATP.lambda_1465
tff(fact_9638_ATP_Olambda__1466,axiom,
    ! [B: $tType,Uu: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_mg(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),Uu) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ).

% ATP.lambda_1466
tff(fact_9639_ATP_Olambda__1467,axiom,
    ! [A: $tType,Uu: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_kq(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Uu) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ).

% ATP.lambda_1467
tff(fact_9640_ATP_Olambda__1468,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_agu(B,option(A)),Uu) = none(A) ).

% ATP.lambda_1468
tff(fact_9641_ATP_Olambda__1469,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_gi(A,option(B)),Uu) = none(B) ).

% ATP.lambda_1469
tff(fact_9642_ATP_Olambda__1470,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,B,aTP_Lamp_aif(A,B),Uu) = undefined(B) ).

% ATP.lambda_1470
tff(fact_9643_ATP_Olambda__1471,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_adj(real,$o),Uu)
    <=> $false ) ).

% ATP.lambda_1471
tff(fact_9644_ATP_Olambda__1472,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_adh(nat,$o),Uu)
    <=> $false ) ).

% ATP.lambda_1472
tff(fact_9645_ATP_Olambda__1473,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_dp(A,$o),Uu)
    <=> $false ) ).

% ATP.lambda_1473
tff(fact_9646_ATP_Olambda__1474,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_adg(nat,$o),Uu)
    <=> $true ) ).

% ATP.lambda_1474
tff(fact_9647_ATP_Olambda__1475,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_do(A,$o),Uu)
    <=> $true ) ).

% ATP.lambda_1475
tff(fact_9648_ATP_Olambda__1476,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_gj(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_1476

% Type constructors (1274)
tff(tcon_VEBT__BuildupMemImp_OVEBTi___Typerep_Otyperep,axiom,
    typerep(vEBT_VEBTi) ).

tff(tcon_VEBT__Definitions_OVEBT___Typerep_Otyperep_1,axiom,
    typerep(vEBT_VEBT) ).

tff(tcon_Heap__Time__Monad_OHeap___Typerep_Otyperep_2,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => typerep(heap_Time_Heap(A10)) ) ).

tff(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_3,axiom,
    typerep(code_integer) ).

tff(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_4,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => typerep(heap_ext(A10)) ) ).

tff(tcon_Product__Type_Ounit___Typerep_Otyperep_5,axiom,
    typerep(product_unit) ).

tff(tcon_Product__Type_Ounit___Enum_Oenum,axiom,
    enum(product_unit) ).

tff(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder,axiom,
    comple5582772986160207858norder(product_unit) ).

tff(tcon_Product__Type_Oprod___Typerep_Otyperep_6,axiom,
    ! [A10: $tType,A11: $tType] :
      ( ( typerep(A10)
        & typerep(A11) )
     => typerep(product_prod(A10,A11)) ) ).

tff(tcon_Product__Type_Oprod___Enum_Oenum_7,axiom,
    ! [A10: $tType,A11: $tType] :
      ( ( enum(A10)
        & enum(A11) )
     => enum(product_prod(A10,A11)) ) ).

tff(tcon_Numeral__Type_Onum1___Typerep_Otyperep_8,axiom,
    typerep(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Enum_Oenum_9,axiom,
    enum(numeral_num1) ).

tff(tcon_Numeral__Type_Onum0___Typerep_Otyperep_10,axiom,
    typerep(numeral_num0) ).

tff(tcon_Numeral__Type_Obit1___Typerep_Otyperep_11,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => typerep(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Enum_Oenum_12,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => enum(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Typerep_Otyperep_13,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => typerep(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Enum_Oenum_14,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => enum(numeral_bit0(A10)) ) ).

tff(tcon_Multiset_Omultiset___Typerep_Otyperep_15,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => typerep(multiset(A10)) ) ).

tff(tcon_Extended__Nat_Oenat___Typerep_Otyperep_16,axiom,
    typerep(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder_17,axiom,
    comple5582772986160207858norder(extended_enat) ).

tff(tcon_Complex_Ocomplex___Typerep_Otyperep_18,axiom,
    typerep(complex) ).

tff(tcon_Assertions_Oassn___Typerep_Otyperep_19,axiom,
    typerep(assn) ).

tff(tcon_Uint32_Ouint32___Typerep_Otyperep_20,axiom,
    typerep(uint32) ).

tff(tcon_Option_Ooption___Typerep_Otyperep_21,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => typerep(option(A10)) ) ).

tff(tcon_Option_Ooption___Enum_Oenum_22,axiom,
    ! [A10: $tType] :
      ( enum(A10)
     => enum(option(A10)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder_23,axiom,
    ! [A10: $tType] :
      ( comple5582772986160207858norder(A10)
     => comple5582772986160207858norder(option(A10)) ) ).

tff(tcon_Filter_Ofilter___Typerep_Otyperep_24,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => typerep(filter(A10)) ) ).

tff(tcon_Enum_Ofinite__3___Typerep_Otyperep_25,axiom,
    typerep(finite_3) ).

tff(tcon_Enum_Ofinite__3___Enum_Oenum_26,axiom,
    enum(finite_3) ).

tff(tcon_Enum_Ofinite__3___Complete__Lattices_Ocomplete__linorder_27,axiom,
    comple5582772986160207858norder(finite_3) ).

tff(tcon_Enum_Ofinite__2___Typerep_Otyperep_28,axiom,
    typerep(finite_2) ).

tff(tcon_Enum_Ofinite__2___Enum_Oenum_29,axiom,
    enum(finite_2) ).

tff(tcon_Enum_Ofinite__2___Complete__Lattices_Ocomplete__linorder_30,axiom,
    comple5582772986160207858norder(finite_2) ).

tff(tcon_Enum_Ofinite__1___Typerep_Otyperep_31,axiom,
    typerep(finite_1) ).

tff(tcon_Enum_Ofinite__1___Enum_Oenum_32,axiom,
    enum(finite_1) ).

tff(tcon_Enum_Ofinite__1___Complete__Lattices_Ocomplete__linorder_33,axiom,
    comple5582772986160207858norder(finite_1) ).

tff(tcon_String_Ochar___Typerep_Otyperep_34,axiom,
    typerep(char) ).

tff(tcon_String_Ochar___Enum_Oenum_35,axiom,
    enum(char) ).

tff(tcon_Heap_Oarray___Typerep_Otyperep_36,axiom,
    ! [A10: $tType] : typerep(array(A10)) ).

tff(tcon_Word_Oword___Typerep_Otyperep_37,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => typerep(word(A10)) ) ).

tff(tcon_Word_Oword___Enum_Oenum_38,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => enum(word(A10)) ) ).

tff(tcon_Real_Oreal___Typerep_Otyperep_39,axiom,
    typerep(real) ).

tff(tcon_List_Olist___Typerep_Otyperep_40,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => typerep(list(A10)) ) ).

tff(tcon_HOL_Obool___Typerep_Otyperep_41,axiom,
    typerep($o) ).

tff(tcon_HOL_Obool___Enum_Oenum_42,axiom,
    enum($o) ).

tff(tcon_Set_Oset___Typerep_Otyperep_43,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => typerep(set(A10)) ) ).

tff(tcon_Set_Oset___Enum_Oenum_44,axiom,
    ! [A10: $tType] :
      ( enum(A10)
     => enum(set(A10)) ) ).

tff(tcon_Rat_Orat___Typerep_Otyperep_45,axiom,
    typerep(rat) ).

tff(tcon_Num_Onum___Typerep_Otyperep_46,axiom,
    typerep(num) ).

tff(tcon_Nat_Onat___Typerep_Otyperep_47,axiom,
    typerep(nat) ).

tff(tcon_Int_Oint___Typerep_Otyperep_48,axiom,
    typerep(int) ).

tff(tcon_fun___Typerep_Otyperep_49,axiom,
    ! [A10: $tType,A11: $tType] :
      ( ( typerep(A10)
        & typerep(A11) )
     => typerep(fun(A10,A11)) ) ).

tff(tcon_fun___Enum_Oenum_50,axiom,
    ! [A10: $tType,A11: $tType] :
      ( ( enum(A10)
        & enum(A11) )
     => enum(fun(A10,A11)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A10: $tType,A11: $tType] :
      ( boolea8198339166811842893lgebra(A11)
     => boolea8198339166811842893lgebra(fun(A10,A11)) ) ).

tff(tcon_fun___Code__Evaluation_Oterm__of,axiom,
    ! [A10: $tType,A11: $tType] :
      ( ( typerep(A10)
        & typerep(A11) )
     => code_term_of(fun(A10,A11)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A10: $tType,A11: $tType] :
      ( order_top(A11)
     => order_top(fun(A10,A11)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A10: $tType,A11: $tType] :
      ( order_bot(A11)
     => order_bot(fun(A10,A11)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A10: $tType,A11: $tType] :
      ( preorder(A11)
     => preorder(fun(A10,A11)) ) ).

