%------------------------------------------------------------------------------
% File     : ITP286_4 : TPTP v8.2.0. Released v8.0.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_SuccPredImperative 00347_018417
% 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_00347_018417 [Des22]

% Status   : Theorem
% Rating   : 1.00 v8.1.0
% Syntax   : Number of formulae    : 12425 (5195 unt;1820 typ;   0 def)
%            Number of atoms       : 18246 (9320 equ)
%            Maximal formula atoms :   47 (   1 avg)
%            Number of connectives : 18705 (2237   ~; 331   |;1635   &)
%                                         (1983 <=>;12519  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   35 (   5 avg)
%            Maximal term depth    :   36 (   2 avg)
%            Number of FOOLs       : 2233 (1070 fml;1163 var)
%            Number of X terms     : 1313 (   0  [];1010 ite; 303 let)
%            Number of types       :   22 (  21 usr)
%            Number of type conns  : 1636 (1373   >; 263   *;   0   +;   0  <<)
%            Number of predicates  :  209 ( 206 usr;   2 prp; 0-13 aty)
%            Number of functors    : 1624 (1624 usr; 124 con; 0-7 aty)
%            Number of variables   : 35715 (32125   !; 857   ?;35715   :)
%                                         (2733  !>;   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 20:22:35.936
%------------------------------------------------------------------------------
% Could-be-implicit typings (36)
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_Code__Evaluation_Oterm,type,
    code_term: $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_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_Heap_Oref,type,
    ref: $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 (1784)
tff(sy_cl_Finite__Set_Ofinite,type,
    finite_finite: 
      !>[A: $tType] : $o ).

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_Groups_Ominus,type,
    minus: 
      !>[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_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_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_Groups_Osemigroup__add,type,
    semigroup_add: 
      !>[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_Groups_Osemigroup__mult,type,
    semigroup_mult: 
      !>[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_Oab__semigroup__add,type,
    ab_semigroup_add: 
      !>[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_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_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_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_Ocancel__ab__semigroup__add,type,
    cancel2418104881723323429up_add: 
      !>[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_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_Oreal__algebra,type,
    real_V6157519004096292374lgebra: 
      !>[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_Euclidean__Division_Oeuclidean__ring,type,
    euclid5891614535332579305n_ring: 
      !>[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_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_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_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_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_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] : fun(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] : fun(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] : ( 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] : ( 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] : ( 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] : ( 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] : ( A > B ) ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

tff(sy_c_ATP_058Lamp__aq____,type,
    aTP_Lamp_aq: 
      !>[A: $tType,B: $tType] : fun(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__ar____,type,
    aTP_Lamp_ar: 
      !>[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__at____,type,
    aTP_Lamp_at: 
      !>[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__av____,type,
    aTP_Lamp_av: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

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

tff(sy_c_ATP_058Lamp__ax____,type,
    aTP_Lamp_ax: 
      !>[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__az____,type,
    aTP_Lamp_az: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ba____,type,
    aTP_Lamp_ba: 
      !>[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__bc____,type,
    aTP_Lamp_bc: 
      !>[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__be____,type,
    aTP_Lamp_be: 
      !>[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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__bn____,type,
    aTP_Lamp_bn: 
      !>[A: $tType,B: $tType] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__bo____,type,
    aTP_Lamp_bo: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__cq____,type,
    aTP_Lamp_cq: 
      !>[A: $tType,B: $tType] : ( 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] : ( 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] : fun(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] : fun(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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__dp____,type,
    aTP_Lamp_dp: 
      !>[A: $tType,B: $tType] : ( 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] : fun(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] : fun(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] : fun(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] : fun(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] : fun(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] : fun(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] : ( 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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fq____,type,
    aTP_Lamp_fq: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fr____,type,
    aTP_Lamp_fr: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__fs____,type,
    aTP_Lamp_fs: 
      !>[A: $tType,B: $tType] : ( 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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gi____,type,
    aTP_Lamp_gi: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gj____,type,
    aTP_Lamp_gj: 
      !>[A: $tType,B: $tType] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gk____,type,
    aTP_Lamp_gk: 
      !>[A: $tType,B: $tType] : ( 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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__gn____,type,
    aTP_Lamp_gn: 
      !>[A: $tType,B: $tType] : ( 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] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__ha____,type,
    aTP_Lamp_ha: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__jw____,type,
    aTP_Lamp_jw: 
      !>[A: $tType,B: $tType] : ( 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] : ( 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] : ( 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] : ( 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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__mb____,type,
    aTP_Lamp_mb: 
      !>[A: $tType,B: $tType] : ( 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] : ( 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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__mh____,type,
    aTP_Lamp_mh: 
      !>[A: $tType,B: $tType] : ( 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] : ( 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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__nm____,type,
    aTP_Lamp_nm: 
      !>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__oz____,type,
    aTP_Lamp_oz: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : ( 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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__rn____,type,
    aTP_Lamp_rn: 
      !>[A: $tType,B: $tType] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__rs____,type,
    aTP_Lamp_rs: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__se____,type,
    aTP_Lamp_se: 
      !>[A: $tType,B: $tType] : fun(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] : fun(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] : fun(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] : ( 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] : fun(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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__vh____,type,
    aTP_Lamp_vh: 
      !>[A: $tType,B: $tType] : fun(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] : ( 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] : fun(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] : fun(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] : ( 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] : fun(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] : ( 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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__ww____,type,
    aTP_Lamp_ww: 
      !>[A: $tType,B: $tType] : ( 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] : fun(A,B) ).

tff(sy_c_ATP_058Lamp__xg____,type,
    aTP_Lamp_xg: 
      !>[A: $tType,B: $tType] : fun(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] : ( A > B ) ).

tff(sy_c_ATP_058Lamp__xo____,type,
    aTP_Lamp_xo: 
      !>[A: $tType,B: $tType] : ( 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] : ( 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] : ( 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__QUERY,type,
    fI_QUERY: ( 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 * 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_Oor__not__num__neg__rel,type,
    bit_or3848514188828904588eg_rel: fun(product_prod(num,num),fun(product_prod(num,num),$o)) ).

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] : ( ( nat * 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] : ( ( nat * 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_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__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_Onegative,type,
    code_negative: fun(num,code_integer) ).

tff(sy_c_Code__Numeral_Onum__of__integer,type,
    code_num_of_integer: code_integer > num ).

tff(sy_c_Code__Numeral_Opositive,type,
    code_positive: fun(num,code_integer) ).

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_Onarrowing__samples__rel,type,
    code_T1710151556404007877es_rel: 
      !>[A: $tType] : ( fun(code_integer,product_prod(A,A)) > fun(code_integer,fun(code_integer,$o)) ) ).

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__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_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] : fun(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(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] : fun(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_OtimeFrame__rel,type,
    heap_T5500966940807335491me_rel: 
      !>[A: $tType] : fun(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),fun(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),$o)) ).

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_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) > fun(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_Oextract,type,
    extract: 
      !>[A: $tType] : ( ( fun(A,$o) * list(A) ) > option(product_prod(list(A),product_prod(A,list(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_Olinorder__class_Osorted__list__of__set,type,
    linord4507533701916653071of_set: 
      !>[A: $tType] : ( set(A) > list(A) ) ).

tff(sy_c_List_Olist_OCons,type,
    cons: 
      !>[A: $tType] : fun(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) > fun(list(A),list(Aa)) ) ).

tff(sy_c_List_Olist_Oset,type,
    set2: 
      !>[A: $tType] : fun(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_Osplice,type,
    splice: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > list(A) ) ).

tff(sy_c_List_Osplice__rel,type,
    splice_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).

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_Oinv__on,type,
    inv_on: 
      !>[A: $tType,B: $tType] : ( ( fun(A,B) * set(A) ) > fun(B,A) ) ).

tff(sy_c_Misc_Omergesort,type,
    mergesort: 
      !>[A: $tType] : fun(list(A),list(A)) ).

tff(sy_c_Misc_Omergesort__by__rel,type,
    mergesort_by_rel: 
      !>[A: $tType] : ( fun(A,fun(A,$o)) > fun(list(A),list(A)) ) ).

tff(sy_c_Misc_Omergesort__by__rel__merge,type,
    merges9089515139780605204_merge: 
      !>[A: $tType] : ( ( fun(A,fun(A,$o)) * list(A) * list(A) ) > list(A) ) ).

tff(sy_c_Misc_Omergesort__by__rel__merge__rel,type,
    merges2244889521215249637ge_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),fun(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o)) ).

tff(sy_c_Misc_Omergesort__by__rel__rel,type,
    mergesort_by_rel_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,$o)),list(A)),fun(product_prod(fun(A,fun(A,$o)),list(A)),$o)) ).

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_Omergesort__by__rel__split__rel,type,
    merges7066485432131860899it_rel: 
      !>[A: $tType] : fun(product_prod(product_prod(list(A),list(A)),list(A)),fun(product_prod(product_prod(list(A),list(A)),list(A)),$o)) ).

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_Opairself__rel,type,
    pairself_rel: 
      !>[A: $tType,B: $tType] : fun(product_prod(fun(A,B),product_prod(A,A)),fun(product_prod(fun(A,B),product_prod(A,A)),$o)) ).

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_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_Multiset_Osubset__eq__mset__impl,type,
    subset_eq_mset_impl: 
      !>[A: $tType] : ( ( list(A) * list(A) ) > option($o) ) ).

tff(sy_c_Multiset_Osubset__eq__mset__impl__rel,type,
    subset751672762298770561pl_rel: 
      !>[A: $tType] : fun(product_prod(list(A),list(A)),fun(product_prod(list(A),list(A)),$o)) ).

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: $tType] : fun(A,fun(A,$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_Oimg__lub,type,
    partial_img_lub: 
      !>[A: $tType,C: $tType,D: $tType,B: $tType] : ( ( fun(A,C) * fun(D,B) * fun(set(C),D) ) > fun(set(A),B) ) ).

tff(sy_c_Partial__Function_Oimg__ord,type,
    partial_img_ord: 
      !>[A: $tType,C: $tType,B: $tType] : ( ( fun(A,C) * fun(C,fun(C,B)) ) > fun(A,fun(A,B)) ) ).

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_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] : ( 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_Ofold__atLeastAtMost__nat__rel,type,
    set_fo1817059534552279752at_rel: 
      !>[A: $tType] : fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),fun(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o)) ).

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_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__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_Omonoseq,type,
    topological_monoseq: 
      !>[A: $tType] : ( fun(nat,A) > $o ) ).

tff(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
    topolo174197925503356063within: 
      !>[A: $tType] : ( ( A * set(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: real > real ).

tff(sy_c_Transcendental_Oarcosh,type,
    arcosh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Oarcsin,type,
    arcsin: real > real ).

tff(sy_c_Transcendental_Oarctan,type,
    arctan: real > real ).

tff(sy_c_Transcendental_Oarsinh,type,
    arsinh: 
      !>[A: $tType] : ( A > A ) ).

tff(sy_c_Transcendental_Oartanh,type,
    artanh: 
      !>[A: $tType] : ( 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] : ( A > A ) ).

tff(sy_c_Transcendental_Odiffs,type,
    diffs: 
      !>[A: $tType] : ( ( fun(nat,A) * nat ) > A ) ).

tff(sy_c_Transcendental_Oexp,type,
    exp: 
      !>[A: $tType] : ( 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 * 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] : ( A > A ) ).

tff(sy_c_Transcendental_Otanh,type,
    tanh: 
      !>[A: $tType] : ( 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_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__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__List__Assn_Olist__assn__rel,type,
    vEBT_L4249061453398456502sn_rel: 
      !>[A: $tType,C: $tType] : fun(product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),fun(product_prod(fun(A,fun(C,assn)),product_prod(list(A),list(C))),$o)) ).

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_Ooption__shift__rel,type,
    vEBT_V459564278314245337ft_rel: 
      !>[A: $tType] : fun(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),fun(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),$o)) ).

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)) > fun(A,$o) ) ).

tff(sy_c_Wellfounded_Ofinite__psubset,type,
    finite_psubset: 
      !>[A: $tType] : set(product_prod(set(A),set(A))) ).

tff(sy_c_Wellfounded_Omeasure,type,
    measure: 
      !>[A: $tType] : ( fun(A,nat) > set(product_prod(A,A)) ) ).

tff(sy_c_Word_OWord,type,
    word2: 
      !>[A: $tType] : ( int > word(A) ) ).

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__cast,type,
    signed_cast: 
      !>[A: $tType,B: $tType] : ( word(A) > word(B) ) ).

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 > fun(word(A),word(B)) ) ).

tff(sy_c_Word_Oslice1,type,
    slice1: 
      !>[A: $tType,B: $tType] : ( nat > fun(word(A),word(B)) ) ).

tff(sy_c_Word_Othe__signed__int,type,
    the_signed_int: 
      !>[A: $tType] : ( word(A) > int ) ).

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 > fun(word(A),word(A)) ) ).

tff(sy_c_Word_Oword__rotr,type,
    word_rotr: 
      !>[A: $tType] : ( nat > fun(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: $tType] : fun(A,fun(A,$o)) ).

tff(sy_c_member,type,
    member: 
      !>[A: $tType] : ( ( A * set(A) ) > $o ) ).

tff(sy_v_ma____,type,
    ma: nat ).

tff(sy_v_mi____,type,
    mi: nat ).

tff(sy_v_na____,type,
    na: nat ).

tff(sy_v_summary____,type,
    summary: vEBT_VEBT ).

tff(sy_v_tia____,type,
    tia: vEBT_VEBTi ).

tff(sy_v_treeList____,type,
    treeList: list(vEBT_VEBT) ).

tff(sy_v_tree__is,type,
    tree_is: list(vEBT_VEBTi) ).

tff(sy_v_va____,type,
    va: nat ).

tff(sy_v_x11______,type,
    x11: option(product_prod(nat,nat)) ).

tff(sy_v_x12______,type,
    x12: nat ).

tff(sy_v_x13______,type,
    x13: array(vEBT_VEBTi) ).

tff(sy_v_x14______,type,
    x14: vEBT_VEBTi ).

tff(sy_v_xa____,type,
    xa: nat ).

% Relevant facts (9281)
tff(fact_0_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_1_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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Xb) ) ) ) ).

% max_in_set_def
tff(fact_2_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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),X) ) ) ) ).

% min_in_set_def
tff(fact_3_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_4_bit__split__inv,axiom,
    ! [Xb: nat,D2: nat] : vEBT_VEBT_bit_concat(vEBT_VEBT_high(Xb,D2),vEBT_VEBT_low(Xb,D2),D2) = Xb ).

% bit_split_inv
tff(fact_5_mimapr,axiom,
    aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),mi),ma) ).

% mimapr
tff(fact_6_high__def,axiom,
    ! [Xb: nat,Nb: nat] : vEBT_VEBT_high(Xb,Nb) = 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))),Nb)) ).

% high_def
tff(fact_7_vebt__minti__h,axiom,
    ! [Ta: 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,Ta),Tib),vEBT_vebt_minti(Tib),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_ar(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Ta),Tib)) ).

% vebt_minti_h
tff(fact_8_vebt__maxti__h,axiom,
    ! [Ta: 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,Ta),Tib),vEBT_vebt_maxti(Tib),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_as(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Ta),Tib)) ).

% vebt_maxti_h
tff(fact_9_lesseq__shift,axiom,
    ! [Xb: nat,Y: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_10__C7_OIH_C_I2_J,axiom,
    ! [Xaa: nat,Xb: nat,Xc: option(nat),Nb: nat,Tib: vEBT_VEBTi] :
      ( ~ aa(nat,$o,ord_less(nat,ma),xa)
     => ( ( Xaa = vEBT_VEBT_low(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
       => ( ( Xb = vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => ( aa(nat,$o,ord_less(nat,Xb),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),treeList))
           => ( ( Xc = vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),Xb)) )
             => ( ~ ( ( Xc != none(nat) )
                    & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),Xaa),Xc) )
               => ( vEBT_invar_vebt(summary,Nb)
                 => hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,summary),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,summary),Tib),Xb),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_at(nat,fun(vEBT_VEBTi,fun(option(nat),assn)),Xb),Tib)) ) ) ) ) ) ) ) ).

% "7.IH"(2)
tff(fact_11__C7_OIH_C_I1_J,axiom,
    ! [Xaa: nat,Xb: nat,Xc: option(nat),Nb: nat,Tib: vEBT_VEBTi] :
      ( ~ aa(nat,$o,ord_less(nat,ma),xa)
     => ( ( Xaa = vEBT_VEBT_low(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
       => ( ( Xb = vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,va))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => ( aa(nat,$o,ord_less(nat,Xb),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),treeList))
           => ( ( Xc = vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),Xb)) )
             => ( ( ( Xc != none(nat) )
                  & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),Xaa),Xc) )
               => ( vEBT_invar_vebt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),Xb),Nb)
                 => hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),Xb)),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,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),Xb)),Tib),Xaa),aa(vEBT_VEBTi,fun(option(nat),assn),aa(nat,fun(vEBT_VEBTi,fun(option(nat),assn)),aTP_Lamp_au(nat,fun(nat,fun(vEBT_VEBTi,fun(option(nat),assn))),Xaa),Xb),Tib)) ) ) ) ) ) ) ) ).

% "7.IH"(1)
tff(fact_12_pred__less__length__list,axiom,
    ! [Deg: nat,Xb: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Mia: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Maa)
       => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
         => ( 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),Mia),Maa)),Deg,TreeLista,Summarya),Xb) = $let(
                l: nat,
                l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),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),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),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),Deg),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,TreeLista),h),l)),
                      $let(
                        pr: option(nat),
                        pr:= vEBT_vebt_pred(Summarya,h),
                        $ite(
                          pr = none(nat),
                          $ite(aa(nat,$o,ord_less(nat,Mia),Xb),aa(nat,option(nat),some(nat),Mia),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),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
tff(fact_13_pred__lesseq__max,axiom,
    ! [Deg: nat,Xb: nat,Maa: nat,Mia: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Maa)
       => ( 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),Mia),Maa)),Deg,TreeLista,Summarya),Xb) = $let(
              l: nat,
              l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),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),Deg),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)),TreeLista)),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),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),Deg),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,TreeLista),h),l)),
                      $let(
                        pr: option(nat),
                        pr:= vEBT_vebt_pred(Summarya,h),
                        $ite(
                          pr = none(nat),
                          $ite(aa(nat,$o,ord_less(nat,Mia),Xb),aa(nat,option(nat),some(nat),Mia),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),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% pred_lesseq_max
tff(fact_14_vebt__maxti__hT,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : time_htt(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_maxti(Tib),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_as(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Ta),Tib),one_one(nat)) ).

% vebt_maxti_hT
tff(fact_15_vebt__minti__hT,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : time_htt(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),vEBT_vebt_minti(Tib),aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_ar(vEBT_VEBT,fun(vEBT_VEBTi,fun(option(nat),assn)),Ta),Tib),one_one(nat)) ).

% vebt_minti_hT
tff(fact_16_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_17_Suc__1,axiom,
    aa(nat,nat,suc,one_one(nat)) = aa(num,nat,numeral_numeral(nat),bit0(one2)) ).

% Suc_1
tff(fact_18_power__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xb: nat,Y: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
          <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Y) ) ) ) ).

% power_increasing_iff
tff(fact_19_power__strict__increasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,Xb: nat,Y: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),B2)
         => ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Y))
          <=> aa(nat,$o,ord_less(nat,Xb),Y) ) ) ) ).

% power_strict_increasing_iff
tff(fact_20_divide__less__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))),A2)
        <=> aa(A,$o,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))) ) ) ).

% divide_less_eq_numeral1(1)
tff(fact_21_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))),B2) ) ) ).

% less_divide_eq_numeral1(1)
tff(fact_22_deg__deg__n,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeLista,Summarya),Nb)
     => ( Deg = Nb ) ) ).

% deg_deg_n
tff(fact_23_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_24_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Tree,aa(nat,nat,suc,aa(nat,nat,suc,Nb)))
     => ? [Info2: option(product_prod(nat,nat)),TreeList: list(vEBT_VEBT),S: vEBT_VEBT] : Tree = vEBT_Node(Info2,aa(nat,nat,suc,aa(nat,nat,suc,Nb)),TreeList,S) ) ).

% deg_SUcn_Node
tff(fact_25_numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: num,Nb: num] :
          ( ( aa(num,A,numeral_numeral(A),M) = aa(num,A,numeral_numeral(A),Nb) )
        <=> ( M = Nb ) ) ) ).

% numeral_eq_iff
tff(fact_26_numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Nb) ) ) ).

% numeral_le_iff
tff(fact_27_numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: num,Nb: num] :
          ( aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(num,$o,ord_less(num,M),Nb) ) ) ).

% numeral_less_iff
tff(fact_28_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_29_numeral__times__numeral,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [M: num,Nb: 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),Nb)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),Nb)) ) ).

% numeral_times_numeral
tff(fact_30_num__double,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(one2)),Nb) = bit0(Nb) ).

% num_double
tff(fact_31_misiz,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,M: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( aa(nat,option(nat),some(nat),M) = vEBT_vebt_mint(Ta) )
       => 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))),Nb)) ) ) ).

% misiz
tff(fact_32_power__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),one_one(A)),Nb) = one_one(A) ) ).

% power_one
tff(fact_33_power__mult__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,Nb: num] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),M))),aa(num,nat,numeral_numeral(nat),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),times_times(num),M),Nb))) ) ).

% power_mult_numeral
tff(fact_34_helpypredd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_pred(Ta,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))),Nb)) ) ) ).

% helpypredd
tff(fact_35_power__one__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),one_one(nat)) = A2 ) ).

% power_one_right
tff(fact_36_mi__ma__2__deg,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: 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),Mia),Maa)),Deg,TreeLista,Summarya),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Maa)
        & aa(nat,$o,ord_less(nat,Maa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)) ) ) ).

% mi_ma_2_deg
tff(fact_37_pred__max,axiom,
    ! [Deg: nat,Maa: nat,Xb: nat,Mia: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,ord_less(nat,Maa),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),Mia),Maa)),Deg,TreeLista,Summarya),Xb) = aa(nat,option(nat),some(nat),Maa) ) ) ) ).

% pred_max
tff(fact_38_summaxma,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),Deg,TreeLista,Summarya),Deg)
     => ( ( Mia != Maa )
       => ( aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(Summarya)) = vEBT_VEBT_high(Maa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% summaxma
tff(fact_39_one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: num] :
          ( ( one_one(A) = aa(num,A,numeral_numeral(A),Nb) )
        <=> ( one2 = Nb ) ) ) ).

% one_eq_numeral_iff
tff(fact_40_numeral__eq__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: num] :
          ( ( aa(num,A,numeral_numeral(A),Nb) = one_one(A) )
        <=> ( Nb = one2 ) ) ) ).

% numeral_eq_one_iff
tff(fact_41_power__inject__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,Nb: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A2)
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) )
          <=> ( M = Nb ) ) ) ) ).

% power_inject_exp
tff(fact_42_pred__list__to__short,axiom,
    ! [Deg: nat,Xb: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Mia: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Maa)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),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),Mia),Maa)),Deg,TreeLista,Summarya),Xb) = none(nat) ) ) ) ) ).

% pred_list_to_short
tff(fact_43_numeral__le__one__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Nb)),one_one(A))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Nb),one2) ) ) ).

% numeral_le_one_iff
tff(fact_44_mem__Collect__eq,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      ( member(A,A2,collect(A,P))
    <=> aa(A,$o,P,A2) ) ).

% mem_Collect_eq
tff(fact_45_Collect__mem__eq,axiom,
    ! [A: $tType,A3: set(A)] : collect(A,aTP_Lamp_av(set(A),fun(A,$o),A3)) = A3 ).

% Collect_mem_eq
tff(fact_46_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_47_HOL_Oext,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),G: fun(A,B)] :
      ( ! [X2: A] : aa(A,B,F,X2) = aa(A,B,G,X2)
     => ( F = G ) ) ).

% HOL.ext
tff(fact_48_one__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] :
          ( aa(A,$o,ord_less(A,one_one(A)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(num,$o,ord_less(num,one2),Nb) ) ) ).

% one_less_numeral_iff
tff(fact_49_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))),B2) ) ) ).

% le_divide_eq_numeral1(1)
tff(fact_50_divide__le__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))) ) ) ).

% divide_le_eq_numeral1(1)
tff(fact_51_mul__def,axiom,
    vEBT_VEBT_mul = vEBT_V2048590022279873568_shift(nat,times_times(nat)) ).

% mul_def
tff(fact_52__C7_Oprems_C,axiom,
    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)),aa(nat,nat,suc,aa(nat,nat,suc,va)),treeList,summary),na) ).

% "7.prems"
tff(fact_53_local_Opower__def,axiom,
    vEBT_VEBT_power = vEBT_V2048590022279873568_shift(nat,power_power(nat)) ).

% local.power_def
tff(fact_54_le__num__One__iff,axiom,
    ! [Xb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Xb),one2)
    <=> ( Xb = one2 ) ) ).

% le_num_One_iff
tff(fact_55_power__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: nat,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),Nb) ) ).

% power_mult
tff(fact_56_div__mult2__eq,axiom,
    ! [M: nat,Nb: 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),Nb),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),Q2) ).

% div_mult2_eq
tff(fact_57_div__mult2__numeral__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: 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),A2),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),A2),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_58_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),Nb))
     => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),I) ) ).

% less_mult_imp_div_less
tff(fact_59_times__div__less__eq__dividend,axiom,
    ! [Nb: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb))),M) ).

% times_div_less_eq_dividend
tff(fact_60_div__times__less__eq__dividend,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$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),Nb)),Nb)),M) ).

% div_times_less_eq_dividend
tff(fact_61_power__even__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power_even_eq
tff(fact_62_div__nat__eqI,axiom,
    ! [Nb: nat,Q2: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)),M)
     => ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Q2)))
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) = Q2 ) ) ) ).

% div_nat_eqI
tff(fact_63_le__numeral__extra_I4_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),one_one(A)) ) ).

% le_numeral_extra(4)
tff(fact_64_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_65_power__commuting__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xb: A,Y: A,Nb: 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),Nb)),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),Nb)) ) ) ) ).

% power_commuting_commutes
tff(fact_66_power__mult__distrib,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ).

% power_mult_distrib
tff(fact_67_power__commutes,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ).

% power_commutes
tff(fact_68_power__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),Nb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ).

% power_divide
tff(fact_69_div__le__dividend,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),M) ).

% div_le_dividend
tff(fact_70_div__le__mono,axiom,
    ! [M: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$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),Nb),K)) ) ).

% div_le_mono
tff(fact_71_one__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% one_le_numeral
tff(fact_72_not__numeral__less__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : ~ aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),Nb)),one_one(A)) ) ).

% not_numeral_less_one
tff(fact_73_numeral__One,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ( aa(num,A,numeral_numeral(A),one2) = one_one(A) ) ) ).

% numeral_One
tff(fact_74_mult__numeral__1__right,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),one2)) = A2 ) ).

% mult_numeral_1_right
tff(fact_75_mult__numeral__1,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),one2)),A2) = A2 ) ).

% mult_numeral_1
tff(fact_76_one__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% one_le_power
tff(fact_77_divide__numeral__1,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),one2)) = A2 ) ).

% divide_numeral_1
tff(fact_78_left__right__inverse__power,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Xb: A,Y: A,Nb: 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),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Nb)) = one_one(A) ) ) ) ).

% left_right_inverse_power
tff(fact_79_power__Suc2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),A2) ) ).

% power_Suc2
tff(fact_80_power__Suc,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ).

% power_Suc
tff(fact_81_power__one__over,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Nb: 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)),A2)),Nb) = 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),A2),Nb)) ) ).

% power_one_over
tff(fact_82_numerals_I1_J,axiom,
    aa(num,nat,numeral_numeral(nat),one2) = one_one(nat) ).

% numerals(1)
tff(fact_83_Suc__div__le__mono,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,M)),Nb)) ).

% Suc_div_le_mono
tff(fact_84_numeral__Bit0__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),bit0(Nb))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(num,A,numeral_numeral(A),Nb) ) ).

% numeral_Bit0_div_2
tff(fact_85_power__less__power__Suc,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A2)
         => aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))) ) ) ).

% power_less_power_Suc
tff(fact_86_power__gt1__lemma,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A2)
         => aa(A,$o,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))) ) ) ).

% power_gt1_lemma
tff(fact_87_power__gt1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A2)
         => aa(A,$o,ord_less(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb))) ) ) ).

% power_gt1
tff(fact_88_power__strict__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N: nat,A2: A] :
          ( aa(nat,$o,ord_less(nat,Nb),N)
         => ( aa(A,$o,ord_less(A,one_one(A)),A2)
           => aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ) ) ).

% power_strict_increasing
tff(fact_89_power__less__imp__less__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,Nb: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A2)
         => ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
           => aa(nat,$o,ord_less(nat,M),Nb) ) ) ) ).

% power_less_imp_less_exp
tff(fact_90_power__increasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)) ) ) ) ).

% power_increasing
tff(fact_91_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_92_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_93_power2__eq__square,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) ) ).

% power2_eq_square
tff(fact_94_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_95_power__le__imp__le__exp,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,M: nat,Nb: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ) ).

% power_le_imp_le_exp
tff(fact_96_power2__nat__le__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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)))),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% power2_nat_le_imp_le
tff(fact_97_power2__nat__le__eq__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% power2_nat_le_eq_le
tff(fact_98_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$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_99_power__odd__eq,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% power_odd_eq
tff(fact_100_vebt__pred_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya),Xb) = $ite(
        aa(nat,$o,ord_less(nat,Maa),Xb),
        aa(nat,option(nat),some(nat),Maa),
        $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,Vaa))),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,Vaa))),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)),TreeLista)),
              $let(
                minlow: option(nat),
                minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),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,Vaa))),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,TreeLista),h),l)),
                  $let(
                    pr: option(nat),
                    pr:= vEBT_vebt_pred(Summarya,h),
                    $ite(
                      pr = none(nat),
                      $ite(aa(nat,$o,ord_less(nat,Mia),Xb),aa(nat,option(nat),some(nat),Mia),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,Vaa))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),pr)),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),pr)))) ) ) ) ),
              none(nat) ) ) ) ) ).

% vebt_pred.simps(7)
tff(fact_101_tdeletemimi_H,axiom,
    ! [Deg: nat,Mia: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => aa(nat,$o,aa(nat,fun(nat,$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),Mia),Mia)),Deg,TreeLista,Summarya),Xb)),one_one(nat)) ) ).

% tdeletemimi'
tff(fact_102__092_060open_062_092_060And_062x22_Ax21_O_Ati_A_061_ALeafi_Ax21_Ax22_A_092_060Longrightarrow_062_A_060vebt__assn__raw_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_062_Avebt__predi_H_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_Ax_A_060_092_060lambda_062r_O_Avebt__assn__raw_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ati_A_K_A_092_060up_062_A_Ir_A_061_Avebt__pred_A_INode_A_ISome_A_Imi_M_Ama_J_J_A_ISuc_A_ISuc_Ava_J_J_AtreeList_Asummary_J_Ax_J_062_092_060close_062,axiom,
    ! [X21: $o,X22: $o] :
      ( ( tia = vEBT_Leafi((X21),(X22)) )
     => hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,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,summary)),tia),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,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,summary)),tia),xa),aTP_Lamp_aw(option(nat),assn)) ) ).

% \<open>\<And>x22 x21. ti = Leafi x21 x22 \<Longrightarrow> <vebt_assn_raw (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti> vebt_predi' (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti x <\<lambda>r. vebt_assn_raw (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) ti * \<up> (r = vebt_pred (Node (Some (mi, ma)) (Suc (Suc va)) treeList summary) x)>\<close>
tff(fact_103_maxt__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta),Xb)
       => ( vEBT_vebt_maxt(Ta) = aa(nat,option(nat),some(nat),Xb) ) ) ) ).

% maxt_sound
tff(fact_104_maxt__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_maxt(Ta) = aa(nat,option(nat),some(nat),Xb) )
       => vEBT_VEBT_max_in_set(vEBT_VEBT_set_vebt(Ta),Xb) ) ) ).

% maxt_corr
tff(fact_105_mint__sound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta),Xb)
       => ( vEBT_vebt_mint(Ta) = aa(nat,option(nat),some(nat),Xb) ) ) ) ).

% mint_sound
tff(fact_106_mint__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_mint(Ta) = aa(nat,option(nat),some(nat),Xb) )
       => vEBT_VEBT_min_in_set(vEBT_VEBT_set_vebt(Ta),Xb) ) ) ).

% mint_corr
tff(fact_107_nested__mint,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat,Vaa: 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),Mia),Maa)),Deg,TreeLista,Summarya),Nb)
     => ( ( Nb = aa(nat,nat,suc,aa(nat,nat,suc,Vaa)) )
       => ( ~ aa(nat,$o,ord_less(nat,Maa),Mia)
         => ( ( Maa != Mia )
           => aa(nat,$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(Summarya))),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),Vaa),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya)))))),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Vaa),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)) ) ) ) ) ).

% nested_mint
tff(fact_108_member__inv,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Deg,TreeLista,Summarya)),Xb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
        & ( ( Xb = Mia )
          | ( Xb = Maa )
          | ( aa(nat,$o,ord_less(nat,Xb),Maa)
            & aa(nat,$o,ord_less(nat,Mia),Xb)
            & aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
            & aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% member_inv
tff(fact_109_succ__list__to__short,axiom,
    ! [Deg: nat,Mia: nat,Xb: nat,TreeLista: list(vEBT_VEBT),Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Xb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista)),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),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),Mia),Maa)),Deg,TreeLista,Summarya),Xb) = none(nat) ) ) ) ) ).

% succ_list_to_short
tff(fact_110_lowi__hT,axiom,
    ! [Xb: nat,Nb: nat] : time_htt(nat,one_one(assn),vEBT_VEBT_lowi(Xb,Nb),aa(nat,fun(nat,assn),aTP_Lamp_ax(nat,fun(nat,fun(nat,assn)),Xb),Nb),one_one(nat)) ).

% lowi_hT
tff(fact_111_highi__hT,axiom,
    ! [Xb: nat,Nb: nat] : time_htt(nat,one_one(assn),vEBT_VEBT_highi(Xb,Nb),aa(nat,fun(nat,assn),aTP_Lamp_ay(nat,fun(nat,fun(nat,assn)),Xb),Nb),one_one(nat)) ).

% highi_hT
tff(fact_112_even__odd__cases,axiom,
    ! [Xb: nat] :
      ( ! [N2: nat] : Xb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),N2)
     => ~ ! [N2: nat] : Xb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(nat,nat,suc,N2)) ) ).

% even_odd_cases
tff(fact_113_set__vebt_H__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_VEBT_set_vebt(Ta) = collect(nat,vEBT_vebt_member(Ta)) ).

% set_vebt'_def
tff(fact_114_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_115_add__def,axiom,
    vEBT_VEBT_add = vEBT_V2048590022279873568_shift(nat,plus_plus(nat)) ).

% add_def
tff(fact_116_mint__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_mint(Ta) = aa(nat,option(nat),some(nat),Maxi) )
       => aa(nat,$o,vEBT_vebt_member(Ta),Maxi) ) ) ).

% mint_member
tff(fact_117_maxt__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_maxt(Ta) = aa(nat,option(nat),some(nat),Maxi) )
       => aa(nat,$o,vEBT_vebt_member(Ta),Maxi) ) ) ).

% maxt_member
tff(fact_118_mint__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Mini: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_mint(Ta) = aa(nat,option(nat),some(nat),Mini) )
       => ( aa(nat,$o,vEBT_vebt_member(Ta),Xb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mini),Xb) ) ) ) ).

% mint_corr_help
tff(fact_119_maxt__corr__help,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Maxi: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_maxt(Ta) = aa(nat,option(nat),some(nat),Maxi) )
       => ( aa(nat,$o,vEBT_vebt_member(Ta),Xb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Maxi) ) ) ) ).

% maxt_corr_help
tff(fact_120_pow__sum,axiom,
    ! [A2: nat,B2: 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),A2),B2))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A2)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2) ).

% pow_sum
tff(fact_121_high__bound__aux,axiom,
    ! [Maa: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Maa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M)))
     => aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Maa,Nb)),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_122_member__bound,axiom,
    ! [Tree: vEBT_VEBT,Xb: nat,Nb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Tree),Xb)
     => ( vEBT_invar_vebt(Tree,Nb)
       => aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ) ) ).

% member_bound
tff(fact_123_geqmaxNone,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: 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),Mia),Maa)),Deg,TreeLista,Summarya),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Maa),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),Mia),Maa)),Deg,TreeLista,Summarya),Xb) = none(nat) ) ) ) ).

% geqmaxNone
tff(fact_124_high__inv,axiom,
    ! [Xb: nat,Nb: 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))),Nb))
     => ( 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))),Nb))),Xb),Nb) = Y ) ) ).

% high_inv
tff(fact_125_low__inv,axiom,
    ! [Xb: nat,Nb: 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))),Nb))
     => ( 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))),Nb))),Xb),Nb) = Xb ) ) ).

% low_inv
tff(fact_126_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_127_numeral__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [M: num,Nb: 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),Nb)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Nb)) ) ).

% numeral_plus_numeral
tff(fact_128_bit__concat__def,axiom,
    ! [H: nat,L: nat,D2: nat] : vEBT_VEBT_bit_concat(H,L,D2) = 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))),D2))),L) ).

% bit_concat_def
tff(fact_129_helpyd,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_succ(Ta,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))),Nb)) ) ) ).

% helpyd
tff(fact_130_succ__min,axiom,
    ! [Deg: nat,Xb: nat,Mia: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,ord_less(nat,Xb),Mia)
       => ( 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),Mia),Maa)),Deg,TreeLista,Summarya),Xb) = aa(nat,option(nat),some(nat),Mia) ) ) ) ).

% succ_min
tff(fact_131_member__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_vebt_member(Ta),Xb)
      <=> member(nat,Xb,vEBT_set_vebt(Ta)) ) ) ).

% member_correct
tff(fact_132_set__vebt__set__vebt_H__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_set_vebt(Ta) = vEBT_VEBT_set_vebt(Ta) ) ) ).

% set_vebt_set_vebt'_valid
tff(fact_133_distrib__left__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [V: num,B2: 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),B2),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)),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),V)),C2)) ) ).

% distrib_left_numeral
tff(fact_134_distrib__right__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & semiring(A) )
     => ! [A2: A,B2: 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),A2),B2)),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),A2),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).

% distrib_right_numeral
tff(fact_135_numeral__plus__one,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Nb)),one_one(A)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),Nb),one2)) ) ).

% numeral_plus_one
tff(fact_136_one__plus__numeral,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Nb)) ) ).

% one_plus_numeral
tff(fact_137_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_138_add__2__eq__Suc,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) = aa(nat,nat,suc,aa(nat,nat,suc,Nb)) ).

% add_2_eq_Suc
tff(fact_139_add__2__eq__Suc_H,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,suc,Nb)) ).

% add_2_eq_Suc'
tff(fact_140_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_141_pred__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat,Y: nat] :
      ( vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta),Xb,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(Ta),Y)
        & aa(nat,$o,ord_less(nat,Y),Xb)
        & ! [Z2: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(Ta),Z2)
              & aa(nat,$o,ord_less(nat,Z2),Xb) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Z2),Y) ) ) ) ).

% pred_member
tff(fact_142_succ__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat,Y: nat] :
      ( vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta),Xb,Y)
    <=> ( aa(nat,$o,vEBT_vebt_member(Ta),Y)
        & aa(nat,$o,ord_less(nat,Xb),Y)
        & ! [Z2: nat] :
            ( ( aa(nat,$o,vEBT_vebt_member(Ta),Z2)
              & aa(nat,$o,ord_less(nat,Xb),Z2) )
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Z2) ) ) ) ).

% succ_member
tff(fact_143_pred__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Px: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = aa(nat,option(nat),some(nat),Px) )
      <=> vEBT_is_pred_in_set(vEBT_VEBT_set_vebt(Ta),Xb,Px) ) ) ).

% pred_corr
tff(fact_144_succ__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_VEBT_set_vebt(Ta),Xb,Sx) ) ) ).

% succ_corr
tff(fact_145_vebt__succi_H__rf__abstr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => hoare_hoare_triple(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),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,Ta),Tib),Xb),aa(nat,fun(option(nat),assn),aa(vEBT_VEBTi,fun(nat,fun(option(nat),assn)),aTP_Lamp_az(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),Ta),Tib),Xb)) ) ).

% vebt_succi'_rf_abstr
tff(fact_146_is__num__normalize_I1_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A2: A,B2: 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% is_num_normalize(1)
tff(fact_147_numeral__Bit0,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] : aa(num,A,numeral_numeral(A),bit0(Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(num,A,numeral_numeral(A),Nb)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% numeral_Bit0
tff(fact_148_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_149_power__add,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: nat,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ).

% power_add
tff(fact_150_numeral__code_I2_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] :
          aa(num,A,numeral_numeral(A),bit0(Nb)) = $let(
            m: A,
            m:= aa(num,A,numeral_numeral(A),Nb),
            aa(A,A,aa(A,fun(A,A),plus_plus(A),m),m) ) ) ).

% numeral_code(2)
tff(fact_151_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_152_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_153_left__add__twice,axiom,
    ! [A: $tType] :
      ( semiring_numeral(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = 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))),A2)),B2) ) ).

% left_add_twice
tff(fact_154_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_155_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_156_ex__power__ivl1,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),K)
       => ? [N2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N2)),K)
            & aa(nat,$o,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat)))) ) ) ) ).

% ex_power_ivl1
tff(fact_157_ex__power__ivl2,axiom,
    ! [B2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
       => ? [N2: nat] :
            ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),N2)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),one_one(nat)))) ) ) ) ).

% ex_power_ivl2
tff(fact_158_succ__greatereq__min,axiom,
    ! [Deg: nat,Mia: nat,Xb: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),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),Mia),Maa)),Deg,TreeLista,Summarya),Xb) = $let(
              l: nat,
              l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),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),Deg),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)),TreeLista)),
                  $let(
                    maxlow: option(nat),
                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),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),Deg),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,TreeLista),h),l)),
                      $let(
                        sc: option(nat),
                        sc:= vEBT_vebt_succ(Summarya,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),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                  none(nat) ) ) ) ) ) ) ).

% succ_greatereq_min
tff(fact_159_succ__less__length__list,axiom,
    ! [Deg: nat,Mia: nat,Xb: nat,TreeLista: list(vEBT_VEBT),Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Xb)
       => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
         => ( 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),Mia),Maa)),Deg,TreeLista,Summarya),Xb) = $let(
                l: nat,
                l:= vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),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),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),
                  $let(
                    maxlow: option(nat),
                    maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),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),Deg),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,TreeLista),h),l)),
                      $let(
                        sc: option(nat),
                        sc:= vEBT_vebt_succ(Summarya,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),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),sc))))) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
tff(fact_160_vebt__member_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya)),Xb)
    <=> $ite(
          Xb = Mia,
          $true,
          $ite(
            Xb = Maa,
            $true,
            $ite(
              aa(nat,$o,ord_less(nat,Xb),Mia),
              $false,
              $ite(
                aa(nat,$o,ord_less(nat,Maa),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,Vaa))),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)),TreeLista)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ).

% vebt_member.simps(5)
tff(fact_161_post__member__pre__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
       => ( aa(nat,$o,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
         => ( aa(nat,$o,vEBT_vebt_member(vEBT_vebt_insert(Ta,Xb)),Y)
           => ( aa(nat,$o,vEBT_vebt_member(Ta),Y)
              | ( Xb = Y ) ) ) ) ) ) ).

% post_member_pre_member
tff(fact_162_vebt__assn__raw_Osimps_I2_J,axiom,
    ! [Mmo: option(product_prod(nat,nat)),Deg: nat,Tree_list: list(vEBT_VEBT),Summarya: vEBT_VEBT,Mmoi: option(product_prod(nat,nat)),Degi: nat,Tree_array: array(vEBT_VEBTi),Summaryi: vEBT_VEBTi] :
      aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_Node(Mmo,Deg,Tree_list,Summarya)),vEBT_Nodei(Mmoi,Degi,Tree_array,Summaryi)) = 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,Summarya),Summaryi))),
        ex_assn(list(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn),aTP_Lamp_ba(list(vEBT_VEBT),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn)),Tree_list),Tree_array))) ).

% vebt_assn_raw.simps(2)
tff(fact_163_insert__simp__mima,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        | ( Xb = Maa ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
       => ( 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),Mia),Maa)),Deg,TreeLista,Summarya),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),Mia),Maa)),Deg,TreeLista,Summarya) ) ) ) ).

% insert_simp_mima
tff(fact_164_mintlistlength,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: 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),Mia),Maa)),Deg,TreeLista,Summarya),Nb)
     => ( ( Mia != Maa )
       => ( aa(nat,$o,ord_less(nat,Mia),Maa)
          & ? [M2: nat] :
              ( ( aa(nat,option(nat),some(nat),M2) = vEBT_vebt_mint(Summarya) )
              & aa(nat,$o,ord_less(nat,M2),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,Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ) ) ).

% mintlistlength
tff(fact_165_nth__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xs: list(A),A2: array(A)] :
          ( aa(nat,$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,A2),Xs),aa(nat,heap_Time_Heap(A),array_nth(A,A2),I),aa(array(A),fun(A,assn),aa(list(A),fun(array(A),fun(A,assn)),aTP_Lamp_bb(nat,fun(list(A),fun(array(A),fun(A,assn))),I),Xs),A2)) ) ) ).

% nth_rule
tff(fact_166_power__2__mult__step__le,axiom,
    ! [N3: nat,Nb: nat,K2: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N3),Nb)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N3)),K2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),K))
       => aa(nat,$o,aa(nat,fun(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))),N3)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K2),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))),Nb)),K)) ) ) ).

% power_2_mult_step_le
tff(fact_167_norm__pre__pure__iff__sng,axiom,
    ! [A: $tType,B2: $o,F: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( hoare_hoare_triple(A,pure_assn((B2)),F,Q)
    <=> ( (B2)
       => hoare_hoare_triple(A,one_one(assn),F,Q) ) ) ).

% norm_pre_pure_iff_sng
tff(fact_168_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [Xb: nat,Deg: nat,TreeLista: list(vEBT_VEBT),Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Deg)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),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),Mia),Maa)),Deg,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
tff(fact_169_norm__pre__pure__iff,axiom,
    ! [A: $tType,P: assn,B2: $o,F: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B2))),F,Q)
    <=> ( (B2)
       => hoare_hoare_triple(A,P,F,Q) ) ) ).

% norm_pre_pure_iff
tff(fact_170_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_171_maxbmo,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( ( vEBT_vebt_maxt(Ta) = aa(nat,option(nat),some(nat),Xb) )
     => aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xb) ) ).

% maxbmo
tff(fact_172_both__member__options__equiv__member,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xb)
      <=> aa(nat,$o,vEBT_vebt_member(Ta),Xb) ) ) ).

% both_member_options_equiv_member
tff(fact_173_valid__member__both__member__options,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xb)
       => aa(nat,$o,vEBT_vebt_member(Ta),Xb) ) ) ).

% valid_member_both_member_options
tff(fact_174_highi__h,axiom,
    ! [Xb: nat,Nb: nat] : hoare_hoare_triple(nat,one_one(assn),vEBT_VEBT_highi(Xb,Nb),aa(nat,fun(nat,assn),aTP_Lamp_ay(nat,fun(nat,fun(nat,assn)),Xb),Nb)) ).

% highi_h
tff(fact_175_lowi__h,axiom,
    ! [Xb: nat,Nb: nat] : hoare_hoare_triple(nat,one_one(assn),vEBT_VEBT_lowi(Xb,Nb),aa(nat,fun(nat,assn),aTP_Lamp_ax(nat,fun(nat,fun(nat,assn)),Xb),Nb)) ).

% lowi_h
tff(fact_176_VEBTi_Oinject_I1_J,axiom,
    ! [X11a: option(product_prod(nat,nat)),X12a: nat,X13a: array(vEBT_VEBTi),X14a: vEBT_VEBTi,Y11: option(product_prod(nat,nat)),Y12: nat,Y13: array(vEBT_VEBTi),Y14: vEBT_VEBTi] :
      ( ( vEBT_Nodei(X11a,X12a,X13a,X14a) = vEBT_Nodei(Y11,Y12,Y13,Y14) )
    <=> ( ( X11a = Y11 )
        & ( X12a = Y12 )
        & ( X13a = Y13 )
        & ( X14a = Y14 ) ) ) ).

% VEBTi.inject(1)
tff(fact_177_VEBTi_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( vEBT_Leafi((X21),(X22)) = vEBT_Leafi((Y21),(Y22)) )
    <=> ( ( (X21)
        <=> (Y21) )
        & ( (X22)
        <=> (Y22) ) ) ) ).

% VEBTi.inject(2)
tff(fact_178_pred__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_pred_in_set(vEBT_set_vebt(Ta),Xb,Sx) ) ) ).

% pred_correct
tff(fact_179_succ__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Sx: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = aa(nat,option(nat),some(nat),Sx) )
      <=> vEBT_is_succ_in_set(vEBT_set_vebt(Ta),Xb,Sx) ) ) ).

% succ_correct
tff(fact_180_valid__insert__both__member__options__pres,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
       => ( aa(nat,$o,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
         => ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xb)
           => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Y)),Xb) ) ) ) ) ).

% valid_insert_both_member_options_pres
tff(fact_181_valid__insert__both__member__options__add,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
       => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_insert(Ta,Xb)),Xb) ) ) ).

% valid_insert_both_member_options_add
tff(fact_182_sumprop,axiom,
    vEBT_invar_vebt(summary,aa(nat,nat,minus_minus(nat,na),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),na),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% sumprop
tff(fact_183_listlength,axiom,
    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))),aa(nat,nat,minus_minus(nat,na),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),na),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% listlength
tff(fact_184_right__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [V: num,B2: 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,B2),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)),B2)),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_185_left__diff__distrib__numeral,axiom,
    ! [A: $tType] :
      ( ( numeral(A)
        & ring(A) )
     => ! [A2: A,B2: A,V: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),aa(num,A,numeral_numeral(A),V)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),V))),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(num,A,numeral_numeral(A),V))) ) ).

% left_diff_distrib_numeral
tff(fact_186_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_bc(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),I),Ts),Tsi)) ) ).

% vebt_mintilist
tff(fact_187_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_bd(nat,fun(list(vEBT_VEBT),fun(list(vEBT_VEBTi),fun(option(nat),assn))),I),Ts),Tsi)) ) ).

% vebt_maxtilist
tff(fact_188_both__member__options__ding,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(vEBT_Node(Info,Deg,TreeLista,Summarya),Nb)
     => ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
         => aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(Info,Deg,TreeLista,Summarya)),Xb) ) ) ) ).

% both_member_options_ding
tff(fact_189_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mia: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Deg)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mia),Maa)),Deg,TreeLista,Summarya)),Xb)
       => ( aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))
          | ( Xb = Mia )
          | ( Xb = Maa ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
tff(fact_190_Suc__numeral,axiom,
    ! [Nb: num] : aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),Nb)) = aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),Nb),one2)) ).

% Suc_numeral
tff(fact_191_power__add__numeral2,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,Nb: num,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),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),A2),aa(num,nat,numeral_numeral(nat),Nb))),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Nb)))),B2) ) ).

% power_add_numeral2
tff(fact_192_power__add__numeral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A,M: num,Nb: num] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),Nb))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Nb))) ) ).

% power_add_numeral
tff(fact_193_vebt__memberi_H__rf__abstr,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : hoare_hoare_triple($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),vEBT_V854960066525838166emberi,Ta),Tib),Xb),aa(nat,fun($o,assn),aa(vEBT_VEBTi,fun(nat,fun($o,assn)),aTP_Lamp_be(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),Ta),Tib),Xb)) ).

% vebt_memberi'_rf_abstr
tff(fact_194_vebt__inserti_H__rf__abstr,axiom,
    ! [Ta: 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,Ta),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,Ta),Tib),Xb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(Ta,Xb))) ).

% vebt_inserti'_rf_abstr
tff(fact_195_diff__diff__less,axiom,
    ! [I: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,I),aa(nat,nat,minus_minus(nat,M),aa(nat,nat,minus_minus(nat,M),Nb)))
    <=> ( aa(nat,$o,ord_less(nat,I),M)
        & aa(nat,$o,ord_less(nat,I),Nb) ) ) ).

% diff_diff_less
tff(fact_196_add__One__commute,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Nb) = aa(num,num,aa(num,fun(num,num),plus_plus(num),Nb),one2) ).

% add_One_commute
tff(fact_197_if__rule,axiom,
    ! [A: $tType,B2: $o,P: assn,F: heap_Time_Heap(A),Q: fun(A,assn),G: heap_Time_Heap(A)] :
      ( ( (B2)
       => hoare_hoare_triple(A,P,F,Q) )
     => ( ( ~ (B2)
         => hoare_hoare_triple(A,P,G,Q) )
       => hoare_hoare_triple(A,P,
            $ite((B2),F,G),
            Q) ) ) ).

% if_rule
tff(fact_198_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_199_Suc__nat__number__of__add,axiom,
    ! [V: num,Nb: nat] : aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),Nb)) = 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))),Nb) ).

% Suc_nat_number_of_add
tff(fact_200_nat__power__less__diff,axiom,
    ! [Nb: nat,Q2: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),Q2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
     => aa(nat,$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),Nb))) ) ).

% nat_power_less_diff
tff(fact_201_nat__le__power__trans,axiom,
    ! [Nb: nat,M: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
       => aa(nat,$o,aa(nat,fun(nat,$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)),Nb)),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_202_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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Y) ) ) ) ) ).

% is_pred_in_set_def
tff(fact_203_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),Nb)),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),Nb))) ) ).

% diff_le_diff_pow
tff(fact_204_nat__less__power__trans,axiom,
    ! [Nb: nat,M: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),K)))
     => ( aa(nat,$o,aa(nat,fun(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)),Nb)),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_205_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_206_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_207_bind__rule,axiom,
    ! [A: $tType,B: $tType,P: assn,F: heap_Time_Heap(A),R: fun(A,assn),G: fun(A,heap_Time_Heap(B)),Q: fun(B,assn)] :
      ( hoare_hoare_triple(A,P,F,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,F,G),Q) ) ) ).

% bind_rule
tff(fact_208_VEBTi_Oexhaust,axiom,
    ! [Y: vEBT_VEBTi] :
      ( ! [X11: option(product_prod(nat,nat)),X12: nat,X13: array(vEBT_VEBTi),X14: vEBT_VEBTi] : Y != vEBT_Nodei(X11,X12,X13,X14)
     => ~ ! [X212: $o,X222: $o] : Y != vEBT_Leafi((X212),(X222)) ) ).

% VEBTi.exhaust
tff(fact_209_VEBTi_Odistinct_I1_J,axiom,
    ! [X11a: option(product_prod(nat,nat)),X12a: nat,X13a: array(vEBT_VEBTi),X14a: vEBT_VEBTi,X21: $o,X22: $o] : vEBT_Nodei(X11a,X12a,X13a,X14a) != vEBT_Leafi((X21),(X22)) ).

% VEBTi.distinct(1)
tff(fact_210_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_bf(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ) ).

% frame_rule
tff(fact_211_norm__pre__ex__rule,axiom,
    ! [A: $tType,B: $tType,P: fun(A,assn),F: heap_Time_Heap(B),Q: fun(B,assn)] :
      ( ! [X2: A] : hoare_hoare_triple(B,aa(A,assn,P,X2),F,Q)
     => hoare_hoare_triple(B,ex_assn(A,P),F,Q) ) ).

% norm_pre_ex_rule
tff(fact_212_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_bg(fun(A,fun(B,assn)),fun(B,fun(A,assn)),Q),Xb))
     => hoare_hoare_triple(A,P,C2,aTP_Lamp_bh(fun(A,fun(B,assn)),fun(A,assn),Q)) ) ).

% post_exI_rule
tff(fact_213_norm__pre__pure__rule1,axiom,
    ! [A: $tType,B2: $o,P: assn,F: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( ( (B2)
       => hoare_hoare_triple(A,P,F,Q) )
     => hoare_hoare_triple(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B2))),F,Q) ) ).

% norm_pre_pure_rule1
tff(fact_214_norm__pre__pure__rule2,axiom,
    ! [A: $tType,B2: $o,F: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( ( (B2)
       => hoare_hoare_triple(A,one_one(assn),F,Q) )
     => hoare_hoare_triple(A,pure_assn((B2)),F,Q) ) ).

% norm_pre_pure_rule2
tff(fact_215_bind__rule_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,F: heap_Time_Heap(B),R: fun(B,assn),G: fun(B,heap_Time_Heap(C)),Q: fun(C,assn)] :
      ( ! [R2: A] : hoare_hoare_triple(B,P,F,aa(fun(B,assn),fun(B,assn),aTP_Lamp_bi(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,F,G),Q) ) ) ).

% bind_rule'
tff(fact_216_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_217_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_bj(assn,fun(A,fun(A,assn)),P),Xb)) ).

% return_sp_rule
tff(fact_218_n__less__equal__power__2,axiom,
    ! [Nb: nat] : aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% n_less_equal_power_2
tff(fact_219_vebt__minti_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: 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),Mia),Maa)),Ux,Uy,Uz)) = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),Mia)) ).

% vebt_minti.simps(3)
tff(fact_220_vebt__maxti_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: 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),Mia),Maa)),Ux,Uy,Uz)) = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),Maa)) ).

% vebt_maxti.simps(3)
tff(fact_221_nat__add__offset__less,axiom,
    ! [Y: nat,Nb: nat,Xb: nat,M: nat,Sz: nat] :
      ( aa(nat,$o,ord_less(nat,Y),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
     => ( aa(nat,$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),Nb) )
         => 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))),Nb))),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_222_vebt__succ_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya),Xb) = $ite(
        aa(nat,$o,ord_less(nat,Xb),Mia),
        aa(nat,option(nat),some(nat),Mia),
        $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,Vaa))),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,Vaa))),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)),TreeLista)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),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,Vaa))),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,TreeLista),h),l)),
                  $let(
                    sc: option(nat),
                    sc:= vEBT_vebt_succ(Summarya,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,Vaa))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
              none(nat) ) ) ) ) ).

% vebt_succ.simps(6)
tff(fact_223_in__children__def,axiom,
    ! [Nb: nat,TreeLista: list(vEBT_VEBT),Xb: nat] :
      ( vEBT_V5917875025757280293ildren(Nb,TreeLista,Xb)
    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,Nb))),vEBT_VEBT_low(Xb,Nb)) ) ).

% in_children_def
tff(fact_224_Suc__diff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
       => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Nb),M)) = aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,minus_minus(nat,M),one_one(nat))) ) ) ) ).

% Suc_diff
tff(fact_225_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_226_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_227_semiring__norm_I76_J,axiom,
    ! [Nb: num] : aa(num,$o,ord_less(num,one2),bit0(Nb)) ).

% semiring_norm(76)
tff(fact_228_semiring__norm_I69_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),one2) ).

% semiring_norm(69)
tff(fact_229_less__two__pow__divI,axiom,
    ! [Xb: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,Nb),M)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
       => aa(nat,$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))),Nb)),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_230_less__two__pow__divD,axiom,
    ! [Xb: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
        & aa(nat,$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,Nb),M))) ) ) ).

% less_two_pow_divD
tff(fact_231_sum__squares__bound,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(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_232_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_233_highsimp,axiom,
    ! [Xb: nat,Nb: nat] : aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),vEBT_VEBT_high(Xb,Nb)) = vEBT_VEBT_highi(Xb,Nb) ).

% highsimp
tff(fact_234_lowsimp,axiom,
    ! [Xb: nat,Nb: nat] : aa(nat,heap_Time_Heap(nat),heap_Time_return(nat),vEBT_VEBT_low(Xb,Nb)) = vEBT_VEBT_lowi(Xb,Nb) ).

% lowsimp
tff(fact_235_semiring__norm_I87_J,axiom,
    ! [M: num,Nb: num] :
      ( ( bit0(M) = bit0(Nb) )
    <=> ( M = Nb ) ) ).

% semiring_norm(87)
tff(fact_236_nat_Oinject,axiom,
    ! [X23: nat,Y2: nat] :
      ( ( aa(nat,nat,suc,X23) = aa(nat,nat,suc,Y2) )
    <=> ( X23 = Y2 ) ) ).

% nat.inject
tff(fact_237_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_238_option_Oinject,axiom,
    ! [A: $tType,X23: A,Y2: A] :
      ( ( aa(A,option(A),some(A),X23) = aa(A,option(A),some(A),Y2) )
    <=> ( X23 = Y2 ) ) ).

% option.inject
tff(fact_239_highi__def,axiom,
    ! [Xb: nat,Nb: nat] : vEBT_VEBT_highi(Xb,Nb) = 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))),Nb))) ).

% highi_def
tff(fact_240_semiring__norm_I85_J,axiom,
    ! [M: num] : bit0(M) != one2 ).

% semiring_norm(85)
tff(fact_241_semiring__norm_I83_J,axiom,
    ! [Nb: num] : one2 != bit0(Nb) ).

% semiring_norm(83)
tff(fact_242_lessI,axiom,
    ! [Nb: nat] : aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,suc,Nb)) ).

% lessI
tff(fact_243_Suc__mono,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => aa(nat,$o,ord_less(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,Nb)) ) ).

% Suc_mono
tff(fact_244_Suc__less__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,ord_less(nat,M),Nb) ) ).

% Suc_less_eq
tff(fact_245_add__Suc__right,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)) ).

% add_Suc_right
tff(fact_246_Suc__le__mono,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,suc,M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M) ) ).

% Suc_le_mono
tff(fact_247_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb))
    <=> aa(nat,$o,ord_less(nat,M),Nb) ) ).

% nat_add_left_cancel_less
tff(fact_248_diff__Suc__Suc,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,Nb)) = aa(nat,nat,minus_minus(nat,M),Nb) ).

% diff_Suc_Suc
tff(fact_249_Suc__diff__diff,axiom,
    ! [M: nat,Nb: nat,K: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Nb)),aa(nat,nat,suc,K)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),Nb)),K) ).

% Suc_diff_diff
tff(fact_250_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% nat_add_left_cancel_le
tff(fact_251_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_252_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_253_diff__diff__cancel,axiom,
    ! [I: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),Nb)
     => ( aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,minus_minus(nat,Nb),I)) = I ) ) ).

% diff_diff_cancel
tff(fact_254_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_255_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_256_nat__1__eq__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( one_one(nat) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb) )
    <=> ( ( M = one_one(nat) )
        & ( Nb = one_one(nat) ) ) ) ).

% nat_1_eq_mult_iff
tff(fact_257_nat__mult__eq__1__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb) = one_one(nat) )
    <=> ( ( M = one_one(nat) )
        & ( Nb = one_one(nat) ) ) ) ).

% nat_mult_eq_1_iff
tff(fact_258_semiring__norm_I6_J,axiom,
    ! [M: num,Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),bit0(Nb)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Nb)) ).

% semiring_norm(6)
tff(fact_259_semiring__norm_I13_J,axiom,
    ! [M: num,Nb: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M)),bit0(Nb)) = bit0(bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),Nb))) ).

% semiring_norm(13)
tff(fact_260_semiring__norm_I12_J,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),times_times(num),one2),Nb) = Nb ).

% semiring_norm(12)
tff(fact_261_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_262_semiring__norm_I71_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),bit0(Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Nb) ) ).

% semiring_norm(71)
tff(fact_263_semiring__norm_I78_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,ord_less(num,bit0(M)),bit0(Nb))
    <=> aa(num,$o,ord_less(num,M),Nb) ) ).

% semiring_norm(78)
tff(fact_264_semiring__norm_I68_J,axiom,
    ! [Nb: num] : aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),one2),Nb) ).

% semiring_norm(68)
tff(fact_265_semiring__norm_I75_J,axiom,
    ! [M: num] : ~ aa(num,$o,ord_less(num,M),one2) ).

% semiring_norm(75)
tff(fact_266_diff__Suc__1,axiom,
    ! [Nb: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,Nb)),one_one(nat)) = Nb ).

% diff_Suc_1
tff(fact_267_mult__Suc__right,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb)) ).

% mult_Suc_right
tff(fact_268_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_269_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_270_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_271_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_272_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list(vEBT_VEBT),Uz: vEBT_VEBT,Vaa: nat] : vEBT_vebt_succ(vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz),Vaa) = none(nat) ).

% vebt_succ.simps(3)
tff(fact_273_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Xb: nat] : ~ aa(nat,$o,vEBT_vebt_member(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)),Xb) ).

% vebt_member.simps(2)
tff(fact_274_ord__eq__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => ( ( C2 = D2 )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),D2) ) ) ) ) ).

% ord_eq_le_eq_trans
tff(fact_275_measure__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X2: A] :
              ( ! [Y4: A] :
                  ( aa(B,$o,ord_less(B,aa(A,B,F,Y4)),aa(A,B,F,X2))
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X2) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct
tff(fact_276_measure__induct__rule,axiom,
    ! [B: $tType,A: $tType] :
      ( wellorder(B)
     => ! [F: fun(A,B),P: fun(A,$o),A2: A] :
          ( ! [X2: A] :
              ( ! [Y4: A] :
                  ( aa(B,$o,ord_less(B,aa(A,B,F,Y4)),aa(A,B,F,X2))
                 => aa(A,$o,P,Y4) )
             => aa(A,$o,P,X2) )
         => aa(A,$o,P,A2) ) ) ).

% measure_induct_rule
tff(fact_277_pairself_Ocases,axiom,
    ! [B: $tType,A: $tType,Xb: product_prod(fun(A,B),product_prod(A,A))] :
      ~ ! [F2: fun(A,B),A4: A,B3: 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)),F2),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),A4),B3)) ).

% pairself.cases
tff(fact_278_bex2I,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,S2: set(product_prod(A,B)),P: fun(A,fun(B,$o))] :
      ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2),S2)
     => ( ( 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),S2)
         => aa(B,$o,aa(A,fun(B,$o),P,A2),B2) )
       => ? [A4: A,B3: B] :
            ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3),S2)
            & aa(B,$o,aa(A,fun(B,$o),P,A4),B3) ) ) ) ).

% bex2I
tff(fact_279_Suc__inject,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( aa(nat,nat,suc,Xb) = aa(nat,nat,suc,Y) )
     => ( Xb = Y ) ) ).

% Suc_inject
tff(fact_280_n__not__Suc__n,axiom,
    ! [Nb: nat] : Nb != aa(nat,nat,suc,Nb) ).

% n_not_Suc_n
tff(fact_281_nat__neq__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M != Nb )
    <=> ( aa(nat,$o,ord_less(nat,M),Nb)
        | aa(nat,$o,ord_less(nat,Nb),M) ) ) ).

% nat_neq_iff
tff(fact_282_less__not__refl,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,ord_less(nat,Nb),Nb) ).

% less_not_refl
tff(fact_283_less__not__refl2,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),M)
     => ( M != Nb ) ) ).

% less_not_refl2
tff(fact_284_less__not__refl3,axiom,
    ! [S3: nat,Ta: nat] :
      ( aa(nat,$o,ord_less(nat,S3),Ta)
     => ( S3 != Ta ) ) ).

% less_not_refl3
tff(fact_285_less__irrefl__nat,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,ord_less(nat,Nb),Nb) ).

% less_irrefl_nat
tff(fact_286_nat__less__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,ord_less(nat,M3),N2)
             => aa(nat,$o,P,M3) )
         => aa(nat,$o,P,N2) )
     => aa(nat,$o,P,Nb) ) ).

% nat_less_induct
tff(fact_287_infinite__descent,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N2: nat] :
          ( ~ aa(nat,$o,P,N2)
         => ? [M3: nat] :
              ( aa(nat,$o,ord_less(nat,M3),N2)
              & ~ aa(nat,$o,P,M3) ) )
     => aa(nat,$o,P,Nb) ) ).

% infinite_descent
tff(fact_288_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_289_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_290_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_291_le__refl,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),Nb) ).

% le_refl
tff(fact_292_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K) ) ) ).

% le_trans
tff(fact_293_eq__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M = Nb )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% eq_imp_le
tff(fact_294_le__antisym,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
       => ( M = Nb ) ) ) ).

% le_antisym
tff(fact_295_nat__le__linear,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
      | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M) ) ).

% nat_le_linear
tff(fact_296_Nat_Oex__has__greatest__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y5: nat] :
            ( aa(nat,$o,P,Y5)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y5),B2) )
       => ? [X2: nat] :
            ( aa(nat,$o,P,X2)
            & ! [Y4: nat] :
                ( aa(nat,$o,P,Y4)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y4),X2) ) ) ) ) ).

% Nat.ex_has_greatest_nat
tff(fact_297_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_298_vebt__pred_Osimps_I4_J,axiom,
    ! [Uy: nat,Uz: list(vEBT_VEBT),Vaa: vEBT_VEBT,Vb: nat] : vEBT_vebt_pred(vEBT_Node(none(product_prod(nat,nat)),Uy,Uz,Vaa),Vb) = none(nat) ).

% vebt_pred.simps(4)
tff(fact_299_VEBT__internal_OminNulli_Ocases,axiom,
    ! [Xb: vEBT_VEBTi] :
      ( ( Xb != vEBT_Leafi($false,$false) )
     => ( ! [Uv2: $o] : Xb != vEBT_Leafi($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leafi((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: array(vEBT_VEBTi),Uy2: vEBT_VEBTi] : Xb != vEBT_Nodei(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod(nat,nat),Va: nat,Vb2: array(vEBT_VEBTi),Vc: vEBT_VEBTi] : Xb != vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va,Vb2,Vc) ) ) ) ) ).

% VEBT_internal.minNulli.cases
tff(fact_300_vebt__minti_Ocases,axiom,
    ! [Xb: vEBT_VEBTi] :
      ( ! [A4: $o,B3: $o] : Xb != vEBT_Leafi((A4),(B3))
     => ( ! [Uu2: nat,Uv2: array(vEBT_VEBTi),Uw2: vEBT_VEBTi] : Xb != vEBT_Nodei(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
       => ~ ! [Mi: nat,Ma: 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),Mi),Ma)),Ux2,Uy2,Uz2) ) ) ).

% vebt_minti.cases
tff(fact_301_vebt__maxti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: array(vEBT_VEBTi),Uw: vEBT_VEBTi] : vEBT_vebt_maxti(vEBT_Nodei(none(product_prod(nat,nat)),Uu,Uv,Uw)) = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)) ).

% vebt_maxti.simps(2)
tff(fact_302_vebt__minti_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: array(vEBT_VEBTi),Uw: vEBT_VEBTi] : vEBT_vebt_minti(vEBT_Nodei(none(product_prod(nat,nat)),Uu,Uv,Uw)) = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)) ).

% vebt_minti.simps(2)
tff(fact_303_option_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType,H: fun(B,A),F1: B,F22: fun(C,B),Option: option(C)] : aa(B,A,H,case_option(B,C,F1,F22,Option)) = case_option(A,C,aa(B,A,H,F1),aa(fun(C,B),fun(C,A),aTP_Lamp_bk(fun(B,A),fun(fun(C,B),fun(C,A)),H),F22),Option) ).

% option.case_distrib
tff(fact_304_z1pdiv2,axiom,
    ! [B2: 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))),B2)),one_one(int))),aa(num,int,numeral_numeral(int),bit0(one2))) = B2 ).

% z1pdiv2
tff(fact_305_le__some__optE,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M: A,Xb: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),aa(A,option(A),some(A),M)),Xb)
         => ~ ! [M4: A] :
                ( ( Xb = aa(A,option(A),some(A),M4) )
               => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),M4) ) ) ) ).

% le_some_optE
tff(fact_306_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_307_Suc__lessD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,M)),Nb)
     => aa(nat,$o,ord_less(nat,M),Nb) ) ).

% Suc_lessD
tff(fact_308_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_309_Suc__lessI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => ( ( aa(nat,nat,suc,M) != Nb )
       => aa(nat,$o,ord_less(nat,aa(nat,nat,suc,M)),Nb) ) ) ).

% Suc_lessI
tff(fact_310_less__SucE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,Nb))
     => ( ~ aa(nat,$o,ord_less(nat,M),Nb)
       => ( M = Nb ) ) ) ).

% less_SucE
tff(fact_311_less__SucI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,Nb)) ) ).

% less_SucI
tff(fact_312_Ex__less__Suc,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,suc,Nb))
          & aa(nat,$o,P,I2) )
    <=> ( aa(nat,$o,P,Nb)
        | ? [I2: nat] :
            ( aa(nat,$o,ord_less(nat,I2),Nb)
            & aa(nat,$o,P,I2) ) ) ) ).

% Ex_less_Suc
tff(fact_313_less__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,Nb))
    <=> ( aa(nat,$o,ord_less(nat,M),Nb)
        | ( M = Nb ) ) ) ).

% less_Suc_eq
tff(fact_314_not__less__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa(nat,$o,ord_less(nat,M),Nb)
    <=> aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,suc,M)) ) ).

% not_less_eq
tff(fact_315_Nat_OAll__less__Suc,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,suc,Nb))
         => aa(nat,$o,P,I2) )
    <=> ( aa(nat,$o,P,Nb)
        & ! [I2: nat] :
            ( aa(nat,$o,ord_less(nat,I2),Nb)
           => aa(nat,$o,P,I2) ) ) ) ).

% Nat.All_less_Suc
tff(fact_316_Suc__less__eq2,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,Nb)),M)
    <=> ? [M5: nat] :
          ( ( M = aa(nat,nat,suc,M5) )
          & aa(nat,$o,ord_less(nat,Nb),M5) ) ) ).

% Suc_less_eq2
tff(fact_317_less__antisym,axiom,
    ! [Nb: nat,M: nat] :
      ( ~ aa(nat,$o,ord_less(nat,Nb),M)
     => ( aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,suc,M))
       => ( M = Nb ) ) ) ).

% less_antisym
tff(fact_318_Suc__less__SucD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,Nb))
     => aa(nat,$o,ord_less(nat,M),Nb) ) ).

% Suc_less_SucD
tff(fact_319_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_320_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( ! [I3: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,I3),aa(nat,nat,suc,I3))
       => ( ! [I3: nat,J2: nat,K3: nat] :
              ( aa(nat,$o,ord_less(nat,I3),J2)
             => ( aa(nat,$o,ord_less(nat,J2),K3)
               => ( aa(nat,$o,aa(nat,fun(nat,$o),P,I3),J2)
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),P,J2),K3)
                   => aa(nat,$o,aa(nat,fun(nat,$o),P,I3),K3) ) ) ) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,I),J) ) ) ) ).

% less_Suc_induct
tff(fact_321_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( ! [I3: nat] :
            ( ( J = aa(nat,nat,suc,I3) )
           => aa(nat,$o,P,I3) )
       => ( ! [I3: nat] :
              ( aa(nat,$o,ord_less(nat,I3),J)
             => ( aa(nat,$o,P,aa(nat,nat,suc,I3))
               => aa(nat,$o,P,I3) ) )
         => aa(nat,$o,P,I) ) ) ) ).

% strict_inc_induct
tff(fact_322_not__less__less__Suc__eq,axiom,
    ! [Nb: nat,M: nat] :
      ( ~ aa(nat,$o,ord_less(nat,Nb),M)
     => ( aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,suc,M))
      <=> ( Nb = M ) ) ) ).

% not_less_less_Suc_eq
tff(fact_323_nat__arith_Osuc1,axiom,
    ! [A3: nat,K: nat,A2: nat] :
      ( ( A3 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),A2) )
     => ( aa(nat,nat,suc,A3) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,suc,A2)) ) ) ).

% nat_arith.suc1
tff(fact_324_add__Suc,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)) ).

% add_Suc
tff(fact_325_add__Suc__shift,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,suc,M)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,suc,Nb)) ).

% add_Suc_shift
tff(fact_326_Suc__leD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% Suc_leD
tff(fact_327_le__SucE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Nb))
     => ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
       => ( M = aa(nat,nat,suc,Nb) ) ) ) ).

% le_SucE
tff(fact_328_le__SucI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Nb)) ) ).

% le_SucI
tff(fact_329_Suc__le__D,axiom,
    ! [Nb: nat,M6: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),M6)
     => ? [M2: nat] : M6 = aa(nat,nat,suc,M2) ) ).

% Suc_le_D
tff(fact_330_le__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
        | ( M = aa(nat,nat,suc,Nb) ) ) ) ).

% le_Suc_eq
tff(fact_331_Suc__n__not__le__n,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),Nb) ).

% Suc_n_not_le_n
tff(fact_332_not__less__eq__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),M) ) ).

% not_less_eq_eq
tff(fact_333_full__nat__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M3)),N2)
             => aa(nat,$o,P,M3) )
         => aa(nat,$o,P,N2) )
     => aa(nat,$o,P,Nb) ) ).

% full_nat_induct
tff(fact_334_nat__induct__at__least,axiom,
    ! [M: nat,Nb: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(nat,$o,P,M)
       => ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),N2)
             => ( aa(nat,$o,P,N2)
               => aa(nat,$o,P,aa(nat,nat,suc,N2)) ) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_induct_at_least
tff(fact_335_transitive__stepwise__le,axiom,
    ! [M: nat,Nb: nat,R: fun(nat,fun(nat,$o))] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( ! [X2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,X2),X2)
       => ( ! [X2: nat,Y5: nat,Z3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),R,X2),Y5)
             => ( aa(nat,$o,aa(nat,fun(nat,$o),R,Y5),Z3)
               => aa(nat,$o,aa(nat,fun(nat,$o),R,X2),Z3) ) )
         => ( ! [N2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),R,N2),aa(nat,nat,suc,N2))
           => aa(nat,$o,aa(nat,fun(nat,$o),R,M),Nb) ) ) ) ) ).

% transitive_stepwise_le
tff(fact_336_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_337_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_338_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_339_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_340_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_341_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_342_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_343_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,K),L)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),L) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb) )
       => aa(nat,$o,ord_less(nat,M),Nb) ) ) ).

% less_add_eq_less
tff(fact_344_exists__leI,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ( ! [N4: nat] :
            ( aa(nat,$o,ord_less(nat,N4),Nb)
           => ~ aa(nat,$o,P,N4) )
       => aa(nat,$o,P,Nb) )
     => ? [N5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N5),Nb)
          & aa(nat,$o,P,N5) ) ) ).

% exists_leI
tff(fact_345_nat__less__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
        & ( M != Nb ) ) ) ).

% nat_less_le
tff(fact_346_less__imp__le__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% less_imp_le_nat
tff(fact_347_le__eq__less__or__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
    <=> ( aa(nat,$o,ord_less(nat,M),Nb)
        | ( M = Nb ) ) ) ).

% le_eq_less_or_eq
tff(fact_348_less__or__eq__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,$o,ord_less(nat,M),Nb)
        | ( M = Nb ) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% less_or_eq_imp_le
tff(fact_349_le__neq__implies__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( ( M != Nb )
       => aa(nat,$o,ord_less(nat,M),Nb) ) ) ).

% le_neq_implies_less
tff(fact_350_less__mono__imp__le__mono,axiom,
    ! [F: fun(nat,nat),I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( aa(nat,$o,ord_less(nat,I3),J2)
         => aa(nat,$o,ord_less(nat,aa(nat,nat,F,I3)),aa(nat,nat,F,J2)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,F,I)),aa(nat,nat,F,J)) ) ) ).

% less_mono_imp_le_mono
tff(fact_351_zero__induct__lemma,axiom,
    ! [P: fun(nat,$o),K: nat,I: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [N2: nat] :
            ( aa(nat,$o,P,aa(nat,nat,suc,N2))
           => aa(nat,$o,P,N2) )
       => aa(nat,$o,P,aa(nat,nat,minus_minus(nat,K),I)) ) ) ).

% zero_induct_lemma
tff(fact_352_diff__less__mono2,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => ( aa(nat,$o,ord_less(nat,M),L)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,L),Nb)),aa(nat,nat,minus_minus(nat,L),M)) ) ) ).

% diff_less_mono2
tff(fact_353_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,J),K)
     => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,J),Nb)),K) ) ).

% less_imp_diff_less
tff(fact_354_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,Nb: 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)),Nb) )
    <=> ( M = Nb ) ) ).

% Suc_mult_cancel1
tff(fact_355_add__leE,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Nb)
     => ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ) ).

% add_leE
tff(fact_356_le__add1,axiom,
    ! [Nb: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M)) ).

% le_add1
tff(fact_357_le__add2,axiom,
    ! [Nb: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)) ).

% le_add2
tff(fact_358_add__leD1,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% add_leD1
tff(fact_359_add__leD2,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ).

% add_leD2
tff(fact_360_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
     => ? [N2: nat] : L = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),N2) ) ).

% le_Suc_ex
tff(fact_361_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$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_362_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$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_363_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),M)) ) ).

% trans_le_add1
tff(fact_364_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),J)) ) ).

% trans_le_add2
tff(fact_365_nat__le__iff__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
    <=> ? [K4: nat] : Nb = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K4) ) ).

% nat_le_iff_add
tff(fact_366_fstE,axiom,
    ! [B: $tType,A: $tType,Xb: product_prod(A,B),A2: A,B2: B,P: fun(A,$o)] :
      ( ( Xb = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) )
     => ( aa(A,$o,P,aa(product_prod(A,B),A,product_fst(A,B),Xb))
       => aa(A,$o,P,A2) ) ) ).

% fstE
tff(fact_367_diff__add__inverse2,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)),Nb) = M ).

% diff_add_inverse2
tff(fact_368_diff__add__inverse,axiom,
    ! [Nb: nat,M: nat] : aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M)),Nb) = M ).

% diff_add_inverse
tff(fact_369_diff__cancel2,axiom,
    ! [M: nat,K: nat,Nb: 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),Nb),K)) = aa(nat,nat,minus_minus(nat,M),Nb) ).

% diff_cancel2
tff(fact_370_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb)) = aa(nat,nat,minus_minus(nat,M),Nb) ).

% Nat.diff_cancel
tff(fact_371_eq__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( ( aa(nat,nat,minus_minus(nat,M),K) = aa(nat,nat,minus_minus(nat,Nb),K) )
        <=> ( M = Nb ) ) ) ) ).

% eq_diff_iff
tff(fact_372_le__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Nb),K))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ) ).

% le_diff_iff
tff(fact_373_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Nb),K)) = aa(nat,nat,minus_minus(nat,M),Nb) ) ) ) ).

% Nat.diff_diff_eq
tff(fact_374_diff__le__mono,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),L)),aa(nat,nat,minus_minus(nat,Nb),L)) ) ).

% diff_le_mono
tff(fact_375_diff__le__self,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),Nb)),M) ).

% diff_le_self
tff(fact_376_le__diff__iff_H,axiom,
    ! [A2: nat,C2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),C2)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,C2),A2)),aa(nat,nat,minus_minus(nat,C2),B2))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A2) ) ) ) ).

% le_diff_iff'
tff(fact_377_diff__le__mono2,axiom,
    ! [M: nat,Nb: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,L),Nb)),aa(nat,nat,minus_minus(nat,L),M)) ) ).

% diff_le_mono2
tff(fact_378_option_Odistinct_I1_J,axiom,
    ! [A: $tType,X23: A] : none(A) != aa(A,option(A),some(A),X23) ).

% option.distinct(1)
tff(fact_379_option_OdiscI,axiom,
    ! [A: $tType,Option: option(A),X23: A] :
      ( ( Option = aa(A,option(A),some(A),X23) )
     => ( Option != none(A) ) ) ).

% option.discI
tff(fact_380_option_Oexhaust,axiom,
    ! [A: $tType,Y: option(A)] :
      ( ( Y != none(A) )
     => ~ ! [X24: A] : Y != aa(A,option(A),some(A),X24) ) ).

% option.exhaust
tff(fact_381_split__option__ex,axiom,
    ! [A: $tType,P: fun(option(A),$o)] :
      ( ? [X_1: option(A)] : aa(option(A),$o,P,X_1)
    <=> ( aa(option(A),$o,P,none(A))
        | ? [X: A] : aa(option(A),$o,P,aa(A,option(A),some(A),X)) ) ) ).

% split_option_ex
tff(fact_382_split__option__all,axiom,
    ! [A: $tType,P: fun(option(A),$o)] :
      ( ! [X_1: option(A)] : aa(option(A),$o,P,X_1)
    <=> ( aa(option(A),$o,P,none(A))
        & ! [X: A] : aa(option(A),$o,P,aa(A,option(A),some(A),X)) ) ) ).

% split_option_all
tff(fact_383_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) )
       => ( ! [A4: A,B3: B] :
              ( ( Xb = aa(A,option(A),some(A),A4) )
             => ( ( Y = aa(B,option(B),some(B),B3) )
               => 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_384_sndE,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(A,B),A2: A,B2: B,P: fun(B,$o)] :
      ( ( Xb = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) )
     => ( aa(B,$o,P,aa(product_prod(A,B),B,product_snd(A,B),Xb))
       => aa(B,$o,P,B2) ) ) ).