tff(tcon_fun___Finite__Set_Ofinite,axiom,
    ! [A10: $tType,A11: $tType] :
      ( ( finite_finite(A10)
        & finite_finite(A11) )
     => finite_finite(fun(A10,A11)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A10: $tType,A11: $tType] :
      ( order(A11)
     => order(fun(A10,A11)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A10: $tType,A11: $tType] :
      ( ord(A11)
     => ord(fun(A10,A11)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A10: $tType,A11: $tType] :
      ( uminus(A11)
     => uminus(fun(A10,A11)) ) ).

tff(tcon_fun___HOL_Oequal,axiom,
    ! [A10: $tType,A11: $tType] :
      ( ( enum(A10)
        & cl_HOL_Oequal(A11) )
     => cl_HOL_Oequal(fun(A10,A11)) ) ).

tff(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
    condit6923001295902523014norder(int) ).

tff(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
    bit_un5681908812861735899ations(int) ).

tff(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
    semiri1453513574482234551roduct(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
    euclid5411537665997757685th_nat(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
    euclid8789492081693882211th_nat(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
    ordere1937475149494474687imp_le(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
    euclid3128863361964157862miring(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
    euclid4440199948858584721cancel(int) ).

tff(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
    unique1627219031080169319umeral(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
    euclid8851590272496341667cancel(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
    semiri6575147826004484403cancel(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
    strict9044650504122735259up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
    ordere580206878836729694up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
    ordere2412721322843649153imp_le(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
    bit_se359711467146920520ations(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
    linord2810124833399127020strict(int) ).

tff(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
    strict7427464778891057005id_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
    ordere8940638589300402666id_add(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
    euclid3725896446679973847miring(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
    topolo4958980785337419405_space(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
    topolo1944317154257567458pology(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Odiscrete__topology,axiom,
    topolo8865339358273720382pology(int) ).

tff(tcon_Int_Oint___Quickcheck__Narrowing_Opartial__term__of,axiom,
    quickc6926020345158392990erm_of(int) ).

tff(tcon_Int_Oint___Limits_Otopological__comm__monoid__mult,axiom,
    topolo4987421752381908075d_mult(int) ).

tff(tcon_Int_Oint___Bit__Comprehension_Obit__comprehension,axiom,
    bit_bi6583157726757044596ension(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
    linord715952674999750819strict(int) ).

tff(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
    topolo5987344860129210374id_add(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__semigroup__add,axiom,
    linord4140545234300271783up_add(int) ).

tff(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
    bit_ri3973907225187159222ations(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
    topolo2564578578187576103pology(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
    semiri2026040879449505780visors(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__nonzero__semiring,axiom,
    linord181362715937106298miring(int) ).

tff(tcon_Int_Oint___Limits_Otopological__semigroup__mult,axiom,
    topolo4211221413907600880p_mult(int) ).

tff(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring,axiom,
    euclid5891614535332579305n_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
    linord8928482502909563296strict(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
    semiri3467727345109120633visors(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
    ordere6658533253407199908up_add(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
    ordere166539214618696060dd_abs(int) ).

tff(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
    semiri6843258321239162965malize(int) ).

tff(tcon_Int_Oint___Limits_Otopological__monoid__mult,axiom,
    topolo1898628316856586783d_mult(int) ).

tff(tcon_Int_Oint___Groups_Oordered__comm__monoid__add,axiom,
    ordere6911136660526730532id_add(int) ).

tff(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
    linord5086331880401160121up_add(int) ).

tff(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
    ring_15535105094025558882visors(int) ).

tff(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
    topolo6943815403480290642id_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
    cancel1802427076303600483id_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
    linord4710134922213307826strict(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
    comm_s4317794764714335236cancel(int) ).

tff(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
    bit_semiring_bits(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
    topological_t2_space(int) ).

tff(tcon_Int_Oint___Topological__Spaces_Ot1__space,axiom,
    topological_t1_space(int) ).

tff(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
    ordere2520102378445227354miring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
    linord6961819062388156250ring_1(int) ).

tff(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
    ordered_ab_group_add(int) ).

tff(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
    cancel_semigroup_add(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
    linordered_semiring(int) ).

tff(tcon_Int_Oint___Least__significant__bit_Olsb,axiom,
    least_6119777620449941438nt_lsb(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
    ordered_semiring_0(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
    linordered_semidom(int) ).

tff(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
    ab_semigroup_mult(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
    semiring_1_cancel(int) ).

tff(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
    algebraic_semidom(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
    comm_monoid_mult(int) ).

tff(tcon_Int_Oint___Generic__set__bit_Oset__bit,axiom,
    generic_set_set_bit(int) ).

tff(tcon_Int_Oint___Code__Evaluation_Oterm__of_51,axiom,
    code_term_of(int) ).

tff(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
    ordered_semiring(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
    ordered_ring_abs(int) ).

tff(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
    semiring_parity(int) ).

tff(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
    comm_monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
    semiring_modulo(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
    linordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
    linordered_idom(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
    comm_semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
    comm_semiring_0(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
    semidom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
    semidom_divide(int) ).

tff(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
    semiring_numeral(int) ).

tff(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
    zero_less_one(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
    comm_semiring(int) ).

tff(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
    semiring_char_0(int) ).

tff(tcon_Int_Oint___Groups_Oab__group__add,axiom,
    ab_group_add(int) ).

tff(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
    zero_neq_one(int) ).

tff(tcon_Int_Oint___Rings_Oordered__ring,axiom,
    ordered_ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
    idom_abs_sgn(int) ).

tff(tcon_Int_Oint___Parity_Oring__parity,axiom,
    ring_parity(int) ).

tff(tcon_Int_Oint___Orderings_Opreorder_52,axiom,
    preorder(int) ).

tff(tcon_Int_Oint___Orderings_Olinorder,axiom,
    linorder(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__mult,axiom,
    monoid_mult(int) ).

tff(tcon_Int_Oint___Rings_Oidom__modulo,axiom,
    idom_modulo(int) ).

tff(tcon_Int_Oint___Rings_Oidom__divide,axiom,
    idom_divide(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring__1,axiom,
    comm_ring_1(int) ).

tff(tcon_Int_Oint___Groups_Omonoid__add,axiom,
    monoid_add(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__1,axiom,
    semiring_1(int) ).

tff(tcon_Int_Oint___Rings_Osemiring__0,axiom,
    semiring_0(int) ).

tff(tcon_Int_Oint___Orderings_Ono__top,axiom,
    no_top(int) ).

tff(tcon_Int_Oint___Orderings_Ono__bot,axiom,
    no_bot(int) ).

tff(tcon_Int_Oint___Groups_Ogroup__add,axiom,
    group_add(int) ).

tff(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
    semiring_gcd(int) ).

tff(tcon_Int_Oint___Rings_Omult__zero,axiom,
    mult_zero(int) ).

tff(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
    comm_ring(int) ).

tff(tcon_Int_Oint___Orderings_Oorder_53,axiom,
    order(int) ).

tff(tcon_Int_Oint___Num_Oneg__numeral,axiom,
    neg_numeral(int) ).

tff(tcon_Int_Oint___Nat_Oring__char__0,axiom,
    ring_char_0(int) ).

tff(tcon_Int_Oint___Rings_Osemiring,axiom,
    semiring(int) ).

tff(tcon_Int_Oint___Rings_Osemidom,axiom,
    semidom(int) ).

tff(tcon_Int_Oint___Orderings_Oord_54,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_55,axiom,
    uminus(int) ).

tff(tcon_Int_Oint___Rings_Oring__1,axiom,
    ring_1(int) ).

tff(tcon_Int_Oint___Rings_Oabs__if,axiom,
    abs_if(int) ).

tff(tcon_Int_Oint___GCD_Oring__gcd,axiom,
    ring_gcd(int) ).

tff(tcon_Int_Oint___Power_Opower,axiom,
    power(int) ).

tff(tcon_Int_Oint___Num_Onumeral,axiom,
    numeral(int) ).

tff(tcon_Int_Oint___Groups_Ozero,axiom,
    zero(int) ).

tff(tcon_Int_Oint___Groups_Oplus,axiom,
    plus(int) ).

tff(tcon_Int_Oint___Rings_Oring,axiom,
    ring(int) ).

tff(tcon_Int_Oint___Rings_Oidom,axiom,
    idom(int) ).

tff(tcon_Int_Oint___Groups_Oone,axiom,
    one(int) ).

tff(tcon_Int_Oint___Rings_Odvd,axiom,
    dvd(int) ).

tff(tcon_Int_Oint___Heap_Oheap,axiom,
    heap(int) ).