% sndE
tff(fact_385_add__mult__distrib,axiom,
    ! [M: nat,Nb: 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),Nb)),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),Nb),K)) ).

% add_mult_distrib
tff(fact_386_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb)) = 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),Nb)) ).

% add_mult_distrib2
tff(fact_387_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_388_le__cube,axiom,
    ! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$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_389_le__square,axiom,
    ! [M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),M)) ).

% le_square
tff(fact_390_mult__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),L)
       => aa(nat,$o,aa(nat,fun(nat,$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_391_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$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_392_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => aa(nat,$o,aa(nat,fun(nat,$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_393_nat__mult__1,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),one_one(nat)),Nb) = Nb ).

% nat_mult_1
tff(fact_394_nat__mult__1__right,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),one_one(nat)) = Nb ).

% nat_mult_1_right
tff(fact_395_diff__mult__distrib,axiom,
    ! [M: nat,Nb: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,M),Nb)),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),Nb),K)) ).

% diff_mult_distrib
tff(fact_396_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),aa(nat,nat,minus_minus(nat,M),Nb)) = 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),Nb)) ).

% diff_mult_distrib2
tff(fact_397_option_Osel,axiom,
    ! [A: $tType,X23: A] : aa(option(A),A,the2(A),aa(A,option(A),some(A),X23)) = X23 ).

% option.sel
tff(fact_398_Ex__prod__contract,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] :
      ( ? [A5: A,X_1: B] : aa(B,$o,aa(A,fun(B,$o),P,A5),X_1)
    <=> ? [Z2: product_prod(A,B)] : aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(A,B),A,product_fst(A,B),Z2)),aa(product_prod(A,B),B,product_snd(A,B),Z2)) ) ).

% Ex_prod_contract
tff(fact_399_All__prod__contract,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] :
      ( ! [A5: A,X_1: B] : aa(B,$o,aa(A,fun(B,$o),P,A5),X_1)
    <=> ! [Z2: product_prod(A,B)] : aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(A,B),A,product_fst(A,B),Z2)),aa(product_prod(A,B),B,product_snd(A,B),Z2)) ) ).

% All_prod_contract
tff(fact_400_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_401_option_Osimps_I5_J,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,A),X23: B] : case_option(A,B,F1,F22,aa(B,option(B),some(B),X23)) = aa(B,A,F22,X23) ).

% option.simps(5)
tff(fact_402_option_Osimps_I4_J,axiom,
    ! [B: $tType,A: $tType,F1: A,F22: fun(B,A)] : case_option(A,B,F1,F22,none(B)) = F1 ).

% option.simps(4)
tff(fact_403_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_404_L2__set__mult__ineq__lemma,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : aa(real,$o,aa(real,fun(real,$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),A2),C2))),aa(real,real,aa(real,fun(real,real),times_times(real),B2),D2))),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),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),D2),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),B2),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_405_lift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(nat,A),Nb: nat,N3: nat] :
          ( ! [N2: nat] : aa(A,$o,ord_less(A,aa(nat,A,F,N2)),aa(nat,A,F,aa(nat,nat,suc,N2)))
         => ( aa(nat,$o,ord_less(nat,Nb),N3)
           => aa(A,$o,ord_less(A,aa(nat,A,F,Nb)),aa(nat,A,F,N3)) ) ) ) ).

% lift_Suc_mono_less
tff(fact_406_lift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(nat,A),Nb: nat,M: nat] :
          ( ! [N2: nat] : aa(A,$o,ord_less(A,aa(nat,A,F,N2)),aa(nat,A,F,aa(nat,nat,suc,N2)))
         => ( aa(A,$o,ord_less(A,aa(nat,A,F,Nb)),aa(nat,A,F,M))
          <=> aa(nat,$o,ord_less(nat,Nb),M) ) ) ) ).

% lift_Suc_mono_less_iff
tff(fact_407_lift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(nat,A),Nb: nat,N3: nat] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,N2)),aa(nat,A,F,aa(nat,nat,suc,N2)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,Nb)),aa(nat,A,F,N3)) ) ) ) ).

% lift_Suc_mono_le
tff(fact_408_lift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [F: fun(nat,A),Nb: nat,N3: nat] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,aa(nat,nat,suc,N2))),aa(nat,A,F,N2))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,N3)),aa(nat,A,F,Nb)) ) ) ) ).

% lift_Suc_antimono_le
tff(fact_409_less__natE,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => ~ ! [Q3: nat] : Nb != aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q3)) ) ).

% less_natE
tff(fact_410_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_411_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_412_less__iff__Suc__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
    <=> ? [K4: nat] : Nb = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K4)) ) ).

% less_iff_Suc_add
tff(fact_413_less__imp__Suc__add,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => ? [K3: nat] : Nb = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K3)) ) ).

% less_imp_Suc_add
tff(fact_414_nat__in__between__eq_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
        & aa(nat,$o,ord_less(nat,B2),aa(nat,nat,suc,A2)) )
    <=> ( B2 = A2 ) ) ).

% nat_in_between_eq(2)
tff(fact_415_nat__in__between__eq_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,$o,ord_less(nat,A2),B2)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),aa(nat,nat,suc,A2)) )
    <=> ( B2 = aa(nat,nat,suc,A2) ) ) ).

% nat_in_between_eq(1)
tff(fact_416_Suc__leI,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Nb) ) ).

% Suc_leI
tff(fact_417_Suc__le__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Nb)
    <=> aa(nat,$o,ord_less(nat,M),Nb) ) ).

% Suc_le_eq
tff(fact_418_dec__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,P,I)
       => ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),N2)
             => ( aa(nat,$o,ord_less(nat,N2),J)
               => ( aa(nat,$o,P,N2)
                 => aa(nat,$o,P,aa(nat,nat,suc,N2)) ) ) )
         => aa(nat,$o,P,J) ) ) ) ).

% dec_induct
tff(fact_419_inc__induct,axiom,
    ! [I: nat,J: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,P,J)
       => ( ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),N2)
             => ( aa(nat,$o,ord_less(nat,N2),J)
               => ( aa(nat,$o,P,aa(nat,nat,suc,N2))
                 => aa(nat,$o,P,N2) ) ) )
         => aa(nat,$o,P,I) ) ) ) ).

% inc_induct
tff(fact_420_Suc__le__lessD,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Nb)
     => aa(nat,$o,ord_less(nat,M),Nb) ) ).

% Suc_le_lessD
tff(fact_421_le__less__Suc__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,suc,M))
      <=> ( Nb = M ) ) ) ).

% le_less_Suc_eq
tff(fact_422_less__Suc__eq__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% less_Suc_eq_le
tff(fact_423_less__eq__Suc__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),M)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),M) ) ).

% less_eq_Suc_le
tff(fact_424_le__imp__less__Suc,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,Nb)) ) ).

% le_imp_less_Suc
tff(fact_425_Suc__diff__Suc,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),M)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,Nb))) = aa(nat,nat,minus_minus(nat,M),Nb) ) ) ).

% Suc_diff_Suc
tff(fact_426_diff__less__Suc,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,M),Nb)),aa(nat,nat,suc,M)) ).

% diff_less_Suc
tff(fact_427_Suc__eq__plus1,axiom,
    ! [Nb: nat] : aa(nat,nat,suc,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)) ).

% Suc_eq_plus1
tff(fact_428_plus__1__eq__Suc,axiom,
    aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)) = suc ).

% plus_1_eq_Suc
tff(fact_429_Suc__eq__plus1__left,axiom,
    ! [Nb: nat] : aa(nat,nat,suc,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),Nb) ).

% Suc_eq_plus1_left
tff(fact_430_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),Nb))
    <=> aa(nat,$o,ord_less(nat,M),Nb) ) ).

% Suc_mult_less_cancel1
tff(fact_431_mono__nat__linear__lb,axiom,
    ! [F: fun(nat,nat),M: nat,K: nat] :
      ( ! [M2: nat,N2: nat] :
          ( aa(nat,$o,ord_less(nat,M2),N2)
         => aa(nat,$o,ord_less(nat,aa(nat,nat,F,M2)),aa(nat,nat,F,N2)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,F,M)),K)),aa(nat,nat,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K))) ) ).

% mono_nat_linear_lb
tff(fact_432_Suc__diff__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Nb) = aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,M),Nb)) ) ) ).

% Suc_diff_le
tff(fact_433_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,Nb)) = aa(nat,nat,minus_minus(nat,aa(nat,nat,minus_minus(nat,M),one_one(nat))),Nb) ).

% diff_Suc_eq_diff_pred
tff(fact_434_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_435_add__diff__inverse__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa(nat,$o,ord_less(nat,M),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,minus_minus(nat,M),Nb)) = M ) ) ).

% add_diff_inverse_nat
tff(fact_436_less__diff__iff,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
       => ( aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,M),K)),aa(nat,nat,minus_minus(nat,Nb),K))
        <=> aa(nat,$o,ord_less(nat,M),Nb) ) ) ) ).

% less_diff_iff
tff(fact_437_diff__less__mono,axiom,
    ! [A2: nat,B2: nat,C2: nat] :
      ( aa(nat,$o,ord_less(nat,A2),B2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),A2)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,A2),C2)),aa(nat,nat,minus_minus(nat,B2),C2)) ) ) ).

% diff_less_mono
tff(fact_438_mult__Suc,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,M)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb)) ).

% mult_Suc
tff(fact_439_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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)),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% Suc_mult_le_cancel1
tff(fact_440_mlex__snd__decrI,axiom,
    ! [A2: nat,A6: nat,B2: nat,B4: nat,N: nat] :
      ( ( A2 = A6 )
     => ( aa(nat,$o,ord_less(nat,B2),B4)
       => aa(nat,$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),A2),N)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A6),N)),B4)) ) ) ).

% mlex_snd_decrI
tff(fact_441_mlex__fst__decrI,axiom,
    ! [A2: nat,A6: nat,B2: nat,N: nat,B4: nat] :
      ( aa(nat,$o,ord_less(nat,A2),A6)
     => ( aa(nat,$o,ord_less(nat,B2),N)
       => ( aa(nat,$o,ord_less(nat,B4),N)
         => aa(nat,$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),A2),N)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A6),N)),B4)) ) ) ) ).

% mlex_fst_decrI
tff(fact_442_mlex__bound,axiom,
    ! [A2: nat,A3: nat,B2: nat,N: nat] :
      ( aa(nat,$o,ord_less(nat,A2),A3)
     => ( aa(nat,$o,ord_less(nat,B2),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),A2),N)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A3),N)) ) ) ).

% mlex_bound
tff(fact_443_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,J),K)),I)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K)) ) ).

% le_diff_conv
tff(fact_444_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),aa(nat,nat,minus_minus(nat,J),K))
      <=> aa(nat,$o,aa(nat,fun(nat,$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_445_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_446_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_447_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_448_obtain__list__from__elements,axiom,
    ! [A: $tType,Nb: nat,P: fun(A,fun(nat,$o))] :
      ( ! [I3: nat] :
          ( aa(nat,$o,ord_less(nat,I3),Nb)
         => ? [Li: A] : aa(nat,$o,aa(A,fun(nat,$o),P,Li),I3) )
     => ~ ! [L2: list(A)] :
            ( ( aa(list(A),nat,size_size(list(A)),L2) = Nb )
           => ~ ! [I4: nat] :
                  ( aa(nat,$o,ord_less(nat,I4),Nb)
                 => aa(nat,$o,aa(A,fun(nat,$o),P,aa(nat,A,nth(A,L2),I4)),I4) ) ) ) ).

% obtain_list_from_elements
tff(fact_449_mlex__leI,axiom,
    ! [A2: nat,A6: nat,B2: nat,B4: nat,N: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),A6)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),B4)
       => aa(nat,$o,aa(nat,fun(nat,$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),A2),N)),B2)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A6),N)),B4)) ) ) ).

% mlex_leI
tff(fact_450_div__mult__le,axiom,
    ! [A2: nat,B2: nat] : aa(nat,$o,aa(nat,fun(nat,$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),A2),B2)),B2)),A2) ).

% div_mult_le
tff(fact_451_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_452_option_Ocase__eq__if,axiom,
    ! [A: $tType,B: $tType,F1: A,F22: fun(B,A),Option: option(B)] :
      case_option(A,B,F1,F22,Option) = $ite(Option = none(B),F1,aa(B,A,F22,aa(option(B),B,the2(B),Option))) ).

% option.case_eq_if
tff(fact_453_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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),X) ) ) ) ) ).

% is_succ_in_set_def
tff(fact_454_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_455_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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)),Nb) )
      <=> ( 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) = Nb ) ) ) ).

% nat_eq_add_iff1
tff(fact_456_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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)),Nb) )
      <=> ( 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)),Nb) ) ) ) ).

% nat_eq_add_iff2
tff(fact_457_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,$o,aa(nat,fun(nat,$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)),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$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)),Nb) ) ) ).

% nat_le_add_iff1
tff(fact_458_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,aa(nat,fun(nat,$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)),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$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)),Nb)) ) ) ).

% nat_le_add_iff2
tff(fact_459_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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)),Nb)) = 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)),Nb) ) ) ).

% nat_diff_add_eq1
tff(fact_460_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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)),Nb)) = 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)),Nb)) ) ) ).

% nat_diff_add_eq2
tff(fact_461_option_Osplit__sel,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F1: A,F22: fun(B,A),Option: option(B)] :
      ( aa(A,$o,P,case_option(A,B,F1,F22,Option))
    <=> ( ( ( Option = none(B) )
         => aa(A,$o,P,F1) )
        & ( ( Option = aa(B,option(B),some(B),aa(option(B),B,the2(B),Option)) )
         => aa(A,$o,P,aa(B,A,F22,aa(option(B),B,the2(B),Option))) ) ) ) ).

% option.split_sel
tff(fact_462_option_Osplit__sel__asm,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F1: A,F22: fun(B,A),Option: option(B)] :
      ( aa(A,$o,P,case_option(A,B,F1,F22,Option))
    <=> ~ ( ( ( Option = none(B) )
            & ~ aa(A,$o,P,F1) )
          | ( ( Option = aa(B,option(B),some(B),aa(option(B),B,the2(B),Option)) )
            & ~ aa(A,$o,P,aa(B,A,F22,aa(option(B),B,the2(B),Option))) ) ) ) ).

% option.split_sel_asm
tff(fact_463_Suc__n__minus__m__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(nat,$o,ord_less(nat,one_one(nat)),M)
       => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Nb),M)) = aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,minus_minus(nat,M),one_one(nat))) ) ) ) ).

% Suc_n_minus_m_eq
tff(fact_464_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> aa(nat,$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)),Nb) ) ) ).

% nat_less_add_iff1
tff(fact_465_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),U)),M)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),J),U)),Nb))
      <=> aa(nat,$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)),Nb)) ) ) ).

% nat_less_add_iff2
tff(fact_466_two__pow__div__gt__le,axiom,
    ! [V: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,V),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M)))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% two_pow_div_gt_le
tff(fact_467_power__minus__is__div,axiom,
    ! [B2: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),A2)
     => ( 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,A2),B2)) = 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))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)) ) ) ).

% power_minus_is_div
tff(fact_468_cnt__bound_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Ta)),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))),Nb)),one_one(real)))) ) ).

% cnt_bound'
tff(fact_469_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_470_le__add__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,minus_minus(A,A2),B2)) = A2 ) ) ) ).

% le_add_diff_inverse
tff(fact_471_le__add__diff__inverse2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A2),B2)),B2) = A2 ) ) ) ).

% le_add_diff_inverse2
tff(fact_472_two__powr__height__bound__deg,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$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,Ta))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ) ).

% two_powr_height_bound_deg
tff(fact_473_setprop,axiom,
    ! [Ta: vEBT_VEBT] :
      ( member(vEBT_VEBT,Ta,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),treeList))
     => vEBT_invar_vebt(Ta,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),na),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% setprop
tff(fact_474_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_bl(assn,fun(fun(A,assn),fun(A,assn)),R),Q)) ).

% norm_assertion_simps(17)
tff(fact_475_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_bf(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ).

% norm_assertion_simps(16)
tff(fact_476_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya),Xb) = $ite(
        aa(nat,$o,ord_less(nat,Xb),Mia),
        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,Vaa))),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,Vaa))),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)),TreeLista)),
              $let(
                maxlow: option(nat),
                maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),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,TreeLista),h),l)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summarya,h)),one_one(nat)) ) ),
              one_one(nat) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(6)
tff(fact_477_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya),Xb) = $ite(
        aa(nat,$o,ord_less(nat,Maa),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,Vaa))),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,Vaa))),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)),TreeLista)),
              $let(
                minlow: option(nat),
                minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),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,TreeLista),h),l)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summarya,h)),one_one(nat)) ) ),
              one_one(nat) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(7)
tff(fact_478_set__n__deg__not__0,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,M: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X2,Nb) )
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb) ) ) ).

% set_n_deg_not_0
tff(fact_479_inthall,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Nb: nat] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X2) )
     => ( aa(nat,$o,ord_less(nat,Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% inthall
tff(fact_480_old_Oprod_Oinject,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B4) )
    <=> ( ( A2 = A6 )
        & ( B2 = B4 ) ) ) ).

% old.prod.inject
tff(fact_481_prod_Oinject,axiom,
    ! [A: $tType,B: $tType,X1: A,X23: B,Y1: A,Y2: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X1),X23) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Y1),Y2) )
    <=> ( ( X1 = Y1 )
        & ( X23 = Y2 ) ) ) ).

% prod.inject
tff(fact_482_real__divide__square__eq,axiom,
    ! [R3: real,A2: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,aa(real,fun(real,real),times_times(real),R3),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),R3),R3)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),R3) ).

% real_divide_square_eq
tff(fact_483_height__compose__summary,axiom,
    ! [Summarya: vEBT_VEBT,Info: option(product_prod(nat,nat)),Deg: nat,TreeLista: list(vEBT_VEBT)] : aa(nat,$o,aa(nat,fun(nat,$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,Summarya))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info,Deg,TreeLista,Summarya))) ).

% height_compose_summary
tff(fact_484_height__compose__child,axiom,
    ! [Ta: vEBT_VEBT,TreeLista: list(vEBT_VEBT),Info: option(product_prod(nat,nat)),Deg: nat,Summarya: vEBT_VEBT] :
      ( member(vEBT_VEBT,Ta,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
     => aa(nat,$o,aa(nat,fun(nat,$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,Ta))),aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Info,Deg,TreeLista,Summarya))) ) ).

% height_compose_child
tff(fact_485_mi__eq__ma__no__ch,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),Deg,TreeLista,Summarya),Deg)
     => ( ( Mia = Maa )
       => ( ! [X3: vEBT_VEBT] :
              ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
             => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) )
          & ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X_12) ) ) ) ).

% mi_eq_ma_no_ch
tff(fact_486_delete__bound__height_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V1232361888498592333_e_t_e(Ta,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))) ) ).

% delete_bound_height'
tff(fact_487_div__by__1,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),one_one(A)) = A2 ) ).

% div_by_1
tff(fact_488_option_Odisc__eq__case_I1_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option = none(A) )
    <=> case_option($o,A,$true,aTP_Lamp_bm(A,$o),Option) ) ).

% option.disc_eq_case(1)
tff(fact_489_option_Odisc__eq__case_I2_J,axiom,
    ! [A: $tType,Option: option(A)] :
      ( ( Option != none(A) )
    <=> case_option($o,A,$false,aTP_Lamp_bn(A,$o),Option) ) ).

% option.disc_eq_case(2)
tff(fact_490_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),Vaa: vEBT_VEBT,Vb: nat] : vEBT_T_p_r_e_d2(vEBT_Node(none(product_prod(nat,nat)),Uy,Uz,Vaa),Vb) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(4)
tff(fact_491_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,Vaa: nat] : vEBT_T_s_u_c_c2(vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz),Vaa) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(3)
tff(fact_492_pred__bound__height_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_p_r_e_d2(Ta,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))) ) ).

% pred_bound_height'
tff(fact_493_succ_H__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_s_u_c_c2(Ta,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))) ) ).

% succ'_bound_height
tff(fact_494_case__optionE,axiom,
    ! [A: $tType,P: $o,Q: fun(A,$o),Xb: option(A)] :
      ( case_option($o,A,(P),Q,Xb)
     => ( ( ( Xb = none(A) )
         => ~ (P) )
       => ~ ! [Y5: A] :
              ( ( Xb = aa(A,option(A),some(A),Y5) )
             => ~ aa(A,$o,Q,Y5) ) ) ) ).

% case_optionE
tff(fact_495_all__set__conv__nth,axiom,
    ! [A: $tType,L: list(A),P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),L))
         => aa(A,$o,P,X) )
    <=> ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(list(A),nat,size_size(list(A)),L))
         => aa(A,$o,P,aa(nat,A,nth(A,L),I2)) ) ) ).

% all_set_conv_nth
tff(fact_496_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_497_prod__induct7,axiom,
    ! [G2: $tType,F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))))] :
      ( ! [A4: A,B3: B,C3: C,D3: D,E2: E,F2: F3,G3: G2] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),B3),aa(product_prod(D,product_prod(E,product_prod(F3,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F3,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),C3),aa(product_prod(E,product_prod(F3,G2)),product_prod(D,product_prod(E,product_prod(F3,G2))),aa(D,fun(product_prod(E,product_prod(F3,G2)),product_prod(D,product_prod(E,product_prod(F3,G2)))),product_Pair(D,product_prod(E,product_prod(F3,G2))),D3),aa(product_prod(F3,G2),product_prod(E,product_prod(F3,G2)),aa(E,fun(product_prod(F3,G2),product_prod(E,product_prod(F3,G2))),product_Pair(E,product_prod(F3,G2)),E2),aa(G2,product_prod(F3,G2),aa(F3,fun(G2,product_prod(F3,G2)),product_Pair(F3,G2),F2),G3)))))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),$o,P,Xb) ) ).

% prod_induct7
tff(fact_498_prod__induct6,axiom,
    ! [F3: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))))] :
      ( ! [A4: A,B3: B,C3: C,D3: D,E2: E,F2: F3] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,F3))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F3))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F3)))),B3),aa(product_prod(D,product_prod(E,F3)),product_prod(C,product_prod(D,product_prod(E,F3))),aa(C,fun(product_prod(D,product_prod(E,F3)),product_prod(C,product_prod(D,product_prod(E,F3)))),product_Pair(C,product_prod(D,product_prod(E,F3))),C3),aa(product_prod(E,F3),product_prod(D,product_prod(E,F3)),aa(D,fun(product_prod(E,F3),product_prod(D,product_prod(E,F3))),product_Pair(D,product_prod(E,F3)),D3),aa(F3,product_prod(E,F3),aa(E,fun(F3,product_prod(E,F3)),product_Pair(E,F3),E2),F2))))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),$o,P,Xb) ) ).

% prod_induct6
tff(fact_499_prod__induct5,axiom,
    ! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ( ! [A4: A,B3: B,C3: C,D3: D,E2: E] : aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o,P,aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A4),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B3),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C3),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D3),E2)))))
     => aa(product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),$o,P,Xb) ) ).

% prod_induct5
tff(fact_500_prod__induct4,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,P: fun(product_prod(A,product_prod(B,product_prod(C,D))),$o),Xb: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ( ! [A4: A,B3: B,C3: C,D3: D] : aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P,aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A4),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D3))))
     => aa(product_prod(A,product_prod(B,product_prod(C,D))),$o,P,Xb) ) ).

% prod_induct4
tff(fact_501_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))] :
      ( ! [A4: A,B3: B,C3: C] : aa(product_prod(A,product_prod(B,C)),$o,P,aa(product_prod(B,C),product_prod(A,product_prod(B,C)),aa(A,fun(product_prod(B,C),product_prod(A,product_prod(B,C))),product_Pair(A,product_prod(B,C)),A4),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B3),C3)))
     => aa(product_prod(A,product_prod(B,C)),$o,P,Xb) ) ).

% prod_induct3
tff(fact_502_prod__cases7,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,G2: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))))] :
      ~ ! [A4: A,B3: B,C3: C,D3: D,E2: E,F2: F3,G3: G2] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),B3),aa(product_prod(D,product_prod(E,product_prod(F3,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),aa(C,fun(product_prod(D,product_prod(E,product_prod(F3,G2))),product_prod(C,product_prod(D,product_prod(E,product_prod(F3,G2))))),product_Pair(C,product_prod(D,product_prod(E,product_prod(F3,G2)))),C3),aa(product_prod(E,product_prod(F3,G2)),product_prod(D,product_prod(E,product_prod(F3,G2))),aa(D,fun(product_prod(E,product_prod(F3,G2)),product_prod(D,product_prod(E,product_prod(F3,G2)))),product_Pair(D,product_prod(E,product_prod(F3,G2))),D3),aa(product_prod(F3,G2),product_prod(E,product_prod(F3,G2)),aa(E,fun(product_prod(F3,G2),product_prod(E,product_prod(F3,G2))),product_Pair(E,product_prod(F3,G2)),E2),aa(G2,product_prod(F3,G2),aa(F3,fun(G2,product_prod(F3,G2)),product_Pair(F3,G2),F2),G3)))))) ).

% prod_cases7
tff(fact_503_prod__cases6,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F3: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))))] :
      ~ ! [A4: A,B3: B,C3: C,D3: D,E2: E,F2: F3] : Y != aa(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),A4),aa(product_prod(C,product_prod(D,product_prod(E,F3))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3)))),aa(B,fun(product_prod(C,product_prod(D,product_prod(E,F3))),product_prod(B,product_prod(C,product_prod(D,product_prod(E,F3))))),product_Pair(B,product_prod(C,product_prod(D,product_prod(E,F3)))),B3),aa(product_prod(D,product_prod(E,F3)),product_prod(C,product_prod(D,product_prod(E,F3))),aa(C,fun(product_prod(D,product_prod(E,F3)),product_prod(C,product_prod(D,product_prod(E,F3)))),product_Pair(C,product_prod(D,product_prod(E,F3))),C3),aa(product_prod(E,F3),product_prod(D,product_prod(E,F3)),aa(D,fun(product_prod(E,F3),product_prod(D,product_prod(E,F3))),product_Pair(D,product_prod(E,F3)),D3),aa(F3,product_prod(E,F3),aa(E,fun(F3,product_prod(E,F3)),product_Pair(E,F3),E2),F2))))) ).

% prod_cases6
tff(fact_504_prod__cases5,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))] :
      ~ ! [A4: A,B3: B,C3: C,D3: D,E2: E] : Y != aa(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E)))),aa(A,fun(product_prod(B,product_prod(C,product_prod(D,E))),product_prod(A,product_prod(B,product_prod(C,product_prod(D,E))))),product_Pair(A,product_prod(B,product_prod(C,product_prod(D,E)))),A4),aa(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E))),aa(B,fun(product_prod(C,product_prod(D,E)),product_prod(B,product_prod(C,product_prod(D,E)))),product_Pair(B,product_prod(C,product_prod(D,E))),B3),aa(product_prod(D,E),product_prod(C,product_prod(D,E)),aa(C,fun(product_prod(D,E),product_prod(C,product_prod(D,E))),product_Pair(C,product_prod(D,E)),C3),aa(E,product_prod(D,E),aa(D,fun(E,product_prod(D,E)),product_Pair(D,E),D3),E2)))) ).

% prod_cases5
tff(fact_505_prod__cases4,axiom,
    ! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod(A,product_prod(B,product_prod(C,D)))] :
      ~ ! [A4: A,B3: B,C3: C,D3: D] : Y != aa(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D))),aa(A,fun(product_prod(B,product_prod(C,D)),product_prod(A,product_prod(B,product_prod(C,D)))),product_Pair(A,product_prod(B,product_prod(C,D))),A4),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),B3),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),C3),D3))) ).

% prod_cases4
tff(fact_506_prod__cases3,axiom,
    ! [A: $tType,B: $tType,C: $tType,Y: product_prod(A,product_prod(B,C))] :
      ~ ! [A4: A,B3: 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)),A4),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),B3),C3)) ).

% prod_cases3
tff(fact_507_Pair__inject,axiom,
    ! [A: $tType,B: $tType,A2: A,B2: B,A6: A,B4: B] :
      ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2) = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A6),B4) )
     => ~ ( ( A2 = A6 )
         => ( B2 != B4 ) ) ) ).

% Pair_inject
tff(fact_508_prod__cases,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o),P2: product_prod(A,B)] :
      ( ! [A4: A,B3: B] : aa(product_prod(A,B),$o,P,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3))
     => aa(product_prod(A,B),$o,P,P2) ) ).

% prod_cases
tff(fact_509_surj__pair,axiom,
    ! [A: $tType,B: $tType,P2: product_prod(A,B)] :
    ? [X2: A,Y5: B] : P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y5) ).

% surj_pair
tff(fact_510_old_Oprod_Oexhaust,axiom,
    ! [A: $tType,B: $tType,Y: product_prod(A,B)] :
      ~ ! [A4: A,B3: B] : Y != aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) ).

% old.prod.exhaust
tff(fact_511_star__assoc,axiom,
    ! [A2: assn,B2: assn,C2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A2),B2)),C2) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),C2)) ).

% star_assoc
tff(fact_512_star__aci_I2_J,axiom,
    ! [A2: assn,B2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A2),B2) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),A2) ).

% star_aci(2)
tff(fact_513_star__aci_I3_J,axiom,
    ! [A2: assn,B2: assn,C2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),C2)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A2),C2)) ).

% star_aci(3)
tff(fact_514_assn__aci_I10_J,axiom,
    ! [A2: assn,B2: assn,C2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A2),B2)),C2) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A2),C2)),B2) ).

% assn_aci(10)
tff(fact_515_is__hoare__triple,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => hoare_hoare_triple(A,P,C2,Q) ) ).

% is_hoare_triple
tff(fact_516_height__node,axiom,
    ! [Mia: nat,Maa: nat,Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Nb: 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),Mia),Maa)),Deg,TreeLista,Summarya),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$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),Mia),Maa)),Deg,TreeLista,Summarya))) ) ).

% height_node
tff(fact_517_combine__common__factor,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,E3: A,B2: 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),A2),E3)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),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),A2),B2)),E3)),C2) ) ).

% combine_common_factor
tff(fact_518_distrib__right,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,B2: 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% distrib_right
tff(fact_519_distrib__left,axiom,
    ! [A: $tType] :
      ( semiring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% distrib_left
tff(fact_520_comm__semiring__class_Odistrib,axiom,
    ! [A: $tType] :
      ( comm_semiring(A)
     => ! [A2: A,B2: 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% comm_semiring_class.distrib
tff(fact_521_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% ring_class.ring_distribs(1)
tff(fact_522_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ring_class.ring_distribs(2)
tff(fact_523_right__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,minus_minus(A,B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% right_diff_distrib'
tff(fact_524_left__diff__distrib_H,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [B2: A,C2: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,B2),C2)),A2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) ) ).

% left_diff_distrib'
tff(fact_525_right__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,minus_minus(A,B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ).

% right_diff_distrib
tff(fact_526_left__diff__distrib,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% left_diff_distrib
tff(fact_527_fst__conv,axiom,
    ! [B: $tType,A: $tType,X1: A,X23: 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),X23)) = X1 ).

% fst_conv
tff(fact_528_fst__eqD,axiom,
    ! [B: $tType,A: $tType,Xb: A,Y: B,A2: 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)) = A2 )
     => ( Xb = A2 ) ) ).

% fst_eqD
tff(fact_529_snd__conv,axiom,
    ! [B: $tType,A: $tType,X1: B,X23: 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),X23)) = X23 ).

% snd_conv
tff(fact_530_snd__eqD,axiom,
    ! [B: $tType,A: $tType,Xb: B,Y: A,A2: 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)) = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
tff(fact_531_prod__eq__iff,axiom,
    ! [B: $tType,A: $tType,S3: product_prod(A,B),Ta: product_prod(A,B)] :
      ( ( S3 = Ta )
    <=> ( ( aa(product_prod(A,B),A,product_fst(A,B),S3) = aa(product_prod(A,B),A,product_fst(A,B),Ta) )
        & ( aa(product_prod(A,B),B,product_snd(A,B),S3) = aa(product_prod(A,B),B,product_snd(A,B),Ta) ) ) ) ).

% prod_eq_iff
tff(fact_532_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_533_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_534_norm__assertion__simps_I1_J,axiom,
    ! [A2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),one_one(assn)),A2) = A2 ).

% norm_assertion_simps(1)
tff(fact_535_norm__assertion__simps_I2_J,axiom,
    ! [A2: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A2),one_one(assn)) = A2 ).

% norm_assertion_simps(2)
tff(fact_536_lambda__one,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ( aTP_Lamp_bo(A,A) = aa(A,fun(A,A),times_times(A),one_one(A)) ) ) ).

% lambda_one
tff(fact_537_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_bp(fun(A,assn),fun(assn,fun(A,assn)),Q),R)) ) ).

% frame_rule_left
tff(fact_538_add__mono1,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),one_one(A))) ) ) ).

% add_mono1
tff(fact_539_less__add__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A] : aa(A,$o,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))) ) ).

% less_add_one
tff(fact_540_less__1__mult,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: A,Nb: A] :
          ( aa(A,$o,ord_less(A,one_one(A)),M)
         => ( aa(A,$o,ord_less(A,one_one(A)),Nb)
           => aa(A,$o,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),M),Nb)) ) ) ) ).

% less_1_mult
tff(fact_541_add__le__add__imp__diff__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Nb: A,J: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Nb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),J),K))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,Nb),K)),J) ) ) ) ) ) ).

% add_le_add_imp_diff_le
tff(fact_542_add__le__imp__le__diff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: A,K: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),I),K)),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),aa(A,A,minus_minus(A,Nb),K)) ) ) ).

% add_le_imp_le_diff
tff(fact_543_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,minus_minus(A,A2),B2)) = A2 ) ) ) ).

% linordered_semidom_class.add_diff_inverse
tff(fact_544_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_545_eq__add__iff2,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D2) )
        <=> ( 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,B2),A2)),E3)),D2) ) ) ) ).

% eq_add_iff2
tff(fact_546_eq__add__iff1,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D2) )
        <=> ( 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,A2),B2)),E3)),C2) = D2 ) ) ) ).

% eq_add_iff1
tff(fact_547_surjective__pairing,axiom,
    ! [B: $tType,A: $tType,Ta: product_prod(A,B)] : Ta = 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),Ta)),aa(product_prod(A,B),B,product_snd(A,B),Ta)) ).

% surjective_pairing
tff(fact_548_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_549_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D2))
        <=> aa(A,$o,aa(A,fun(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,B2),A2)),E3)),D2)) ) ) ).

% ordered_ring_class.le_add_iff2
tff(fact_550_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D2))
        <=> aa(A,$o,aa(A,fun(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,A2),B2)),E3)),C2)),D2) ) ) ).

% ordered_ring_class.le_add_iff1
tff(fact_551_less__add__iff2,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
          ( aa(A,$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),A2),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D2))
        <=> 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,B2),A2)),E3)),D2)) ) ) ).

% less_add_iff2
tff(fact_552_less__add__iff1,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,E3: A,C2: A,B2: A,D2: A] :
          ( aa(A,$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),A2),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),E3)),D2))
        <=> 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,A2),B2)),E3)),C2)),D2) ) ) ).

% less_add_iff1
tff(fact_553_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_554_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,M: nat,Deg: nat,Mia: nat,Maa: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X2,Nb) )
     => ( vEBT_invar_vebt(Summarya,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = 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,Nb) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M) )
             => ( ! [I3: nat] :
                    ( aa(nat,$o,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                   => ( ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),X_1)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),I3) ) )
               => ( ( ( Mia = Maa )
                   => ! [X2: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                       => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_13) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Maa)
                   => ( aa(nat,$o,ord_less(nat,Maa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                     => ( ( ( Mia != Maa )
                         => ! [I3: nat] :
                              ( aa(nat,$o,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                             => ( ( ( vEBT_VEBT_high(Maa,Nb) = I3 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),vEBT_VEBT_low(Maa,Nb)) )
                                & ! [X2: nat] :
                                    ( ( ( vEBT_VEBT_high(X2,Nb) = I3 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),vEBT_VEBT_low(X2,Nb)) )
                                   => ( aa(nat,$o,ord_less(nat,Mia),X2)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Maa) ) ) ) ) )
                       => 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),Mia),Maa)),Deg,TreeLista,Summarya),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
tff(fact_555_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,M: nat,Deg: nat,Mia: nat,Maa: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X2,Nb) )
     => ( vEBT_invar_vebt(Summarya,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = Nb )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M) )
             => ( ! [I3: nat] :
                    ( aa(nat,$o,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                   => ( ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),X_1)
                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),I3) ) )
               => ( ( ( Mia = Maa )
                   => ! [X2: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                       => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_13) ) )
                 => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Maa)
                   => ( aa(nat,$o,ord_less(nat,Maa),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg))
                     => ( ( ( Mia != Maa )
                         => ! [I3: nat] :
                              ( aa(nat,$o,ord_less(nat,I3),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M))
                             => ( ( ( vEBT_VEBT_high(Maa,Nb) = I3 )
                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),vEBT_VEBT_low(Maa,Nb)) )
                                & ! [X2: nat] :
                                    ( ( ( vEBT_VEBT_high(X2,Nb) = I3 )
                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),I3)),vEBT_VEBT_low(X2,Nb)) )
                                   => ( aa(nat,$o,ord_less(nat,Mia),X2)
                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),Maa) ) ) ) ) )
                       => 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),Mia),Maa)),Deg,TreeLista,Summarya),Deg) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
tff(fact_556_real__average__minus__first,axiom,
    ! [A2: real,B2: 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),A2),B2)),aa(num,real,numeral_numeral(real),bit0(one2)))),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,B2),A2)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% real_average_minus_first
tff(fact_557_real__average__minus__second,axiom,
    ! [B2: real,A2: 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),B2),A2)),aa(num,real,numeral_numeral(real),bit0(one2)))),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,B2),A2)),aa(num,real,numeral_numeral(real),bit0(one2))) ).

% real_average_minus_second
tff(fact_558_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X2,Nb) )
     => ( vEBT_invar_vebt(Summarya,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = 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,Nb) )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M) )
             => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X_13)
               => ( ! [X2: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_13) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeLista,Summarya),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
tff(fact_559_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeLista: list(vEBT_VEBT),Nb: nat,Summarya: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X2: vEBT_VEBT] :
          ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
         => vEBT_invar_vebt(X2,Nb) )
     => ( vEBT_invar_vebt(Summarya,M)
       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M) )
         => ( ( M = Nb )
           => ( ( Deg = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M) )
             => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X_13)
               => ( ! [X2: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X2),X_13) )
                 => vEBT_invar_vebt(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeLista,Summarya),Deg) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
tff(fact_560_less__eq__option__Some__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A] : ~ aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),aa(A,option(A),some(A),Xb)),none(A)) ) ).

% less_eq_option_Some_None
tff(fact_561_count__buildup,axiom,
    ! [Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb))),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))),Nb))) ).

% count_buildup
tff(fact_562_less__eq__option__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(option(A),$o,aa(option(A),fun(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,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% less_eq_option_Some
tff(fact_563_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_564_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_565_VEBT_Oinject_I1_J,axiom,
    ! [X11a: option(product_prod(nat,nat)),X12a: nat,X13a: list(vEBT_VEBT),X14a: vEBT_VEBT,Y11: option(product_prod(nat,nat)),Y12: nat,Y13: list(vEBT_VEBT),Y14: vEBT_VEBT] :
      ( ( vEBT_Node(X11a,X12a,X13a,X14a) = vEBT_Node(Y11,Y12,Y13,Y14) )
    <=> ( ( X11a = Y11 )
        & ( X12a = Y12 )
        & ( X13a = Y13 )
        & ( X14a = Y14 ) ) ) ).

% VEBT.inject(1)
tff(fact_566_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_567_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_568_less__eq__option__None__code,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: option(A)] : aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),none(A)),Xb) ) ).

% less_eq_option_None_code
tff(fact_569_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_br(option(A),fun(A,$o),Xb),Y) ) ) ).

% less_option_def
tff(fact_570_mult__commute__abs,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [C2: A] : aTP_Lamp_bs(A,fun(A,A),C2) = aa(A,fun(A,A),times_times(A),C2) ) ).

% mult_commute_abs
tff(fact_571_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_572_less__option__None__is__Some,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: option(A)] :
          ( aa(option(A),$o,ord_less(option(A),none(A)),Xb)
         => ? [Z3: A] : Xb = aa(A,option(A),some(A),Z3) ) ) ).

% less_option_None_is_Some
tff(fact_573_less__eq__option__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: option(A)] : aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),none(A)),Xb) ) ).

% less_eq_option_None
tff(fact_574_less__eq__option__None__is__None,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),Xb),none(A))
         => ( Xb = none(A) ) ) ) ).

% less_eq_option_None_is_None
tff(fact_575_less__eq__option__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: option(A),Y: option(A)] :
          ( aa(option(A),$o,aa(option(A),fun(option(A),$o),ord_less_eq(option(A)),Xb),Y)
        <=> case_option($o,A,$true,aTP_Lamp_bt(option(A),fun(A,$o),Y),Xb) ) ) ).

% less_eq_option_def
tff(fact_576_set__vebt__def,axiom,
    ! [Ta: vEBT_VEBT] : vEBT_set_vebt(Ta) = collect(nat,vEBT_V8194947554948674370ptions(Ta)) ).

% set_vebt_def
tff(fact_577_discrete,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),B2) ) ) ).

% discrete
tff(fact_578_builupi_Hcorr,axiom,
    ! [Nb: nat] : hoare_hoare_triple(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(Nb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb))) ).

% builupi'corr
tff(fact_579_builupicorr,axiom,
    ! [Nb: nat] : hoare_hoare_triple(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(Nb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb))) ).

% builupicorr
tff(fact_580_divmod__step__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [L: num,Q2: A,R3: A] :
          unique1321980374590559556d_step(A,L,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R3)) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),L)),R3),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,R3),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)),R3)) ) ).

% divmod_step_eq
tff(fact_581_cnt__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Ta)),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))),Nb)),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_582_TBOUND__vebt__memberi,axiom,
    ! [Ta: 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,Ta),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,Ta)))) ).

% TBOUND_vebt_memberi
tff(fact_583_vebt__insert_Osimps_I4_J,axiom,
    ! [V: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V)),TreeLista,Summarya),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)),TreeLista,Summarya) ).

% vebt_insert.simps(4)
tff(fact_584_two__realpow__ge__one,axiom,
    ! [Nb: nat] : aa(real,$o,aa(real,fun(real,$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))),Nb)) ).

% two_realpow_ge_one
tff(fact_585_count__buildup_H,axiom,
    ! [Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb))),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))),Nb)))) ).

% count_buildup'
tff(fact_586_heaphelp,axiom,
    ! [A: $tType,Xaa: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: vEBT_VEBTi,Nb: 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,TreeLista),Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summarya),Xb))),
                pure_assn(( ( none(A) = none(A) )
                  & ( Nb = Nb ) )))),
            pure_assn(Xc = vEBT_Nodei(none(product_prod(nat,nat)),Nb,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)),Nb,TreeLista,Summarya)),Xc),H) ) ).

% heaphelp
tff(fact_587_div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,Nb: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),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))),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),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),Nb))) ) ).

% div_exp_eq
tff(fact_588_semiring__norm_I90_J,axiom,
    ! [M: num,Nb: num] :
      ( ( aa(num,num,bit1,M) = aa(num,num,bit1,Nb) )
    <=> ( M = Nb ) ) ).

% semiring_norm(90)
tff(fact_589_of__nat__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [M: nat,Nb: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),M) = aa(nat,A,semiring_1_of_nat(A),Nb) )
        <=> ( M = Nb ) ) ) ).

% of_nat_eq_iff
tff(fact_590_two__realpow__ge__two,axiom,
    ! [Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(nat,real,semiring_1_of_nat(real),Nb))
     => aa(real,$o,aa(real,fun(real,$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))),Nb)) ) ).

% two_realpow_ge_two
tff(fact_591_bits__div__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),one_one(A)) = A2 ) ).

% bits_div_by_1
tff(fact_592_semiring__norm_I88_J,axiom,
    ! [M: num,Nb: num] : bit0(M) != aa(num,num,bit1,Nb) ).

% semiring_norm(88)
tff(fact_593_semiring__norm_I89_J,axiom,
    ! [M: num,Nb: num] : aa(num,num,bit1,M) != bit0(Nb) ).

% semiring_norm(89)
tff(fact_594_semiring__norm_I84_J,axiom,
    ! [Nb: num] : one2 != aa(num,num,bit1,Nb) ).

% semiring_norm(84)
tff(fact_595_semiring__norm_I86_J,axiom,
    ! [M: num] : aa(num,num,bit1,M) != one2 ).

% semiring_norm(86)
tff(fact_596_semiring__norm_I73_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit1,M)),aa(num,num,bit1,Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Nb) ) ).

% semiring_norm(73)
tff(fact_597_semiring__norm_I80_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,ord_less(num,aa(num,num,bit1,M)),aa(num,num,bit1,Nb))
    <=> aa(num,$o,ord_less(num,M),Nb) ) ).

% semiring_norm(80)
tff(fact_598_of__nat__numeral,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: num] : aa(nat,A,semiring_1_of_nat(A),aa(num,nat,numeral_numeral(nat),Nb)) = aa(num,A,numeral_numeral(A),Nb) ) ).

% of_nat_numeral
tff(fact_599_of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Nb: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,ord_less(nat,M),Nb) ) ) ).

% of_nat_less_iff
tff(fact_600_of__nat__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ).

% of_nat_le_iff
tff(fact_601_of__nat__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)) = 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),Nb)) ) ).

% of_nat_add
tff(fact_602_of__nat__eq__1__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( ( aa(nat,A,semiring_1_of_nat(A),Nb) = one_one(A) )
        <=> ( Nb = one_one(nat) ) ) ) ).

% of_nat_eq_1_iff
tff(fact_603_of__nat__1__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( ( one_one(A) = aa(nat,A,semiring_1_of_nat(A),Nb) )
        <=> ( Nb = one_one(nat) ) ) ) ).

% of_nat_1_eq_iff
tff(fact_604_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_605_of__nat__mult,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb)) = 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),Nb)) ) ).

% of_nat_mult
tff(fact_606_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Xb: nat,B2: 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),B2)),W) )
        <=> ( Xb = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W) ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
tff(fact_607_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [B2: nat,W: nat,Xb: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W) = aa(nat,A,semiring_1_of_nat(A),Xb) )
        <=> ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W) = Xb ) ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
tff(fact_608_semiring__1__class_Oof__nat__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [M: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),M),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),M)),Nb) ) ).

% semiring_1_class.of_nat_power
tff(fact_609_semiring__norm_I7_J,axiom,
    ! [M: num,Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bit0(M)),aa(num,num,bit1,Nb)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Nb)) ).

% semiring_norm(7)
tff(fact_610_semiring__norm_I9_J,axiom,
    ! [M: num,Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),bit0(Nb)) = aa(num,num,bit1,aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Nb)) ).

% semiring_norm(9)
tff(fact_611_semiring__norm_I15_J,axiom,
    ! [M: num,Nb: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),bit0(Nb)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),Nb)) ).

% semiring_norm(15)
tff(fact_612_semiring__norm_I14_J,axiom,
    ! [M: num,Nb: num] : aa(num,num,aa(num,fun(num,num),times_times(num),bit0(M)),aa(num,num,bit1,Nb)) = bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),aa(num,num,bit1,Nb))) ).

% semiring_norm(14)
tff(fact_613_semiring__norm_I81_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,ord_less(num,aa(num,num,bit1,M)),bit0(Nb))
    <=> aa(num,$o,ord_less(num,M),Nb) ) ).

% semiring_norm(81)
tff(fact_614_semiring__norm_I72_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),bit0(M)),aa(num,num,bit1,Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Nb) ) ).

% semiring_norm(72)
tff(fact_615_semiring__norm_I77_J,axiom,
    ! [Nb: num] : aa(num,$o,ord_less(num,one2),aa(num,num,bit1,Nb)) ).

% semiring_norm(77)
tff(fact_616_semiring__norm_I70_J,axiom,
    ! [M: num] : ~ aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit1,M)),one2) ).

% semiring_norm(70)
tff(fact_617_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_618_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_619_real__of__nat__less__numeral__iff,axiom,
    ! [Nb: nat,W: num] :
      ( aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),Nb)),aa(num,real,numeral_numeral(real),W))
    <=> aa(nat,$o,ord_less(nat,Nb),aa(num,nat,numeral_numeral(nat),W)) ) ).

% real_of_nat_less_numeral_iff
tff(fact_620_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,Nb: nat] :
      ( aa(real,$o,ord_less(real,aa(num,real,numeral_numeral(real),W)),aa(nat,real,semiring_1_of_nat(real),Nb))
    <=> aa(nat,$o,ord_less(nat,aa(num,nat,numeral_numeral(nat),W)),Nb) ) ).

% numeral_less_real_of_nat_iff
tff(fact_621_semiring__norm_I3_J,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bit0(Nb)) = aa(num,num,bit1,Nb) ).

% semiring_norm(3)
tff(fact_622_semiring__norm_I4_J,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),aa(num,num,bit1,Nb)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),Nb),one2)) ).

% semiring_norm(4)
tff(fact_623_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_624_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_625_semiring__norm_I10_J,axiom,
    ! [M: num,Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,bit1,M)),aa(num,num,bit1,Nb)) = bit0(aa(num,num,aa(num,fun(num,num),plus_plus(num),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Nb)),one2)) ).

% semiring_norm(10)
tff(fact_626_semiring__norm_I16_J,axiom,
    ! [M: num,Nb: num] : aa(num,num,aa(num,fun(num,num),times_times(num),aa(num,num,bit1,M)),aa(num,num,bit1,Nb)) = 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),Nb)),bit0(aa(num,num,aa(num,fun(num,num),times_times(num),M),Nb)))) ).

% semiring_norm(16)
tff(fact_627_semiring__norm_I79_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,ord_less(num,bit0(M)),aa(num,num,bit1,Nb))
    <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Nb) ) ).

% semiring_norm(79)
tff(fact_628_semiring__norm_I74_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),aa(num,num,bit1,M)),bit0(Nb))
    <=> aa(num,$o,ord_less(num,M),Nb) ) ).

% semiring_norm(74)
tff(fact_629_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Y: nat,Xb: num,Nb: 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)),Nb) )
        <=> ( Y = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb) ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
tff(fact_630_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Xb: num,Nb: nat,Y: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb) = 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)),Nb) = Y ) ) ) ).

% numeral_power_eq_of_nat_cancel_iff
tff(fact_631_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,B2: nat,W: nat] :
          ( aa(A,$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),B2)),W))
        <=> aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
tff(fact_632_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,Xb: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),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),B2),W)),Xb) ) ) ).

% of_nat_less_of_nat_power_cancel_iff
tff(fact_633_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,B2: nat,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
tff(fact_634_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: nat,W: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(nat,A,semiring_1_of_nat(A),B2)),W)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),W)),Xb) ) ) ).

% of_nat_le_of_nat_power_cancel_iff
tff(fact_635_numeral__le__real__of__nat__iff,axiom,
    ! [Nb: num,M: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),Nb)),aa(nat,real,semiring_1_of_nat(real),M))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),Nb)),M) ) ).

% numeral_le_real_of_nat_iff
tff(fact_636_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_637_div__Suc__eq__div__add3,axiom,
    ! [M: nat,Nb: 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,Nb)))) = 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))),Nb)) ).

% div_Suc_eq_div_add3
tff(fact_638_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,I: num,Nb: nat] :
          ( aa(A,$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)),Nb))
        <=> 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)),Nb)) ) ) ).

% of_nat_less_numeral_power_cancel_iff
tff(fact_639_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Nb: nat,Xb: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb)),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)),Nb)),Xb) ) ) ).

% numeral_power_less_of_nat_cancel_iff
tff(fact_640_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,I: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb))
        <=> aa(nat,$o,aa(nat,fun(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)),Nb)) ) ) ).

% of_nat_le_numeral_power_cancel_iff
tff(fact_641_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [I: num,Nb: nat,Xb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),I)),Nb)),aa(nat,A,semiring_1_of_nat(A),Xb))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),I)),Nb)),Xb) ) ) ).

% numeral_power_le_of_nat_cancel_iff
tff(fact_642_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_643_hoare__triple__preI,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( ! [H2: product_prod(heap_ext(product_unit),set(nat))] :
          ( rep_assn(P,H2)
         => hoare_hoare_triple(A,P,C2,Q) )
     => hoare_hoare_triple(A,P,C2,Q) ) ).

% hoare_triple_preI
tff(fact_644_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) )
     => ( ! [N2: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),bit0(N2))
       => ( ! [N2: 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,N2))
         => ( ! [M2: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M2)),one2)
           => ( ! [M2: num,N2: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M2)),bit0(N2))
             => ( ! [M2: num,N2: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(M2)),aa(num,num,bit1,N2))
               => ( ! [M2: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M2)),one2)
                 => ( ! [M2: num,N2: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M2)),bit0(N2))
                   => ~ ! [M2: num,N2: num] : Xb != aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,M2)),aa(num,num,bit1,N2)) ) ) ) ) ) ) ) ) ).

% xor_num.cases
tff(fact_645_of__nat__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Nb: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Nb))
         => aa(nat,$o,ord_less(nat,M),Nb) ) ) ).

% of_nat_less_imp_less
tff(fact_646_less__imp__of__nat__less,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,ord_less(nat,M),Nb)
         => aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ).

% less_imp_of_nat_less
tff(fact_647_div__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,M: nat,Nb: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),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),Nb))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),M))),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% div_mult2_eq'
tff(fact_648_of__nat__mono,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [I: nat,J: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
         => aa(A,$o,aa(A,fun(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_649_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)) = 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),Nb)) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
tff(fact_650_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one2 )
     => ( ! [X24: num] : Y != bit0(X24)
       => ~ ! [X32: num] : Y != aa(num,num,bit1,X32) ) ) ).

% num.exhaust
tff(fact_651_real__of__nat__div4,axiom,
    ! [Nb: nat,Xb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),Xb))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,semiring_1_of_nat(real),Xb))) ).

% real_of_nat_div4
tff(fact_652_nat__le__real__less,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
    <=> aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),Nb)),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_653_nat__less__real__le,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),M)
    <=> aa(real,$o,aa(real,fun(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),Nb)),one_one(real))),aa(nat,real,semiring_1_of_nat(real),M)) ) ).

% nat_less_real_le
tff(fact_654_of__nat__diff,axiom,
    ! [A: $tType] :
      ( semiring_1_cancel(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,minus_minus(nat,M),Nb)) = aa(A,A,minus_minus(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ) ) ).

% of_nat_diff
tff(fact_655_numeral__Bit1,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] : aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)) = 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),Nb)),aa(num,A,numeral_numeral(A),Nb))),one_one(A)) ) ).

% numeral_Bit1
tff(fact_656_eval__nat__numeral_I3_J,axiom,
    ! [Nb: num] : aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,Nb)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bit0(Nb))) ).

% eval_nat_numeral(3)
tff(fact_657_complete__real,axiom,
    ! [S2: set(real)] :
      ( ? [X3: real] : member(real,X3,S2)
     => ( ? [Z4: real] :
          ! [X2: real] :
            ( member(real,X2,S2)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Z4) )
       => ? [Y5: real] :
            ( ! [X3: real] :
                ( member(real,X3,S2)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X3),Y5) )
            & ! [Z4: real] :
                ( ! [X2: real] :
                    ( member(real,X2,S2)
                   => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),Z4) )
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y5),Z4) ) ) ) ) ).

% complete_real
tff(fact_658_less__eq__real__def,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
    <=> ( aa(real,$o,ord_less(real,Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_real_def
tff(fact_659_of__nat__less__two__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) ) ).

% of_nat_less_two_power
tff(fact_660_numeral__code_I3_J,axiom,
    ! [A: $tType] :
      ( numeral(A)
     => ! [Nb: num] :
          aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb)) = $let(
            m: A,
            m:= aa(num,A,numeral_numeral(A),Nb),
            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_661_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_662_real__of__nat__div3,axiom,
    ! [Nb: nat,Xb: nat] : aa(real,$o,aa(real,fun(real,$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),Nb)),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),Nb),Xb)))),one_one(real)) ).

% real_of_nat_div3
tff(fact_663_numeral__Bit1__div__2,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(num,A,numeral_numeral(A),Nb) ) ).

% numeral_Bit1_div_2
tff(fact_664_power3__eq__cube,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),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),A2),A2)),A2) ) ).

% power3_eq_cube
tff(fact_665_Suc3__eq__add__3,axiom,
    ! [Nb: nat] : aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,suc,Nb))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),Nb) ).

% Suc3_eq_add_3
tff(fact_666_Suc__div__eq__add3__div,axiom,
    ! [M: nat,Nb: 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)))),Nb) = 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)),Nb) ).

% Suc_div_eq_add3_div
tff(fact_667_add__diff__add,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,C2: A,B2: A,D2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A2),B2)),aa(A,A,minus_minus(A,C2),D2)) ) ).

% add_diff_add
tff(fact_668_real__arch__pow,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xb)
     => ? [N2: nat] : aa(real,$o,ord_less(real,Y),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N2)) ) ).

% real_arch_pow
tff(fact_669_small__powers__of__2,axiom,
    ! [Xb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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_670_mult__diff__mult,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [Xb: A,Y: A,A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,minus_minus(A,Y),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Xb),A2)),B2)) ) ).

% mult_diff_mult
tff(fact_671_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_672_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_673_space__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space,Ta)),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_674_cnt__cnt__eq,axiom,
    ! [Ta: vEBT_VEBT] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Ta) = aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Ta)) ).

% cnt_cnt_eq
tff(fact_675_space__2__pow__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Ta))),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))),Nb)),one_one(real)))) ) ).

% space_2_pow_bound
tff(fact_676_space__cnt,axiom,
    ! [Ta: vEBT_VEBT] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Ta))),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,Ta))) ).

% space_cnt
tff(fact_677_space_H__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Ta)),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_678_TBOUND__replicate,axiom,
    ! [A: $tType,Xb: heap_Time_Heap(A),C2: nat,Nb: nat] :
      ( time_TBOUND(A,Xb,C2)
     => time_TBOUND(list(A),vEBT_VEBT_replicatei(A,Nb,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)),Nb))) ) ).

% TBOUND_replicate
tff(fact_679_delete__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_d_e_l_e_t_e(Ta,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,Ta))),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,aa(num,num,bit1,bit0(bit0(bit0(one2))))))))) ) ).

% delete_bound_height
tff(fact_680_t__build__cnt,axiom,
    ! [Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V8646137997579335489_i_l_d(Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))) ).

% t_build_cnt
tff(fact_681_t__buildup__cnt,axiom,
    ! [Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V8346862874174094_d_u_p(Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_vebt_buildup(Nb))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,bit0(aa(num,num,bit1,one2)))))) ).

% t_buildup_cnt
tff(fact_682_tdeletemimi,axiom,
    ! [Deg: nat,Mia: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
     => aa(nat,$o,aa(nat,fun(nat,$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),Mia),Mia)),Deg,TreeLista,Summarya),Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))) ) ).

% tdeletemimi
tff(fact_683_TBOUND__highi,axiom,
    ! [Xb: nat,Nb: nat] : time_TBOUND(nat,vEBT_VEBT_highi(Xb,Nb),one_one(nat)) ).

% TBOUND_highi
tff(fact_684_TBOUND__lowi,axiom,
    ! [Xb: nat,Nb: nat] : time_TBOUND(nat,vEBT_VEBT_lowi(Xb,Nb),one_one(nat)) ).

% TBOUND_lowi
tff(fact_685_TBOUND__vebt__maxti,axiom,
    ! [Ta: vEBT_VEBTi] : time_TBOUND(option(nat),vEBT_vebt_maxti(Ta),one_one(nat)) ).

% TBOUND_vebt_maxti
tff(fact_686_TBOUND__vebt__minti,axiom,
    ! [Ta: vEBT_VEBTi] : time_TBOUND(option(nat),vEBT_vebt_minti(Ta),one_one(nat)) ).

% TBOUND_vebt_minti
tff(fact_687_buildup__build__time,axiom,
    ! [Nb: nat] : aa(nat,$o,ord_less(nat,vEBT_V8346862874174094_d_u_p(Nb)),vEBT_V8646137997579335489_i_l_d(Nb)) ).

% buildup_build_time
tff(fact_688_space__space_H,axiom,
    ! [Ta: vEBT_VEBT] : aa(nat,$o,ord_less(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_space,Ta)),aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Ta)) ).

% space_space'
tff(fact_689_vebt__buildup__bound,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V8346862874174094_d_u_p(Nb)),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_690_TBOUND__vebt__succi,axiom,
    ! [Ta: 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,Ta),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,Ta)))) ).

% TBOUND_vebt_succi
tff(fact_691_zdiv__int,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% zdiv_int
tff(fact_692_Tb_H__cnt,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_VEBT_Tb2(Nb)),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(Nb)))) ).

% Tb'_cnt
tff(fact_693_htt__vebt__buildupi_H__univ,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) )
     => time_htt(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(Nb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb)),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_694_htt__vebt__buildupi__univ,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) )
     => time_htt(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(Nb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb)),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_695_T__vebt__buildupi__cnt_H,axiom,
    ! [Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V441764108873111860ildupi(Nb))),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(Nb)))) ).

% T_vebt_buildupi_cnt'
tff(fact_696_T__vebt__buildupi__univ,axiom,
    ! [U: nat,Nb: nat] :
      ( ( U = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V441764108873111860ildupi(Nb)),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_697_TBOUND__minNulli,axiom,
    ! [Ta: vEBT_VEBTi] : time_TBOUND($o,vEBT_VEBT_minNulli(Ta),one_one(nat)) ).

% TBOUND_minNulli
tff(fact_698_mod__pure__star__dist,axiom,
    ! [P: assn,B2: $o,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B2))),H)
    <=> ( rep_assn(P,H)
        & (B2) ) ) ).

% mod_pure_star_dist
tff(fact_699_TBOUND__buildupi,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => time_TBOUND(vEBT_VEBTi,vEBT_vebt_buildupi(Nb),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))),Nb))) ) ).

% TBOUND_buildupi
tff(fact_700_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,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
        $ite(
          Xb = Mia,
          one_one(nat),
          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
            $ite(
              Xb = Maa,
              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),Mia),
                  one_one(nat),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                    $ite(
                      aa(nat,$o,ord_less(nat,Maa),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,Vaa))),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)),TreeLista)),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,TreeLista),h),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),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_701_TBOUND__vebt__inserti,axiom,
    ! [Ta: 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,Ta),Tib),Xb),
        $ite(vEBT_VEBT_minNull(Ta),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,Ta))))) ).

% TBOUND_vebt_inserti
tff(fact_702_valid__0__not,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_0_not
tff(fact_703_valid__tree__deg__neq__0,axiom,
    ! [Ta: vEBT_VEBT] : ~ vEBT_invar_vebt(Ta,zero_zero(nat)) ).

% valid_tree_deg_neq_0
tff(fact_704_not__min__Null__member,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Ta)
     => ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),X_13) ) ).

% not_min_Null_member
tff(fact_705_min__Null__member,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_minNull(Ta)
     => ~ aa(nat,$o,vEBT_vebt_member(Ta),Xb) ) ).

% min_Null_member
tff(fact_706_deg__not__0,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ).

% deg_not_0
tff(fact_707_T__vebt__buildupi__gq__0,axiom,
    ! [Nb: nat] : aa(nat,$o,ord_less(nat,zero_zero(nat)),vEBT_V441764108873111860ildupi(Nb)) ).

% T_vebt_buildupi_gq_0
tff(fact_708_minminNull,axiom,
    ! [Ta: vEBT_VEBT] :
      ( ( vEBT_vebt_mint(Ta) = none(nat) )
     => vEBT_VEBT_minNull(Ta) ) ).

% minminNull
tff(fact_709_minNullmin,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Ta)
     => ( vEBT_vebt_mint(Ta) = none(nat) ) ) ).

% minNullmin
tff(fact_710_TBOUND__vebt__buildupi,axiom,
    ! [Nb: nat] : time_TBOUND(vEBT_VEBTi,vEBT_V739175172307565963ildupi(Nb),vEBT_V441764108873111860ildupi(Nb)) ).

% TBOUND_vebt_buildupi
tff(fact_711_buildup__gives__valid,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => vEBT_invar_vebt(vEBT_vebt_buildup(Nb),Nb) ) ).

% buildup_gives_valid
tff(fact_712_pure__assn__eq__conv,axiom,
    ! [P: $o,Q: $o] :
      ( ( pure_assn((P)) = pure_assn((Q)) )
    <=> ( (P)
      <=> (Q) ) ) ).

% pure_assn_eq_conv
tff(fact_713_TBOUND__minNull,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] :
      ( vEBT_VEBT_minNull(Ta)
     => 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,Ta),Tib),Xb),one_one(nat)) ) ).

% TBOUND_minNull
tff(fact_714_ex__assn__const,axiom,
    ! [A: $tType,C2: assn] : ex_assn(A,aTP_Lamp_bu(assn,fun(A,assn),C2)) = C2 ).

% ex_assn_const
tff(fact_715_Tb__T__vebt__buildupi_H_H,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V441764108873111860ildupi(Nb)),aa(nat,nat,minus_minus(nat,vEBT_VEBT_Tb2(Nb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ).

% Tb_T_vebt_buildupi''
tff(fact_716_htt__vebt__buildupi,axiom,
    ! [Nb: nat] : time_htt(vEBT_VEBTi,one_one(assn),vEBT_vebt_buildupi(Nb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb)),vEBT_V441764108873111860ildupi(Nb)) ).

% htt_vebt_buildupi
tff(fact_717_htt__vebt__buildupi_H,axiom,
    ! [Nb: nat] : time_htt(vEBT_VEBTi,one_one(assn),vEBT_V739175172307565963ildupi(Nb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb)),vEBT_V441764108873111860ildupi(Nb)) ).

% htt_vebt_buildupi'
tff(fact_718_mult__cancel__right,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_right
tff(fact_719_mult__cancel__left,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% mult_cancel_left
tff(fact_720_mult__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% mult_eq_0_iff
tff(fact_721_mult__zero__right,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% mult_zero_right
tff(fact_722_mult__zero__left,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% mult_zero_left
tff(fact_723_bits__div__by__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% bits_div_by_0
tff(fact_724_bits__div__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% bits_div_0
tff(fact_725_div__by__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% div_by_0
tff(fact_726_div__0,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% div_0
tff(fact_727_less__nat__zero__code,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,ord_less(nat,Nb),zero_zero(nat)) ).

% less_nat_zero_code
tff(fact_728_neq0__conv,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ).

% neq0_conv
tff(fact_729_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero(nat) )
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),A2) ) ).

% bot_nat_0.not_eq_extremum
tff(fact_730_le0,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Nb) ).

% le0
tff(fact_731_bot__nat__0_Oextremum,axiom,
    ! [A2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),A2) ).

% bot_nat_0.extremum
tff(fact_732_add__is__0,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        & ( Nb = zero_zero(nat) ) ) ) ).

% add_is_0
tff(fact_733_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_734_diff__0__eq__0,axiom,
    ! [Nb: nat] : aa(nat,nat,minus_minus(nat,zero_zero(nat)),Nb) = zero_zero(nat) ).

% diff_0_eq_0
tff(fact_735_diff__self__eq__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),M) = zero_zero(nat) ).

% diff_self_eq_0
tff(fact_736_minNulli__rule,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : hoare_hoare_triple($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),vEBT_VEBT_minNulli(Tib),aa(vEBT_VEBTi,fun($o,assn),aTP_Lamp_bv(vEBT_VEBT,fun(vEBT_VEBTi,fun($o,assn)),Ta),Tib)) ).

% minNulli_rule
tff(fact_737_mult__cancel2,axiom,
    ! [M: nat,K: nat,Nb: 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),Nb),K) )
    <=> ( ( M = Nb )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel2
tff(fact_738_mult__cancel1,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb) )
    <=> ( ( M = Nb )
        | ( K = zero_zero(nat) ) ) ) ).

% mult_cancel1
tff(fact_739_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_740_mult__is__0,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb) = zero_zero(nat) )
    <=> ( ( M = zero_zero(nat) )
        | ( Nb = zero_zero(nat) ) ) ) ).

% mult_is_0
tff(fact_741_merge__pure__star,axiom,
    ! [A2: $o,B2: $o] :
      aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn((A2))),pure_assn((B2))) = pure_assn(( (A2)
        & (B2) )) ).

% merge_pure_star
tff(fact_742_pure__assn__eq__emp__iff,axiom,
    ! [P: $o] :
      ( ( pure_assn((P)) = one_one(assn) )
    <=> (P) ) ).

% pure_assn_eq_emp_iff
tff(fact_743_pure__true,axiom,
    pure_assn($true) = one_one(assn) ).

% pure_true
tff(fact_744_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_745_zero__comp__diff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,minus_minus(A,A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% zero_comp_diff_simps(1)
tff(fact_746_zero__comp__diff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,minus_minus(A,A2),B2))
        <=> aa(A,$o,ord_less(A,B2),A2) ) ) ).

% zero_comp_diff_simps(2)
tff(fact_747_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_748_mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_right2
tff(fact_749_mult__cancel__right1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B2: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_right1
tff(fact_750_mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = C2 )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = one_one(A) ) ) ) ) ).

% mult_cancel_left2
tff(fact_751_mult__cancel__left1,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [C2: A,B2: A] :
          ( ( C2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( B2 = one_one(A) ) ) ) ) ).

% mult_cancel_left1
tff(fact_752_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_753_div__self,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = one_one(A) ) ) ) ).

% div_self
tff(fact_754_nonzero__mult__div__cancel__right,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != 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),A2),B2)),B2) = A2 ) ) ) ).

% nonzero_mult_div_cancel_right
tff(fact_755_nonzero__mult__div__cancel__left,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != 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),A2),B2)),A2) = B2 ) ) ) ).

% nonzero_mult_div_cancel_left
tff(fact_756_div__mult__mult1__if,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% div_mult_mult1_if
tff(fact_757_div__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: 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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% div_mult_mult2
tff(fact_758_div__mult__mult1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: 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),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% div_mult_mult1
tff(fact_759_power__0__Suc,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),aa(nat,nat,suc,Nb)) = zero_zero(A) ) ).

% power_0_Suc
tff(fact_760_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_761_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_762_of__nat__0__eq__iff,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( ( zero_zero(A) = aa(nat,A,semiring_1_of_nat(A),Nb) )
        <=> ( zero_zero(nat) = Nb ) ) ) ).

% of_nat_0_eq_iff
tff(fact_763_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_764_power__Suc0__right,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% power_Suc0_right
tff(fact_765_zero__less__Suc,axiom,
    ! [Nb: nat] : aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,suc,Nb)) ).

% zero_less_Suc
tff(fact_766_less__Suc0,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))
    <=> ( Nb = zero_zero(nat) ) ) ).

% less_Suc0
tff(fact_767_add__gr__0,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
        | aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ) ).

% add_gr_0
tff(fact_768_less__one,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),one_one(nat))
    <=> ( Nb = zero_zero(nat) ) ) ).

% less_one
tff(fact_769_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_770_zero__less__diff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,minus_minus(nat,Nb),M))
    <=> aa(nat,$o,ord_less(nat,M),Nb) ) ).

% zero_less_diff
tff(fact_771_mult__eq__1__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% mult_eq_1_iff
tff(fact_772_one__eq__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb) )
    <=> ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
        & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% one_eq_mult_iff
tff(fact_773_div__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) = zero_zero(nat) ) ) ).

% div_less
tff(fact_774_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_775_power__Suc__0,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) = aa(nat,nat,suc,zero_zero(nat)) ).

% power_Suc_0
tff(fact_776_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),K)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
        & aa(nat,$o,ord_less(nat,M),Nb) ) ) ).

% mult_less_cancel2
tff(fact_777_nat__0__less__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
        & aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ) ).

% nat_0_less_mult_iff
tff(fact_778_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
        & aa(nat,$o,ord_less(nat,M),Nb) ) ) ).

% nat_mult_less_cancel_disj
tff(fact_779_diff__is__0__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),Nb) = zero_zero(nat) )
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% diff_is_0_eq
tff(fact_780_diff__is__0__eq_H,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(nat,nat,minus_minus(nat,M),Nb) = zero_zero(nat) ) ) ).

% diff_is_0_eq'
tff(fact_781_nat__zero__less__power__iff,axiom,
    ! [Xb: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Xb),Nb))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Xb)
        | ( Nb = zero_zero(nat) ) ) ) ).

% nat_zero_less_power_iff
tff(fact_782_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb)) = $ite(K = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)) ).

% nat_mult_div_cancel_disj
tff(fact_783_vebt__buildupi__rule,axiom,
    ! [Nb: nat] : time_htt(vEBT_VEBTi,pure_assn(aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)),vEBT_vebt_buildupi(Nb),aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_buildup(Nb)),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))),Nb))) ).

% vebt_buildupi_rule
tff(fact_784_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W: num] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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),A2),aa(num,A,numeral_numeral(A),W)) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(1)
tff(fact_785_divide__eq__eq__numeral1_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),W)) = A2 )
        <=> $ite(aa(num,A,numeral_numeral(A),W) != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W)),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(1)
tff(fact_786_div__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != 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),B2),C2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self4
tff(fact_787_div__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != 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),B2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self3
tff(fact_788_div__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != 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),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self2
tff(fact_789_div__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != 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),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_mult_self1
tff(fact_790_of__nat__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [M: nat] :
          ( aa(A,$o,aa(A,fun(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_791_power__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,Nb: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            & aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ) ) ).

% power_eq_0_iff
tff(fact_792_Suc__pred,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))) = Nb ) ) ).

% Suc_pred
tff(fact_793_one__le__mult__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),M)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) ) ) ).

% one_le_mult_iff
tff(fact_794_div__eq__dividend__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) = M )
      <=> ( Nb = one_one(nat) ) ) ) ).

% div_eq_dividend_iff
tff(fact_795_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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),Nb))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ).

% nat_mult_le_cancel_disj
tff(fact_796_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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),Nb),K))
    <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ).

% mult_le_cancel2
tff(fact_797_div__mult__self1__is__m,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),M)),Nb) = M ) ) ).

% div_mult_self1_is_m
tff(fact_798_div__mult__self__is__m,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb)),Nb) = M ) ) ).

% div_mult_self_is_m
tff(fact_799_zero__eq__power2,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_power2
tff(fact_800_power__strict__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,Nb: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
         => ( aa(A,$o,ord_less(A,B2),one_one(A))
           => ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
            <=> aa(nat,$o,ord_less(nat,Nb),M) ) ) ) ) ).

% power_strict_decreasing_iff
tff(fact_801_power__mono__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
              <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ) ).

% power_mono_iff
tff(fact_802_of__nat__0__less__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ) ).

% of_nat_0_less_iff
tff(fact_803_Suc__diff__1,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Nb),one_one(nat))) = Nb ) ) ).

% Suc_diff_1
tff(fact_804_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_805_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_806_power2__eq__iff__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(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_807_power2__less__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% power2_less_eq_zero_iff
tff(fact_808_zero__less__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_power2
tff(fact_809_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_810_power__decreasing__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [B2: A,M: nat,Nb: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
         => ( aa(A,$o,ord_less(A,B2),one_one(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
            <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M) ) ) ) ) ).

% power_decreasing_iff
tff(fact_811_of__nat__zero__less__power__iff,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: nat,Nb: nat] :
          ( aa(A,$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)),Nb))
        <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Xb)
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% of_nat_zero_less_power_iff
tff(fact_812_htt__vebt__inserti,axiom,
    ! [Ta: 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,Ta),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(Ta,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,Ta)))) ).

% htt_vebt_inserti
tff(fact_813_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_814_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_815_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_816_power__0__left,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Nb) = $ite(Nb = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% power_0_left
tff(fact_817_zero__power,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),zero_zero(A)),Nb) = zero_zero(A) ) ) ) ).

% zero_power
tff(fact_818_le__numeral__extra_I3_J,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),zero_zero(A)) ) ).

% le_numeral_extra(3)
tff(fact_819_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_820_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_821_zero__neq__numeral,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: num] : zero_zero(A) != aa(num,A,numeral_numeral(A),Nb) ) ).

% zero_neq_numeral
tff(fact_822_zero__neq__one,axiom,
    ! [A: $tType] :
      ( zero_neq_one(A)
     => ( zero_zero(A) != one_one(A) ) ) ).

% zero_neq_one
tff(fact_823_mult__right__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_right_cancel
tff(fact_824_mult__left__cancel,axiom,
    ! [A: $tType] :
      ( semiri6575147826004484403cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) )
          <=> ( A2 = B2 ) ) ) ) ).

% mult_left_cancel
tff(fact_825_no__zero__divisors,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) != zero_zero(A) ) ) ) ) ).

% no_zero_divisors
tff(fact_826_divisors__zero,axiom,
    ! [A: $tType] :
      ( semiri3467727345109120633visors(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = zero_zero(A) )
         => ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divisors_zero
tff(fact_827_mult__not__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) != zero_zero(A) )
         => ( ( A2 != zero_zero(A) )
            & ( B2 != zero_zero(A) ) ) ) ) ).

% mult_not_zero
tff(fact_828_semiring__1__no__zero__divisors__class_Opower__not__zero,axiom,
    ! [A: $tType] :
      ( semiri2026040879449505780visors(A)
     => ! [A2: A,Nb: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) != zero_zero(A) ) ) ) ).

% semiring_1_no_zero_divisors_class.power_not_zero
tff(fact_829_num_Osize_I4_J,axiom,
    aa(num,nat,size_size(num),one2) = zero_zero(nat) ).

% num.size(4)
tff(fact_830_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)
     => ~ ! [N2: 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,N2)),X2) ) ).

% VEBT_internal.replicatei.cases
tff(fact_831_not0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => ? [M2: nat] : Nb = aa(nat,nat,suc,M2) ) ).

% not0_implies_Suc
tff(fact_832_Zero__not__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_not_Suc
tff(fact_833_Zero__neq__Suc,axiom,
    ! [M: nat] : zero_zero(nat) != aa(nat,nat,suc,M) ).

% Zero_neq_Suc
tff(fact_834_Suc__neq__Zero,axiom,
    ! [M: nat] : aa(nat,nat,suc,M) != zero_zero(nat) ).

% Suc_neq_Zero
tff(fact_835_zero__induct,axiom,
    ! [P: fun(nat,$o),K: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [N2: nat] :
            ( aa(nat,$o,P,aa(nat,nat,suc,N2))
           => aa(nat,$o,P,N2) )
       => aa(nat,$o,P,zero_zero(nat)) ) ) ).

% zero_induct
tff(fact_836_diff__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),M: nat,Nb: nat] :
      ( ! [X2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,X2),zero_zero(nat))
     => ( ! [Y5: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,zero_zero(nat)),aa(nat,nat,suc,Y5))
       => ( ! [X2: nat,Y5: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,X2),Y5)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,aa(nat,nat,suc,X2)),aa(nat,nat,suc,Y5)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,M),Nb) ) ) ) ).

% diff_induct
tff(fact_837_nat__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N2: nat] :
            ( aa(nat,$o,P,N2)
           => aa(nat,$o,P,aa(nat,nat,suc,N2)) )
       => aa(nat,$o,P,Nb) ) ) ).

% nat_induct
tff(fact_838_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero(nat) )
     => ~ ! [Nat3: nat] : Y != aa(nat,nat,suc,Nat3) ) ).

% old.nat.exhaust
tff(fact_839_nat_OdiscI,axiom,
    ! [Nat: nat,X23: nat] :
      ( ( Nat = aa(nat,nat,suc,X23) )
     => ( Nat != zero_zero(nat) ) ) ).

% nat.discI
tff(fact_840_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).

% old.nat.distinct(1)
tff(fact_841_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] : aa(nat,nat,suc,Nat2) != zero_zero(nat) ).

% old.nat.distinct(2)
tff(fact_842_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] : zero_zero(nat) != aa(nat,nat,suc,X23) ).

% nat.distinct(1)
tff(fact_843_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_844_infinite__descent0,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N2: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N2)
           => ( ~ aa(nat,$o,P,N2)
             => ? [M3: nat] :
                  ( aa(nat,$o,ord_less(nat,M3),N2)
                  & ~ aa(nat,$o,P,M3) ) ) )
       => aa(nat,$o,P,Nb) ) ) ).

% infinite_descent0
tff(fact_845_gr__implies__not0,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => ( Nb != zero_zero(nat) ) ) ).

% gr_implies_not0
tff(fact_846_less__zeroE,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,ord_less(nat,Nb),zero_zero(nat)) ).

% less_zeroE
tff(fact_847_not__less0,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,ord_less(nat,Nb),zero_zero(nat)) ).

% not_less0
tff(fact_848_not__gr0,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
    <=> ( Nb = zero_zero(nat) ) ) ).

% not_gr0
tff(fact_849_gr0I,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ).

% gr0I
tff(fact_850_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] : ~ aa(nat,$o,ord_less(nat,A2),zero_zero(nat)) ).

% bot_nat_0.extremum_strict
tff(fact_851_le__0__eq,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),zero_zero(nat))
    <=> ( Nb = zero_zero(nat) ) ) ).

% le_0_eq
tff(fact_852_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),zero_zero(nat))
     => ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_uniqueI
tff(fact_853_bot__nat__0_Oextremum__unique,axiom,
    ! [A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),zero_zero(nat))
    <=> ( A2 = zero_zero(nat) ) ) ).

% bot_nat_0.extremum_unique
tff(fact_854_less__eq__nat_Osimps_I1_J,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),zero_zero(nat)),Nb) ).

% less_eq_nat.simps(1)
tff(fact_855_plus__nat_Oadd__0,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),zero_zero(nat)),Nb) = Nb ).

% plus_nat.add_0
tff(fact_856_add__eq__self__zero,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb) = M )
     => ( Nb = zero_zero(nat) ) ) ).

% add_eq_self_zero
tff(fact_857_minus__nat_Odiff__0,axiom,
    ! [M: nat] : aa(nat,nat,minus_minus(nat,M),zero_zero(nat)) = M ).

% minus_nat.diff_0
tff(fact_858_diffs0__imp__equal,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,nat,minus_minus(nat,M),Nb) = zero_zero(nat) )
     => ( ( aa(nat,nat,minus_minus(nat,Nb),M) = zero_zero(nat) )
       => ( M = Nb ) ) ) ).

% diffs0_imp_equal
tff(fact_859_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb) )
    <=> ( ( K = zero_zero(nat) )
        | ( M = Nb ) ) ) ).

% nat_mult_eq_cancel_disj
tff(fact_860_mult__0,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),zero_zero(nat)),Nb) = zero_zero(nat) ).

% mult_0
tff(fact_861_VEBTi_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] : aa(vEBT_VEBTi,nat,size_size(vEBT_VEBTi),vEBT_Leafi((X21),(X22))) = zero_zero(nat) ).

% VEBTi.size(4)
tff(fact_862_power__eq__imp__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
               => ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_imp_eq_base
tff(fact_863_power__eq__iff__eq__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb) )
              <=> ( A2 = B2 ) ) ) ) ) ) ).

% power_eq_iff_eq_base
tff(fact_864_lambda__zero,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ( aTP_Lamp_bw(A,A) = aa(A,fun(A,A),times_times(A),zero_zero(A)) ) ) ).

% lambda_zero
tff(fact_865_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_866_VEBT__internal_OminNulli_Osimps_I1_J,axiom,
    vEBT_VEBT_minNulli(vEBT_Leafi($false,$false)) = aa($o,heap_Time_Heap($o),heap_Time_return($o),$true) ).

% VEBT_internal.minNulli.simps(1)
tff(fact_867_VEBT__internal_OminNulli_Osimps_I2_J,axiom,
    ! [Uv: $o] : vEBT_VEBT_minNulli(vEBT_Leafi($true,(Uv))) = aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) ).

% VEBT_internal.minNulli.simps(2)
tff(fact_868_VEBT__internal_OminNulli_Osimps_I3_J,axiom,
    ! [Uu: $o] : vEBT_VEBT_minNulli(vEBT_Leafi((Uu),$true)) = aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) ).

% VEBT_internal.minNulli.simps(3)
tff(fact_869_power__strict__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
             => aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ) ).

% power_strict_mono
tff(fact_870_not__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Nb)),zero_zero(A)) ) ).

% not_numeral_le_zero
tff(fact_871_zero__le__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% zero_le_numeral
tff(fact_872_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod(nat,nat),Vaa: nat,Vb: list(vEBT_VEBT),Vc2: vEBT_VEBT] : ~ vEBT_VEBT_minNull(vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Vaa,Vb,Vc2)) ).

% VEBT_internal.minNull.simps(5)
tff(fact_873_not__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : ~ aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),Nb)),zero_zero(A)) ) ).