tff(tcon_Int_Oint___HOL_Oequal_56,axiom,
    cl_HOL_Oequal(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_57,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_58,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_59,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_60,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_61,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_62,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_63,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_64,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_65,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_66,axiom,
    strict9044650504122735259up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__diff,axiom,
    ordere1170586879665033532d_diff(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add_67,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_68,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_69,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_70,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_71,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_72,axiom,
    ordere8940638589300402666id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
    canoni5634975068530333245id_add(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring_73,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_74,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_75,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_76,axiom,
    topolo8865339358273720382pology(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Narrowing_Opartial__term__of_77,axiom,
    quickc6926020345158392990erm_of(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_78,axiom,
    topolo4987421752381908075d_mult(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_79,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_80,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_81,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_82,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_83,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_84,axiom,
    topolo4211221413907600880p_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_85,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_86,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_87,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_88,axiom,
    semiri6843258321239162965malize(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__mult_89,axiom,
    topolo1898628316856586783d_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_90,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__monoid__add_91,axiom,
    topolo6943815403480290642id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_92,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_93,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_94,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_95,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot1__space_96,axiom,
    topological_t1_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_97,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_98,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_99,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_100,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_101,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_102,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_103,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_104,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_105,axiom,
    comm_monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
    comm_monoid_diff(nat) ).

tff(tcon_Nat_Onat___Code__Evaluation_Oterm__of_106,axiom,
    code_term_of(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_107,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_108,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_109,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_110,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_111,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_112,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_113,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_114,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_115,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_116,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_117,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_118,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_119,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_120,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_121,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_122,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_123,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_124,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_125,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_126,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_127,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_128,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_129,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_130,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_131,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom_132,axiom,
    semidom(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_133,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Power_Opower_134,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_135,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_136,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_137,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_138,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_139,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Heap_Oheap_140,axiom,
    heap(nat) ).

tff(tcon_Nat_Onat___HOL_Oequal_141,axiom,
    cl_HOL_Oequal(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Quickcheck__Narrowing_Opartial__term__of_142,axiom,
    quickc6926020345158392990erm_of(num) ).

tff(tcon_Num_Onum___Code__Evaluation_Oterm__of_143,axiom,
    code_term_of(num) ).

tff(tcon_Num_Onum___Orderings_Opreorder_144,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_145,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_146,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_147,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_148,axiom,
    plus(num) ).

tff(tcon_Num_Onum___HOL_Oequal_149,axiom,
    cl_HOL_Oequal(num) ).

tff(tcon_Num_Onum___Nat_Osize_150,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_151,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_152,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_153,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_154,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_155,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_156,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_157,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_158,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_159,axiom,
    ordere8940638589300402666id_add(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Narrowing_Opartial__term__of_160,axiom,
    quickc6926020345158392990erm_of(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Oarchimedean__field,axiom,
    archim462609752435547400_field(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1__strict_161,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
    unboun7993243217541854897norder(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_162,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_163,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_164,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_165,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_166,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_167,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_168,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_169,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_170,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_171,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_172,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_173,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_174,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_175,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_176,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_177,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_178,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_179,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_180,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_181,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_182,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_183,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_184,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Code__Evaluation_Oterm__of_185,axiom,
    code_term_of(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_186,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_187,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_188,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_189,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_190,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_191,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_192,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_193,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_194,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield__abs__sgn,axiom,
    field_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
    division_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__less__one_195,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_196,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_197,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_198,axiom,
    ab_group_add(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
    field_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Ozero__neq__one_199,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_200,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_201,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_202,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_203,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_204,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_205,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_206,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_207,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_208,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_209,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_210,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_211,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_212,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_213,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_214,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_215,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_216,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_217,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_218,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Fields_Oinverse,axiom,
    inverse(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom_219,axiom,
    semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_220,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_221,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_222,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_223,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_224,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_225,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_226,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_227,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_228,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_229,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_230,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_231,axiom,
    dvd(rat) ).

tff(tcon_Rat_Orat___HOL_Oequal_232,axiom,
    cl_HOL_Oequal(rat) ).

tff(tcon_Set_Oset___Quickcheck__Narrowing_Opartial__term__of_233,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => quickc6926020345158392990erm_of(set(A10)) ) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_234,axiom,
    ! [A10: $tType] : boolea8198339166811842893lgebra(set(A10)) ).

tff(tcon_Set_Oset___Code__Evaluation_Oterm__of_235,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => code_term_of(set(A10)) ) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_236,axiom,
    ! [A10: $tType] : order_top(set(A10)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_237,axiom,
    ! [A10: $tType] : order_bot(set(A10)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_238,axiom,
    ! [A10: $tType] : preorder(set(A10)) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_239,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => finite_finite(set(A10)) ) ).

tff(tcon_Set_Oset___Orderings_Oorder_240,axiom,
    ! [A10: $tType] : order(set(A10)) ).

tff(tcon_Set_Oset___Orderings_Oord_241,axiom,
    ! [A10: $tType] : ord(set(A10)) ).

tff(tcon_Set_Oset___Groups_Ouminus_242,axiom,
    ! [A10: $tType] : uminus(set(A10)) ).

tff(tcon_Set_Oset___HOL_Oequal_243,axiom,
    ! [A10: $tType] :
      ( cl_HOL_Oequal(A10)
     => cl_HOL_Oequal(set(A10)) ) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_244,axiom,
    topolo4958980785337419405_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_245,axiom,
    topolo1944317154257567458pology($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_246,axiom,
    topolo8865339358273720382pology($o) ).

tff(tcon_HOL_Obool___Quickcheck__Narrowing_Opartial__term__of_247,axiom,
    quickc6926020345158392990erm_of($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_248,axiom,
    topolo2564578578187576103pology($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_249,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_250,axiom,
    topological_t2_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot1__space_251,axiom,
    topological_t1_space($o) ).

tff(tcon_HOL_Obool___Code__Evaluation_Oterm__of_252,axiom,
    code_term_of($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_253,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_254,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_255,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_256,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_257,axiom,
    finite_finite($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_258,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_259,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_260,axiom,
    uminus($o) ).

tff(tcon_HOL_Obool___Heap_Oheap_261,axiom,
    heap($o) ).

tff(tcon_HOL_Obool___HOL_Oequal_262,axiom,
    cl_HOL_Oequal($o) ).

tff(tcon_List_Olist___Quickcheck__Narrowing_Opartial__term__of_263,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => quickc6926020345158392990erm_of(list(A10)) ) ).

tff(tcon_List_Olist___Code__Evaluation_Oterm__of_264,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => code_term_of(list(A10)) ) ).

tff(tcon_List_Olist___Heap_Oheap_265,axiom,
    ! [A10: $tType] :
      ( heap(A10)
     => heap(list(A10)) ) ).

tff(tcon_List_Olist___HOL_Oequal_266,axiom,
    ! [A10: $tType] : cl_HOL_Oequal(list(A10)) ).

tff(tcon_List_Olist___Nat_Osize_267,axiom,
    ! [A10: $tType] : size(list(A10)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_268,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_269,axiom,
    semiri1453513574482234551roduct(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_270,axiom,
    ordere1937475149494474687imp_le(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinear__continuum__topology,axiom,
    topolo8458572112393995274pology(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ofirst__countable__topology,axiom,
    topolo3112930676232923870pology(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__div__algebra,axiom,
    real_V8999393235501362500lgebra(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra__1,axiom,
    real_V2822296259951069270ebra_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_271,axiom,
    semiri6575147826004484403cancel(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra,axiom,
    real_V4412858255891104859lgebra(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oordered__real__vector,axiom,
    real_V5355595471888546746vector(real) ).

tff(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add_272,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_273,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_274,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_275,axiom,
    linord2810124833399127020strict(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
    real_V822414075346904944vector(real) ).

tff(tcon_Real_Oreal___Groups_Ostrict__ordered__comm__monoid__add_276,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_277,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_278,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_279,axiom,
    topolo1944317154257567458pology(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
    real_V3459762299906320749_field(real) ).