% not_numeral_less_zero
tff(fact_874_zero__less__numeral,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: num] : aa(A,$o,ord_less(A,zero_zero(A)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% zero_less_numeral
tff(fact_875_not__one__le__zero,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),zero_zero(A)) ) ).

% not_one_le_zero
tff(fact_876_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% linordered_nonzero_semiring_class.zero_le_one
tff(fact_877_zero__less__one__class_Ozero__le__one,axiom,
    ! [A: $tType] :
      ( zero_less_one(A)
     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),one_one(A)) ) ).

% zero_less_one_class.zero_le_one
tff(fact_878_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordere2520102378445227354miring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
tff(fact_879_zero__le__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_le_mult_iff
tff(fact_880_mult__nonneg__nonpos2,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos2
tff(fact_881_mult__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_nonpos_nonneg
tff(fact_882_mult__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_nonneg_nonpos
tff(fact_883_mult__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_nonneg_nonneg
tff(fact_884_split__mult__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_semiring_0(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ).

% split_mult_neg_le
tff(fact_885_mult__le__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ) ) ).

% mult_le_0_iff
tff(fact_886_mult__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_right_mono
tff(fact_887_mult__right__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_right_mono_neg
tff(fact_888_mult__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_left_mono
tff(fact_889_mult__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_nonpos_nonpos
tff(fact_890_mult__left__mono__neg,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_left_mono_neg
tff(fact_891_split__mult__pos__le,axiom,
    ! [A: $tType] :
      ( ordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ).

% split_mult_pos_le
tff(fact_892_zero__le__square,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2)) ) ).

% zero_le_square
tff(fact_893_mult__mono_H,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_mono'
tff(fact_894_mult__mono,axiom,
    ! [A: $tType] :
      ( ordered_semiring(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_mono
tff(fact_895_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_896_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_897_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_898_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_899_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord2810124833399127020strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( 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),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
tff(fact_900_mult__less__cancel__right__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
              & aa(A,$o,ord_less(A,A2),B2) )
            | ( aa(A,$o,ord_less(A,C2),zero_zero(A))
              & aa(A,$o,ord_less(A,B2),A2) ) ) ) ) ).

% mult_less_cancel_right_disj
tff(fact_901_mult__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( 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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_strict_right_mono
tff(fact_902_mult__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
         => ( 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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% mult_strict_right_mono_neg
tff(fact_903_mult__less__cancel__left__disj,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
              & aa(A,$o,ord_less(A,A2),B2) )
            | ( aa(A,$o,ord_less(A,C2),zero_zero(A))
              & aa(A,$o,ord_less(A,B2),A2) ) ) ) ) ).

% mult_less_cancel_left_disj
tff(fact_904_mult__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( 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),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_strict_left_mono
tff(fact_905_mult__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
         => ( 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),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ).

% mult_strict_left_mono_neg
tff(fact_906_mult__less__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,ord_less(A,A2),B2) ) ) ) ).

% mult_less_cancel_left_pos
tff(fact_907_mult__less__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,ord_less(A,B2),A2) ) ) ) ).

% mult_less_cancel_left_neg
tff(fact_908_zero__less__mult__pos2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
           => aa(A,$o,ord_less(A,zero_zero(A)),B2) ) ) ) ).

% zero_less_mult_pos2
tff(fact_909_zero__less__mult__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
           => aa(A,$o,ord_less(A,zero_zero(A)),B2) ) ) ) ).

% zero_less_mult_pos
tff(fact_910_zero__less__mult__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
              & aa(A,$o,ord_less(A,zero_zero(A)),B2) )
            | ( aa(A,$o,ord_less(A,A2),zero_zero(A))
              & aa(A,$o,ord_less(A,B2),zero_zero(A)) ) ) ) ) ).

% zero_less_mult_iff
tff(fact_911_mult__pos__neg2,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,B2),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),zero_zero(A)) ) ) ) ).

% mult_pos_neg2
tff(fact_912_mult__pos__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_pos_pos
tff(fact_913_mult__pos__neg,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,B2),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_pos_neg
tff(fact_914_mult__neg__pos,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A)) ) ) ) ).

% mult_neg_pos
tff(fact_915_mult__less__0__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
              & aa(A,$o,ord_less(A,B2),zero_zero(A)) )
            | ( aa(A,$o,ord_less(A,A2),zero_zero(A))
              & aa(A,$o,ord_less(A,zero_zero(A)),B2) ) ) ) ) ).

% mult_less_0_iff
tff(fact_916_not__square__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A] : ~ aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2)),zero_zero(A)) ) ).

% not_square_less_zero
tff(fact_917_mult__neg__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,B2),zero_zero(A))
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ) ) ).

% mult_neg_neg
tff(fact_918_zero__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% zero_le_power
tff(fact_919_power__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ).

% power_mono
tff(fact_920_zero__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% zero_less_power
tff(fact_921_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : vEBT_VEBT_minNull(vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)) ).

% VEBT_internal.minNull.simps(4)
tff(fact_922_of__nat__0__le__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% of_nat_0_le_iff
tff(fact_923_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_924_semiring__char__0__class_Oof__nat__neq__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Nb)) != zero_zero(A) ) ).

% semiring_char_0_class.of_nat_neq_0
tff(fact_925_power__0,axiom,
    ! [A: $tType] :
      ( power(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),zero_zero(nat)) = one_one(A) ) ).

% power_0
tff(fact_926_less__Suc__eq__0__disj,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,suc,Nb))
    <=> ( ( M = zero_zero(nat) )
        | ? [J3: nat] :
            ( ( M = aa(nat,nat,suc,J3) )
            & aa(nat,$o,ord_less(nat,J3),Nb) ) ) ) ).

% less_Suc_eq_0_disj
tff(fact_927_gr0__implies__Suc,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ? [M2: nat] : Nb = aa(nat,nat,suc,M2) ) ).

% gr0_implies_Suc
tff(fact_928_All__less__Suc2,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,suc,Nb))
         => aa(nat,$o,P,I2) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I2: nat] :
            ( aa(nat,$o,ord_less(nat,I2),Nb)
           => aa(nat,$o,P,aa(nat,nat,suc,I2)) ) ) ) ).

% All_less_Suc2
tff(fact_929_gr0__conv__Suc,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
    <=> ? [M7: nat] : Nb = aa(nat,nat,suc,M7) ) ).

% gr0_conv_Suc
tff(fact_930_Ex__less__Suc2,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,suc,Nb))
          & aa(nat,$o,P,I2) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        | ? [I2: nat] :
            ( aa(nat,$o,ord_less(nat,I2),Nb)
            & aa(nat,$o,P,aa(nat,nat,suc,I2)) ) ) ) ).

% Ex_less_Suc2
tff(fact_931_nat__compl__induct_H,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nn),N2)
               => aa(nat,$o,P,Nn) )
           => aa(nat,$o,P,aa(nat,nat,suc,N2)) )
       => aa(nat,$o,P,Nb) ) ) ).

% nat_compl_induct'
tff(fact_932_nat__compl__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N2: nat] :
            ( ! [Nn: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nn),N2)
               => aa(nat,$o,P,Nn) )
           => aa(nat,$o,P,aa(nat,nat,suc,N2)) )
       => aa(nat,$o,P,Nb) ) ) ).

% nat_compl_induct
tff(fact_933_add__is__1,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( Nb = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% add_is_1
tff(fact_934_one__is__add,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,nat,suc,zero_zero(nat)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb) )
    <=> ( ( ( M = aa(nat,nat,suc,zero_zero(nat)) )
          & ( Nb = zero_zero(nat) ) )
        | ( ( M = zero_zero(nat) )
          & ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ) ).

% one_is_add
tff(fact_935_One__nat__def,axiom,
    one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).

% One_nat_def
tff(fact_936_option_Osize_I4_J,axiom,
    ! [A: $tType,X23: A] : aa(option(A),nat,size_size(option(A)),aa(A,option(A),some(A),X23)) = aa(nat,nat,suc,zero_zero(nat)) ).

% option.size(4)
tff(fact_937_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_938_ex__least__nat__le,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K3: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K3),Nb)
            & ! [I4: nat] :
                ( aa(nat,$o,ord_less(nat,I4),K3)
               => ~ aa(nat,$o,P,I4) )
            & aa(nat,$o,P,K3) ) ) ) ).

% ex_least_nat_le
tff(fact_939_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ? [K3: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K3)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I),K3) = J ) ) ) ).

% less_imp_add_positive
tff(fact_940_Suc__to__right,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa(nat,nat,suc,Nb) = M )
     => ( Nb = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,zero_zero(nat))) ) ) ).

% Suc_to_right
tff(fact_941_diff__less,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,M),Nb)),M) ) ) ).

% diff_less
tff(fact_942_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) = zero_zero(nat) )
    <=> ( aa(nat,$o,ord_less(nat,M),Nb)
        | ( Nb = zero_zero(nat) ) ) ) ).

% Euclidean_Division.div_eq_0_iff
tff(fact_943_nat__geq__1__eq__neqz,axiom,
    ! [Xb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Xb)
    <=> ( Xb != zero_zero(nat) ) ) ).

% nat_geq_1_eq_neqz
tff(fact_944_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_945_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_946_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_947_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb) )
      <=> ( M = Nb ) ) ) ).

% nat_mult_eq_cancel1
tff(fact_948_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb))
      <=> aa(nat,$o,ord_less(nat,M),Nb) ) ) ).

% nat_mult_less_cancel1
tff(fact_949_diff__add__0,axiom,
    ! [Nb: nat,M: nat] : aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M)) = zero_zero(nat) ).

% diff_add_0
tff(fact_950_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,Nb: 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),Nb))
       => aa(nat,$o,ord_less(nat,M),Nb) ) ) ).

% nat_power_less_imp_less
tff(fact_951_mult__eq__self__implies__10,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb) )
     => ( ( Nb = one_one(nat) )
        | ( M = zero_zero(nat) ) ) ) ).

% mult_eq_self_implies_10
tff(fact_952_vebt__insert_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,zero_zero(nat),Ts,S3),Xb) = vEBT_Node(Info,zero_zero(nat),Ts,S3) ).

% vebt_insert.simps(2)
tff(fact_953_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_954_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_955_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_956_mult__less__le__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( 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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_less_le_imp_less
tff(fact_957_mult__le__less__imp__less,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,ord_less(A,C2),D2)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_le_less_imp_less
tff(fact_958_mult__right__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_right_le_imp_le
tff(fact_959_mult__left__le__imp__le,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_left_le_imp_le
tff(fact_960_mult__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% mult_le_cancel_left_pos
tff(fact_961_mult__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% mult_le_cancel_left_neg
tff(fact_962_mult__less__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B2),A2) ) ) ) ) ).

% mult_less_cancel_right
tff(fact_963_mult__strict__mono_H,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,ord_less(A,C2),D2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
             => ( aa(A,$o,aa(A,fun(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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_strict_mono'
tff(fact_964_mult__right__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,ord_less(A,A2),B2) ) ) ) ).

% mult_right_less_imp_less
tff(fact_965_mult__less__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A2),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B2),A2) ) ) ) ) ).

% mult_less_cancel_left
tff(fact_966_mult__strict__mono,axiom,
    ! [A: $tType] :
      ( linord8928482502909563296strict(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,ord_less(A,C2),D2)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
             => ( aa(A,$o,aa(A,fun(A,$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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ) ).

% mult_strict_mono
tff(fact_967_mult__left__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_semiring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,ord_less(A,A2),B2) ) ) ) ).

% mult_left_less_imp_less
tff(fact_968_mult__le__cancel__right,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% mult_le_cancel_right
tff(fact_969_mult__le__cancel__left,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% mult_le_cancel_left
tff(fact_970_sum__squares__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(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_971_sum__squares__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linord4710134922213307826strict(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(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_972_mult__left__le,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),A2) ) ) ) ).

% mult_left_le
tff(fact_973_mult__le__one,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A)) ) ) ) ) ).

% mult_le_one
tff(fact_974_mult__right__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
             => aa(A,$o,aa(A,fun(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_975_mult__left__le__one__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),one_one(A))
             => aa(A,$o,aa(A,fun(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_976_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_977_div__positive,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% div_positive
tff(fact_978_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,A2),B2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
tff(fact_979_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_980_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_981_power__less__imp__less__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,ord_less(A,A2),B2) ) ) ) ).

% power_less_imp_less_base
tff(fact_982_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
tff(fact_983_power__le__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),one_one(A)) ) ) ) ).

% power_le_one
tff(fact_984_eq__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B2: A,C2: A] :
          ( ( aa(num,A,numeral_numeral(A),W) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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) = B2,aa(num,A,numeral_numeral(A),W) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(1)
tff(fact_985_divide__eq__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(num,A,numeral_numeral(A),W) )
        <=> $ite(C2 != zero_zero(A),B2 = 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_986_power__le__imp__le__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,Nb)))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% power_le_imp_le_base
tff(fact_987_power__inject__base,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat,B2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),aa(nat,nat,suc,Nb)) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% power_inject_base
tff(fact_988_div__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != 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),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% div_add_self2
tff(fact_989_div__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != 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),B2),A2)),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% div_add_self1
tff(fact_990_VEBT__internal_OminNulli_Osimps_I5_J,axiom,
    ! [Uz: product_prod(nat,nat),Vaa: nat,Vb: array(vEBT_VEBTi),Vc2: vEBT_VEBTi] : vEBT_VEBT_minNulli(vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz),Vaa,Vb,Vc2)) = aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) ).

% VEBT_internal.minNulli.simps(5)
tff(fact_991_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_992_VEBT__internal_OminNulli_Osimps_I4_J,axiom,
    ! [Uw: nat,Ux: array(vEBT_VEBTi),Uy: vEBT_VEBTi] : vEBT_VEBT_minNulli(vEBT_Nodei(none(product_prod(nat,nat)),Uw,Ux,Uy)) = aa($o,heap_Time_Heap($o),heap_Time_return($o),$true) ).

% VEBT_internal.minNulli.simps(4)
tff(fact_993_num_Osize_I5_J,axiom,
    ! [X23: num] : aa(num,nat,size_size(num),bit0(X23)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,size_size(num),X23)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size(5)
tff(fact_994_ex__least__nat__less,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ? [K3: nat] :
            ( aa(nat,$o,ord_less(nat,K3),Nb)
            & ! [I4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I4),K3)
               => ~ aa(nat,$o,P,I4) )
            & aa(nat,$o,P,aa(nat,nat,suc,K3)) ) ) ) ).

% ex_least_nat_less
tff(fact_995_nat__induct__non__zero,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N2: nat] :
              ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N2)
             => ( aa(nat,$o,P,N2)
               => aa(nat,$o,P,aa(nat,nat,suc,N2)) ) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_induct_non_zero
tff(fact_996_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_997_diff__Suc__less,axiom,
    ! [Nb: nat,I: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,I))),Nb) ) ).

% diff_Suc_less
tff(fact_998_n__less__n__mult__m,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),M)) ) ) ).

% n_less_n_mult_m
tff(fact_999_n__less__m__mult__n,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),M)
       => aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb)) ) ) ).

% n_less_m_mult_n
tff(fact_1000_one__less__mult,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),Nb)
     => ( aa(nat,$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),Nb)) ) ) ).

% one_less_mult
tff(fact_1001_power__gt__expt,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),K)) ) ).

% power_gt_expt
tff(fact_1002_nat__diff__split__asm,axiom,
    ! [P: fun(nat,$o),A2: nat,B2: nat] :
      ( aa(nat,$o,P,aa(nat,nat,minus_minus(nat,A2),B2))
    <=> ~ ( ( aa(nat,$o,ord_less(nat,A2),B2)
            & ~ aa(nat,$o,P,zero_zero(nat)) )
          | ? [D4: nat] :
              ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D4) )
              & ~ aa(nat,$o,P,D4) ) ) ) ).

% nat_diff_split_asm
tff(fact_1003_nat__diff__split,axiom,
    ! [P: fun(nat,$o),A2: nat,B2: nat] :
      ( aa(nat,$o,P,aa(nat,nat,minus_minus(nat,A2),B2))
    <=> ( ( aa(nat,$o,ord_less(nat,A2),B2)
         => aa(nat,$o,P,zero_zero(nat)) )
        & ! [D4: nat] :
            ( ( A2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),D4) )
           => aa(nat,$o,P,D4) ) ) ) ).

% nat_diff_split
tff(fact_1004_div__greater__zero__iff,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
        & aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ) ).

% div_greater_zero_iff
tff(fact_1005_div__le__mono2,axiom,
    ! [M: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),K),M)) ) ) ).

% div_le_mono2
tff(fact_1006_div__less__dividend,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,one_one(nat)),Nb)
     => ( aa(nat,$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),Nb)),M) ) ) ).

% div_less_dividend
tff(fact_1007_nat__one__le__power,axiom,
    ! [I: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),I)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Nb)) ) ).

% nat_one_le_power
tff(fact_1008_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ).

% nat_mult_le_cancel1
tff(fact_1009_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M: nat,Nb: 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)),Nb)
      <=> aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q2)) ) ) ).

% div_less_iff_less_mult
tff(fact_1010_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) ) ) ).

% nat_mult_div_cancel1
tff(fact_1011_td__gal__lt,axiom,
    ! [C2: nat,A2: nat,B2: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => ( aa(nat,$o,ord_less(nat,A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),C2))
      <=> aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),C2)),B2) ) ) ).

% td_gal_lt
tff(fact_1012_assn__times__assoc,axiom,
    ! [P: assn,Q: assn,R: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q)),R) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),R)) ).

% assn_times_assoc
tff(fact_1013_assn__times__comm,axiom,
    ! [P: assn,Q: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),P) ).

% assn_times_comm
tff(fact_1014_vebt__insert_Osimps_I3_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT,Xb: nat] : vEBT_vebt_insert(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S3),Xb) = vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S3) ).

% vebt_insert.simps(3)
tff(fact_1015_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_1016_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_1017_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_1018_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),Vc2: 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,Vc2),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_1019_mult__less__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,ord_less(A,one_one(A)),A2) ) ) ) ) ).

% mult_less_cancel_right2
tff(fact_1020_mult__less__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,ord_less(A,C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_right1
tff(fact_1021_mult__less__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),C2)
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A2),one_one(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,ord_less(A,one_one(A)),A2) ) ) ) ) ).

% mult_less_cancel_left2
tff(fact_1022_mult__less__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,ord_less(A,C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,one_one(A)),B2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B2),one_one(A)) ) ) ) ) ).

% mult_less_cancel_left1
tff(fact_1023_mult__le__cancel__right2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),C2)
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A)) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2) ) ) ) ) ).

% mult_le_cancel_right2
tff(fact_1024_mult__le__cancel__right1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_right1
tff(fact_1025_mult__le__cancel__left2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),C2)
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A)) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2) ) ) ) ) ).

% mult_le_cancel_left2
tff(fact_1026_mult__le__cancel__left1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),B2) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),one_one(A)) ) ) ) ) ).

% mult_le_cancel_left1
tff(fact_1027_convex__bound__le,axiom,
    ! [A: $tType] :
      ( linord6961819062388156250ring_1(A)
     => ! [Xb: A,A2: A,Y: A,U: A,V: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(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,aa(A,fun(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))),A2) ) ) ) ) ) ) ).

% convex_bound_le
tff(fact_1028_less__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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)),B2),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).

% less_divide_eq_numeral(1)
tff(fact_1029_divide__less__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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,B2),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,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),aa(A,$o,ord_less(A,zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).

% divide_less_eq_numeral(1)
tff(fact_1030_power__Suc__less,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,A2),one_one(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ).

% power_Suc_less
tff(fact_1031_power__Suc__le__self,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb))),A2) ) ) ) ).

% power_Suc_le_self
tff(fact_1032_power__Suc__less__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,A2),one_one(A))
           => aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,Nb))),one_one(A)) ) ) ) ).

% power_Suc_less_one
tff(fact_1033_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_1034_power__strict__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N: nat,A2: A] :
          ( aa(nat,$o,ord_less(nat,Nb),N)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
           => ( aa(A,$o,ord_less(A,A2),one_one(A))
             => aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ) ).

% power_strict_decreasing
tff(fact_1035_power__decreasing,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,N: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),N)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ) ).

% power_decreasing
tff(fact_1036_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_1037_self__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),A2)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ).

% self_le_power
tff(fact_1038_pos2,axiom,
    aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% pos2
tff(fact_1039_one__less__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,ord_less(A,one_one(A)),A2)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
           => aa(A,$o,ord_less(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ).

% one_less_power
tff(fact_1040_power__diff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,Nb: nat,M: nat] :
          ( ( A2 != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,minus_minus(nat,M),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ) ).

% power_diff
tff(fact_1041_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_1042_nz__le__conv__less,axiom,
    ! [K: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
     => ( aa(nat,$o,aa(nat,fun(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_1043_Suc__diff__eq__diff__pred,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Nb) = aa(nat,nat,minus_minus(nat,M),aa(nat,nat,minus_minus(nat,Nb),one_one(nat))) ) ) ).

% Suc_diff_eq_diff_pred
tff(fact_1044_Suc__pred_H,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( Nb = aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Nb),one_one(nat))) ) ) ).

% Suc_pred'
tff(fact_1045_div__if,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) = $ite(
        ( aa(nat,$o,ord_less(nat,M),Nb)
        | ( Nb = 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),Nb)),Nb)) ) ).

% div_if
tff(fact_1046_div__geq,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( ~ aa(nat,$o,ord_less(nat,M),Nb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,M),Nb)),Nb)) ) ) ) ).

% div_geq
tff(fact_1047_add__eq__if,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb) = $ite(M = zero_zero(nat),Nb,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))),Nb))) ).

% add_eq_if
tff(fact_1048_msrevs_I1_J,axiom,
    ! [Nb: nat,K: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( 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),Nb)),M)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),K) ) ) ).

% msrevs(1)
tff(fact_1049_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Q2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),Q2))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)),Nb) ) ) ).

% less_eq_div_iff_mult_less_eq
tff(fact_1050_dividend__less__times__div,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)))) ) ).

% dividend_less_times_div
tff(fact_1051_dividend__less__div__times,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),Nb))) ) ).

% dividend_less_div_times
tff(fact_1052_split__div,axiom,
    ! [P: fun(nat,$o),M: nat,Nb: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(nat,$o,P,zero_zero(nat)) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I2: nat,J3: nat] :
              ( aa(nat,$o,ord_less(nat,J3),Nb)
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),I2)),J3) )
               => aa(nat,$o,P,I2) ) ) ) ) ) ).

% split_div
tff(fact_1053_td__gal,axiom,
    ! [C2: nat,B2: nat,A2: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),C2)),A2)
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A2),C2)) ) ) ).

% td_gal
tff(fact_1054_mult__eq__if,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb) = $ite(M = zero_zero(nat),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,M),one_one(nat))),Nb))) ).

% mult_eq_if
tff(fact_1055_nat__mult__power__less__eq,axiom,
    ! [B2: nat,A2: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),B2)
     => ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),M))
      <=> aa(nat,$o,ord_less(nat,A2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,minus_minus(nat,M),Nb))) ) ) ).

% nat_mult_power_less_eq
tff(fact_1056_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vb: list(vEBT_VEBT),Vc2: 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,Vc2)),Xb) ).

% vebt_member.simps(4)
tff(fact_1057_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_1058_vebt__succ_Osimps_I4_J,axiom,
    ! [V: product_prod(nat,nat),Vc2: 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),Vc2,Vd),Ve) = none(nat) ).

% vebt_succ.simps(4)
tff(fact_1059_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_1060_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),Vc2: 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),Vc2,Vd),Ve) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(4)
tff(fact_1061_convex__bound__lt,axiom,
    ! [A: $tType] :
      ( linord715952674999750819strict(A)
     => ! [Xb: A,A2: A,Y: A,U: A,V: A] :
          ( aa(A,$o,ord_less(A,Xb),A2)
         => ( aa(A,$o,ord_less(A,Y),A2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),U)
             => ( aa(A,$o,aa(A,fun(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))),A2) ) ) ) ) ) ) ).

% convex_bound_lt
tff(fact_1062_le__divide__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),W)),zero_zero(A))) ) ) ) ).

% le_divide_eq_numeral(1)
tff(fact_1063_divide__le__eq__numeral_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(num,A,numeral_numeral(A),W))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),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,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(num,A,numeral_numeral(A),W))) ) ) ) ).

% divide_le_eq_numeral(1)
tff(fact_1064_half__gt__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A2) ) ) ).

% half_gt_zero_iff
tff(fact_1065_half__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ) ).

% half_gt_zero
tff(fact_1066_power2__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% power2_le_imp_le
tff(fact_1067_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,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
             => ( Xb = Y ) ) ) ) ) ).

% power2_eq_imp_eq
tff(fact_1068_zero__le__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% zero_le_power2
tff(fact_1069_power2__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : ~ aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),zero_zero(A)) ) ).

% power2_less_0
tff(fact_1070_exp__add__not__zero__imp__right,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Nb: 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),Nb)) != zero_zero(A) )
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) != zero_zero(A) ) ) ) ).

% exp_add_not_zero_imp_right
tff(fact_1071_exp__add__not__zero__imp__left,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Nb: 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),Nb)) != 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_1072_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat,M: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) != 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,Nb),M)) != zero_zero(A) ) ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
tff(fact_1073_power__diff__power__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,M: nat,Nb: nat] :
          ( ( A2 != 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),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,minus_minus(nat,M),Nb)),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),A2),aa(nat,nat,minus_minus(nat,Nb),M)))) ) ) ) ).

% power_diff_power_eq
tff(fact_1074_less__2__cases__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))
    <=> ( ( Nb = zero_zero(nat) )
        | ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases_iff
tff(fact_1075_less__2__cases,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))
     => ( ( Nb = zero_zero(nat) )
        | ( Nb = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).

% less_2_cases
tff(fact_1076_inverse__of__nat__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
         => ( ( Nb != zero_zero(nat) )
           => aa(A,$o,aa(A,fun(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),Nb))) ) ) ) ).

% inverse_of_nat_le
tff(fact_1077_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_1078_power__minus__mult,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Nb: nat,A2: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))),A2) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) ) ) ) ).

% power_minus_mult
tff(fact_1079_le__div__geq,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,minus_minus(nat,M),Nb)),Nb)) ) ) ) ).

% le_div_geq
tff(fact_1080_split__div_H,axiom,
    ! [P: fun(nat,$o),M: nat,Nb: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
          & aa(nat,$o,P,zero_zero(nat)) )
        | ? [Q4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q4)),M)
            & aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Q4)))
            & aa(nat,$o,P,Q4) ) ) ) ).

% split_div'
tff(fact_1081_power__sub,axiom,
    ! [Nb: nat,M: nat,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),A2)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A2),aa(nat,nat,minus_minus(nat,M),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A2),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),A2),Nb)) ) ) ) ).

% power_sub
tff(fact_1082_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_1083_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_1084_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_1085_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_1086_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,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,ord_less(A,Xb),Y) ) ) ) ).

% power2_less_imp_less
tff(fact_1087_sum__power2__le__zero__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(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_1088_sum__power2__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] : aa(A,$o,aa(A,fun(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_1089_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_1090_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_1091_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) ) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leafi($true,(Uv2))
           => ( 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) ) )
           => ( ( ? [Uw2: nat,Ux2: array(vEBT_VEBTi),Uy2: vEBT_VEBTi] : Xb = vEBT_Nodei(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
               => ( Y != aa($o,heap_Time_Heap($o),heap_Time_return($o),$true) ) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va: nat,Vb2: array(vEBT_VEBTi),Vc: vEBT_VEBTi] : Xb = vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va,Vb2,Vc)
                 => ( Y != aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.elims
tff(fact_1092_zero__le__even__power_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ) ).

% zero_le_even_power'
tff(fact_1093_Suc__n__div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% Suc_n_div_2_gt_zero
tff(fact_1094_div__2__gt__zero,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% div_2_gt_zero
tff(fact_1095_nat__bit__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( ! [N2: nat] :
            ( aa(nat,$o,P,N2)
           => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N2)
             => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)) ) )
       => ( ! [N2: nat] :
              ( aa(nat,$o,P,N2)
             => aa(nat,$o,P,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N2))) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_bit_induct
tff(fact_1096_vebt__minti_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_minti(vEBT_Leafi((A2),(B2))) = $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))) ) ).

% vebt_minti.simps(1)
tff(fact_1097_vebt__maxti_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_maxti(vEBT_Leafi((A2),(B2))) = $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))) ) ).

% vebt_maxti.simps(1)
tff(fact_1098_mod__starE,axiom,
    ! [A2: assn,B2: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A2),B2),H)
     => ~ ( ? [X_13: product_prod(heap_ext(product_unit),set(nat))] : rep_assn(A2,X_13)
         => ! [H_2: product_prod(heap_ext(product_unit),set(nat))] : ~ rep_assn(B2,H_2) ) ) ).

% mod_starE
tff(fact_1099_mod__starD,axiom,
    ! [A3: assn,B5: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B5),H)
     => ? [H1: product_prod(heap_ext(product_unit),set(nat)),H22: product_prod(heap_ext(product_unit),set(nat))] :
          ( rep_assn(A3,H1)
          & rep_assn(B5,H22) ) ) ).

% mod_starD
tff(fact_1100_odd__0__le__power__imp__0__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% odd_0_le_power_imp_0_le
tff(fact_1101_odd__power__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,ord_less(A,A2),zero_zero(A))
         => aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))),zero_zero(A)) ) ) ).

% odd_power_less_zero
tff(fact_1102_assn__one__left,axiom,
    ! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),one_one(assn)),P) = P ).

% assn_one_left
tff(fact_1103_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [Xb: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M)))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
         => aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,Nb)),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_1104_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [Xb: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Xb),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M)))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
         => aa(nat,$o,ord_less(nat,vEBT_VEBT_low(Xb,Nb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
tff(fact_1105_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,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Xb: nat] : vEBT_T_m_e_m_b_e_r(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw),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_1106_nat__div__eq__Suc__0__iff,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),M) = aa(nat,nat,suc,zero_zero(nat)) )
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
        & aa(nat,$o,ord_less(nat,Nb),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_1107_ex__distrib__star,axiom,
    ! [A: $tType,P: fun(A,assn),Q: assn] : ex_assn(A,aa(assn,fun(A,assn),aTP_Lamp_bf(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_1108_ex__one__point__gen,axiom,
    ! [A: $tType,P: fun(A,assn),V: A] :
      ( ! [H2: product_prod(heap_ext(product_unit),set(nat)),X2: A] :
          ( rep_assn(aa(A,assn,P,X2),H2)
         => ( X2 = V ) )
     => ( ex_assn(A,P) = aa(A,assn,P,V) ) ) ).

% ex_one_point_gen
tff(fact_1109_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_1110_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_1111_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,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
             => aa(A,$o,aa(A,fun(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_1112_vebt__minti_Oelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_Time_Heap(option(nat))] :
      ( ( vEBT_vebt_minti(Xb) = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leafi((A4),(B3)) )
           => ( Y != $ite(
                  (A4),
                  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))) ) ) )
       => ( ( ? [Uu2: nat,Uv2: array(vEBT_VEBTi),Uw2: vEBT_VEBTi] : Xb = vEBT_Nodei(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)) ) )
         => ~ ! [Mi: nat] :
                ( ? [Ma: 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),Mi),Ma)),Ux2,Uy2,Uz2)
               => ( Y != aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),Mi)) ) ) ) ) ) ).

% vebt_minti.elims
tff(fact_1113_vebt__maxti_Oelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_Time_Heap(option(nat))] :
      ( ( vEBT_vebt_maxti(Xb) = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leafi((A4),(B3)) )
           => ( Y != $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((A4),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,Uv2: array(vEBT_VEBTi),Uw2: vEBT_VEBTi] : Xb = vEBT_Nodei(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)) ) )
         => ~ ! [Mi: nat,Ma: 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),Mi),Ma)),Ux2,Uy2,Uz2)
               => ( Y != aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),Ma)) ) ) ) ) ) ).

% vebt_maxti.elims
tff(fact_1114_member__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_e_m_b_e_r(Ta,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,Ta))),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))) ) ).

% member_bound_height
tff(fact_1115_minNrulli__ruleT,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi] : time_htt($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),vEBT_VEBT_minNulli(Tib),aa(vEBT_VEBTi,fun($o,assn),aTP_Lamp_bv(vEBT_VEBT,fun(vEBT_VEBTi,fun($o,assn)),Ta),Tib),one_one(nat)) ).

% minNrulli_ruleT
tff(fact_1116_Tb__T__vebt__buildupi,axiom,
    ! [Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),vEBT_V441764108873111860ildupi(Nb))),aa(int,int,minus_minus(int,vEBT_VEBT_Tb(Nb)),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% Tb_T_vebt_buildupi
tff(fact_1117_le__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% le_divide_eq_1_pos
tff(fact_1118_le__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% le_divide_eq_1_neg
tff(fact_1119_divide__le__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% divide_le_eq_1_pos
tff(fact_1120_divide__le__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% divide_le_eq_1_neg
tff(fact_1121_T__vebt__buildupi,axiom,
    ! [Nb: nat,H: heap_ext(product_unit)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(vEBT_VEBTi,vEBT_V739175172307565963ildupi(Nb),H)),vEBT_V441764108873111860ildupi(Nb)) ).

% T_vebt_buildupi
tff(fact_1122_nonzero__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ) ) ).

% nonzero_divide_mult_cancel_right
tff(fact_1123_nonzero__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ).

% nonzero_divide_mult_cancel_left
tff(fact_1124_zero__less__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A2) ) ) ).

% zero_less_divide_1_iff
tff(fact_1125_cnt__non__neg,axiom,
    ! [Ta: vEBT_VEBT] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Ta)) ).

% cnt_non_neg
tff(fact_1126_Tb__Tb_H,axiom,
    ! [Ta: nat] : vEBT_VEBT_Tb(Ta) = aa(nat,int,semiring_1_of_nat(int),vEBT_VEBT_Tb2(Ta)) ).

% Tb_Tb'
tff(fact_1127_time__replicate,axiom,
    ! [A: $tType,Xb: heap_Time_Heap(A),C2: nat,Nb: nat,H: heap_ext(product_unit)] :
      ( ! [H2: heap_ext(product_unit)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,Xb,H2)),C2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(list(A),vEBT_VEBT_replicatei(A,Nb,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)),Nb))) ) ).

% time_replicate
tff(fact_1128_divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
        <=> ( ( A2 = zero_zero(A) )
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_eq_0_iff
tff(fact_1129_divide__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_left
tff(fact_1130_divide__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> ( ( C2 = zero_zero(A) )
            | ( A2 = B2 ) ) ) ) ).

% divide_cancel_right
tff(fact_1131_division__ring__divide__zero,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),zero_zero(A)) = zero_zero(A) ) ).

% division_ring_divide_zero
tff(fact_1132_times__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [B2: A,C2: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),C2) ) ).

% times_divide_eq_left
tff(fact_1133_divide__divide__eq__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% divide_divide_eq_left
tff(fact_1134_divide__divide__eq__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ).

% divide_divide_eq_right
tff(fact_1135_times__divide__eq__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ).

% times_divide_eq_right
tff(fact_1136_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_1137_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$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_1138_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$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_1139_int__div__same__is__1,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),A2)
     => ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = A2 )
      <=> ( B2 = one_one(int) ) ) ) ).

% int_div_same_is_1
tff(fact_1140_divide__eq__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = one_one(A) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_1_iff
tff(fact_1141_one__eq__divide__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% one_eq_divide_iff
tff(fact_1142_divide__self,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = one_one(A) ) ) ) ).

% divide_self
tff(fact_1143_divide__self__if,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),A2) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% divide_self_if
tff(fact_1144_divide__eq__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = one_one(A) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% divide_eq_eq_1
tff(fact_1145_eq__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( one_one(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) )
        <=> ( ( A2 != zero_zero(A) )
            & ( A2 = B2 ) ) ) ) ).

% eq_divide_eq_1
tff(fact_1146_one__divide__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% one_divide_eq_0_iff
tff(fact_1147_zero__eq__1__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% zero_eq_1_divide_iff
tff(fact_1148_mult__divide__mult__cancel__left__if,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = $ite(C2 = zero_zero(A),zero_zero(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% mult_divide_mult_cancel_left_if
tff(fact_1149_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: 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),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
tff(fact_1150_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: 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),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
tff(fact_1151_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: 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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
tff(fact_1152_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: A,A2: A,B2: 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),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
tff(fact_1153_divide__le__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% divide_le_0_1_iff
tff(fact_1154_zero__le__divide__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% zero_le_divide_1_iff
tff(fact_1155_divide__less__0__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),zero_zero(A))
        <=> aa(A,$o,ord_less(A,A2),zero_zero(A)) ) ) ).

% divide_less_0_1_iff
tff(fact_1156_divide__less__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,ord_less(A,A2),B2) ) ) ) ).

% divide_less_eq_1_neg
tff(fact_1157_divide__less__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
          <=> aa(A,$o,ord_less(A,B2),A2) ) ) ) ).

% divide_less_eq_1_pos
tff(fact_1158_less__divide__eq__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,ord_less(A,B2),A2) ) ) ) ).

% less_divide_eq_1_neg
tff(fact_1159_less__divide__eq__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
          <=> aa(A,$o,ord_less(A,A2),B2) ) ) ) ).

% less_divide_eq_1_pos
tff(fact_1160_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_1161_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$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,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% half_nonnegative_int_iff
tff(fact_1162_htt__vebt__memberi,axiom,
    ! [Ta: vEBT_VEBT,Tib: vEBT_VEBTi,Xb: nat] : time_htt($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),aa(nat,heap_Time_Heap($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($o,assn)),aTP_Lamp_be(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),Ta),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,Ta)))) ).

% htt_vebt_memberi
tff(fact_1163_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))
      <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),A2)
          & aa(int,$o,ord_less(int,zero_zero(int)),B2) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
tff(fact_1164_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2) ) ) ).

% pos_imp_zdiv_nonneg_iff
tff(fact_1165_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,ord_less(int,B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int)) ) ) ).

% neg_imp_zdiv_nonneg_iff
tff(fact_1166_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,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I) ) ) ).

% pos_imp_zdiv_pos_iff
tff(fact_1167_pos__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
     => ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int))
      <=> aa(int,$o,ord_less(int,A2),zero_zero(int)) ) ) ).

% pos_imp_zdiv_neg_iff
tff(fact_1168_neg__imp__zdiv__neg__iff,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,ord_less(int,B2),zero_zero(int))
     => ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int))
      <=> aa(int,$o,ord_less(int,zero_zero(int)),A2) ) ) ).

% neg_imp_zdiv_neg_iff
tff(fact_1169_div__nonpos__pos__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_nonpos_pos_le0
tff(fact_1170_div__nonneg__neg__le0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,ord_less(int,B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_nonneg_neg_le0
tff(fact_1171_div__neg__pos__less0,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,ord_less(int,A2),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
       => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),zero_zero(int)) ) ) ).

% div_neg_pos_less0
tff(fact_1172_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$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_1173_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$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,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
          & aa(int,$o,aa(int,fun(int,$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_1174_zdiv__mono2__neg,axiom,
    ! [A2: int,B4: int,B2: int] :
      ( aa(int,$o,ord_less(int,A2),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B4)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B4),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B4)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) ) ) ).

% zdiv_mono2_neg
tff(fact_1175_zdiv__mono1__neg,axiom,
    ! [A2: int,A6: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),A6)
     => ( aa(int,$o,ord_less(int,B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A6),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) ) ).

% zdiv_mono1_neg
tff(fact_1176_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,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I)
          & aa(int,$o,ord_less(int,I),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int))
          & aa(int,$o,ord_less(int,K),I) ) ) ) ).

% zdiv_eq_0_iff
tff(fact_1177_zdiv__mono2,axiom,
    ! [A2: int,B4: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B4)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B4),B2)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B4)) ) ) ) ).

% zdiv_mono2
tff(fact_1178_zdiv__mono1,axiom,
    ! [A2: int,A6: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),A6)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A6),B2)) ) ) ).

% zdiv_mono1
tff(fact_1179_zdiv__le__dividend,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),A2) ) ) ).

% zdiv_le_dividend
tff(fact_1180_q__pos__lemma,axiom,
    ! [B4: int,Q5: int,R4: int] :
      ( aa(int,$o,aa(int,fun(int,$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)),R4))
     => ( aa(int,$o,ord_less(int,R4),B4)
       => ( aa(int,$o,ord_less(int,zero_zero(int)),B4)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Q5) ) ) ) ).

% q_pos_lemma
tff(fact_1181_zdiv__mono2__lemma,axiom,
    ! [B2: int,Q2: int,R3: int,B4: int,Q5: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q5)),R4) )
     => ( aa(int,$o,aa(int,fun(int,$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)),R4))
       => ( aa(int,$o,ord_less(int,R4),B4)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R3)
           => ( aa(int,$o,ord_less(int,zero_zero(int)),B4)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B4),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q2),Q5) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
tff(fact_1182_zdiv__mono2__neg__lemma,axiom,
    ! [B2: int,Q2: int,R3: int,B4: int,Q5: int,R4: int] :
      ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R3) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q5)),R4) )
     => ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B4),Q5)),R4)),zero_zero(int))
       => ( aa(int,$o,ord_less(int,R3),B2)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R4)
           => ( aa(int,$o,ord_less(int,zero_zero(int)),B4)
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B4),B2)
               => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q2) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
tff(fact_1183_unique__quotient__lemma,axiom,
    ! [B2: int,Q5: int,R4: int,Q2: int,R3: int] :
      ( aa(int,$o,aa(int,fun(int,$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),B2),Q5)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R3))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R4)
       => ( aa(int,$o,ord_less(int,R4),B2)
         => ( aa(int,$o,ord_less(int,R3),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q5),Q2) ) ) ) ) ).

% unique_quotient_lemma
tff(fact_1184_unique__quotient__lemma__neg,axiom,
    ! [B2: int,Q5: int,R4: int,Q2: int,R3: int] :
      ( aa(int,$o,aa(int,fun(int,$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),B2),Q5)),R4)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R3))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R3),zero_zero(int))
       => ( aa(int,$o,ord_less(int,B2),R3)
         => ( aa(int,$o,ord_less(int,B2),R4)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Q2),Q5) ) ) ) ) ).

% unique_quotient_lemma_neg
tff(fact_1185_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_1186_zdiv__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),C2) ) ) ).

% zdiv_zmult2_eq
tff(fact_1187_linordered__field__no__lb,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X3: A] :
        ? [Y5: A] : aa(A,$o,ord_less(A,Y5),X3) ) ).

% linordered_field_no_lb
tff(fact_1188_linordered__field__no__ub,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [X3: A] :
        ? [X_13: A] : aa(A,$o,ord_less(A,X3),X_13) ) ).

% linordered_field_no_ub
tff(fact_1189_not__exp__less__eq__0__int,axiom,
    ! [Nb: nat] : ~ aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),zero_zero(int)) ).

% not_exp_less_eq_0_int
tff(fact_1190_realpow__pos__nth2,axiom,
    ! [A2: real,Nb: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ? [R2: real] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
          & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R2),aa(nat,nat,suc,Nb)) = A2 ) ) ) ).

% realpow_pos_nth2
tff(fact_1191_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))
       => ? [N2: nat] : aa(real,$o,ord_less(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),N2)),Y) ) ) ).

% real_arch_pow_inv
tff(fact_1192_reals__Archimedean3,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ! [Y4: real] :
        ? [N2: nat] : aa(real,$o,ord_less(real,Y4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),Xb)) ) ).

% reals_Archimedean3
tff(fact_1193_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(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_1194_split__zdiv,axiom,
    ! [P: fun(int,$o),Nb: int,K: int] :
      ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,zero_zero(int)) )
        & ( aa(int,$o,ord_less(int,zero_zero(int)),K)
         => ! [I2: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,ord_less(int,J3),K)
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),J3) ) )
             => aa(int,$o,P,I2) ) )
        & ( aa(int,$o,ord_less(int,K),zero_zero(int))
         => ! [I2: int,J3: int] :
              ( ( aa(int,$o,ord_less(int,K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),J3) ) )
             => aa(int,$o,P,I2) ) ) ) ) ).

% split_zdiv
tff(fact_1195_int__div__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R3: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R3) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R3),zero_zero(int))
       => ( aa(int,$o,ord_less(int,B2),R3)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q2 ) ) ) ) ).

% int_div_neg_eq
tff(fact_1196_int__div__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R3: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R3) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R3)
       => ( aa(int,$o,ord_less(int,R3),B2)
         => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = Q2 ) ) ) ) ).

% int_div_pos_eq
tff(fact_1197_zdiv__mult__self,axiom,
    ! [M: int,A2: int,Nb: 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),A2),aa(int,int,aa(int,fun(int,int),times_times(int),M),Nb))),M) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),M)),Nb) ) ) ).

% zdiv_mult_self
tff(fact_1198_realpow__pos__nth__unique,axiom,
    ! [Nb: nat,A2: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
       => ? [X2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),X2)
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),X2),Nb) = A2 )
            & ! [Y4: real] :
                ( ( aa(real,$o,ord_less(real,zero_zero(real)),Y4)
                  & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y4),Nb) = A2 ) )
               => ( Y4 = X2 ) ) ) ) ) ).

% realpow_pos_nth_unique
tff(fact_1199_realpow__pos__nth,axiom,
    ! [Nb: nat,A2: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
       => ? [R2: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),R2)
            & ( aa(nat,real,aa(real,fun(nat,real),power_power(real),R2),Nb) = A2 ) ) ) ) ).

% realpow_pos_nth
tff(fact_1200_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_1201_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_1202_real__archimedian__rdiv__eq__0,axiom,
    ! [Xb: real,C2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),C2)
       => ( ! [M2: nat] :
              ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M2)
             => aa(real,$o,aa(real,fun(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),M2)),Xb)),C2) )
         => ( Xb = zero_zero(real) ) ) ) ) ).

% real_archimedian_rdiv_eq_0
tff(fact_1203_real__of__nat__div2,axiom,
    ! [Nb: nat,Xb: nat] : aa(real,$o,aa(real,fun(real,$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),Nb)),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),Nb),Xb)))) ).

% real_of_nat_div2
tff(fact_1204_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_1205_add__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ).

% add_divide_distrib
tff(fact_1206_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_1207_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_1208_divide__divide__eq__left_H,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% divide_divide_eq_left'
tff(fact_1209_diff__divide__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ).

% diff_divide_distrib
tff(fact_1210_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_1211_pos__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( 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))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),B2),A2) ) ) ).

% pos_zdiv_mult_2
tff(fact_1212_neg__zdiv__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),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))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),one_one(int))),A2) ) ) ).

% neg_zdiv_mult_2
tff(fact_1213_divide__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ) ) ).

% divide_le_0_iff
tff(fact_1214_divide__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% divide_right_mono
tff(fact_1215_zero__le__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))
        <=> ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) ) ) ) ).

% zero_le_divide_iff
tff(fact_1216_divide__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(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_1217_divide__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(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_1218_divide__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(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_1219_divide__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(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_1220_divide__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)) ) ) ) ).

% divide_right_mono_neg
tff(fact_1221_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_1222_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_1223_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_1224_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_1225_divide__less__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),zero_zero(A))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
              & aa(A,$o,ord_less(A,B2),zero_zero(A)) )
            | ( aa(A,$o,ord_less(A,A2),zero_zero(A))
              & aa(A,$o,ord_less(A,zero_zero(A)),B2) ) ) ) ) ).

% divide_less_0_iff
tff(fact_1226_divide__less__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,ord_less(A,A2),B2) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,ord_less(A,B2),A2) )
            & ( C2 != zero_zero(A) ) ) ) ) ).

% divide_less_cancel
tff(fact_1227_zero__less__divide__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
              & aa(A,$o,ord_less(A,zero_zero(A)),B2) )
            | ( aa(A,$o,ord_less(A,A2),zero_zero(A))
              & aa(A,$o,ord_less(A,B2),zero_zero(A)) ) ) ) ) ).

% zero_less_divide_iff
tff(fact_1228_divide__strict__right__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( 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),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% divide_strict_right_mono
tff(fact_1229_divide__strict__right__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
         => ( 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),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% divide_strict_right_mono_neg
tff(fact_1230_right__inverse__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = one_one(A) )
          <=> ( A2 = B2 ) ) ) ) ).

% right_inverse_eq
tff(fact_1231_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_1232_divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 )
        <=> $ite(C2 != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq
tff(fact_1233_eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq
tff(fact_1234_divide__eq__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) )
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 ) ) ) ) ).

% divide_eq_imp
tff(fact_1235_eq__divide__imp,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 )
           => ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) ) ) ) ) ).

% eq_divide_imp
tff(fact_1236_nonzero__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = A2 )
          <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) ) ) ) ) ).

% nonzero_divide_eq_eq
tff(fact_1237_nonzero__eq__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: A,A2: A,B2: A] :
          ( ( C2 != zero_zero(A) )
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = B2 ) ) ) ) ).

% nonzero_eq_divide_eq
tff(fact_1238_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,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),E2)) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% field_le_epsilon
tff(fact_1239_frac__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Y: A,Xb: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( aa(A,$o,ord_less(A,zero_zero(A)),W)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),W),Z)
               => aa(A,$o,aa(A,fun(A,$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_1240_frac__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A,W: A,Z: A] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(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_1241_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,$o,aa(A,fun(A,$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_1242_divide__le__cancel,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(A,$o,ord_less(A,C2),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% divide_le_cancel
tff(fact_1243_divide__nonneg__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,ord_less(A,Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$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_1244_divide__nonneg__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$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_1245_divide__nonpos__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,ord_less(A,Y),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$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_1246_divide__nonpos__pos,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),Y)
           => aa(A,$o,aa(A,fun(A,$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_1247_divide__less__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
              & aa(A,$o,ord_less(A,B2),A2) )
            | ( aa(A,$o,ord_less(A,A2),zero_zero(A))
              & aa(A,$o,ord_less(A,A2),B2) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_less_eq_1
tff(fact_1248_less__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
              & aa(A,$o,ord_less(A,A2),B2) )
            | ( aa(A,$o,ord_less(A,A2),zero_zero(A))
              & aa(A,$o,ord_less(A,B2),A2) ) ) ) ) ).

% less_divide_eq_1
tff(fact_1249_divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),aa(A,$o,ord_less(A,zero_zero(A)),A2)) ) ) ) ).

% divide_less_eq
tff(fact_1250_less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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),A2),C2)),B2),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,ord_less(A,A2),zero_zero(A))) ) ) ) ).

% less_divide_eq
tff(fact_1251_neg__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% neg_divide_less_eq
tff(fact_1252_neg__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% neg_less_divide_eq
tff(fact_1253_pos__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$o,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% pos_divide_less_eq
tff(fact_1254_pos__less__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% pos_less_divide_eq
tff(fact_1255_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_1256_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_1257_divide__strict__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
         => ( 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),A2),B2))
             => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_strict_left_mono
tff(fact_1258_divide__strict__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( 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),A2),B2))
             => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_strict_left_mono_neg
tff(fact_1259_add__divide__eq__if__simps_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = $ite(Z = zero_zero(A),B2,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(2)
tff(fact_1260_add__divide__eq__if__simps_I1_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,Z: A] :
          aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = $ite(Z = zero_zero(A),A2,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),A2),Z)),B2)),Z)) ) ).

% add_divide_eq_if_simps(1)
tff(fact_1261_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_1262_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_1263_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_1264_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_1265_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_1266_gt__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => 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),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))),B2) ) ) ).

% gt_half_sum
tff(fact_1267_less__half__sum,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => aa(A,$o,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)))) ) ) ).

% less_half_sum
tff(fact_1268_add__divide__eq__if__simps_I4_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,Z: A] :
          aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),Z)) = $ite(Z = zero_zero(A),A2,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),A2),Z)),B2)),Z)) ) ).

% add_divide_eq_if_simps(4)
tff(fact_1269_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_1270_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_1271_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_1272_field__le__mult__one__interval,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A,Y: A] :
          ( ! [Z3: A] :
              ( aa(A,$o,ord_less(A,zero_zero(A)),Z3)
             => ( aa(A,$o,ord_less(A,Z3),one_one(A))
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z3),Xb)),Y) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% field_le_mult_one_interval
tff(fact_1273_divide__le__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),one_one(A))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) )
            | ( aa(A,$o,ord_less(A,A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            | ( A2 = zero_zero(A) ) ) ) ) ).

% divide_le_eq_1
tff(fact_1274_le__divide__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            | ( aa(A,$o,ord_less(A,A2),zero_zero(A))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% le_divide_eq_1
tff(fact_1275_divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% divide_le_eq
tff(fact_1276_le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ).

% le_divide_eq
tff(fact_1277_divide__left__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(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),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_left_mono
tff(fact_1278_neg__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% neg_divide_le_eq
tff(fact_1279_neg__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% neg_le_divide_eq
tff(fact_1280_pos__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% pos_divide_le_eq
tff(fact_1281_pos__le__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% pos_le_divide_eq
tff(fact_1282_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,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y))
           => aa(A,$o,aa(A,fun(A,$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_1283_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,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),Z),Y)),Xb)
           => aa(A,$o,aa(A,fun(A,$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_1284_divide__left__mono__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(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),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2)) ) ) ) ) ).

% divide_left_mono_neg
tff(fact_1285_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,aa(A,fun(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,aa(A,fun(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_1286_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_1287_scaling__mono,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [U: A,V: A,R3: A,S3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),U),V)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R3)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R3),S3)
             => aa(A,$o,aa(A,fun(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),R3),aa(A,A,minus_minus(A,V),U))),S3))),V) ) ) ) ) ).

% scaling_mono
tff(fact_1288_Tb__T__vebt__buildupi_H,axiom,
    ! [Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),vEBT_V9176841429113362141ildupi(Nb)),aa(int,int,minus_minus(int,vEBT_VEBT_Tb(Nb)),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% Tb_T_vebt_buildupi'
tff(fact_1289_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,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          Xb = Mia,
          zero_zero(nat),
          $ite(
            Xb = Maa,
            zero_zero(nat),
            $ite(
              aa(nat,$o,ord_less(nat,Xb),Mia),
              zero_zero(nat),
              $ite(
                aa(nat,$o,ord_less(nat,Maa),Xb),
                zero_zero(nat),
                $ite(
                  ( aa(nat,$o,ord_less(nat,Mia),Xb)
                  & aa(nat,$o,ord_less(nat,Xb),Maa) ),
                  $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,Vaa))),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)),TreeLista)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),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_1290_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$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_1291_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,aa(int,fun(int,$o),ord_less_eq(int),W),Z) ) ).

% zle_add1_eq_le
tff(fact_1292_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_1293_vebt__succ_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = Y )
     => ( ! [Uu2: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((Uu2),(B3)) )
           => ( ( Xaa = zero_zero(nat) )
             => ( Y != $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xb = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N2: nat] : Xaa = aa(nat,nat,suc,N2)
             => ( 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),Vc: 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),Vc,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) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                     => ( Y != $ite(
                            aa(nat,$o,ord_less(nat,Xaa),Mi),
                            aa(nat,option(nat),some(nat),Mi),
                            $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)),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,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,TreeList),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,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                                  none(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
tff(fact_1294_vebt__pred_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(Xb,Xaa) = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != none(nat) ) ) )
       => ( ! [A4: $o] :
              ( ? [Uw2: $o] : Xb = vEBT_Leaf((A4),(Uw2))
             => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( Xb = vEBT_Leaf((A4),(B3)) )
               => ( ? [Va2: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va2))
                 => ( Y != $ite(
                        (B3),
                        aa(nat,option(nat),some(nat),one_one(nat)),
                        $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va)
               => ( 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) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                       => ( Y != $ite(
                              aa(nat,$o,ord_less(nat,Ma),Xaa),
                              aa(nat,option(nat),some(nat),Ma),
                              $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)),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,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,TreeList),h),l)),
                                        $let(
                                          pr: option(nat),
                                          pr:= vEBT_vebt_pred(Summary,h),
                                          $ite(
                                            pr = none(nat),
                                            $ite(aa(nat,$o,ord_less(nat,Mi),Xaa),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,Va2))),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.elims
tff(fact_1295_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,
    ! [Mia: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,zero_zero(nat)),TreeLista,Summarya),Xb) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(6)
tff(fact_1296_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,
    ! [Mia: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,zero_zero(nat)),TreeLista,Summarya),Xb) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(6)
tff(fact_1297_Leaf__0__not,axiom,
    ! [A2: $o,B2: $o] : ~ vEBT_invar_vebt(vEBT_Leaf((A2),(B2)),zero_zero(nat)) ).

% Leaf_0_not
tff(fact_1298_deg__1__Leafy,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( Nb = one_one(nat) )
       => ? [A4: $o,B3: $o] : Ta = vEBT_Leaf((A4),(B3)) ) ) ).

% deg_1_Leafy
tff(fact_1299_deg__1__Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
     => ? [A4: $o,B3: $o] : Ta = vEBT_Leaf((A4),(B3)) ) ).

% deg_1_Leaf
tff(fact_1300_deg1Leaf,axiom,
    ! [Ta: vEBT_VEBT] :
      ( vEBT_invar_vebt(Ta,one_one(nat))
    <=> ? [A5: $o,B6: $o] : Ta = vEBT_Leaf((A5),(B6)) ) ).

% deg1Leaf
tff(fact_1301_Tbuildupi__buildupi_H,axiom,
    ! [Nb: nat] : aa(nat,int,semiring_1_of_nat(int),vEBT_V441764108873111860ildupi(Nb)) = vEBT_V9176841429113362141ildupi(Nb) ).

% Tbuildupi_buildupi'
tff(fact_1302_double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% double_eq_0_iff
tff(fact_1303_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,
    ! [A2: $o,B2: $o,Xb: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Leaf((A2),(B2)),Xb) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(1)
tff(fact_1304_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Leaf((X21),(X22))) = zero_zero(nat) ).

% VEBT.size(4)
tff(fact_1305_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $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),(Uv2))),D3)
     => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList: 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,Deg2,TreeList,Summary)),Deg3) ) ).

% VEBT_internal.valid'.cases
tff(fact_1306_VEBT_Odistinct_I1_J,axiom,
    ! [X11a: option(product_prod(nat,nat)),X12a: nat,X13a: list(vEBT_VEBT),X14a: vEBT_VEBT,X21: $o,X22: $o] : vEBT_Node(X11a,X12a,X13a,X14a) != vEBT_Leaf((X21),(X22)) ).

% VEBT.distinct(1)
tff(fact_1307_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X11: option(product_prod(nat,nat)),X12: nat,X13: list(vEBT_VEBT),X14: vEBT_VEBT] : Y != vEBT_Node(X11,X12,X13,X14)
     => ~ ! [X212: $o,X222: $o] : Y != vEBT_Leaf((X212),(X222)) ) ).

% VEBT.exhaust
tff(fact_1308_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull(vEBT_Leaf($false,$false)) ).

% VEBT_internal.minNull.simps(1)
tff(fact_1309_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] : ~ vEBT_VEBT_minNull(vEBT_Leaf($true,(Uv))) ).

% VEBT_internal.minNull.simps(2)
tff(fact_1310_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] : ~ vEBT_VEBT_minNull(vEBT_Leaf((Uu),$true)) ).

% VEBT_internal.minNull.simps(3)
tff(fact_1311_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,
    ! [A2: $o,B2: $o,Nb: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Nb))) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(3)
tff(fact_1312_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,
    ! [A2: $o,B2: $o,Nb: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Nb))) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(3)
tff(fact_1313_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,
    ! [A2: $o,B2: $o] : vEBT_V1232361888498592333_e_t_e(vEBT_Leaf((A2),(B2)),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_1314_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,
    ! [A2: $o,B2: $o] : vEBT_T_d_e_l_e_t_e(vEBT_Leaf((A2),(B2)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(1)
tff(fact_1315_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,
    ! [A2: $o,B2: $o] : vEBT_V1232361888498592333_e_t_e(vEBT_Leaf((A2),(B2)),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_1316_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,
    ! [A2: $o,B2: $o] : vEBT_T_d_e_l_e_t_e(vEBT_Leaf((A2),(B2)),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_1317_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [A4: $o,B3: $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((A4),(B3))),X2)
     => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: 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),Uv2,Uw2)),Ux2)
       => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),S: 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),TreeList,S)),X2) ) ) ).

% VEBT_internal.naive_member.cases
tff(fact_1318_invar__vebt_Ointros_I1_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_invar_vebt(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,zero_zero(nat))) ).

% invar_vebt.intros(1)
tff(fact_1319_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_1320_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) )
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
           => ~ ! [Uz2: product_prod(nat,nat),Va: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va,Vb2,Vc) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.cases
tff(fact_1321_vebt__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] :
      ( aa(nat,$o,vEBT_vebt_member(vEBT_Leaf((A2),(B2))),Xb)
    <=> $ite(
          Xb = zero_zero(nat),
          (A2),
          $ite(Xb = one_one(nat),(B2),$false) ) ) ).

% vebt_member.simps(1)
tff(fact_1322_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_1323_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xb)
     => ( ! [Uv2: $o] : Xb != vEBT_Leaf($true,(Uv2))
       => ( ! [Uu2: $o] : Xb != vEBT_Leaf((Uu2),$true)
         => ~ ! [Uz2: product_prod(nat,nat),Va: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : Xb != vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va,Vb2,Vc) ) ) ) ).

% VEBT_internal.minNull.elims(3)
tff(fact_1324_vebt__insert_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] :
      vEBT_vebt_insert(vEBT_Leaf((A2),(B2)),Xb) = $ite(
        Xb = zero_zero(nat),
        vEBT_Leaf($true,(B2)),
        $ite(Xb = one_one(nat),vEBT_Leaf((A2),$true),vEBT_Leaf((A2),(B2))) ) ).

% vebt_insert.simps(1)
tff(fact_1325_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : vEBT_vebt_succ(vEBT_Leaf((Uv),(Uw)),aa(nat,nat,suc,Nb)) = none(nat) ).

% vebt_succ.simps(2)
tff(fact_1326_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] : vEBT_vebt_pred(vEBT_Leaf((Uu),(Uv)),zero_zero(nat)) = none(nat) ).

% vebt_pred.simps(1)
tff(fact_1327_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xb)
     => ( ( Xb != vEBT_Leaf($false,$false) )
       => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) ) ) ).

% VEBT_internal.minNull.elims(2)
tff(fact_1328_vebt__assn__raw_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,vEBT_VEBTi)] :
      ( ! [A4: $o,B3: $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((A4),(B3))),vEBT_Leafi((Ai),(Bi)))
     => ( ! [Mmo2: option(product_prod(nat,nat)),Deg2: nat,Tree_list2: list(vEBT_VEBT),Summary: vEBT_VEBT,Mmoi2: option(product_prod(nat,nat)),Degi2: nat,Tree_array2: array(vEBT_VEBTi),Summaryi2: 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(Mmo2,Deg2,Tree_list2,Summary)),vEBT_Nodei(Mmoi2,Degi2,Tree_array2,Summaryi2))
       => ( ! [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_1329_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Vaa: nat] : vEBT_T_p_r_e_d2(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(3)
tff(fact_1330_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] : vEBT_T_p_r_e_d2(vEBT_Leaf((Uu),(Uv)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(1)
tff(fact_1331_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : vEBT_T_s_u_c_c2(vEBT_Leaf((Uv),(Uw)),aa(nat,nat,suc,Nb)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(2)
tff(fact_1332_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_H_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] : vEBT_T_s_u_c_c2(vEBT_Leaf((Uu),(B2)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.simps(1)
tff(fact_1333_vebt__assn__raw_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Ai2: $o,Bi2: $o] :
      aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_Leaf((A2),(B2))),vEBT_Leafi((Ai2),(Bi2))) = pure_assn(( ( (Ai2) = (A2) )
        & ( (Bi2) = (B2) ) )) ).

% vebt_assn_raw.simps(1)
tff(fact_1334_less__eq__int__code_I1_J,axiom,
    aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),zero_zero(int)) ).

% less_eq_int_code(1)
tff(fact_1335_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_1336_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_1337_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_1338_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_1339_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_1340_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_1341_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A4: $o,B3: $o] : Xb != vEBT_Leaf((A4),(B3))
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
       => ~ ! [Mi: nat,Ma: 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),Mi),Ma)),Ux2,Uy2,Uz2) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.cases
tff(fact_1342_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_H_Osimps_I2_J,axiom,
    ! [A2: $o,Uw: $o] : vEBT_T_p_r_e_d2(vEBT_Leaf((A2),(Uw)),aa(nat,nat,suc,zero_zero(nat))) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.simps(2)
tff(fact_1343_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xb)
      <=> (Y) )
     => ( ( ( Xb = vEBT_Leaf($false,$false) )
         => ~ (Y) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => (Y) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => (Y) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
               => ~ (Y) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va,Vb2,Vc)
                 => (Y) ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
tff(fact_1344_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)] :
      ( ! [A4: $o,B3: $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((A4),(B3))),zero_zero(nat))
     => ( ! [A4: $o,B3: $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((A4),(B3))),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A4: $o,B3: $o,N2: 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((A4),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,N2)))
         => ( ! [Deg2: nat,TreeList: 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)),Deg2,TreeList,Summary)),Uu2)
           => ( ! [Mi: nat,Ma: nat,TreeList: 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),Mi),Ma)),zero_zero(nat),TreeList,Summary)),X2)
             => ( ! [Mi: nat,Ma: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary)),X2)
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),X2) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.cases
tff(fact_1345_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_1346_vebt__pred_Osimps_I2_J,axiom,
    ! [A2: $o,Uw: $o] :
      vEBT_vebt_pred(vEBT_Leaf((A2),(Uw)),aa(nat,nat,suc,zero_zero(nat))) = $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ).

% vebt_pred.simps(2)
tff(fact_1347_vebt__succ_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] :
      vEBT_vebt_succ(vEBT_Leaf((Uu),(B2)),zero_zero(nat)) = $ite((B2),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ).

% vebt_succ.simps(1)
tff(fact_1348_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,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Xb: nat] : vEBT_T_m_e_m_b_e_r2(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw),Xb) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(2)
tff(fact_1349_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)] :
      ( ! [A4: $o,B3: $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((A4),(B3))),X2)
     => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: 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,Uv2,Uw2)),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),Vc: 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,Vc)),X2)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),X2) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.cases
tff(fact_1350_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)] :
      ( ! [A4: $o,B3: $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((A4),(B3))),X2)
     => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S: 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(Info2,zero_zero(nat),Ts2,S)),X2)
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S: 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(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S)),X2)
         => ( ! [V3: nat,TreeList: 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)),TreeList,Summary)),X2)
           => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),X2) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.cases
tff(fact_1351_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,B3: $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),(B3))),zero_zero(nat))
     => ( ! [Uv2: $o,Uw2: $o,N2: 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((Uv2),(Uw2))),aa(nat,nat,suc,N2))
       => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT,Va: 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)),Va)
         => ( ! [V3: product_prod(nat,nat),Vc: 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),Vc,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)
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),X2) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.cases
tff(fact_1352_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,Uv2: $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),(Uv2))),zero_zero(nat))
     => ( ! [A4: $o,Uw2: $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((A4),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))
       => ( ! [A4: $o,B3: $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((A4),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))
         => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va: 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,Va)),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)
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),X2) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.cases
tff(fact_1353_VEBT__internal_Omembermima_Ocases,axiom,
    ! [Xb: product_prod(vEBT_VEBT,nat)] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: 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),(Uv2))),Uw2)
     => ( ! [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)
       => ( ! [Mi: nat,Ma: nat,Va: 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),Mi),Ma)),zero_zero(nat),Va,Vb2)),X2)
         => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list(vEBT_VEBT),Vc: 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),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc)),X2)
           => ~ ! [V3: nat,TreeList: 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),TreeList,Vd2)),X2) ) ) ) ) ).

% VEBT_internal.membermima.cases
tff(fact_1354_vebt__pred_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Vaa: nat] :
      vEBT_vebt_pred(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))) = $ite(
        (B2),
        aa(nat,option(nat),some(nat),one_one(nat)),
        $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ).

% vebt_pred.simps(3)
tff(fact_1355_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,
    ! [A2: $o,B2: $o,Xb: nat] :
      vEBT_T_m_e_m_b_e_r(vEBT_Leaf((A2),(B2)),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_1356_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_1357_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$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_1358_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_1359_int__gr__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,K),I)
     => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int)))
       => ( ! [I3: int] :
              ( aa(int,$o,ord_less(int,K),I3)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_gr_induct
tff(fact_1360_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_1361_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_1362_pos__zmult__eq__1__iff,axiom,
    ! [M: int,Nb: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),M)
     => ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),Nb) = one_one(int) )
      <=> ( ( M = one_one(int) )
          & ( Nb = one_one(int) ) ) ) ) ).

% pos_zmult_eq_1_iff
tff(fact_1363_zero__le__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ? [N2: nat] : K = aa(nat,int,semiring_1_of_nat(int),N2) ) ).

% zero_le_imp_eq_int
tff(fact_1364_nonneg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ~ ! [N2: nat] : K != aa(nat,int,semiring_1_of_nat(int),N2) ) ).

% nonneg_int_cases
tff(fact_1365_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)),W) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W)) ).

% int_distrib(1)
tff(fact_1366_int__distrib_I2_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z22)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z22)) ).

% int_distrib(2)
tff(fact_1367_int__less__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,I),K)
     => ( aa(int,$o,P,aa(int,int,minus_minus(int,K),one_one(int)))
       => ( ! [I3: int] :
              ( aa(int,$o,ord_less(int,I3),K)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I3),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_less_induct
tff(fact_1368_int__distrib_I4_J,axiom,
    ! [W: int,Z1: int,Z22: int] : aa(int,int,aa(int,fun(int,int),times_times(int),W),aa(int,int,minus_minus(int,Z1),Z22)) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),W),Z1)),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z22)) ).

% int_distrib(4)
tff(fact_1369_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,minus_minus(int,Z1),Z22)),W) = aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),Z1),W)),aa(int,int,aa(int,fun(int,int),times_times(int),Z22),W)) ).

% int_distrib(3)
tff(fact_1370_int__diff__cases,axiom,
    ! [Z: int] :
      ~ ! [M2: nat,N2: nat] : Z != aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),M2)),aa(nat,int,semiring_1_of_nat(int),N2)) ).

% int_diff_cases
tff(fact_1371_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),Vc2: 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,Vc2),Xb) = one_one(nat) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.simps(4)
tff(fact_1372_member__bound__height_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_e_m_b_e_r2(Ta,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))) ) ).

% member_bound_height'
tff(fact_1373_pos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ~ ! [N2: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),N2) ) ) ).

% pos_int_cases
tff(fact_1374_zero__less__imp__eq__int,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ? [N2: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N2)
          & ( K = aa(nat,int,semiring_1_of_nat(int),N2) ) ) ) ).

% zero_less_imp_eq_int
tff(fact_1375_zless__iff__Suc__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less(int,W),Z)
    <=> ? [N6: 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,N6))) ) ).

% zless_iff_Suc_zadd
tff(fact_1376_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 )
     => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( 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),Vc: 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,Vc)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                          $ite(
                            Xaa = Mi,
                            zero_zero(nat),
                            $ite(
                              Xaa = Ma,
                              zero_zero(nat),
                              $ite(
                                aa(nat,$o,ord_less(nat,Xaa),Mi),
                                zero_zero(nat),
                                $ite(
                                  aa(nat,$o,ord_less(nat,Ma),Xaa),
                                  zero_zero(nat),
                                  $ite(
                                    ( aa(nat,$o,ord_less(nat,Mi),Xaa)
                                    & aa(nat,$o,ord_less(nat,Xaa),Ma) ),
                                    $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)),TreeList)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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_1377_zle__int,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(nat,int,semiring_1_of_nat(int),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% zle_int
tff(fact_1378_zadd__int__left,axiom,
    ! [M: nat,Nb: nat,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),M)),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),Z)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb))),Z) ).

% zadd_int_left
tff(fact_1379_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,ord_less(int,W),Z)
     => aa(int,$o,aa(int,fun(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_1380_int__ge__induct,axiom,
    ! [K: int,I: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I)
     => ( aa(int,$o,P,K)
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I3)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_ge_induct
tff(fact_1381_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$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_1382_le__imp__0__less,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$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_1383_zle__iff__zadd,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z)
    <=> ? [N6: nat] : Z = aa(int,int,aa(int,fun(int,int),plus_plus(int),W),aa(nat,int,semiring_1_of_nat(int),N6)) ) ).

% zle_iff_zadd
tff(fact_1384_int__le__induct,axiom,
    ! [I: int,K: int,P: fun(int,$o)] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),K)
     => ( aa(int,$o,P,K)
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I3),K)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I3),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_le_induct
tff(fact_1385_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_1386_int__induct,axiom,
    ! [P: fun(int,$o),K: int,I: int] :
      ( aa(int,$o,P,K)
     => ( ! [I3: int] :
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),I3)
           => ( aa(int,$o,P,I3)
             => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))) ) )
       => ( ! [I3: int] :
              ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I3),K)
             => ( aa(int,$o,P,I3)
               => aa(int,$o,P,aa(int,int,minus_minus(int,I3),one_one(int))) ) )
         => aa(int,$o,P,I) ) ) ) ).

% int_induct
tff(fact_1387_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,
    ! [Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Uu: nat] : vEBT_V1232361888498592333_e_t_e(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeLista,Summarya),Uu) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(4)
tff(fact_1388_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,
    ! [Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Uu: nat] : vEBT_T_d_e_l_e_t_e(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeLista,Summarya),Uu) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(4)
tff(fact_1389_vebt__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A4),
                  $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)
             => ~ $ite(
                    Xaa = Mi,
                    $true,
                    $ite(
                      Xaa = Ma,
                      $true,
                      $ite(
                        aa(nat,$o,ord_less(nat,Xaa),Mi),
                        $false,
                        $ite(
                          aa(nat,$o,ord_less(nat,Ma),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)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
tff(fact_1390_vebt__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => (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),Vc: 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,Vc)
               => (Y) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)
                   => ( (Y)
                    <=> ~ $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa(nat,$o,ord_less(nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa(nat,$o,ord_less(nat,Ma),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)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
tff(fact_1391_vebt__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A4),
                $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
         => ( ! [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),Vc: 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,Vc)
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)
                   => $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa(nat,$o,ord_less(nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa(nat,$o,ord_less(nat,Ma),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)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
tff(fact_1392_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 )
     => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
         => ( 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,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( 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),Vc: 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,Vc)
               => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)
                   => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $ite(
                            Xaa = Mi,
                            one_one(nat),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xaa = 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,Xaa),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),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)),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(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_1393_invar__vebt_Osimps,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
    <=> ( ( ? [A5: $o,B6: $o] : A1 = vEBT_Leaf((A5),(B6))
          & ( A22 = aa(nat,nat,suc,zero_zero(nat)) ) )
        | ? [TreeList2: list(vEBT_VEBT),N6: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList2,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X,N6) )
            & vEBT_invar_vebt(Summary2,N6)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N6) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N6),N6) )
            & ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_1)
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
        | ? [TreeList2: list(vEBT_VEBT),N6: nat,Summary2: vEBT_VEBT] :
            ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),A22,TreeList2,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X,N6) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N6))
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N6)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N6),aa(nat,nat,suc,N6)) )
            & ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),X_1)
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
        | ? [TreeList2: list(vEBT_VEBT),N6: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
            ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi2),Ma2)),A22,TreeList2,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X,N6) )
            & vEBT_invar_vebt(Summary2,N6)
            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList2) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N6) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N6),N6) )
            & ! [I2: nat] :
                ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N6))
               => ( ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I2)),X_1)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I2) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                 => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
            & aa(nat,$o,aa(nat,fun(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))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I2: nat] :
                  ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),N6))
                 => ( ( ( vEBT_VEBT_high(Ma2,N6) = I2 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I2)),vEBT_VEBT_low(Ma2,N6)) )
                    & ! [X: nat] :
                        ( ( ( vEBT_VEBT_high(X,N6) = I2 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I2)),vEBT_VEBT_low(X,N6)) )
                       => ( aa(nat,$o,ord_less(nat,Mi2),X)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma2) ) ) ) ) ) )
        | ? [TreeList2: list(vEBT_VEBT),N6: nat,Summary2: vEBT_VEBT,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)),A22,TreeList2,Summary2) )
            & ! [X: vEBT_VEBT] :
                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
               => vEBT_invar_vebt(X,N6) )
            & vEBT_invar_vebt(Summary2,aa(nat,nat,suc,N6))
            & ( 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))),aa(nat,nat,suc,N6)) )
            & ( A22 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N6),aa(nat,nat,suc,N6)) )
            & ! [I2: nat] :
                ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N6)))
               => ( ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I2)),X_1)
                <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary2),I2) ) )
            & ( ( Mi2 = Ma2 )
             => ! [X: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList2))
                 => ~ ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X_1) ) )
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi2),Ma2)
            & aa(nat,$o,ord_less(nat,Ma2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),A22))
            & ( ( Mi2 != Ma2 )
             => ! [I2: nat] :
                  ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,N6)))
                 => ( ( ( vEBT_VEBT_high(Ma2,N6) = I2 )
                     => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I2)),vEBT_VEBT_low(Ma2,N6)) )
                    & ! [X: nat] :
                        ( ( ( vEBT_VEBT_high(X,N6) = I2 )
                          & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList2),I2)),vEBT_VEBT_low(X,N6)) )
                       => ( aa(nat,$o,ord_less(nat,Mi2),X)
                          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Ma2) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
tff(fact_1394_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( vEBT_invar_vebt(A1,A22)
     => ( ( ? [A4: $o,B3: $o] : A1 = vEBT_Leaf((A4),(B3))
         => ( A22 != aa(nat,nat,suc,zero_zero(nat)) ) )
       => ( ! [TreeList: list(vEBT_VEBT),N2: nat,Summary: vEBT_VEBT,M2: nat,Deg2: nat] :
              ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary) )
             => ( ( A22 = Deg2 )
               => ( ! [X3: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                     => vEBT_invar_vebt(X3,N2) )
                 => ( vEBT_invar_vebt(Summary,M2)
                   => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2) )
                     => ( ( M2 = N2 )
                       => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M2) )
                         => ( ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_12)
                           => ~ ! [X3: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list(vEBT_VEBT),N2: nat,Summary: vEBT_VEBT,M2: nat,Deg2: nat] :
                ( ( A1 = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary) )
               => ( ( A22 = Deg2 )
                 => ( ! [X3: vEBT_VEBT] :
                        ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                       => vEBT_invar_vebt(X3,N2) )
                   => ( vEBT_invar_vebt(Summary,M2)
                     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2) )
                       => ( ( M2 = aa(nat,nat,suc,N2) )
                         => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M2) )
                           => ( ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X_12)
                             => ~ ! [X3: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list(vEBT_VEBT),N2: nat,Summary: vEBT_VEBT,M2: nat,Deg2: nat,Mi: nat,Ma: nat] :
                  ( ( A1 = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Deg2,TreeList,Summary) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X3: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                         => vEBT_invar_vebt(X3,N2) )
                     => ( vEBT_invar_vebt(Summary,M2)
                       => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2) )
                         => ( ( M2 = N2 )
                           => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M2) )
                             => ( ! [I4: nat] :
                                    ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2))
                                   => ( ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),X_1)
                                    <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I4) ) )
                               => ( ( ( Mi = Ma )
                                   => ! [X3: vEBT_VEBT] :
                                        ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                       => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) )
                                 => ( aa(nat,$o,aa(nat,fun(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 )
                                         => ! [I4: nat] :
                                              ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2))
                                             => ( ( ( vEBT_VEBT_high(Ma,N2) = I4 )
                                                 => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(Ma,N2)) )
                                                & ! [X3: nat] :
                                                    ( ( ( vEBT_VEBT_high(X3,N2) = I4 )
                                                      & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(X3,N2)) )
                                                   => ( aa(nat,$o,ord_less(nat,Mi),X3)
                                                      & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list(vEBT_VEBT),N2: nat,Summary: vEBT_VEBT,M2: nat,Deg2: nat,Mi: nat,Ma: 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),Mi),Ma)),Deg2,TreeList,Summary) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X3: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                           => vEBT_invar_vebt(X3,N2) )
                       => ( vEBT_invar_vebt(Summary,M2)
                         => ( ( 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))),M2) )
                           => ( ( M2 = aa(nat,nat,suc,N2) )
                             => ( ( Deg2 = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),M2) )
                               => ( ! [I4: nat] :
                                      ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2))
                                     => ( ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),X_1)
                                      <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),I4) ) )
                                 => ( ( ( Mi = Ma )
                                     => ! [X3: vEBT_VEBT] :
                                          ( member(vEBT_VEBT,X3,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                         => ~ ? [X_12: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X3),X_12) ) )
                                   => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mi),Ma)
                                     => ( aa(nat,$o,ord_less(nat,Ma),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg2))
                                       => ~ ( ( Mi != Ma )
                                           => ! [I4: nat] :
                                                ( aa(nat,$o,ord_less(nat,I4),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),M2))
                                               => ( ( ( vEBT_VEBT_high(Ma,N2) = I4 )
                                                   => aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(Ma,N2)) )
                                                  & ! [X3: nat] :
                                                      ( ( ( vEBT_VEBT_high(X3,N2) = I4 )
                                                        & aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),I4)),vEBT_VEBT_low(X3,N2)) )
                                                     => ( aa(nat,$o,ord_less(nat,Mi),X3)
                                                        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Ma) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
tff(fact_1395_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,
    ! [Mia: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),zero_zero(nat),TreeLista,Summarya),Xb) = one_one(nat) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.simps(5)
tff(fact_1396_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,
    ! [Mia: nat,Maa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),zero_zero(nat),TreeLista,Summarya),Xb) = one_one(nat) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.simps(5)
tff(fact_1397_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,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,Uw2: $o] : Xb = vEBT_Leaf((A4),(Uw2))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
             => ( ? [Va2: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va2))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list(vEBT_VEBT),Va: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va)
               => ( 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) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                       => ( Y != $ite(
                              aa(nat,$o,ord_less(nat,Ma),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)),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(Summary,h)),one_one(nat)) ) ),
                                    one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.elims
tff(fact_1398_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,B3: $o] : Xb = vEBT_Leaf((Uu2),(B3))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xb = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N2: nat] : Xaa = aa(nat,nat,suc,N2)
             => ( 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),Vc: 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),Vc,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) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                     => ( Y != $ite(
                            aa(nat,$o,ord_less(nat,Xaa),Mi),
                            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)),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(Summary,h)),one_one(nat)) ) ),
                                  one_one(nat) ) ) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.elims
tff(fact_1399_vebt__maxt_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ( Y != $ite(
                  (B3),
                  aa(nat,option(nat),some(nat),one_one(nat)),
                  $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != none(nat) ) )
         => ~ ! [Mi: nat,Ma: 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),Mi),Ma)),Ux2,Uy2,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Ma) ) ) ) ) ) ).

% vebt_maxt.elims
tff(fact_1400_vebt__mint_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ( Y != $ite(
                  (A4),
                  aa(nat,option(nat),some(nat),zero_zero(nat)),
                  $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != none(nat) ) )
         => ~ ! [Mi: nat] :
                ( ? [Ma: 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),Mi),Ma)),Ux2,Uy2,Uz2)
               => ( Y != aa(nat,option(nat),some(nat),Mi) ) ) ) ) ) ).

% vebt_mint.elims
tff(fact_1401_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 )
     => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
         => ( Y != one_one(nat) ) )
       => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S: vEBT_VEBT] : Xb = vEBT_Node(Info2,zero_zero(nat),Ts2,S)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S: vEBT_VEBT] : Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary)
               => ( Y != one_one(nat) ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,ord_less(nat,Xaa),Mi),Mi,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)),TreeList))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = 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,Va2))),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(Summary,h),one_one(nat))),
                              one_one(nat) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.elims
tff(fact_1402_linear__plus__1__le__power,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$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),Nb)),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))),Nb)) ) ).

% linear_plus_1_le_power
tff(fact_1403_diff__add__zero,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = zero_zero(A) ) ).

% diff_add_zero
tff(fact_1404_diff__gt__0__iff__gt,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,minus_minus(A,A2),B2))
        <=> aa(A,$o,ord_less(A,B2),A2) ) ) ).

% diff_gt_0_iff_gt
tff(fact_1405_diff__ge__0__iff__ge,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,minus_minus(A,A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% diff_ge_0_iff_ge
tff(fact_1406_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A2) ) ) ).

% zero_less_double_add_iff_zero_less_single_add
tff(fact_1407_add__right__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
        <=> ( B2 = C2 ) ) ) ).

% add_right_cancel
tff(fact_1408_add__left__cancel,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add_left_cancel
tff(fact_1409_le__zero__eq,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),zero_zero(A))
        <=> ( Nb = zero_zero(A) ) ) ) ).

% le_zero_eq
tff(fact_1410_not__gr__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( ~ aa(A,$o,ord_less(A,zero_zero(A)),Nb)
        <=> ( Nb = zero_zero(A) ) ) ) ).