tff(tcon_Real_Oreal___Quickcheck__Narrowing_Opartial__term__of_280,axiom,
    quickc6926020345158392990erm_of(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
    real_V5047593784448816457lgebra(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_281,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_282,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_283,axiom,
    unboun7993243217541854897norder(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_284,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_285,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_286,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_287,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_288,axiom,
    linord181362715937106298miring(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra__1,axiom,
    real_V2191834092415804123ebra_1(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space,axiom,
    real_V8037385150606011577_space(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__semigroup__mult_289,axiom,
    topolo4211221413907600880p_mult(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
    topolo8386298272705272623_space(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_290,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_291,axiom,
    semiri3467727345109120633visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
    real_V7819770556892013058_space(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__ab__group__add,axiom,
    topolo1287966508704411220up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_292,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_293,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_294,axiom,
    archim2362893244070406136eiling(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
    real_V4867850818363320053vector(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add_295,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_296,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_297,axiom,
    ring_15535105094025558882visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__field,axiom,
    real_V7773925162809079976_field(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__monoid__add_298,axiom,
    topolo6943815403480290642id_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_299,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_300,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_301,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
    topolo1633459387980952147up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_302,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot1__space_303,axiom,
    topological_t1_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_304,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_305,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_306,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_307,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_308,axiom,
    linordered_semiring(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Obanach,axiom,
    real_Vector_banach(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring__0_309,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_310,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_311,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_312,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_313,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_314,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_315,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Code__Evaluation_Oterm__of_316,axiom,
    code_term_of(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_317,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_318,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_319,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_320,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_321,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_322,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_323,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_324,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_325,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_326,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__abs__sgn_327,axiom,
    field_abs_sgn(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_328,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_329,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_330,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_331,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_332,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_333,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_334,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_335,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_336,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_337,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_338,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_339,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_340,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_341,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_342,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_343,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_344,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_345,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_346,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_347,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_348,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_349,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_350,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_351,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_352,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_353,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Fields_Oinverse_354,axiom,
    inverse(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom_355,axiom,
    semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_356,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_357,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_358,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_359,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_360,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_361,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_362,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_363,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_364,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_365,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_366,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_367,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_368,axiom,
    dvd(real) ).

tff(tcon_Real_Oreal___HOL_Oequal_369,axiom,
    cl_HOL_Oequal(real) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bit__operations_370,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => bit_se359711467146920520ations(word(A10)) ) ).

tff(tcon_Word_Oword___Quickcheck__Narrowing_Opartial__term__of_371,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => quickc6926020345158392990erm_of(word(A10)) ) ).

tff(tcon_Word_Oword___Bit__Comprehension_Obit__comprehension_372,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => bit_bi6583157726757044596ension(word(A10)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Oring__bit__operations_373,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => bit_ri3973907225187159222ations(word(A10)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__comm__monoid__add_374,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => cancel1802427076303600483id_add(word(A10)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1__cancel_375,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => comm_s4317794764714335236cancel(word(A10)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bits_376,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => bit_semiring_bits(word(A10)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__semigroup__add_377,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => cancel_semigroup_add(word(A10)) ) ).

tff(tcon_Word_Oword___Least__significant__bit_Olsb_378,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => least_6119777620449941438nt_lsb(word(A10)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__mult_379,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => ab_semigroup_mult(word(A10)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1__cancel_380,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => semiring_1_cancel(word(A10)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__mult_381,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => comm_monoid_mult(word(A10)) ) ).

tff(tcon_Word_Oword___Generic__set__bit_Oset__bit_382,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => generic_set_set_bit(word(A10)) ) ).

tff(tcon_Word_Oword___Code__Evaluation_Oterm__of_383,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => code_term_of(word(A10)) ) ).

tff(tcon_Word_Oword___Parity_Osemiring__parity_384,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => semiring_parity(word(A10)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__add_385,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => comm_monoid_add(word(A10)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__modulo_386,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => semiring_modulo(word(A10)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1_387,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => comm_semiring_1(word(A10)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__0_388,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => comm_semiring_0(word(A10)) ) ).

tff(tcon_Word_Oword___Num_Osemiring__numeral_389,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => semiring_numeral(word(A10)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring_390,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => comm_semiring(word(A10)) ) ).

tff(tcon_Word_Oword___Orderings_Owellorder_391,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => wellorder(word(A10)) ) ).

tff(tcon_Word_Oword___Groups_Oab__group__add_392,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => ab_group_add(word(A10)) ) ).

tff(tcon_Word_Oword___Rings_Ozero__neq__one_393,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => zero_neq_one(word(A10)) ) ).

tff(tcon_Word_Oword___Parity_Oring__parity_394,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => ring_parity(word(A10)) ) ).

tff(tcon_Word_Oword___Orderings_Opreorder_395,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => preorder(word(A10)) ) ).

tff(tcon_Word_Oword___Orderings_Olinorder_396,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => linorder(word(A10)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__mult_397,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => monoid_mult(word(A10)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring__1_398,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => comm_ring_1(word(A10)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__add_399,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => monoid_add(word(A10)) ) ).

tff(tcon_Word_Oword___Finite__Set_Ofinite_400,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => finite_finite(word(A10)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1_401,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => semiring_1(word(A10)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__0_402,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => semiring_0(word(A10)) ) ).

tff(tcon_Word_Oword___Groups_Ogroup__add_403,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => group_add(word(A10)) ) ).

tff(tcon_Word_Oword___Rings_Omult__zero_404,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => mult_zero(word(A10)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring_405,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => comm_ring(word(A10)) ) ).

tff(tcon_Word_Oword___Orderings_Oorder_406,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => order(word(A10)) ) ).

tff(tcon_Word_Oword___Num_Oneg__numeral_407,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => neg_numeral(word(A10)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring_408,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => semiring(word(A10)) ) ).

tff(tcon_Word_Oword___Orderings_Oord_409,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => ord(word(A10)) ) ).

tff(tcon_Word_Oword___Groups_Ouminus_410,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => uminus(word(A10)) ) ).

tff(tcon_Word_Oword___Rings_Oring__1_411,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => ring_1(word(A10)) ) ).

tff(tcon_Word_Oword___Power_Opower_412,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => power(word(A10)) ) ).

tff(tcon_Word_Oword___Num_Onumeral_413,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => numeral(word(A10)) ) ).

tff(tcon_Word_Oword___Groups_Ozero_414,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => zero(word(A10)) ) ).

tff(tcon_Word_Oword___Groups_Oplus_415,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => plus(word(A10)) ) ).

tff(tcon_Word_Oword___Rings_Oring_416,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => ring(word(A10)) ) ).

tff(tcon_Word_Oword___Groups_Oone_417,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => one(word(A10)) ) ).

tff(tcon_Word_Oword___Rings_Odvd_418,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => dvd(word(A10)) ) ).

tff(tcon_Word_Oword___HOL_Oequal_419,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => cl_HOL_Oequal(word(A10)) ) ).

tff(tcon_Word_Oword___Nat_Osize_420,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => size(word(A10)) ) ).

tff(tcon_Heap_Oarray___Quickcheck__Narrowing_Opartial__term__of_421,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => quickc6926020345158392990erm_of(array(A10)) ) ).

tff(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_422,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => code_term_of(array(A10)) ) ).

tff(tcon_Heap_Oarray___Heap_Oheap_423,axiom,
    ! [A10: $tType] : heap(array(A10)) ).

tff(tcon_Heap_Oarray___HOL_Oequal_424,axiom,
    ! [A10: $tType] : cl_HOL_Oequal(array(A10)) ).

tff(tcon_Heap_Oarray___Nat_Osize_425,axiom,
    ! [A10: $tType] : size(array(A10)) ).

tff(tcon_String_Ochar___Quickcheck__Narrowing_Opartial__term__of_426,axiom,
    quickc6926020345158392990erm_of(char) ).

tff(tcon_String_Ochar___Code__Evaluation_Oterm__of_427,axiom,
    code_term_of(char) ).

tff(tcon_String_Ochar___Orderings_Opreorder_428,axiom,
    preorder(char) ).

tff(tcon_String_Ochar___Orderings_Olinorder_429,axiom,
    linorder(char) ).

tff(tcon_String_Ochar___Finite__Set_Ofinite_430,axiom,
    finite_finite(char) ).

tff(tcon_String_Ochar___Orderings_Oorder_431,axiom,
    order(char) ).

tff(tcon_String_Ochar___Orderings_Oord_432,axiom,
    ord(char) ).

tff(tcon_String_Ochar___Heap_Oheap_433,axiom,
    heap(char) ).

tff(tcon_String_Ochar___HOL_Oequal_434,axiom,
    cl_HOL_Oequal(char) ).

tff(tcon_String_Ochar___Nat_Osize_435,axiom,
    size(char) ).

tff(tcon_Enum_Ofinite__1___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_436,axiom,
    condit6923001295902523014norder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__monoid__add__imp__le_437,axiom,
    ordere1937475149494474687imp_le(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors__cancel_438,axiom,
    semiri6575147826004484403cancel(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__ab__semigroup__add_439,axiom,
    strict9044650504122735259up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__diff_440,axiom,
    ordere1170586879665033532d_diff(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__ab__semigroup__add_441,axiom,
    ordere580206878836729694up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add__imp__le_442,axiom,
    ordere2412721322843649153imp_le(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__comm__semiring__strict_443,axiom,
    linord2810124833399127020strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__comm__monoid__add_444,axiom,
    strict7427464778891057005id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__add_445,axiom,
    ordere8940638589300402666id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocanonically__ordered__monoid__add_446,axiom,
    canoni5634975068530333245id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Quickcheck__Narrowing_Opartial__term__of_447,axiom,
    quickc6926020345158392990erm_of(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__semigroup__add_448,axiom,
    linord4140545234300271783up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring__strict_449,axiom,
    linord8928482502909563296strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Boolean__Algebras_Oboolean__algebra_450,axiom,
    boolea8198339166811842893lgebra(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors_451,axiom,
    semiri3467727345109120633visors(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add_452,axiom,
    ordere6658533253407199908up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add__abs_453,axiom,
    ordere166539214618696060dd_abs(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__comm__monoid__add_454,axiom,
    ordere6911136660526730532id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__group__add_455,axiom,
    linord5086331880401160121up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__comm__monoid__add_456,axiom,
    cancel1802427076303600483id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__ring__strict_457,axiom,
    linord4710134922213307826strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__comm__semiring_458,axiom,
    ordere2520102378445227354miring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add_459,axiom,
    ordered_ab_group_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__semigroup__add_460,axiom,
    cancel_semigroup_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring_461,axiom,
    linordered_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__semiring__0_462,axiom,
    ordered_semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Odense__linorder_463,axiom,
    dense_linorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__mult_464,axiom,
    ab_semigroup_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__mult_465,axiom,
    comm_monoid_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__diff_466,axiom,
    comm_monoid_diff(finite_1) ).

tff(tcon_Enum_Ofinite__1___Code__Evaluation_Oterm__of_467,axiom,
    code_term_of(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__semiring_468,axiom,
    ordered_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__ring__abs_469,axiom,
    ordered_ring_abs(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__add_470,axiom,
    comm_monoid_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__ring_471,axiom,
    linordered_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring__0_472,axiom,
    comm_semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Odense__order_473,axiom,
    dense_order(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring_474,axiom,
    comm_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Owellorder_475,axiom,
    wellorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder__top_476,axiom,
    order_top(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder__bot_477,axiom,
    order_bot(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__group__add_478,axiom,
    ab_group_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__ring_479,axiom,
    ordered_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Opreorder_480,axiom,
    preorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Olinorder_481,axiom,
    linorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Omonoid__mult_482,axiom,
    monoid_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Omonoid__add_483,axiom,
    monoid_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Finite__Set_Ofinite_484,axiom,
    finite_finite(finite_1) ).

tff(tcon_Enum_Ofinite__1___Type__Length_Olen0,axiom,
    type_len0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__0_485,axiom,
    semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ogroup__add_486,axiom,
    group_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Type__Length_Olen,axiom,
    type_len(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Omult__zero_487,axiom,
    mult_zero(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__ring_488,axiom,
    comm_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder_489,axiom,
    order(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring_490,axiom,
    semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Fields_Oinverse_491,axiom,
    inverse(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oord_492,axiom,
    ord(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ouminus_493,axiom,
    uminus(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oabs__if_494,axiom,
    abs_if(finite_1) ).

tff(tcon_Enum_Ofinite__1___Power_Opower_495,axiom,
    power(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ozero_496,axiom,
    zero(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oplus_497,axiom,
    plus(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oring_498,axiom,
    ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oone_499,axiom,
    one(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Odvd_500,axiom,
    dvd(finite_1) ).

tff(tcon_Enum_Ofinite__1___HOL_Oequal_501,axiom,
    cl_HOL_Oequal(finite_1) ).

tff(tcon_Enum_Ofinite__2___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_502,axiom,
    condit6923001295902523014norder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_503,axiom,
    semiri1453513574482234551roduct(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Ounique__euclidean__semiring_504,axiom,
    euclid3128863361964157862miring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring__cancel_505,axiom,
    euclid4440199948858584721cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors__cancel_506,axiom,
    semiri6575147826004484403cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring_507,axiom,
    euclid3725896446679973847miring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Quickcheck__Narrowing_Opartial__term__of_508,axiom,
    quickc6926020345158392990erm_of(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1__no__zero__divisors_509,axiom,
    semiri2026040879449505780visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors_510,axiom,
    semiri3467727345109120633visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring__1__no__zero__divisors_511,axiom,
    ring_15535105094025558882visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__comm__monoid__add_512,axiom,
    cancel1802427076303600483id_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1__cancel_513,axiom,
    comm_s4317794764714335236cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__semigroup__add_514,axiom,
    cancel_semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__mult_515,axiom,
    ab_semigroup_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1__cancel_516,axiom,
    semiring_1_cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oalgebraic__semidom_517,axiom,
    algebraic_semidom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__mult_518,axiom,
    comm_monoid_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Code__Evaluation_Oterm__of_519,axiom,
    code_term_of(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__add_520,axiom,
    comm_monoid_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__modulo_521,axiom,
    semiring_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1_522,axiom,
    comm_semiring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__0_523,axiom,
    comm_semiring_0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom__modulo_524,axiom,
    semidom_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom__divide_525,axiom,
    semidom_divide(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Osemiring__numeral_526,axiom,
    semiring_numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Odivision__ring_527,axiom,
    division_ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring_528,axiom,
    comm_semiring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Owellorder_529,axiom,
    wellorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder__top_530,axiom,
    order_top(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder__bot_531,axiom,
    order_bot(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__group__add_532,axiom,
    ab_group_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ozero__neq__one_533,axiom,
    zero_neq_one(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__abs__sgn_534,axiom,
    idom_abs_sgn(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Opreorder_535,axiom,
    preorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Olinorder_536,axiom,
    linorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Omonoid__mult_537,axiom,
    monoid_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__modulo_538,axiom,
    idom_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__divide_539,axiom,
    idom_divide(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__ring__1_540,axiom,
    comm_ring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Omonoid__add_541,axiom,
    monoid_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Finite__Set_Ofinite_542,axiom,
    finite_finite(finite_2) ).

tff(tcon_Enum_Ofinite__2___Type__Length_Olen0_543,axiom,
    type_len0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1_544,axiom,
    semiring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__0_545,axiom,
    semiring_0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ogroup__add_546,axiom,
    group_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Type__Length_Olen_547,axiom,
    type_len(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Omult__zero_548,axiom,
    mult_zero(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__ring_549,axiom,
    comm_ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder_550,axiom,
    order(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Oneg__numeral_551,axiom,
    neg_numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring_552,axiom,
    semiring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Oinverse_553,axiom,
    inverse(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom_554,axiom,
    semidom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oord_555,axiom,
    ord(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ouminus_556,axiom,
    uminus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring__1_557,axiom,
    ring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Ofield_558,axiom,
    field(finite_2) ).

tff(tcon_Enum_Ofinite__2___Power_Opower_559,axiom,
    power(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Onumeral_560,axiom,
    numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ozero_561,axiom,
    zero(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oplus_562,axiom,
    plus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring_563,axiom,
    ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom_564,axiom,
    idom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oone_565,axiom,
    one(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Odvd_566,axiom,
    dvd(finite_2) ).

tff(tcon_Enum_Ofinite__2___HOL_Oequal_567,axiom,
    cl_HOL_Oequal(finite_2) ).

tff(tcon_Enum_Ofinite__3___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_568,axiom,
    condit6923001295902523014norder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_569,axiom,
    semiri1453513574482234551roduct(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Ounique__euclidean__semiring_570,axiom,
    euclid3128863361964157862miring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring__cancel_571,axiom,
    euclid4440199948858584721cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors__cancel_572,axiom,
    semiri6575147826004484403cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring_573,axiom,
    euclid3725896446679973847miring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Quickcheck__Narrowing_Opartial__term__of_574,axiom,
    quickc6926020345158392990erm_of(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1__no__zero__divisors_575,axiom,
    semiri2026040879449505780visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors_576,axiom,
    semiri3467727345109120633visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring__1__no__zero__divisors_577,axiom,
    ring_15535105094025558882visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__comm__monoid__add_578,axiom,
    cancel1802427076303600483id_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1__cancel_579,axiom,
    comm_s4317794764714335236cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__semigroup__add_580,axiom,
    cancel_semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__mult_581,axiom,
    ab_semigroup_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1__cancel_582,axiom,
    semiring_1_cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oalgebraic__semidom_583,axiom,
    algebraic_semidom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__mult_584,axiom,
    comm_monoid_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Code__Evaluation_Oterm__of_585,axiom,
    code_term_of(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__add_586,axiom,
    comm_monoid_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__modulo_587,axiom,
    semiring_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1_588,axiom,
    comm_semiring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__0_589,axiom,
    comm_semiring_0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom__modulo_590,axiom,
    semidom_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom__divide_591,axiom,
    semidom_divide(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Osemiring__numeral_592,axiom,
    semiring_numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Odivision__ring_593,axiom,
    division_ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring_594,axiom,
    comm_semiring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Owellorder_595,axiom,
    wellorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder__top_596,axiom,
    order_top(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder__bot_597,axiom,
    order_bot(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__group__add_598,axiom,
    ab_group_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ozero__neq__one_599,axiom,
    zero_neq_one(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__abs__sgn_600,axiom,
    idom_abs_sgn(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Opreorder_601,axiom,
    preorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Olinorder_602,axiom,
    linorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Omonoid__mult_603,axiom,
    monoid_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__modulo_604,axiom,
    idom_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__divide_605,axiom,
    idom_divide(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__ring__1_606,axiom,
    comm_ring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Omonoid__add_607,axiom,
    monoid_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Finite__Set_Ofinite_608,axiom,
    finite_finite(finite_3) ).