% not_gr_zero
tff(fact_1411_add__0,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% add_0
tff(fact_1412_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_1413_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_1414_add__cancel__right__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_right
tff(fact_1415_add__cancel__right__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_right_left
tff(fact_1416_add__cancel__left__right,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = A2 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_right
tff(fact_1417_add__cancel__left__left,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = A2 )
        <=> ( B2 = zero_zero(A) ) ) ) ).

% add_cancel_left_left
tff(fact_1418_double__zero__sym,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( ( zero_zero(A) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% double_zero_sym
tff(fact_1419_add_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% add.right_neutral
tff(fact_1420_add__le__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_cancel_left
tff(fact_1421_add__le__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_cancel_right
tff(fact_1422_add__less__cancel__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
        <=> aa(A,$o,ord_less(A,A2),B2) ) ) ).

% add_less_cancel_left
tff(fact_1423_add__less__cancel__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
        <=> aa(A,$o,ord_less(A,A2),B2) ) ) ).

% add_less_cancel_right
tff(fact_1424_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,A2),A2) = zero_zero(A) ) ).

% cancel_comm_monoid_add_class.diff_cancel
tff(fact_1425_diff__zero,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,A2),zero_zero(A)) = A2 ) ).

% diff_zero
tff(fact_1426_zero__diff,axiom,
    ! [A: $tType] :
      ( comm_monoid_diff(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,zero_zero(A)),A2) = zero_zero(A) ) ).

% zero_diff
tff(fact_1427_diff__0__right,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,A2),zero_zero(A)) = A2 ) ).

% diff_0_right
tff(fact_1428_diff__self,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,A2),A2) = zero_zero(A) ) ).

% diff_self
tff(fact_1429_mult_Oright__neutral,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),one_one(A)) = A2 ) ).

% mult.right_neutral
tff(fact_1430_mult__1,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A2) = A2 ) ).

% mult_1
tff(fact_1431_add__diff__cancel__right_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = A2 ) ).

% add_diff_cancel_right'
tff(fact_1432_add__diff__cancel__right,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,minus_minus(A,A2),B2) ) ).

% add_diff_cancel_right
tff(fact_1433_add__diff__cancel__left_H,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),A2) = B2 ) ).

% add_diff_cancel_left'
tff(fact_1434_add__diff__cancel__left,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [C2: A,A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) = aa(A,A,minus_minus(A,A2),B2) ) ).

% add_diff_cancel_left
tff(fact_1435_diff__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A2),B2)),B2) = A2 ) ).

% diff_add_cancel
tff(fact_1436_add__diff__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = A2 ) ).

% add_diff_cancel
tff(fact_1437_add__le__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% add_le_same_cancel1
tff(fact_1438_add__le__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% add_le_same_cancel2
tff(fact_1439_le__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).

% le_add_same_cancel1
tff(fact_1440_le__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) ) ) ).

% le_add_same_cancel2
tff(fact_1441_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% double_add_le_zero_iff_single_add_le_zero
tff(fact_1442_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% zero_le_double_add_iff_zero_le_single_add
tff(fact_1443_add__less__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2)
        <=> aa(A,$o,ord_less(A,A2),zero_zero(A)) ) ) ).

% add_less_same_cancel1
tff(fact_1444_add__less__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2)
        <=> aa(A,$o,ord_less(A,A2),zero_zero(A)) ) ) ).

% add_less_same_cancel2
tff(fact_1445_less__add__same__cancel1,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),B2) ) ) ).

% less_add_same_cancel1
tff(fact_1446_less__add__same__cancel2,axiom,
    ! [A: $tType] :
      ( ordere1937475149494474687imp_le(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),B2) ) ) ).

% less_add_same_cancel2
tff(fact_1447_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2)),zero_zero(A))
        <=> aa(A,$o,ord_less(A,A2),zero_zero(A)) ) ) ).

% double_add_less_zero_iff_single_add_less_zero
tff(fact_1448_add__right__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
         => ( B2 = C2 ) ) ) ).

% add_right_imp_eq
tff(fact_1449_add__left__imp__eq,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
         => ( B2 = C2 ) ) ) ).

% add_left_imp_eq
tff(fact_1450_ab__semigroup__add__class_Oadd_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% ab_semigroup_add_class.add.left_commute
tff(fact_1451_ab__semigroup__add__class_Oadd_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) ) ).

% ab_semigroup_add_class.add.commute
tff(fact_1452_add_Oright__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) )
        <=> ( B2 = C2 ) ) ) ).

% add.right_cancel
tff(fact_1453_add_Oleft__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) )
        <=> ( B2 = C2 ) ) ) ).

% add.left_cancel
tff(fact_1454_add_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_add(A)
     => ! [A2: A,B2: 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% add.assoc
tff(fact_1455_group__cancel_Oadd2,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [B5: A,K: A,B2: A,A2: A] :
          ( ( B5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add2
tff(fact_1456_group__cancel_Oadd1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: A,K: A,A2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% group_cancel.add1
tff(fact_1457_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( ( I = J )
            & ( K = L ) )
         => ( 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(4)
tff(fact_1458_one__reorient,axiom,
    ! [A: $tType] :
      ( one(A)
     => ! [Xb: A] :
          ( ( one_one(A) = Xb )
        <=> ( Xb = one_one(A) ) ) ) ).

% one_reorient
tff(fact_1459_mult_Oassoc,axiom,
    ! [A: $tType] :
      ( semigroup_mult(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% mult.assoc
tff(fact_1460_ab__semigroup__mult__class_Omult_Ocommute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2) ) ).

% ab_semigroup_mult_class.mult.commute
tff(fact_1461_ab__semigroup__mult__class_Omult_Oleft__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [B2: A,A2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% ab_semigroup_mult_class.mult.left_commute
tff(fact_1462_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A2),C2)),B2) = aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A2),B2)),C2) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
tff(fact_1463_diff__eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,minus_minus(A,C2),D2) )
         => ( ( A2 = B2 )
          <=> ( C2 = D2 ) ) ) ) ).

% diff_eq_diff_eq
tff(fact_1464_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,
    ! [A2: $o,B2: $o,Xb: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Leaf((A2),(B2)),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(1)
tff(fact_1465_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,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT,Xb: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,zero_zero(nat),Ts,S3),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(2)
tff(fact_1466_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,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT,Xb: nat] : vEBT_T_i_n_s_e_r_t2(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S3),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(3)
tff(fact_1467_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,TreeLista: list(vEBT_VEBT),Summarya: 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)),TreeLista,Summarya),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(4)
tff(fact_1468_insersimp_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),X_13)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(Ta,Y)),one_one(nat)) ) ) ).

% insersimp'
tff(fact_1469_insertsimp_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,L: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_minNull(Ta)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(Ta,L)),one_one(nat)) ) ) ).

% insertsimp'
tff(fact_1470_zero__le,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb) ) ).

% zero_le
tff(fact_1471_gr__zeroI,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( ( Nb != zero_zero(A) )
         => aa(A,$o,ord_less(A,zero_zero(A)),Nb) ) ) ).

% gr_zeroI
tff(fact_1472_not__less__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] : ~ aa(A,$o,ord_less(A,Nb),zero_zero(A)) ) ).

% not_less_zero
tff(fact_1473_gr__implies__not__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [M: A,Nb: A] :
          ( aa(A,$o,ord_less(A,M),Nb)
         => ( Nb != zero_zero(A) ) ) ) ).

% gr_implies_not_zero
tff(fact_1474_zero__less__iff__neq__zero,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [Nb: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Nb)
        <=> ( Nb != zero_zero(A) ) ) ) ).

% zero_less_iff_neq_zero
tff(fact_1475_add_Ogroup__left__neutral,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% add.group_left_neutral
tff(fact_1476_add_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% add.comm_neutral
tff(fact_1477_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A2) = A2 ) ).

% comm_monoid_add_class.add_0
tff(fact_1478_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & ( K = L ) )
         => aa(A,$o,aa(A,fun(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_1479_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,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(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_1480_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),J)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),K),L) )
         => aa(A,$o,aa(A,fun(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_1481_add__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_mono
tff(fact_1482_add__left__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% add_left_mono
tff(fact_1483_less__eqE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ~ ! [C3: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3) ) ) ).

% less_eqE
tff(fact_1484_add__right__mono,axiom,
    ! [A: $tType] :
      ( ordere6658533253407199908up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ).

% add_right_mono
tff(fact_1485_le__iff__add,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ? [C4: A] : B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C4) ) ) ).

% le_iff_add
tff(fact_1486_add__le__imp__le__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_imp_le_left
tff(fact_1487_add__le__imp__le__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% add_le_imp_le_right
tff(fact_1488_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_1489_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_1490_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_1491_add__strict__mono,axiom,
    ! [A: $tType] :
      ( strict9044650504122735259up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,ord_less(A,C2),D2)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_strict_mono
tff(fact_1492_add__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% add_strict_left_mono
tff(fact_1493_add__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ).

% add_strict_right_mono
tff(fact_1494_add__less__imp__less__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2))
         => aa(A,$o,ord_less(A,A2),B2) ) ) ).

% add_less_imp_less_left
tff(fact_1495_add__less__imp__less__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
         => aa(A,$o,ord_less(A,A2),B2) ) ) ).

% add_less_imp_less_right
tff(fact_1496_eq__iff__diff__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,A,minus_minus(A,A2),B2) = zero_zero(A) ) ) ) ).

% eq_iff_diff_eq_0
tff(fact_1497_diff__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),D2)) ) ) ) ).

% diff_mono
tff(fact_1498_diff__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,C2),A2)),aa(A,A,minus_minus(A,C2),B2)) ) ) ).

% diff_left_mono
tff(fact_1499_diff__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),C2)) ) ) ).

% diff_right_mono
tff(fact_1500_diff__eq__diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,minus_minus(A,C2),D2) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2) ) ) ) ).

% diff_eq_diff_less_eq
tff(fact_1501_diff__strict__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,D2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,ord_less(A,D2),C2)
           => aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),D2)) ) ) ) ).

% diff_strict_mono
tff(fact_1502_diff__eq__diff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,minus_minus(A,C2),D2) )
         => ( aa(A,$o,ord_less(A,A2),B2)
          <=> aa(A,$o,ord_less(A,C2),D2) ) ) ) ).

% diff_eq_diff_less
tff(fact_1503_diff__strict__left__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
         => aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,C2),A2)),aa(A,A,minus_minus(A,C2),B2)) ) ) ).

% diff_strict_left_mono
tff(fact_1504_diff__strict__right__mono,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A2),C2)),aa(A,A,minus_minus(A,B2),C2)) ) ) ).

% diff_strict_right_mono
tff(fact_1505_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),one_one(A)),A2) = A2 ) ).

% comm_monoid_mult_class.mult_1
tff(fact_1506_mult_Ocomm__neutral,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),one_one(A)) = A2 ) ).

% mult.comm_neutral
tff(fact_1507_diff__diff__eq,axiom,
    ! [A: $tType] :
      ( cancel2418104881723323429up_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% diff_diff_eq
tff(fact_1508_add__implies__diff,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) = A2 )
         => ( C2 = aa(A,A,minus_minus(A,A2),B2) ) ) ) ).

% add_implies_diff
tff(fact_1509_diff__add__eq__diff__diff__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,minus_minus(A,A2),C2)),B2) ) ).

% diff_add_eq_diff_diff_swap
tff(fact_1510_diff__add__eq,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_add_eq
tff(fact_1511_diff__diff__eq2,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,minus_minus(A,B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% diff_diff_eq2
tff(fact_1512_add__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,minus_minus(A,B2),C2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ).

% add_diff_eq
tff(fact_1513_eq__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( A2 = aa(A,A,minus_minus(A,C2),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = C2 ) ) ) ).

% eq_diff_eq
tff(fact_1514_diff__eq__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( aa(A,A,minus_minus(A,A2),B2) = C2 )
        <=> ( A2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2) ) ) ) ).

% diff_eq_eq
tff(fact_1515_group__cancel_Osub1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,K: A,A2: A,B2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,minus_minus(A,A3),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),aa(A,A,minus_minus(A,A2),B2)) ) ) ) ).

% group_cancel.sub1
tff(fact_1516_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,$o)),Uv2: option(A)] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$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,$o)),product_prod(option(A),option(A))),Uu2),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Uv2))
     => ( ! [Uw2: fun(A,fun(A,$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))),Uw2),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)))
       => ~ ! [F2: fun(A,fun(A,$o)),X2: A,Y5: A] : Xb != aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,$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))),F2),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),Y5))) ) ) ).

% VEBT_internal.option_comp_shift.cases
tff(fact_1517_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)),Uv2: 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)),Uv2))
     => ( ! [Uw2: 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))),Uw2),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)))
       => ~ ! [F2: fun(A,fun(A,A)),A4: A,B3: 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))),F2),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),A4)),aa(A,option(A),some(A),B3))) ) ) ).

% VEBT_internal.option_shift.cases
tff(fact_1518_insert_H__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t2(Ta,Xb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))) ) ).

% insert'_bound_height
tff(fact_1519_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [A: $tType,F: fun(A,fun(A,A)),A2: A,B2: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,F),aa(A,option(A),some(A),A2)),aa(A,option(A),some(A),B2)) = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),F,A2),B2)) ).

% VEBT_internal.option_shift.simps(3)
tff(fact_1520_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,A)),Uv: option(A)] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Uu),none(A)),Uv) = none(A) ).

% VEBT_internal.option_shift.simps(1)
tff(fact_1521_add__decreasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ) ) ).

% add_decreasing
tff(fact_1522_add__increasing,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_increasing
tff(fact_1523_add__decreasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ) ) ).

% add_decreasing2
tff(fact_1524_add__increasing2,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_increasing2
tff(fact_1525_add__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_nonneg_nonneg
tff(fact_1526_add__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_nonpos_nonpos
tff(fact_1527_add__nonneg__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,$o,aa(A,fun(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_1528_add__nonpos__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(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_1529_add__neg__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,B2),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_neg_neg
tff(fact_1530_add__pos__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_pos_pos
tff(fact_1531_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: $tType] :
      ( canoni5634975068530333245id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ~ ! [C3: A] :
                ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C3) )
               => ( C3 = zero_zero(A) ) ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
tff(fact_1532_pos__add__strict,axiom,
    ! [A: $tType] :
      ( strict7427464778891057005id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,B2),C2)
           => aa(A,$o,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% pos_add_strict
tff(fact_1533_add__mono__thms__linordered__field_I4_J,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [I: A,J: A,K: A,L: A] :
          ( ( aa(A,$o,aa(A,fun(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_1534_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,$o,aa(A,fun(A,$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_1535_add__le__less__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,ord_less(A,C2),D2)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_le_less_mono
tff(fact_1536_add__less__le__mono,axiom,
    ! [A: $tType] :
      ( ordere580206878836729694up_add(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),D2)) ) ) ) ).

% add_less_le_mono
tff(fact_1537_le__iff__diff__le__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A2),B2)),zero_zero(A)) ) ) ).

% le_iff_diff_le_0
tff(fact_1538_less__iff__diff__less__0,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
        <=> aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A2),B2)),zero_zero(A)) ) ) ).

% less_iff_diff_less_0
tff(fact_1539_diff__le__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A2),B2)),C2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% diff_le_eq
tff(fact_1540_le__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,minus_minus(A,C2),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ) ).

% le_diff_eq
tff(fact_1541_ordered__cancel__comm__monoid__diff__class_Odiff__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B2),A2)),A2) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add
tff(fact_1542_le__add__diff,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2)) ) ) ).

% le_add_diff
tff(fact_1543_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,minus_minus(A,B2),A2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
tff(fact_1544_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,minus_minus(A,B2),A2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
tff(fact_1545_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),aa(A,A,minus_minus(A,B2),A2)) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
tff(fact_1546_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B2),A2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
tff(fact_1547_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B2),A2)),C2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
tff(fact_1548_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,minus_minus(A,C2),aa(A,A,minus_minus(A,B2),A2)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)),B2) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
tff(fact_1549_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,minus_minus(A,B2),A2)) = B2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
tff(fact_1550_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: $tType] :
      ( ordere1170586879665033532d_diff(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( ( aa(A,A,minus_minus(A,B2),A2) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2) ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
tff(fact_1551_diff__less__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A2),B2)),C2)
        <=> aa(A,$o,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),B2)) ) ) ).

% diff_less_eq
tff(fact_1552_less__diff__eq,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),aa(A,A,minus_minus(A,C2),B2))
        <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ) ).

% less_diff_eq
tff(fact_1553_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [A: $tType,Uw: fun(A,fun(A,A)),V: A] : aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Uw),aa(A,option(A),some(A),V)),none(A)) = none(A) ).

% VEBT_internal.option_shift.simps(2)
tff(fact_1554_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) ) ) )
         => ~ ! [A4: A] :
                ( ( Xaa = aa(A,option(A),some(A),A4) )
               => ! [B3: A] :
                    ( ( Xba = aa(A,option(A),some(A),B3) )
                   => ( Y != aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),Xb,A4),B3)) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
tff(fact_1555_add__neg__nonpos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_neg_nonpos
tff(fact_1556_add__nonneg__pos,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_nonneg_pos
tff(fact_1557_add__nonpos__neg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,B2),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),zero_zero(A)) ) ) ) ).

% add_nonpos_neg
tff(fact_1558_add__pos__nonneg,axiom,
    ! [A: $tType] :
      ( ordere6911136660526730532id_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% add_pos_nonneg
tff(fact_1559_add__strict__increasing,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_strict_increasing
tff(fact_1560_add__strict__increasing2,axiom,
    ! [A: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,B2),C2)
           => aa(A,$o,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)) ) ) ) ).

% add_strict_increasing2
tff(fact_1561_Bolzano,axiom,
    ! [A2: real,B2: real,P: fun(real,fun(real,$o))] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [A4: real,B3: real,C3: real] :
            ( aa(real,$o,aa(real,fun(real,$o),P,A4),B3)
           => ( aa(real,$o,aa(real,fun(real,$o),P,B3),C3)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A4),B3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B3),C3)
                 => aa(real,$o,aa(real,fun(real,$o),P,A4),C3) ) ) ) )
       => ( ! [X2: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
             => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2)
               => ? [D5: real] :
                    ( aa(real,$o,ord_less(real,zero_zero(real)),D5)
                    & ! [A4: real,B3: real] :
                        ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A4),X2)
                          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B3)
                          & aa(real,$o,ord_less(real,aa(real,real,minus_minus(real,B3),A4)),D5) )
                       => aa(real,$o,aa(real,fun(real,$o),P,A4),B3) ) ) ) )
         => aa(real,$o,aa(real,fun(real,$o),P,A2),B2) ) ) ) ).

% Bolzano
tff(fact_1562_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : vEBT_vebt_mint(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) = none(nat) ).

% vebt_mint.simps(2)
tff(fact_1563_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : vEBT_vebt_maxt(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) = none(nat) ).

% vebt_maxt.simps(2)
tff(fact_1564_vebt__mint_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: 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),Mia),Maa)),Ux,Uy,Uz)) = aa(nat,option(nat),some(nat),Mia) ).

% vebt_mint.simps(3)
tff(fact_1565_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: 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),Mia),Maa)),Ux,Uy,Uz)) = aa(nat,option(nat),some(nat),Maa) ).

% vebt_maxt.simps(3)
tff(fact_1566_vebt__mint_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_mint(vEBT_Leaf((A2),(B2))) = $ite(
        (A2),
        aa(nat,option(nat),some(nat),zero_zero(nat)),
        $ite((B2),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) ).

% vebt_mint.simps(1)
tff(fact_1567_vebt__maxt_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_vebt_maxt(vEBT_Leaf((A2),(B2))) = $ite(
        (B2),
        aa(nat,option(nat),some(nat),one_one(nat)),
        $ite((A2),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) ).

% vebt_maxt.simps(1)
tff(fact_1568_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,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,ord_less(nat,Xb),Mia),Mia,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,Vaa))),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)),TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            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,TreeLista),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),vEBT_T_i_n_s_e_r_t2(Summarya,h),one_one(nat))),
            one_one(nat) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.simps(5)
tff(fact_1569_vebt__predi_Osimps,axiom,
    ! [Ta: vEBT_VEBTi,Xb: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_vebt_predi,Ta),Xb) = vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aTP_Lamp_cg(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),Xb),aTP_Lamp_ch(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Xb),Ta) ).

% vebt_predi.simps
tff(fact_1570_delete__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_d_e_l_e_t_e(Ta,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)))))))),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U))))) ) ) ).

% delete_bound_size_univ
tff(fact_1571_vebt__succi_Osimps,axiom,
    ! [Ta: vEBT_VEBTi,Xb: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_vebt_succi,Ta),Xb) = vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aTP_Lamp_cp(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),Xb),aTP_Lamp_cq(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Xb),Ta) ).

% vebt_succi.simps
tff(fact_1572_inrange,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),vEBT_VEBT_set_vebt(Ta)),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))),Nb)),one_one(nat)))) ) ).

% inrange
tff(fact_1573_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_1574_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_1575_height__double__log__univ__size,axiom,
    ! [U: real,Deg: nat,Ta: vEBT_VEBT] :
      ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Deg) )
     => ( vEBT_invar_vebt(Ta,Deg)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta))),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U)))) ) ) ).

% height_double_log_univ_size
tff(fact_1576_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,$o,aa(nat,fun(nat,$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_1577_delete__bound__size__univ_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_V1232361888498592333_e_t_e(Ta,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U)))) ) ) ).

% delete_bound_size_univ'
tff(fact_1578_TBOUND__VEBT__case,axiom,
    ! [A: $tType,Tib: vEBT_VEBTi,F: fun($o,fun($o,heap_Time_Heap(A))),Bnd: fun($o,fun($o,nat)),F4: 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))))] :
      ( ! [A4: $o,B3: $o] :
          ( ( Tib = vEBT_Leafi((A4),(B3)) )
         => time_TBOUND(A,aa($o,heap_Time_Heap(A),aa($o,fun($o,heap_Time_Heap(A)),F,(A4)),(B3)),aa($o,nat,aa($o,fun($o,nat),Bnd,(A4)),(B3))) )
     => ( ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeArray: array(vEBT_VEBTi),Summary: vEBT_VEBTi] :
            ( ( Tib = vEBT_Nodei(Info2,Deg2,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)))),F4,Info2),Deg2),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,Info2),Deg2),TreeArray),Summary)) )
       => time_TBOUND(A,vEBT_case_VEBTi(heap_Time_Heap(A),F4,F,Tib),vEBT_case_VEBTi(nat,Bnd2,Bnd,Tib)) ) ) ).

% TBOUND_VEBT_case
tff(fact_1579_subset__Collect__conv,axiom,
    ! [A: $tType,S2: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),collect(A,P))
    <=> ! [X: A] :
          ( member(A,X,S2)
         => aa(A,$o,P,X) ) ) ).

% subset_Collect_conv
tff(fact_1580_VEBTi_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),F22: fun($o,fun($o,B)),VEBTi: vEBT_VEBTi] : aa(B,A,H,vEBT_case_VEBTi(B,F1,F22,VEBTi)) = vEBT_case_VEBTi(A,aa(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),aTP_Lamp_cr(fun(B,A),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A))))),H),F1),aa(fun($o,fun($o,B)),fun($o,fun($o,A)),aTP_Lamp_cs(fun(B,A),fun(fun($o,fun($o,B)),fun($o,fun($o,A))),H),F22),VEBTi) ).

% VEBTi.case_distrib
tff(fact_1581_VEBTi_Osimps_I5_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),F22: fun($o,fun($o,A)),X11a: option(product_prod(nat,nat)),X12a: nat,X13a: array(vEBT_VEBTi),X14a: vEBT_VEBTi] : vEBT_case_VEBTi(A,F1,F22,vEBT_Nodei(X11a,X12a,X13a,X14a)) = 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,X11a),X12a),X13a),X14a) ).

% VEBTi.simps(5)
tff(fact_1582_VEBTi_Osimps_I6_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,A)))),F22: fun($o,fun($o,A)),X21: $o,X22: $o] : vEBT_case_VEBTi(A,F1,F22,vEBT_Leafi((X21),(X22))) = aa($o,A,aa($o,fun($o,A),F22,(X21)),(X22)) ).

% VEBTi.simps(6)
tff(fact_1583_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,$o,Q6,X2) ) )
           => ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,ord_less(A,Z3),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_1584_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,$o,Q6,X2) ) )
           => ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,ord_less(A,Z3),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_1585_pinf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z3),X3)
         => ( X3 != Ta ) ) ) ).

% pinf(3)
tff(fact_1586_pinf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z3),X3)
         => ( X3 != Ta ) ) ) ).

% pinf(4)
tff(fact_1587_pinf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z3),X3)
         => ~ aa(A,$o,ord_less(A,X3),Ta) ) ) ).

% pinf(5)
tff(fact_1588_pinf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z3),X3)
         => aa(A,$o,ord_less(A,Ta),X3) ) ) ).

% pinf(7)
tff(fact_1589_pinf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F5: B] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z3),X3)
         => ( F5 = F5 ) ) ) ).

% pinf(11)
tff(fact_1590_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,$o,Q6,X2) ) )
           => ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,ord_less(A,X3),Z3)
               => ( ( aa(A,$o,P,X3)
                    & aa(A,$o,Q,X3) )
                <=> ( aa(A,$o,P3,X3)
                    & aa(A,$o,Q6,X3) ) ) ) ) ) ) ).

% minf(1)
tff(fact_1591_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,$o,Q6,X2) ) )
           => ? [Z3: A] :
              ! [X3: A] :
                ( aa(A,$o,ord_less(A,X3),Z3)
               => ( ( aa(A,$o,P,X3)
                    | aa(A,$o,Q,X3) )
                <=> ( aa(A,$o,P3,X3)
                    | aa(A,$o,Q6,X3) ) ) ) ) ) ) ).

% minf(2)
tff(fact_1592_minf_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z3)
         => ( X3 != Ta ) ) ) ).

% minf(3)
tff(fact_1593_minf_I4_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z3)
         => ( X3 != Ta ) ) ) ).

% minf(4)
tff(fact_1594_minf_I5_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z3)
         => aa(A,$o,ord_less(A,X3),Ta) ) ) ).

% minf(5)
tff(fact_1595_minf_I7_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z3)
         => ~ aa(A,$o,ord_less(A,Ta),X3) ) ) ).

% minf(7)
tff(fact_1596_minf_I11_J,axiom,
    ! [A: $tType,B: $tType] :
      ( ord(A)
     => ! [F5: B] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z3)
         => ( F5 = F5 ) ) ) ).

% minf(11)
tff(fact_1597_pinf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z3),X3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Ta) ) ) ).

% pinf(6)
tff(fact_1598_pinf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z3),X3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),X3) ) ) ).

% pinf(8)
tff(fact_1599_minf_I6_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Ta) ) ) ).

% minf(6)
tff(fact_1600_minf_I8_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Ta: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z3)
         => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),X3) ) ) ).

% minf(8)
tff(fact_1601_inf__period_I1_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D6: A,Q: fun(A,$o)] :
          ( ! [X2: A,K3: A] :
              ( aa(A,$o,P,X2)
            <=> aa(A,$o,P,aa(A,A,minus_minus(A,X2),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D6))) )
         => ( ! [X2: A,K3: A] :
                ( aa(A,$o,Q,X2)
              <=> aa(A,$o,Q,aa(A,A,minus_minus(A,X2),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D6))) )
           => ! [X3: A,K5: A] :
                ( ( aa(A,$o,P,X3)
                  & aa(A,$o,Q,X3) )
              <=> ( aa(A,$o,P,aa(A,A,minus_minus(A,X3),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D6)))
                  & aa(A,$o,Q,aa(A,A,minus_minus(A,X3),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D6))) ) ) ) ) ) ).

% inf_period(1)
tff(fact_1602_inf__period_I2_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [P: fun(A,$o),D6: A,Q: fun(A,$o)] :
          ( ! [X2: A,K3: A] :
              ( aa(A,$o,P,X2)
            <=> aa(A,$o,P,aa(A,A,minus_minus(A,X2),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D6))) )
         => ( ! [X2: A,K3: A] :
                ( aa(A,$o,Q,X2)
              <=> aa(A,$o,Q,aa(A,A,minus_minus(A,X2),aa(A,A,aa(A,fun(A,A),times_times(A),K3),D6))) )
           => ! [X3: A,K5: A] :
                ( ( aa(A,$o,P,X3)
                  | aa(A,$o,Q,X3) )
              <=> ( aa(A,$o,P,aa(A,A,minus_minus(A,X3),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D6)))
                  | aa(A,$o,Q,aa(A,A,minus_minus(A,X3),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D6))) ) ) ) ) ) ).

% inf_period(2)
tff(fact_1603_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_1604_imp__le__cong,axiom,
    ! [Xb: int,X4: int,P: $o,P3: $o] :
      ( ( Xb = X4 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X4)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
           => (P) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X4)
           => (P3) ) ) ) ) ).

% imp_le_cong
tff(fact_1605_conj__le__cong,axiom,
    ! [Xb: int,X4: int,P: $o,P3: $o] :
      ( ( Xb = X4 )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X4)
         => ( (P)
          <=> (P3) ) )
       => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
            & (P) )
        <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X4)
            & (P3) ) ) ) ) ).

% conj_le_cong
tff(fact_1606_pred__bound__size__univ_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d2(Ta,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U)))) ) ) ).

% pred_bound_size_univ'
tff(fact_1607_succ__bound__size__univ_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c2(Ta,Xb))),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(num,real,numeral_numeral(real),bit0(one2))),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U)))) ) ) ).

% succ_bound_size_univ'
tff(fact_1608_member__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_m_e_m_b_e_r(Ta,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))))),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U))))) ) ) ).

% member_bound_size_univ
tff(fact_1609_VEBT__internal_Ocnt_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_Leaf((A2),(B2))) = one_one(real) ).

% VEBT_internal.cnt.simps(1)
tff(fact_1610_VEBT__internal_Ocnt_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_Leaf((A2),(B2))) = one_one(nat) ).

% VEBT_internal.cnt'.simps(1)
tff(fact_1611_plusinfinity,axiom,
    ! [D2: int,P3: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D2)
     => ( ! [X2: int,K3: int] :
            ( aa(int,$o,P3,X2)
          <=> aa(int,$o,P3,aa(int,int,minus_minus(int,X2),aa(int,int,aa(int,fun(int,int),times_times(int),K3),D2))) )
       => ( ? [Z4: int] :
            ! [X2: int] :
              ( aa(int,$o,ord_less(int,Z4),X2)
             => ( aa(int,$o,P,X2)
              <=> aa(int,$o,P3,X2) ) )
         => ( ? [X_12: int] : aa(int,$o,P3,X_12)
           => ? [X_13: int] : aa(int,$o,P,X_13) ) ) ) ) ).

% plusinfinity
tff(fact_1612_minusinfinity,axiom,
    ! [D2: int,P1: fun(int,$o),P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D2)
     => ( ! [X2: int,K3: int] :
            ( aa(int,$o,P1,X2)
          <=> aa(int,$o,P1,aa(int,int,minus_minus(int,X2),aa(int,int,aa(int,fun(int,int),times_times(int),K3),D2))) )
       => ( ? [Z4: int] :
            ! [X2: int] :
              ( aa(int,$o,ord_less(int,X2),Z4)
             => ( aa(int,$o,P,X2)
              <=> aa(int,$o,P1,X2) ) )
         => ( ? [X_12: int] : aa(int,$o,P1,X_12)
           => ? [X_13: int] : aa(int,$o,P,X_13) ) ) ) ) ).

% minusinfinity
tff(fact_1613_incr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D2)
     => ( ! [X2: int] :
            ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$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),D2))) ) ) ) ) ).

% incr_mult_lemma
tff(fact_1614_decr__mult__lemma,axiom,
    ! [D2: int,P: fun(int,$o),K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D2)
     => ( ! [X2: int] :
            ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X2),D2)) )
       => ( aa(int,$o,aa(int,fun(int,$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),D2))) ) ) ) ) ).

% decr_mult_lemma
tff(fact_1615_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_1616_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_1617_VEBT__internal_Ospace_H_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_space2,vEBT_Leaf((A2),(B2))) = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ).

% VEBT_internal.space'.simps(1)
tff(fact_1618_VEBT__internal_Ospace_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_space,vEBT_Leaf((A2),(B2))) = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ).

% VEBT_internal.space.simps(1)
tff(fact_1619_log__pow__cancel,axiom,
    ! [A2: real,B2: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( log(A2,aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),B2)) = aa(nat,real,semiring_1_of_nat(real),B2) ) ) ) ).

% log_pow_cancel
tff(fact_1620_log__le__cancel__iff,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),log(A2,Xb)),log(A2,Y))
          <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ) ).

% log_le_cancel_iff
tff(fact_1621_log__le__one__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),log(A2,Xb)),one_one(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),A2) ) ) ) ).

% log_le_one_cancel_iff
tff(fact_1622_one__le__log__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),log(A2,Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Xb) ) ) ) ).

% one_le_log_cancel_iff
tff(fact_1623_log__le__zero__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),log(A2,Xb)),zero_zero(real))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real)) ) ) ) ).

% log_le_zero_cancel_iff
tff(fact_1624_zero__le__log__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),log(A2,Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb) ) ) ) ).

% zero_le_log_cancel_iff
tff(fact_1625_log2__of__power__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$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))),Nb))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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),Nb)) ) ) ).

% log2_of_power_le
tff(fact_1626_log__eq__one,axiom,
    ! [A2: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( log(A2,A2) = one_one(real) ) ) ) ).

% log_eq_one
tff(fact_1627_log__one,axiom,
    ! [A2: real] : log(A2,one_one(real)) = zero_zero(real) ).

% log_one
tff(fact_1628_zero__less__log__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),log(A2,Xb))
        <=> aa(real,$o,ord_less(real,one_one(real)),Xb) ) ) ) ).

% zero_less_log_cancel_iff
tff(fact_1629_log__less__zero__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,log(A2,Xb)),zero_zero(real))
        <=> aa(real,$o,ord_less(real,Xb),one_one(real)) ) ) ) ).

% log_less_zero_cancel_iff
tff(fact_1630_one__less__log__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,one_one(real)),log(A2,Xb))
        <=> aa(real,$o,ord_less(real,A2),Xb) ) ) ) ).

% one_less_log_cancel_iff
tff(fact_1631_log__less__one__cancel__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,log(A2,Xb)),one_one(real))
        <=> aa(real,$o,ord_less(real,Xb),A2) ) ) ) ).

% log_less_one_cancel_iff
tff(fact_1632_log__less__cancel__iff,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
         => ( aa(real,$o,ord_less(real,log(A2,Xb)),log(A2,Y))
          <=> aa(real,$o,ord_less(real,Xb),Y) ) ) ) ) ).

% log_less_cancel_iff
tff(fact_1633_log__base__change,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( log(B2,Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),log(A2,Xb)),log(A2,B2)) ) ) ) ).

% log_base_change
tff(fact_1634_less__log__of__power,axiom,
    ! [B2: real,Nb: nat,M: real] :
      ( aa(real,$o,ord_less(real,aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Nb)),M)
     => ( aa(real,$o,ord_less(real,one_one(real)),B2)
       => aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),Nb)),log(B2,M)) ) ) ).

% less_log_of_power
tff(fact_1635_log__of__power__eq,axiom,
    ! [M: nat,B2: real,Nb: nat] :
      ( ( aa(nat,real,semiring_1_of_nat(real),M) = aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Nb) )
     => ( aa(real,$o,ord_less(real,one_one(real)),B2)
       => ( aa(nat,real,semiring_1_of_nat(real),Nb) = log(B2,aa(nat,real,semiring_1_of_nat(real),M)) ) ) ) ).

% log_of_power_eq
tff(fact_1636_log__mult,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
           => ( log(A2,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),log(A2,Xb)),log(A2,Y)) ) ) ) ) ) ).

% log_mult
tff(fact_1637_le__log__of__power,axiom,
    ! [B2: real,Nb: nat,M: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Nb)),M)
     => ( aa(real,$o,ord_less(real,one_one(real)),B2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Nb)),log(B2,M)) ) ) ).

% le_log_of_power
tff(fact_1638_log__divide,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),Y)
           => ( log(A2,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(real,real,minus_minus(real,log(A2,Xb)),log(A2,Y)) ) ) ) ) ) ).

% log_divide
tff(fact_1639_log__base__pow,axiom,
    ! [A2: real,Nb: nat,Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( log(aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),Nb),Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),log(A2,Xb)),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ).

% log_base_pow
tff(fact_1640_log__nat__power,axiom,
    ! [Xb: real,B2: real,Nb: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( log(B2,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),log(B2,Xb)) ) ) ).

% log_nat_power
tff(fact_1641_log2__of__power__eq,axiom,
    ! [M: nat,Nb: nat] :
      ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) )
     => ( aa(nat,real,semiring_1_of_nat(real),Nb) = log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),M)) ) ) ).

% log2_of_power_eq
tff(fact_1642_log__of__power__less,axiom,
    ! [M: nat,B2: real,Nb: nat] :
      ( aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),M)),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),Nb))
     => ( aa(real,$o,ord_less(real,one_one(real)),B2)
       => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
         => aa(real,$o,ord_less(real,log(B2,aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% log_of_power_less
tff(fact_1643_log__of__power__le,axiom,
    ! [M: nat,B2: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$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),B2),Nb))
     => ( aa(real,$o,ord_less(real,one_one(real)),B2)
       => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),log(B2,aa(nat,real,semiring_1_of_nat(real),M))),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% log_of_power_le
tff(fact_1644_less__log2__of__power,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),M)
     => aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),Nb)),log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% less_log2_of_power
tff(fact_1645_le__log2__of__power,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),M)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Nb)),log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),M))) ) ).

% le_log2_of_power
tff(fact_1646_log2__of__power__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
       => aa(real,$o,ord_less(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),Nb)) ) ) ).

% log2_of_power_less
tff(fact_1647_atLeastatMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% atLeastatMost_subset_iff
tff(fact_1648_arcosh__1,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( arcosh(A,one_one(A)) = zero_zero(A) ) ) ).

% arcosh_1
tff(fact_1649_minNull__delete__time__bound,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_minNull(vEBT_vebt_delete(Ta,Xb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_d_e_l_e_t_e(Ta,Xb)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(bit0(one2))))) ) ) ).

% minNull_delete_time_bound
tff(fact_1650_nat__induct2,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,zero_zero(nat))
     => ( aa(nat,$o,P,one_one(nat))
       => ( ! [N2: nat] :
              ( aa(nat,$o,P,N2)
             => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) )
         => aa(nat,$o,P,Nb) ) ) ) ).

% nat_induct2
tff(fact_1651_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xb,Xaa)
     => ( ! [Uu2: $o,Uv2: $o] : Xb != vEBT_Leaf((Uu2),(Uv2))
       => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb != vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va: 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),Mi),Ma)),zero_zero(nat),Va,Vb2)
               => ( ( Xaa = Mi )
                  | ( Xaa = Ma ) ) )
           => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list(vEBT_VEBT)] :
                  ( ? [Vc: 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),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc)
                 => ( ( Xaa = Mi )
                    | ( Xaa = Ma )
                    | $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)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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,TreeList: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,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)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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_1652_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xb,Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => (Y) )
       => ( ( ? [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)
           => (Y) )
         => ( ! [Mi: nat,Ma: nat] :
                ( ? [Va: 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),Mi),Ma)),zero_zero(nat),Va,Vb2)
               => ( (Y)
                <=> ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
           => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list(vEBT_VEBT)] :
                  ( ? [Vc: 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),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc)
                 => ( (Y)
                  <=> ~ ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $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)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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,TreeList: list(vEBT_VEBT)] :
                    ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,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)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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_1653_nat__approx__posE,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [E3: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),E3)
         => ~ ! [N2: 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,N2)))),E3) ) ) ).

% nat_approx_posE
tff(fact_1654_del__single__cont,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        & ( Xb = Maa ) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
       => ( 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),Mia),Maa)),Deg,TreeLista,Summarya),Xb) = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeLista,Summarya) ) ) ) ).

% del_single_cont
tff(fact_1655_delete__pres__valid,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => vEBT_invar_vebt(vEBT_vebt_delete(Ta,Xb),Nb) ) ).

% delete_pres_valid
tff(fact_1656_buildup__nothing__in__min__max,axiom,
    ! [Nb: nat,Xb: nat] : ~ vEBT_VEBT_membermima(vEBT_vebt_buildup(Nb),Xb) ).

% buildup_nothing_in_min_max
tff(fact_1657_dele__bmo__cont__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_V8194947554948674370ptions(vEBT_vebt_delete(Ta,Xb)),Y)
      <=> ( ( Xb != Y )
          & aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Y) ) ) ) ).

% dele_bmo_cont_corr
tff(fact_1658_dele__member__cont__corr,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,$o,vEBT_vebt_member(vEBT_vebt_delete(Ta,Xb)),Y)
      <=> ( ( Xb != Y )
          & aa(nat,$o,vEBT_vebt_member(Ta),Y) ) ) ) ).

% dele_member_cont_corr
tff(fact_1659_minNull__delete__time__bound_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_minNull(vEBT_vebt_delete(Ta,Xb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_V1232361888498592333_e_t_e(Ta,Xb)),one_one(nat)) ) ) ).

% minNull_delete_time_bound'
tff(fact_1660_Icc__eq__Icc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [L: A,H: A,L3: A,H3: A] :
          ( ( set_or1337092689740270186AtMost(A,L,H) = set_or1337092689740270186AtMost(A,L3,H3) )
        <=> ( ( ( L = L3 )
              & ( H = H3 ) )
            | ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
              & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L3),H3) ) ) ) ) ).

% Icc_eq_Icc
tff(fact_1661_atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or1337092689740270186AtMost(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),U) ) ) ) ).

% atLeastAtMost_iff
tff(fact_1662_delt__out__of__range,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( ( aa(nat,$o,ord_less(nat,Xb),Mia)
        | aa(nat,$o,ord_less(nat,Maa),Xb) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
       => ( 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),Mia),Maa)),Deg,TreeLista,Summarya),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),Mia),Maa)),Deg,TreeLista,Summarya) ) ) ) ).

% delt_out_of_range
tff(fact_1663_aset_I2_J,axiom,
    ! [D6: int,A3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D6))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A3)
                 => ( X2 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D6)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D6))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X2 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D6)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A3)
                   => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa2) ) ) )
           => ( ( aa(int,$o,P,X3)
                | aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6))
                | aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6)) ) ) ) ) ) ).