tff(tcon_Enum_Ofinite__3___Type__Length_Olen0_609,axiom,
    type_len0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1_610,axiom,
    semiring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__0_611,axiom,
    semiring_0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ogroup__add_612,axiom,
    group_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Type__Length_Olen_613,axiom,
    type_len(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Omult__zero_614,axiom,
    mult_zero(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__ring_615,axiom,
    comm_ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder_616,axiom,
    order(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Oneg__numeral_617,axiom,
    neg_numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring_618,axiom,
    semiring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Oinverse_619,axiom,
    inverse(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom_620,axiom,
    semidom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oord_621,axiom,
    ord(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ouminus_622,axiom,
    uminus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring__1_623,axiom,
    ring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Ofield_624,axiom,
    field(finite_3) ).

tff(tcon_Enum_Ofinite__3___Power_Opower_625,axiom,
    power(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Onumeral_626,axiom,
    numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ozero_627,axiom,
    zero(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oplus_628,axiom,
    plus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring_629,axiom,
    ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom_630,axiom,
    idom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oone_631,axiom,
    one(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Odvd_632,axiom,
    dvd(finite_3) ).

tff(tcon_Enum_Ofinite__3___HOL_Oequal_633,axiom,
    cl_HOL_Oequal(finite_3) ).

tff(tcon_Filter_Ofilter___Quickcheck__Narrowing_Opartial__term__of_634,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => quickc6926020345158392990erm_of(filter(A10)) ) ).

tff(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_635,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => code_term_of(filter(A10)) ) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_636,axiom,
    ! [A10: $tType] : order_top(filter(A10)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_637,axiom,
    ! [A10: $tType] : order_bot(filter(A10)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_638,axiom,
    ! [A10: $tType] : preorder(filter(A10)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_639,axiom,
    ! [A10: $tType] : order(filter(A10)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_640,axiom,
    ! [A10: $tType] : ord(filter(A10)) ).

tff(tcon_Filter_Ofilter___HOL_Oequal_641,axiom,
    ! [A10: $tType] :
      ( cl_HOL_Oequal(A10)
     => cl_HOL_Oequal(filter(A10)) ) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_642,axiom,
    ! [A10: $tType] :
      ( comple5582772986160207858norder(A10)
     => condit6923001295902523014norder(option(A10)) ) ).

tff(tcon_Option_Ooption___Quickcheck__Narrowing_Opartial__term__of_643,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => quickc6926020345158392990erm_of(option(A10)) ) ).

tff(tcon_Option_Ooption___Code__Evaluation_Oterm__of_644,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => code_term_of(option(A10)) ) ).

tff(tcon_Option_Ooption___Orderings_Owellorder_645,axiom,
    ! [A10: $tType] :
      ( wellorder(A10)
     => wellorder(option(A10)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__top_646,axiom,
    ! [A10: $tType] :
      ( order_top(A10)
     => order_top(option(A10)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__bot_647,axiom,
    ! [A10: $tType] :
      ( order(A10)
     => order_bot(option(A10)) ) ).

tff(tcon_Option_Ooption___Orderings_Opreorder_648,axiom,
    ! [A10: $tType] :
      ( preorder(A10)
     => preorder(option(A10)) ) ).

tff(tcon_Option_Ooption___Orderings_Olinorder_649,axiom,
    ! [A10: $tType] :
      ( linorder(A10)
     => linorder(option(A10)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_650,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => finite_finite(option(A10)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder_651,axiom,
    ! [A10: $tType] :
      ( order(A10)
     => order(option(A10)) ) ).

tff(tcon_Option_Ooption___Orderings_Oord_652,axiom,
    ! [A10: $tType] :
      ( preorder(A10)
     => ord(option(A10)) ) ).

tff(tcon_Option_Ooption___Heap_Oheap_653,axiom,
    ! [A10: $tType] :
      ( heap(A10)
     => heap(option(A10)) ) ).

tff(tcon_Option_Ooption___HOL_Oequal_654,axiom,
    ! [A10: $tType] : cl_HOL_Oequal(option(A10)) ).

tff(tcon_Option_Ooption___Nat_Osize_655,axiom,
    ! [A10: $tType] : size(option(A10)) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bit__operations_656,axiom,
    bit_se359711467146920520ations(uint32) ).

tff(tcon_Uint32_Ouint32___Quickcheck__Narrowing_Opartial__term__of_657,axiom,
    quickc6926020345158392990erm_of(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Comprehension_Obit__comprehension_658,axiom,
    bit_bi6583157726757044596ension(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Oring__bit__operations_659,axiom,
    bit_ri3973907225187159222ations(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__comm__monoid__add_660,axiom,
    cancel1802427076303600483id_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1__cancel_661,axiom,
    comm_s4317794764714335236cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bits_662,axiom,
    bit_semiring_bits(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__semigroup__add_663,axiom,
    cancel_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Least__significant__bit_Olsb_664,axiom,
    least_6119777620449941438nt_lsb(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__mult_665,axiom,
    ab_semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1__cancel_666,axiom,
    semiring_1_cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__mult_667,axiom,
    comm_monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Generic__set__bit_Oset__bit_668,axiom,
    generic_set_set_bit(uint32) ).

tff(tcon_Uint32_Ouint32___Code__Evaluation_Oterm__of_669,axiom,
    code_term_of(uint32) ).

tff(tcon_Uint32_Ouint32___Parity_Osemiring__parity_670,axiom,
    semiring_parity(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__add_671,axiom,
    comm_monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__modulo_672,axiom,
    semiring_modulo(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1_673,axiom,
    comm_semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__0_674,axiom,
    comm_semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Osemiring__numeral_675,axiom,
    semiring_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring_676,axiom,
    comm_semiring(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__group__add_677,axiom,
    ab_group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ozero__neq__one_678,axiom,
    zero_neq_one(uint32) ).

tff(tcon_Uint32_Ouint32___Parity_Oring__parity_679,axiom,
    ring_parity(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Opreorder_680,axiom,
    preorder(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Olinorder_681,axiom,
    linorder(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__mult_682,axiom,
    monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring__1_683,axiom,
    comm_ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__add_684,axiom,
    monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1_685,axiom,
    semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__0_686,axiom,
    semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ogroup__add_687,axiom,
    group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Omult__zero_688,axiom,
    mult_zero(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring_689,axiom,
    comm_ring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oorder_690,axiom,
    order(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Oneg__numeral_691,axiom,
    neg_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring_692,axiom,
    semiring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oord_693,axiom,
    ord(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ouminus_694,axiom,
    uminus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring__1_695,axiom,
    ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Power_Opower_696,axiom,
    power(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Onumeral_697,axiom,
    numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ozero_698,axiom,
    zero(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oplus_699,axiom,
    plus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring_700,axiom,
    ring(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oone_701,axiom,
    one(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Odvd_702,axiom,
    dvd(uint32) ).

tff(tcon_Uint32_Ouint32___HOL_Oequal_703,axiom,
    cl_HOL_Oequal(uint32) ).

tff(tcon_Uint32_Ouint32___Nat_Osize_704,axiom,
    size(uint32) ).

tff(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_705,axiom,
    boolea8198339166811842893lgebra(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_706,axiom,
    ab_semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_707,axiom,
    comm_monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__top_708,axiom,
    order_top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__bot_709,axiom,
    order_bot(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Opreorder_710,axiom,
    preorder(assn) ).

tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_711,axiom,
    monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder_712,axiom,
    order(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oord_713,axiom,
    ord(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ouminus_714,axiom,
    uminus(assn) ).

tff(tcon_Assertions_Oassn___Power_Opower_715,axiom,
    power(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oone_716,axiom,
    one(assn) ).

tff(tcon_Assertions_Oassn___Rings_Odvd_717,axiom,
    dvd(assn) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_718,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_719,axiom,
    topolo3112930676232923870pology(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_720,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_721,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_722,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_723,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_724,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_725,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_726,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Quickcheck__Narrowing_Opartial__term__of_727,axiom,
    quickc6926020345158392990erm_of(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_728,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_729,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_730,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_731,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_732,axiom,
    real_V8037385150606011577_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_733,axiom,
    topolo4211221413907600880p_mult(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_734,axiom,
    topolo8386298272705272623_space(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_735,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_736,axiom,
    real_V7819770556892013058_space(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_737,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_738,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_739,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_740,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_741,axiom,
    topolo6943815403480290642id_add(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_742,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_743,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__group__add_744,axiom,
    topolo1633459387980952147up_add(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_745,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_746,axiom,
    topological_t1_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_747,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_748,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_749,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_750,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_751,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Code__Evaluation_Oterm__of_752,axiom,
    code_term_of(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_753,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_754,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_755,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_756,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_757,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_758,axiom,
    field_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_759,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_760,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_761,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_762,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_763,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_764,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_765,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_766,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_767,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_768,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_769,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_770,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_771,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_772,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_773,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_774,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_775,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_776,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_777,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Oinverse_778,axiom,
    inverse(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom_779,axiom,
    semidom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_780,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_781,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_782,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_783,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_784,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_785,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_786,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_787,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_788,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_789,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_790,axiom,
    dvd(complex) ).

tff(tcon_Complex_Ocomplex___HOL_Oequal_791,axiom,
    cl_HOL_Oequal(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_792,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_793,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_794,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_795,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Quickcheck__Narrowing_Opartial__term__of_796,axiom,
    quickc6926020345158392990erm_of(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_797,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_798,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_799,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_800,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_801,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_802,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_803,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_804,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Code__Evaluation_Oterm__of_805,axiom,
    code_term_of(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_806,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_807,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_808,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_809,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_810,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_811,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_812,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_813,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_814,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_815,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_816,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_817,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_818,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_819,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_820,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_821,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_822,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_823,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_824,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_825,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_826,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_827,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_828,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_829,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_830,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_831,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_832,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_833,axiom,
    dvd(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___HOL_Oequal_834,axiom,
    cl_HOL_Oequal(extended_enat) ).

tff(tcon_Multiset_Omultiset___Quickcheck__Narrowing_Opartial__term__of_835,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => quickc6926020345158392990erm_of(multiset(A10)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_836,axiom,
    ! [A10: $tType] :
      ( preorder(A10)
     => ordere6658533253407199908up_add(multiset(A10)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_837,axiom,
    ! [A10: $tType] : cancel1802427076303600483id_add(multiset(A10)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_838,axiom,
    ! [A10: $tType] : cancel_semigroup_add(multiset(A10)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_839,axiom,
    ! [A10: $tType] : comm_monoid_diff(multiset(A10)) ).

tff(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_840,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => code_term_of(multiset(A10)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_841,axiom,
    ! [A10: $tType] : comm_monoid_add(multiset(A10)) ).

tff(tcon_Multiset_Omultiset___Orderings_Opreorder_842,axiom,
    ! [A10: $tType] :
      ( preorder(A10)
     => preorder(multiset(A10)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_843,axiom,
    ! [A10: $tType] : monoid_add(multiset(A10)) ).

tff(tcon_Multiset_Omultiset___Orderings_Oorder_844,axiom,
    ! [A10: $tType] :
      ( preorder(A10)
     => order(multiset(A10)) ) ).

tff(tcon_Multiset_Omultiset___Orderings_Oord_845,axiom,
    ! [A10: $tType] :
      ( preorder(A10)
     => ord(multiset(A10)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ozero_846,axiom,
    ! [A10: $tType] : zero(multiset(A10)) ).

tff(tcon_Multiset_Omultiset___Groups_Oplus_847,axiom,
    ! [A10: $tType] : plus(multiset(A10)) ).

tff(tcon_Multiset_Omultiset___HOL_Oequal_848,axiom,
    ! [A10: $tType] :
      ( cl_HOL_Oequal(A10)
     => cl_HOL_Oequal(multiset(A10)) ) ).

tff(tcon_Multiset_Omultiset___Nat_Osize_849,axiom,
    ! [A10: $tType] : size(multiset(A10)) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__comm__monoid__add_850,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => cancel1802427076303600483id_add(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1__cancel_851,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => comm_s4317794764714335236cancel(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__semigroup__add_852,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => cancel_semigroup_add(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__mult_853,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => ab_semigroup_mult(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1__cancel_854,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => semiring_1_cancel(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__mult_855,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => comm_monoid_mult(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Code__Evaluation_Oterm__of_856,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => code_term_of(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__add_857,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => comm_monoid_add(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1_858,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => comm_semiring_1(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__0_859,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => comm_semiring_0(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Osemiring__numeral_860,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => semiring_numeral(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring_861,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => comm_semiring(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Owellorder_862,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => wellorder(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__group__add_863,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => ab_group_add(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ozero__neq__one_864,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => zero_neq_one(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Opreorder_865,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => preorder(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Olinorder_866,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => linorder(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__mult_867,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => monoid_mult(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring__1_868,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => comm_ring_1(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__add_869,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => monoid_add(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Finite__Set_Ofinite_870,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => finite_finite(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen0_871,axiom,
    ! [A10: $tType] :
      ( type_len0(A10)
     => type_len0(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1_872,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => semiring_1(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__0_873,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => semiring_0(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ogroup__add_874,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => group_add(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen_875,axiom,
    ! [A10: $tType] :
      ( type_len(A10)
     => type_len(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Omult__zero_876,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => mult_zero(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring_877,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => comm_ring(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oorder_878,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => order(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Oneg__numeral_879,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => neg_numeral(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring_880,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => semiring(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oord_881,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => ord(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ouminus_882,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => uminus(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring__1_883,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => ring_1(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Power_Opower_884,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => power(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Onumeral_885,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => numeral(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ozero_886,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => zero(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oplus_887,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => plus(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring_888,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => ring(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oone_889,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => one(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Odvd_890,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => dvd(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit0___HOL_Oequal_891,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => cl_HOL_Oequal(numeral_bit0(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__comm__monoid__add_892,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => cancel1802427076303600483id_add(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1__cancel_893,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => comm_s4317794764714335236cancel(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__semigroup__add_894,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => cancel_semigroup_add(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__mult_895,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => ab_semigroup_mult(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1__cancel_896,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => semiring_1_cancel(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__mult_897,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => comm_monoid_mult(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Code__Evaluation_Oterm__of_898,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => code_term_of(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__add_899,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => comm_monoid_add(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1_900,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => comm_semiring_1(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__0_901,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => comm_semiring_0(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Osemiring__numeral_902,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => semiring_numeral(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring_903,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => comm_semiring(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Owellorder_904,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => wellorder(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__group__add_905,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => ab_group_add(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ozero__neq__one_906,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => zero_neq_one(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Opreorder_907,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => preorder(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Olinorder_908,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => linorder(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__mult_909,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => monoid_mult(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring__1_910,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => comm_ring_1(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__add_911,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => monoid_add(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Finite__Set_Ofinite_912,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => finite_finite(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen0_913,axiom,
    ! [A10: $tType] :
      ( type_len0(A10)
     => type_len0(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1_914,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => semiring_1(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__0_915,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => semiring_0(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ogroup__add_916,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => group_add(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen_917,axiom,
    ! [A10: $tType] :
      ( type_len0(A10)
     => type_len(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Omult__zero_918,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => mult_zero(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring_919,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => comm_ring(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oorder_920,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => order(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Oneg__numeral_921,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => neg_numeral(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring_922,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => semiring(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oord_923,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => ord(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ouminus_924,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => uminus(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring__1_925,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => ring_1(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Power_Opower_926,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => power(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Onumeral_927,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => numeral(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ozero_928,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => zero(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oplus_929,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => plus(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring_930,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => ring(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oone_931,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => one(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Odvd_932,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => dvd(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Obit1___HOL_Oequal_933,axiom,
    ! [A10: $tType] :
      ( finite_finite(A10)
     => cl_HOL_Oequal(numeral_bit1(A10)) ) ).

tff(tcon_Numeral__Type_Onum0___Quickcheck__Narrowing_Opartial__term__of_934,axiom,
    quickc6926020345158392990erm_of(numeral_num0) ).

tff(tcon_Numeral__Type_Onum0___Code__Evaluation_Oterm__of_935,axiom,
    code_term_of(numeral_num0) ).

tff(tcon_Numeral__Type_Onum0___Type__Length_Olen0_936,axiom,
    type_len0(numeral_num0) ).

tff(tcon_Numeral__Type_Onum0___HOL_Oequal_937,axiom,
    cl_HOL_Oequal(numeral_num0) ).

tff(tcon_Numeral__Type_Onum1___Quickcheck__Narrowing_Opartial__term__of_938,axiom,
    quickc6926020345158392990erm_of(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__comm__monoid__add_939,axiom,
    cancel1802427076303600483id_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__semigroup__add_940,axiom,
    cancel_semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__mult_941,axiom,
    ab_semigroup_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__mult_942,axiom,
    comm_monoid_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Code__Evaluation_Oterm__of_943,axiom,
    code_term_of(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__add_944,axiom,
    comm_monoid_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring__0_945,axiom,
    comm_semiring_0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring_946,axiom,
    comm_semiring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Owellorder_947,axiom,
    wellorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__group__add_948,axiom,
    ab_group_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Opreorder_949,axiom,
    preorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Olinorder_950,axiom,
    linorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Omonoid__mult_951,axiom,
    monoid_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Omonoid__add_952,axiom,
    monoid_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Finite__Set_Ofinite_953,axiom,
    finite_finite(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Type__Length_Olen0_954,axiom,
    type_len0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Osemiring__0_955,axiom,
    semiring_0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ogroup__add_956,axiom,
    group_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Type__Length_Olen_957,axiom,
    type_len(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Omult__zero_958,axiom,
    mult_zero(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__ring_959,axiom,
    comm_ring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Oorder_960,axiom,
    order(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Osemiring_961,axiom,
    semiring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Oord_962,axiom,
    ord(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ouminus_963,axiom,
    uminus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Power_Opower_964,axiom,
    power(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Num_Onumeral_965,axiom,
    numeral(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ozero_966,axiom,
    zero(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oplus_967,axiom,
    plus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Oring_968,axiom,
    ring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oone_969,axiom,
    one(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Odvd_970,axiom,
    dvd(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___HOL_Oequal_971,axiom,
    cl_HOL_Oequal(numeral_num1) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_972,axiom,
    ! [A10: $tType,A11: $tType] :
      ( ( topolo4958980785337419405_space(A10)
        & topolo4958980785337419405_space(A11) )
     => topolo4958980785337419405_space(product_prod(A10,A11)) ) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Narrowing_Opartial__term__of_973,axiom,
    ! [A10: $tType,A11: $tType] :
      ( ( typerep(A10)
        & typerep(A11) )
     => quickc6926020345158392990erm_of(product_prod(A10,A11)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_974,axiom,
    ! [A10: $tType,A11: $tType] :
      ( ( topological_t2_space(A10)
        & topological_t2_space(A11) )
     => topological_t2_space(product_prod(A10,A11)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_975,axiom,
    ! [A10: $tType,A11: $tType] :
      ( ( topological_t1_space(A10)
        & topological_t1_space(A11) )
     => topological_t1_space(product_prod(A10,A11)) ) ).

tff(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_976,axiom,
    ! [A10: $tType,A11: $tType] :
      ( ( typerep(A10)
        & typerep(A11) )
     => code_term_of(product_prod(A10,A11)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_977,axiom,
    ! [A10: $tType,A11: $tType] :
      ( ( finite_finite(A10)
        & finite_finite(A11) )
     => finite_finite(product_prod(A10,A11)) ) ).

tff(tcon_Product__Type_Oprod___Heap_Oheap_978,axiom,
    ! [A10: $tType,A11: $tType] :
      ( ( heap(A10)
        & heap(A11) )
     => heap(product_prod(A10,A11)) ) ).

tff(tcon_Product__Type_Oprod___HOL_Oequal_979,axiom,
    ! [A10: $tType,A11: $tType] : cl_HOL_Oequal(product_prod(A10,A11)) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_980,axiom,
    ! [A10: $tType,A11: $tType] : size(product_prod(A10,A11)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_981,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Quickcheck__Narrowing_Opartial__term__of_982,axiom,
    quickc6926020345158392990erm_of(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_983,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_984,axiom,
    code_term_of(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_985,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_986,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_987,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_988,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_989,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Finite__Set_Ofinite_990,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_991,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_992,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_993,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Heap_Oheap_994,axiom,
    heap(product_unit) ).

tff(tcon_Product__Type_Ounit___HOL_Oequal_995,axiom,
    cl_HOL_Oequal(product_unit) ).

tff(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Narrowing_Opartial__term__of_996,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => quickc6926020345158392990erm_of(heap_ext(A10)) ) ).

tff(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_997,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => code_term_of(heap_ext(A10)) ) ).

tff(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_998,axiom,
    ! [A10: $tType] : cl_HOL_Oequal(heap_ext(A10)) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_999,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_1000,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_1001,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_1002,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_1003,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_1004,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_1005,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_1006,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_1007,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_1008,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_1009,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_1010,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_1011,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_1012,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_1013,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_1014,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_1015,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_1016,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Narrowing_Opartial__term__of_1017,axiom,
    quickc6926020345158392990erm_of(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_1018,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_1019,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_1020,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_1021,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_1022,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_1023,axiom,
    euclid5891614535332579305n_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_1024,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_1025,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_1026,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_1027,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_1028,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_1029,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_1030,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_1031,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_1032,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_1033,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_1034,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_1035,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_1036,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_1037,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_1038,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_1039,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Least__significant__bit_Olsb_1040,axiom,
    least_6119777620449941438nt_lsb(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_1041,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_1042,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_1043,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_1044,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_1045,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_1046,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Generic__set__bit_Oset__bit_1047,axiom,
    generic_set_set_bit(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_1048,axiom,
    code_term_of(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_1049,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_1050,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_1051,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_1052,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_1053,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_1054,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_1055,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_1056,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_1057,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_1058,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_1059,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_1060,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_1061,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_1062,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_1063,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_1064,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_1065,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_1066,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_1067,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_1068,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_1069,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_1070,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_1071,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_1072,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_1073,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_1074,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_1075,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_1076,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_1077,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_1078,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_1079,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_1080,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_1081,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_1082,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_1083,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_1084,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom_1085,axiom,
    semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_1086,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_1087,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_1088,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_1089,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_1090,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_1091,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_1092,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_1093,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_1094,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_1095,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_1096,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_1097,axiom,
    dvd(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___HOL_Oequal_1098,axiom,
    cl_HOL_Oequal(code_integer) ).

tff(tcon_Heap__Time__Monad_OHeap___Quickcheck__Narrowing_Opartial__term__of_1099,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => quickc6926020345158392990erm_of(heap_Time_Heap(A10)) ) ).

tff(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of_1100,axiom,
    ! [A10: $tType] :
      ( typerep(A10)
     => code_term_of(heap_Time_Heap(A10)) ) ).

tff(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_1101,axiom,
    ! [A10: $tType] : cl_HOL_Oequal(heap_Time_Heap(A10)) ).

tff(tcon_Heap__Time__Monad_OHeap___Nat_Osize_1102,axiom,
    ! [A10: $tType] : size(heap_Time_Heap(A10)) ).

tff(tcon_VEBT__Definitions_OVEBT___Quickcheck__Narrowing_Opartial__term__of_1103,axiom,
    quickc6926020345158392990erm_of(vEBT_VEBT) ).

tff(tcon_VEBT__Definitions_OVEBT___Code__Evaluation_Oterm__of_1104,axiom,
    code_term_of(vEBT_VEBT) ).

tff(tcon_VEBT__Definitions_OVEBT___HOL_Oequal_1105,axiom,
    cl_HOL_Oequal(vEBT_VEBT) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_1106,axiom,
    size(vEBT_VEBT) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Quickcheck__Narrowing_Opartial__term__of_1107,axiom,
    quickc6926020345158392990erm_of(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Code__Evaluation_Oterm__of_1108,axiom,
    code_term_of(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Heap_Oheap_1109,axiom,
    heap(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___HOL_Oequal_1110,axiom,
    cl_HOL_Oequal(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Nat_Osize_1111,axiom,
    size(vEBT_VEBTi) ).

% Helper facts (3)
tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X16: A,Y: A] :
      ( ( X16 != Y )
      | aa(A,$o,fequal(A,X16),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X16: A,Y: A] :
      ( ~ aa(A,$o,fequal(A,X16),Y)
      | ( X16 = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
      = ( ? [X6: A] : aa(A,$o,P,X6) ) ) ).

% Free types (2)
tff(tfree_0,hypothesis,
    semiring_1(a) ).

tff(tfree_1,hypothesis,
    linorder(b) ).

% Conjectures (2)
tff(conj_0,hypothesis,
    ! [T7: vEBT_VEBT,Ti3: vEBT_VEBTi,X3: nat] : refine_Imp_refines(option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_vebt_predi,Ti3),X3),f(T7,Ti3,X3)) ).

tff(conj_1,conjecture,
    refine_Imp_refines(option(nat),vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aTP_Lamp_ai(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_aj($o,fun($o,heap_Time_Heap(option(nat)))),tia),vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aTP_Lamp_bc(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_aj($o,fun($o,heap_Time_Heap(option(nat)))),tia)) ).