% aset(2)
tff(fact_1664_aset_I1_J,axiom,
    ! [D6: int,A3: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D6))
             => ! [Xb2: int] :
                  ( member(int,Xb2,A3)
                 => ( X2 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D6)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D6))
               => ! [Xb2: int] :
                    ( member(int,Xb2,A3)
                   => ( X2 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D6)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A3)
                   => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa2) ) ) )
           => ( ( aa(int,$o,P,X3)
                & aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6))
                & aa(int,$o,Q,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6)) ) ) ) ) ) ).

% aset(1)
tff(fact_1665_bset_I2_J,axiom,
    ! [D6: int,B5: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D6))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B5)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X2),D6)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D6))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B5)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,minus_minus(int,X2),D6)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa2) ) ) )
           => ( ( aa(int,$o,P,X3)
                | aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,minus_minus(int,X3),D6))
                | aa(int,$o,Q,aa(int,int,minus_minus(int,X3),D6)) ) ) ) ) ) ).

% bset(2)
tff(fact_1666_bset_I1_J,axiom,
    ! [D6: int,B5: set(int),P: fun(int,$o),Q: fun(int,$o)] :
      ( ! [X2: int] :
          ( ! [Xa: int] :
              ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D6))
             => ! [Xb2: int] :
                  ( member(int,Xb2,B5)
                 => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
         => ( aa(int,$o,P,X2)
           => aa(int,$o,P,aa(int,int,minus_minus(int,X2),D6)) ) )
     => ( ! [X2: int] :
            ( ! [Xa: int] :
                ( member(int,Xa,set_or1337092689740270186AtMost(int,one_one(int),D6))
               => ! [Xb2: int] :
                    ( member(int,Xb2,B5)
                   => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
           => ( aa(int,$o,Q,X2)
             => aa(int,$o,Q,aa(int,int,minus_minus(int,X2),D6)) ) )
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa2) ) ) )
           => ( ( aa(int,$o,P,X3)
                & aa(int,$o,Q,X3) )
             => ( aa(int,$o,P,aa(int,int,minus_minus(int,X3),D6))
                & aa(int,$o,Q,aa(int,int,minus_minus(int,X3),D6)) ) ) ) ) ) ).

% bset(1)
tff(fact_1667_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,aa(nat,fun(nat,$o),ord_less_eq(nat),X2),M8) )
       => ~ ! [M2: nat] :
              ( aa(nat,$o,P,M2)
             => ~ ! [X3: nat] :
                    ( aa(nat,$o,P,X3)
                   => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),M2) ) ) ) ) ).

% bounded_Max_nat
tff(fact_1668_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))] :
      ~ ! [F2: fun(nat,fun(A,A)),A4: nat,B3: 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))),F2),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)),A4),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B3),Acc))) ).

% fold_atLeastAtMost_nat.cases
tff(fact_1669_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_1670_aset_I7_J,axiom,
    ! [D6: int,A3: set(int),Ta: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D6)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A3)
                 => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa2) ) ) )
         => ( aa(int,$o,ord_less(int,Ta),X3)
           => aa(int,$o,ord_less(int,Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6)) ) ) ) ).

% aset(7)
tff(fact_1671_aset_I5_J,axiom,
    ! [D6: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D6)
     => ( member(int,Ta,A3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A3)
                   => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa2) ) ) )
           => ( aa(int,$o,ord_less(int,X3),Ta)
             => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6)),Ta) ) ) ) ) ).

% aset(5)
tff(fact_1672_aset_I4_J,axiom,
    ! [D6: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D6)
     => ( member(int,Ta,A3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A3)
                   => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa2) ) ) )
           => ( ( X3 != Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6) != Ta ) ) ) ) ) ).

% aset(4)
tff(fact_1673_aset_I3_J,axiom,
    ! [D6: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D6)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int)),A3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A3)
                   => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa2) ) ) )
           => ( ( X3 = Ta )
             => ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6) = Ta ) ) ) ) ) ).

% aset(3)
tff(fact_1674_bset_I7_J,axiom,
    ! [D6: int,Ta: int,B5: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D6)
     => ( member(int,Ta,B5)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa2) ) ) )
           => ( aa(int,$o,ord_less(int,Ta),X3)
             => aa(int,$o,ord_less(int,Ta),aa(int,int,minus_minus(int,X3),D6)) ) ) ) ) ).

% bset(7)
tff(fact_1675_bset_I5_J,axiom,
    ! [D6: int,B5: set(int),Ta: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D6)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B5)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa2) ) ) )
         => ( aa(int,$o,ord_less(int,X3),Ta)
           => aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,X3),D6)),Ta) ) ) ) ).

% bset(5)
tff(fact_1676_bset_I4_J,axiom,
    ! [D6: int,Ta: int,B5: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D6)
     => ( member(int,Ta,B5)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa2) ) ) )
           => ( ( X3 != Ta )
             => ( aa(int,int,minus_minus(int,X3),D6) != Ta ) ) ) ) ) ).

% bset(4)
tff(fact_1677_bset_I3_J,axiom,
    ! [D6: int,Ta: int,B5: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D6)
     => ( member(int,aa(int,int,minus_minus(int,Ta),one_one(int)),B5)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa2) ) ) )
           => ( ( X3 = Ta )
             => ( aa(int,int,minus_minus(int,X3),D6) = Ta ) ) ) ) ) ).

% bset(3)
tff(fact_1678_periodic__finite__ex,axiom,
    ! [D2: int,P: fun(int,$o)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D2)
     => ( ! [X2: int,K3: 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),K3),D2))) )
       => ( ? [X_1: int] : aa(int,$o,P,X_1)
        <=> ? [X: int] :
              ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D2))
              & aa(int,$o,P,X) ) ) ) ) ).

% periodic_finite_ex
tff(fact_1679_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: 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),Mia),Maa)),zero_zero(nat),Vaa,Vb),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa ) ) ) ).

% VEBT_internal.membermima.simps(3)
tff(fact_1680_aset_I8_J,axiom,
    ! [D6: int,A3: set(int),Ta: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D6)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A3)
                 => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa2) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),X3)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6)) ) ) ) ).

% aset(8)
tff(fact_1681_aset_I6_J,axiom,
    ! [D6: int,Ta: int,A3: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D6)
     => ( member(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Ta),one_one(int)),A3)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
               => ! [Xb3: int] :
                    ( member(int,Xb3,A3)
                   => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa2) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X3),Ta)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6)),Ta) ) ) ) ) ).

% aset(6)
tff(fact_1682_bset_I8_J,axiom,
    ! [D6: int,Ta: int,B5: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D6)
     => ( member(int,aa(int,int,minus_minus(int,Ta),one_one(int)),B5)
       => ! [X3: int] :
            ( ! [Xa2: int] :
                ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
               => ! [Xb3: int] :
                    ( member(int,Xb3,B5)
                   => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa2) ) ) )
           => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),X3)
             => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Ta),aa(int,int,minus_minus(int,X3),D6)) ) ) ) ) ).

% bset(8)
tff(fact_1683_bset_I6_J,axiom,
    ! [D6: int,B5: set(int),Ta: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D6)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B5)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa2) ) ) )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),X3),Ta)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,minus_minus(int,X3),D6)),Ta) ) ) ) ).

% bset(6)
tff(fact_1684_cpmi,axiom,
    ! [D6: int,P: fun(int,$o),P3: fun(int,$o),B5: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D6)
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa(int,$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),D6))
                 => ! [Xb2: int] :
                      ( member(int,Xb2,B5)
                     => ( X2 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb2),Xa) ) ) )
             => ( aa(int,$o,P,X2)
               => aa(int,$o,P,aa(int,int,minus_minus(int,X2),D6)) ) )
         => ( ! [X2: int,K3: 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),K3),D6))) )
           => ( ? [X_1: int] : aa(int,$o,P,X_1)
            <=> ( ? [X: int] :
                    ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D6))
                    & aa(int,$o,P3,X) )
                | ? [X: int] :
                    ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D6))
                    & ? [Xa3: int] :
                        ( member(int,Xa3,B5)
                        & aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xa3),X)) ) ) ) ) ) ) ) ) ).

% cpmi
tff(fact_1685_cppi,axiom,
    ! [D6: int,P: fun(int,$o),P3: fun(int,$o),A3: set(int)] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D6)
     => ( ? [Z4: int] :
          ! [X2: int] :
            ( aa(int,$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),D6))
                 => ! [Xb2: int] :
                      ( member(int,Xb2,A3)
                     => ( X2 != aa(int,int,minus_minus(int,Xb2),Xa) ) ) )
             => ( aa(int,$o,P,X2)
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),X2),D6)) ) )
         => ( ! [X2: int,K3: 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),K3),D6))) )
           => ( ? [X_1: int] : aa(int,$o,P,X_1)
            <=> ( ? [X: int] :
                    ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D6))
                    & aa(int,$o,P3,X) )
                | ? [X: int] :
                    ( member(int,X,set_or1337092689740270186AtMost(int,one_one(int),D6))
                    & ? [Xa3: int] :
                        ( member(int,Xa3,A3)
                        & aa(int,$o,P,aa(int,int,minus_minus(int,Xa3),X)) ) ) ) ) ) ) ) ) ).

% cppi
tff(fact_1686_exists__least__lemma,axiom,
    ! [P: fun(nat,$o)] :
      ( ~ aa(nat,$o,P,zero_zero(nat))
     => ( ? [X_12: nat] : aa(nat,$o,P,X_12)
       => ? [N2: nat] :
            ( ~ aa(nat,$o,P,N2)
            & aa(nat,$o,P,aa(nat,nat,suc,N2)) ) ) ) ).

% exists_least_lemma
tff(fact_1687_real__arch__simple,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(nat,A,semiring_1_of_nat(A),N2)) ) ).

% real_arch_simple
tff(fact_1688_reals__Archimedean2,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [N2: nat] : aa(A,$o,ord_less(A,Xb),aa(nat,A,semiring_1_of_nat(A),N2)) ) ).

% reals_Archimedean2
tff(fact_1689_atLeastatMost__psubset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,ord_less(set(A),set_or1337092689740270186AtMost(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
              | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
                & ( aa(A,$o,ord_less(A,C2),A2)
                  | aa(A,$o,ord_less(A,B2),D2) ) ) )
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2) ) ) ) ).

% atLeastatMost_psubset_iff
tff(fact_1690_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)
         => ? [N2: 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),N2)),Xb)) ) ) ).

% ex_less_of_nat_mult
tff(fact_1691_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeLista: list(vEBT_VEBT),Vd: vEBT_VEBT,Xb: nat] :
      ( vEBT_VEBT_membermima(vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V),TreeLista,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)),TreeLista)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),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_1692_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mia: nat,Maa: nat,V: nat,TreeLista: list(vEBT_VEBT),Vc2: 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),Mia),Maa)),aa(nat,nat,suc,V),TreeLista,Vc2),Xb)
    <=> ( ( Xb = Mia )
        | ( Xb = Maa )
        | $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)),TreeLista)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),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_1693_double__not__eq__Suc__double,axiom,
    ! [M: nat,Nb: 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))),Nb)) ).

% double_not_eq_Suc_double
tff(fact_1694_Suc__double__not__eq__double,axiom,
    ! [M: nat,Nb: 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))),Nb) ).

% Suc_double_not_eq_double
tff(fact_1695_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xb,Xaa)
     => ( ! [Mi: nat,Ma: nat] :
            ( ? [Va: 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),Mi),Ma)),zero_zero(nat),Va,Vb2)
           => ~ ( ( Xaa = Mi )
                | ( Xaa = Ma ) ) )
       => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [Vc: 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),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc)
             => ~ ( ( Xaa = Mi )
                  | ( Xaa = Ma )
                  | $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)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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,TreeList: list(vEBT_VEBT)] :
                ( ? [Vd2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,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)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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_1696_del__x__mia,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        & aa(nat,$o,ord_less(nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( 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),Mia),Maa)),Deg,TreeLista,Summarya),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(Summarya))),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),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya)))))),
                $let(
                  h: nat,
                  h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),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)),TreeLista)),
                    $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                      $let(
                        newlist: list(vEBT_VEBT),
                        newlist:= list_update(vEBT_VEBT,TreeLista,h,newnode),
                        $ite(
                          vEBT_VEBT_minNull(newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summarya,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 = Maa,
                                    $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),Deg),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)))))) ),
                                    Maa ))),Deg,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 = Maa,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),Deg),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)))),Maa))),Deg,newlist,Summarya) ) ) ),
                    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),Mia),Maa)),Deg,TreeLista,Summarya) ) ) ) ) ) ) ) ).

% del_x_mia
tff(fact_1697_del__x__mi__lets__in__minNull,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Deg: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT),Sn: vEBT_VEBT] :
      ( ( ( Xb = Mia )
        & aa(nat,$o,ord_less(nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeLista,H,Newnode) )
                     => ( vEBT_VEBT_minNull(Newnode)
                       => ( ( Sn = vEBT_vebt_delete(Summarya,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),Mia),Maa)),Deg,TreeLista,Summarya),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 = Maa,
                                      $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),Deg),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)))))) ),
                                      Maa ))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
tff(fact_1698_del__x__mi__lets__in,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Deg: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( Xb = Mia )
        & aa(nat,$o,ord_less(nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeLista,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),Mia),Maa)),Deg,TreeLista,Summarya),Xb) = $ite(
                            vEBT_VEBT_minNull(Newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summarya,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 = Maa,
                                      $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),Deg),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)))))) ),
                                      Maa ))),Deg,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 = Maa,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),Deg),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)))),Maa))),Deg,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
tff(fact_1699_del__x__mi,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Deg: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list(vEBT_VEBT),L: nat] :
      ( ( ( Xb = Mia )
        & aa(nat,$o,ord_less(nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
                 => ( 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),Mia),Maa)),Deg,TreeLista,Summarya),Xb) = $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),H),L),
                        $let(
                          newlist: list(vEBT_VEBT),
                          newlist:= list_update(vEBT_VEBT,TreeLista,H,newnode),
                          $ite(
                            vEBT_VEBT_minNull(newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summarya,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 = Maa,
                                      $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),Deg),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)))))) ),
                                      Maa ))),Deg,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 = Maa,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),Deg),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)))),Maa))),Deg,newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
tff(fact_1700_del__in__range,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Mia),Xb)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( 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),Mia),Maa)),Deg,TreeLista,Summarya),Xb) = $let(
                xn: nat,
                xn:= 
                  $ite(Xb = Mia,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(Summarya))),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),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya)))))),Xb),
                $let(
                  minn: nat,
                  minn:= 
                    $ite(Xb = Mia,xn,Mia),
                  $let(
                    h: nat,
                    h:= vEBT_VEBT_high(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),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)),TreeLista)),
                      $let(
                        newnode: vEBT_VEBT,
                        newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                        $let(
                          newlist: list(vEBT_VEBT),
                          newlist:= list_update(vEBT_VEBT,TreeLista,h,newnode),
                          $ite(
                            vEBT_VEBT_minNull(newnode),
                            $let(
                              sn: vEBT_VEBT,
                              sn:= vEBT_vebt_delete(Summarya,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 = Maa,
                                      $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),Deg),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)))))) ),
                                      Maa ))),Deg,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 = Maa,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),Deg),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)))),Maa))),Deg,newlist,Summarya) ) ) ),
                      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),Mia),Maa)),Deg,TreeLista,Summarya) ) ) ) ) ) ) ) ) ).

% del_in_range
tff(fact_1701_del__x__not__mia,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Deg: nat,H: nat,L: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( ( aa(nat,$o,ord_less(nat,Mia),Xb)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
               => ( 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),Mia),Maa)),Deg,TreeLista,Summarya),Xb) = $let(
                      newnode: vEBT_VEBT,
                      newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),H),L),
                      $let(
                        newlist: list(vEBT_VEBT),
                        newlist:= list_update(vEBT_VEBT,TreeLista,H,newnode),
                        $ite(
                          vEBT_VEBT_minNull(newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summarya,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),Mia),
                                  $ite(
                                    Xb = Maa,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),Mia,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),Deg),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)))))) ),
                                    Maa ))),Deg,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),Mia),
                                $ite(Xb = Maa,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),Deg),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)))),Maa))),Deg,newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
tff(fact_1702_del__x__not__mi__new__node__nil,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeLista: list(vEBT_VEBT),Sn: vEBT_VEBT,Summarya: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( aa(nat,$o,ord_less(nat,Mia),Xb)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),H),L) )
               => ( vEBT_VEBT_minNull(Newnode)
                 => ( ( Sn = vEBT_vebt_delete(Summarya,H) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeLista,H,Newnode) )
                     => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
                       => ( 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),Mia),Maa)),Deg,TreeLista,Summarya),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),Mia),
                                  $ite(
                                    Xb = Maa,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(Sn),
                                      $ite(maxs = none(nat),Mia,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),Deg),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)))))) ),
                                    Maa ))),Deg,Newlist,Sn) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
tff(fact_1703_del__x__not__mi,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeLista: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( ( aa(nat,$o,ord_less(nat,Mia),Xb)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),H),L) )
               => ( ( Newlist = list_update(vEBT_VEBT,TreeLista,H,Newnode) )
                 => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
                   => ( 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),Mia),Maa)),Deg,TreeLista,Summarya),Xb) = $ite(
                          vEBT_VEBT_minNull(Newnode),
                          $let(
                            sn: vEBT_VEBT,
                            sn:= vEBT_vebt_delete(Summarya,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),Mia),
                                  $ite(
                                    Xb = Maa,
                                    $let(
                                      maxs: option(nat),
                                      maxs:= vEBT_vebt_maxt(sn),
                                      $ite(maxs = none(nat),Mia,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),Deg),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)))))) ),
                                    Maa ))),Deg,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),Mia),
                                $ite(Xb = Maa,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),Deg),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)))),Maa))),Deg,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
tff(fact_1704_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_1705_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mia: nat,Xb: nat,Maa: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeLista: list(vEBT_VEBT),Newlist: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( ( aa(nat,$o,ord_less(nat,Mia),Xb)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
             => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),H),L) )
               => ( ~ vEBT_VEBT_minNull(Newnode)
                 => ( ( Newlist = list_update(vEBT_VEBT,TreeLista,H,Newnode) )
                   => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
                     => ( 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),Mia),Maa)),Deg,TreeLista,Summarya),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),Mia),
                                $ite(Xb = Maa,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),Deg),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)))),Maa))),Deg,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
tff(fact_1706_del__x__mi__lets__in__not__minNull,axiom,
    ! [Xb: nat,Mia: nat,Maa: nat,Deg: nat,Xn: nat,H: nat,Summarya: vEBT_VEBT,TreeLista: list(vEBT_VEBT),L: nat,Newnode: vEBT_VEBT,Newlist: list(vEBT_VEBT)] :
      ( ( ( Xb = Mia )
        & aa(nat,$o,ord_less(nat,Xb),Maa) )
     => ( ( Mia != Maa )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = H )
           => ( ( Xn = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya)))))) )
             => ( ( vEBT_VEBT_low(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = L )
               => ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
                 => ( ( Newnode = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),H),L) )
                   => ( ( Newlist = list_update(vEBT_VEBT,TreeLista,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),Mia),Maa)),Deg,TreeLista,Summarya),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 = Maa,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),Deg),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)))),Maa))),Deg,Newlist,Summarya) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
tff(fact_1707_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,aa(list(A),set(A),set2(A),list_update(A,L,I,Y)))
      <=> ( ( Xb = Y )
          | ( member(A,Xb,aa(list(A),set(A),set2(A),L))
            & ! [Y3: A] : member(A,Xb,aa(list(A),set(A),set2(A),list_update(A,L,I,Y3))) ) ) ) ) ).

% in_set_upd_eq
tff(fact_1708_in__set__upd__cases,axiom,
    ! [A: $tType,Xb: A,L: list(A),I: nat,Y: A] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),list_update(A,L,I,Y)))
     => ( ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
         => ( Xb != Y ) )
       => member(A,Xb,aa(list(A),set(A),set2(A),L)) ) ) ).

% in_set_upd_cases
tff(fact_1709_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,aa(list(A),set(A),set2(A),list_update(A,L,I,Y)))
      <=> ( ( Xb = Y )
          | ! [Y3: A] : member(A,Xb,aa(list(A),set(A),set2(A),list_update(A,L,I,Y3))) ) ) ) ).

% in_set_upd_eq_aux
tff(fact_1710_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_1711_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,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya),Xb) = $ite(
        ( aa(nat,$o,ord_less(nat,Xb),Mia)
        | aa(nat,$o,ord_less(nat,Maa),Xb) ),
        one_one(nat),
        $ite(
          ( ( Xb = Mia )
          & ( Xb = Maa ) ),
          one_one(nat),
          $let(
            xn: nat,
            xn:= 
              $ite(Xb = Mia,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(Summarya))),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,Vaa))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya)))))),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,Vaa))),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,Vaa))),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)),TreeLista)),
                  aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V1232361888498592333_e_t_e(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),
                    $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l)),vEBT_V1232361888498592333_e_t_e(Summarya,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_1712_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 )
     => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
             => ( ? [N2: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N2))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary)
               => ( Y != one_one(nat) ) )
             => ( ( ? [Mi: nat,Ma: nat,TreeList: 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),Mi),Ma)),zero_zero(nat),TreeList,Summary)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [Mi: nat,Ma: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                       => ( Y != $ite(
                              ( aa(nat,$o,ord_less(nat,Xaa),Mi)
                              | aa(nat,$o,ord_less(nat,Ma),Xaa) ),
                              one_one(nat),
                              $ite(
                                ( ( Xaa = Mi )
                                & ( Xaa = Ma ) ),
                                one_one(nat),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = 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(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,TreeList),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)),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(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_1713_vebt__delete_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xb,Xaa) = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ( ( Xaa = zero_zero(nat) )
             => ( Y != vEBT_Leaf($false,(B3)) ) ) )
       => ( ! [A4: $o] :
              ( ? [B3: $o] : Xb = vEBT_Leaf((A4),(B3))
             => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
               => ( Y != vEBT_Leaf((A4),$false) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( Xb = vEBT_Leaf((A4),(B3)) )
               => ( ? [N2: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N2))
                 => ( Y != vEBT_Leaf((A4),(B3)) ) ) )
           => ( ! [Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary) )
                 => ( Y != vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary) ) )
             => ( ! [Mi: nat,Ma: nat,TrLst: list(vEBT_VEBT),Smry: 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),Mi),Ma)),zero_zero(nat),TrLst,Smry) )
                   => ( 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),Mi),Ma)),zero_zero(nat),TrLst,Smry) ) )
               => ( ! [Mi: nat,Ma: nat,Tr: list(vEBT_VEBT),Sm: 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),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) )
                     => ( 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),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                       => ( Y != $ite(
                              ( aa(nat,$o,ord_less(nat,Xaa),Mi)
                              | aa(nat,$o,ord_less(nat,Ma),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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),
                              $ite(
                                ( ( Xaa = Mi )
                                & ( Xaa = Ma ) ),
                                vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),
                                $let(
                                  xn: nat,
                                  xn:= 
                                    $ite(Xaa = 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(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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),Xaa),
                                  $let(
                                    minn: nat,
                                    minn:= 
                                      $ite(Xaa = 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,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)),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,Va2))),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(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 = 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,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)))))) ),
                                                        Ma ))),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 = 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,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)))),Ma))),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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
tff(fact_1714_vebt__delete_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya),Xb) = $ite(
        ( aa(nat,$o,ord_less(nat,Xb),Mia)
        | aa(nat,$o,ord_less(nat,Maa),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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya),
        $ite(
          ( ( Xb = Mia )
          & ( Xb = Maa ) ),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya),
          $let(
            xn: nat,
            xn:= 
              $ite(Xb = Mia,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(Summarya))),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,Vaa))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya)))))),Xb),
            $let(
              minn: nat,
              minn:= 
                $ite(Xb = Mia,xn,Mia),
              $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,Vaa))),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)),TreeLista)),
                  $let(
                    newnode: vEBT_VEBT,
                    newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),
                    $let(
                      newlist: list(vEBT_VEBT),
                      newlist:= list_update(vEBT_VEBT,TreeLista,h,newnode),
                      $ite(
                        vEBT_VEBT_minNull(newnode),
                        $let(
                          sn: vEBT_VEBT,
                          sn:= vEBT_vebt_delete(Summarya,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 = Maa,
                                  $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,Vaa))),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)))))) ),
                                  Maa ))),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),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 = Maa,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,Vaa))),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)))),Maa))),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),newlist,Summarya) ) ) ),
                  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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
tff(fact_1715_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))
       => ( aa(list(A),set(A),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))) = aa(list(A),set(A),set2(A),Xs) ) ) ) ).

% set_swap
tff(fact_1716_insert__simp__excp,axiom,
    ! [Mia: nat,Deg: nat,TreeLista: list(vEBT_VEBT),Xb: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => ( aa(nat,$o,ord_less(nat,Xb),Mia)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( Xb != Maa )
           => ( 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),Mia),Maa)),Deg,TreeLista,Summarya),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,Mia),Maa))),Deg,list_update(vEBT_VEBT,TreeLista,vEBT_VEBT_high(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summarya,vEBT_VEBT_high(Mia,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summarya)) ) ) ) ) ) ).

% insert_simp_excp
tff(fact_1717_insert__simp__norm,axiom,
    ! [Xb: nat,Deg: nat,TreeLista: list(vEBT_VEBT),Mia: nat,Maa: nat,Summarya: vEBT_VEBT] :
      ( aa(nat,$o,ord_less(nat,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => ( aa(nat,$o,ord_less(nat,Mia),Xb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Deg)
         => ( ( Xb != Maa )
           => ( 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),Mia),Maa)),Deg,TreeLista,Summarya),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),Mia),aa(nat,nat,ord_max(nat,Xb),Maa))),Deg,list_update(vEBT_VEBT,TreeLista,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))),vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),vEBT_VEBT_low(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_vebt_insert(Summarya,vEBT_VEBT_high(Xb,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))))),Summarya)) ) ) ) ) ) ).

% insert_simp_norm
tff(fact_1718_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_1719_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Tree,Nb)
     => ( aa(nat,$o,vEBT_vebt_member(Tree),Xb)
       => ( vEBT_V5719532721284313246member(Tree,Xb)
          | vEBT_VEBT_membermima(Tree,Xb) ) ) ) ).

% member_valid_both_member_options
tff(fact_1720_buildup__nothing__in__leaf,axiom,
    ! [Nb: nat,Xb: nat] : ~ vEBT_V5719532721284313246member(vEBT_vebt_buildup(Nb),Xb) ).

% buildup_nothing_in_leaf
tff(fact_1721_both__member__options__def,axiom,
    ! [Ta: vEBT_VEBT,Xb: nat] :
      ( aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),Xb)
    <=> ( vEBT_V5719532721284313246member(Ta,Xb)
        | vEBT_VEBT_membermima(Ta,Xb) ) ) ).

% both_member_options_def
tff(fact_1722_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_1723_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_1724_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_1725_max__Suc__Suc,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,ord_max(nat,aa(nat,nat,suc,M)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,M),Nb)) ).

% max_Suc_Suc
tff(fact_1726_max__0R,axiom,
    ! [Nb: nat] : aa(nat,nat,ord_max(nat,Nb),zero_zero(nat)) = Nb ).

% max_0R
tff(fact_1727_max__0L,axiom,
    ! [Nb: nat] : aa(nat,nat,ord_max(nat,zero_zero(nat)),Nb) = Nb ).

% max_0L
tff(fact_1728_max__nat_Oright__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,ord_max(nat,A2),zero_zero(nat)) = A2 ).

% max_nat.right_neutral
tff(fact_1729_max__nat_Oneutr__eq__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( zero_zero(nat) = aa(nat,nat,ord_max(nat,A2),B2) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.neutr_eq_iff
tff(fact_1730_max__nat_Oleft__neutral,axiom,
    ! [A2: nat] : aa(nat,nat,ord_max(nat,zero_zero(nat)),A2) = A2 ).

% max_nat.left_neutral
tff(fact_1731_max__nat_Oeq__neutr__iff,axiom,
    ! [A2: nat,B2: nat] :
      ( ( aa(nat,nat,ord_max(nat,A2),B2) = zero_zero(nat) )
    <=> ( ( A2 = zero_zero(nat) )
        & ( B2 = zero_zero(nat) ) ) ) ).

% max_nat.eq_neutr_iff
tff(fact_1732_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_1733_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_1734_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,aa(A,fun(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_1735_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_1736_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_1737_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_1738_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_1739_list__update__beyond,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xb: A] :
      ( aa(nat,$o,aa(nat,fun(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_1740_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_1741_max__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(A,A,ord_max(A,Y),Z)) = aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z)) ) ).

% max_add_distrib_right
tff(fact_1742_max__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,ord_max(A,Xb),Y)),Z) = aa(A,A,ord_max(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z)),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% max_add_distrib_left
tff(fact_1743_max__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,minus_minus(A,aa(A,A,ord_max(A,Xb),Y)),Z) = aa(A,A,ord_max(A,aa(A,A,minus_minus(A,Xb),Z)),aa(A,A,minus_minus(A,Y),Z)) ) ).

% max_diff_distrib_left
tff(fact_1744_nat__add__max__left,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,ord_max(nat,M),Nb)),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),Nb),Q2)) ).

% nat_add_max_left
tff(fact_1745_nat__add__max__right,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,ord_max(nat,Nb),Q2)) = aa(nat,nat,ord_max(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Q2)) ).

% nat_add_max_right
tff(fact_1746_nat__mult__max__right,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),aa(nat,nat,ord_max(nat,Nb),Q2)) = aa(nat,nat,ord_max(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)) ).

% nat_mult_max_right
tff(fact_1747_nat__mult__max__left,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,ord_max(nat,M),Nb)),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),Nb),Q2)) ).

% nat_mult_max_left
tff(fact_1748_nat__minus__add__max,axiom,
    ! [Nb: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,Nb),M)),M) = aa(nat,nat,ord_max(nat,Nb),M) ).

% nat_minus_add_max
tff(fact_1749_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option(product_prod(nat,nat)),Uv: list(vEBT_VEBT),Uw: vEBT_VEBT,Ux: nat] : ~ vEBT_V5719532721284313246member(vEBT_Node(Uu,zero_zero(nat),Uv,Uw),Ux) ).

% VEBT_internal.naive_member.simps(2)
tff(fact_1750_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_1751_Ex__list__of__length,axiom,
    ! [A: $tType,Nb: nat] :
    ? [Xs2: list(A)] : aa(list(A),nat,size_size(list(A)),Xs2) = Nb ).

% Ex_list_of_length
tff(fact_1752_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o,Xb: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Leaf((A2),(B2)),Xb)
    <=> $ite(
          Xb = zero_zero(nat),
          (A2),
          $ite(Xb = one_one(nat),(B2),$false) ) ) ).

% VEBT_internal.naive_member.simps(1)
tff(fact_1753_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_1754_subset__code_I1_J,axiom,
    ! [A: $tType,Xs: list(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),B5)
    <=> ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => member(A,X,B5) ) ) ).

% subset_code(1)
tff(fact_1755_list__eq__iff__nth__eq,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs = Ys )
    <=> ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
        & ! [I2: nat] :
            ( aa(nat,$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) ) ) ) ) ).

% list_eq_iff_nth_eq
tff(fact_1756_Skolem__list__nth,axiom,
    ! [A: $tType,K: nat,P: fun(nat,fun(A,$o))] :
      ( ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),K)
         => ? [X_1: A] : aa(A,$o,aa(nat,fun(A,$o),P,I2),X_1) )
    <=> ? [Xs3: list(A)] :
          ( ( aa(list(A),nat,size_size(list(A)),Xs3) = K )
          & ! [I2: nat] :
              ( aa(nat,$o,ord_less(nat,I2),K)
             => aa(A,$o,aa(nat,fun(A,$o),P,I2),aa(nat,A,nth(A,Xs3),I2)) ) ) ) ).

% Skolem_list_nth
tff(fact_1757_nth__equalityI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Ys) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,ord_less(nat,I3),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,nth(A,Ys),I3) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
tff(fact_1758_all__nat__less,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [M7: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),Nb)
         => aa(nat,$o,P,M7) )
    <=> ! [X: nat] :
          ( member(nat,X,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P,X) ) ) ).

% all_nat_less
tff(fact_1759_ex__nat__less,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [M7: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),Nb)
          & aa(nat,$o,P,M7) )
    <=> ? [X: nat] :
          ( member(nat,X,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P,X) ) ) ).

% ex_nat_less
tff(fact_1760_set__update__subsetI,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A),Xb: A,I: nat] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A3)
     => ( member(A,Xb,A3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I,Xb))),A3) ) ) ).

% set_update_subsetI
tff(fact_1761_vebt__delete_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Nb: nat] : vEBT_vebt_delete(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Nb))) = vEBT_Leaf((A2),(B2)) ).

% vebt_delete.simps(3)
tff(fact_1762_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Uu: nat] : vEBT_vebt_delete(vEBT_Node(none(product_prod(nat,nat)),Deg,TreeLista,Summarya),Uu) = vEBT_Node(none(product_prod(nat,nat)),Deg,TreeLista,Summarya) ).

% vebt_delete.simps(4)
tff(fact_1763_length__pos__if__in__set,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(list(A),nat,size_size(list(A)),Xs)) ) ).

% length_pos_if_in_set
tff(fact_1764_all__set__conv__all__nth,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X) )
    <=> ! [I2: nat] :
          ( aa(nat,$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)) ) ) ).

% all_set_conv_all_nth
tff(fact_1765_all__nth__imp__all__set,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o),Xb: A] :
      ( ! [I3: nat] :
          ( aa(nat,$o,ord_less(nat,I3),aa(list(A),nat,size_size(list(A)),Xs))
         => aa(A,$o,P,aa(nat,A,nth(A,Xs),I3)) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => aa(A,$o,P,Xb) ) ) ).

% all_nth_imp_all_set
tff(fact_1766_in__set__conv__nth,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
    <=> ? [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(list(A),nat,size_size(list(A)),Xs))
          & ( aa(nat,A,nth(A,Xs),I2) = Xb ) ) ) ).

% in_set_conv_nth
tff(fact_1767_list__ball__nth,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),P: fun(A,$o)] :
      ( aa(nat,$o,ord_less(nat,Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( ! [X2: A] :
            ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
           => aa(A,$o,P,X2) )
       => aa(A,$o,P,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% list_ball_nth
tff(fact_1768_nth__mem,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,ord_less(nat,Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => member(A,aa(nat,A,nth(A,Xs),Nb),aa(list(A),set(A),set2(A),Xs)) ) ).

% nth_mem
tff(fact_1769_set__update__memI,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Xb: A] :
      ( aa(nat,$o,ord_less(nat,Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => member(A,Xb,aa(list(A),set(A),set2(A),list_update(A,Xs,Nb,Xb))) ) ).

% set_update_memI
tff(fact_1770_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_1771_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_1772_vebt__delete_Osimps_I2_J,axiom,
    ! [A2: $o,B2: $o] : vEBT_vebt_delete(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,zero_zero(nat))) = vEBT_Leaf((A2),$false) ).

% vebt_delete.simps(2)
tff(fact_1773_vebt__insert_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya),Xb) = $let(
        xn: nat,
        xn:= 
          $ite(aa(nat,$o,ord_less(nat,Xb),Mia),Mia,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,Vaa))),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)),TreeLista))
            & ~ ( ( Xb = Mia )
                | ( Xb = Maa ) ) ),
            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),Mia),Xb,Mia)),
                  aa(nat,nat,ord_max(nat,xn),Maa))),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),list_update(vEBT_VEBT,TreeLista,h,vEBT_vebt_insert(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
              $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),vEBT_vebt_insert(Summarya,h),Summarya)),
            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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya) ) ) ) ).

% vebt_insert.simps(5)
tff(fact_1774_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option(product_prod(nat,nat)),V: nat,TreeLista: list(vEBT_VEBT),S3: vEBT_VEBT,Xb: nat] :
      ( vEBT_V5719532721284313246member(vEBT_Node(Uy,aa(nat,nat,suc,V),TreeLista,S3),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)),TreeLista)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),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_1775_vebt__delete_Osimps_I5_J,axiom,
    ! [Mia: nat,Maa: nat,TrLst2: list(vEBT_VEBT),Smry2: 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),Mia),Maa)),zero_zero(nat),TrLst2,Smry2),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),Mia),Maa)),zero_zero(nat),TrLst2,Smry2) ).

% vebt_delete.simps(5)
tff(fact_1776_vebt__insert_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ( Y != $ite(
                  Xaa = zero_zero(nat),
                  vEBT_Leaf($true,(B3)),
                  $ite(Xaa = one_one(nat),vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B3))) ) ) )
       => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,zero_zero(nat),Ts2,S) )
             => ( Y != vEBT_Node(Info2,zero_zero(nat),Ts2,S) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S) )
               => ( Y != vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S) ) )
           => ( ! [V3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,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)),TreeList,Summary) ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                   => ( Y != $let(
                          xn: nat,
                          xn:= 
                            $ite(aa(nat,$o,ord_less(nat,Xaa),Mi),Mi,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)),TreeList))
                              & ~ ( ( Xaa = Mi )
                                  | ( Xaa = 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,Xaa),Mi),Xaa,Mi)),
                                    aa(nat,nat,ord_max(nat,xn),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),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,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
tff(fact_1777_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xb,Xaa)
      <=> (Y) )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ( (Y)
            <=> ~ $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
       => ( ( ? [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
           => (Y) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S)
               => ( (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)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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_1778_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xb,Xaa)
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ~ $ite(
                  Xaa = zero_zero(nat),
                  (A4),
                  $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT)] :
              ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S)
             => ~ $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)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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_1779_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xb,Xaa)
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => $ite(
                Xaa = zero_zero(nat),
                (A4),
                $ite(Xaa = one_one(nat),(B3),$false) ) )
       => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb != vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT)] :
                ( ? [S: vEBT_VEBT] : Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S)
               => $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)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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_1780_vebt__delete_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Tr2: list(vEBT_VEBT),Sm2: 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),Mia),Maa)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2),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),Mia),Maa)),aa(nat,nat,suc,zero_zero(nat)),Tr2,Sm2) ).

% vebt_delete.simps(6)
tff(fact_1781_httI,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Ta: nat] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( ! [H2: heap_ext(product_unit),As: set(nat)] :
            ( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H2),As))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,C2,H2)),Ta) )
       => time_htt(A,P,C2,Q,Ta) ) ) ).

% httI
tff(fact_1782_htt__def,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Ta: nat] :
      ( time_htt(A,P,C2,Q,Ta)
    <=> ( hoare_hoare_triple(A,P,C2,Q)
        & ! [H4: heap_ext(product_unit),As2: set(nat)] :
            ( rep_assn(P,aa(set(nat),product_prod(heap_ext(product_unit),set(nat)),aa(heap_ext(product_unit),fun(set(nat),product_prod(heap_ext(product_unit),set(nat))),product_Pair(heap_ext(product_unit),set(nat)),H4),As2))
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,C2,H4)),Ta) ) ) ) ).

% htt_def
tff(fact_1783_succ__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_succ(Ta,Xb) = none(nat) )
      <=> ( collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ct(vEBT_VEBT,fun(nat,fun(nat,$o)),Ta),Xb)) = bot_bot(set(nat)) ) ) ) ).

% succ_empty
tff(fact_1784_pred__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_pred(Ta,Xb) = none(nat) )
      <=> ( collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_cu(vEBT_VEBT,fun(nat,fun(nat,$o)),Ta),Xb)) = bot_bot(set(nat)) ) ) ) ).

% pred_empty
tff(fact_1785_vebt__succ_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_succ(Xb,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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,B3: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),(B3))),zero_zero(nat))) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N2: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N2) )
                   => ( ( Y = none(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((Uv2),(Uw2))),aa(nat,nat,suc,N2))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = none(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_succ_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xaa)) ) )
             => ( ! [V3: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc,Vd2) )
                   => ( ( Y = none(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Vc,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) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                       => ( ( Y = $ite(
                                aa(nat,$o,ord_less(nat,Xaa),Mi),
                                aa(nat,option(nat),some(nat),Mi),
                                $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)),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,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,TreeList),h),l)),
                                          $let(
                                            sc: option(nat),
                                            sc:= vEBT_vebt_succ(Summary,h),
                                            $ite(sc = none(nat),none(nat),aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_add,aa(option(nat),option(nat),aa(option(nat),fun(option(nat),option(nat)),vEBT_VEBT_mul,aa(nat,option(nat),some(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),sc)),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc))))) ) ) ),
                                      none(nat) ) ) ) ) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
tff(fact_1786_psubsetI,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
     => ( ( A3 != B5 )
       => aa(set(A),$o,ord_less(set(A),A3),B5) ) ) ).

% psubsetI
tff(fact_1787_int__ops_I6_J,axiom,
    ! [A2: nat,B2: nat] :
      aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,A2),B2)) = $ite(aa(int,$o,ord_less(int,aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)),zero_zero(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) ).

% int_ops(6)
tff(fact_1788_buildup__gives__empty,axiom,
    ! [Nb: nat] : vEBT_VEBT_set_vebt(vEBT_vebt_buildup(Nb)) = bot_bot(set(nat)) ).

% buildup_gives_empty
tff(fact_1789_verit__eq__simplify_I8_J,axiom,
    ! [X23: num,Y2: num] :
      ( ( bit0(X23) = bit0(Y2) )
    <=> ( X23 = Y2 ) ) ).

% verit_eq_simplify(8)
tff(fact_1790_Diff__cancel,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),A3) = bot_bot(set(A)) ).

% Diff_cancel
tff(fact_1791_empty__Diff,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),minus_minus(set(A),bot_bot(set(A))),A3) = bot_bot(set(A)) ).

% empty_Diff
tff(fact_1792_Diff__empty,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),bot_bot(set(A))) = A3 ).

% Diff_empty
tff(fact_1793_subset__antisym,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
       => ( A3 = B5 ) ) ) ).

% subset_antisym
tff(fact_1794_subsetI,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( ! [X2: A] :
          ( member(A,X2,A3)
         => member(A,X2,B5) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5) ) ).

% subsetI
tff(fact_1795_mint__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_mint(Ta) = none(nat) )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot(set(nat)) ) ) ) ).

% mint_corr_help_empty
tff(fact_1796_maxt__corr__help__empty,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( vEBT_vebt_maxt(Ta) = none(nat) )
       => ( vEBT_VEBT_set_vebt(Ta) = bot_bot(set(nat)) ) ) ) ).

% maxt_corr_help_empty
tff(fact_1797_Diff__eq__empty__iff,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( ( aa(set(A),set(A),minus_minus(set(A),A3),B5) = bot_bot(set(A)) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5) ) ).

% Diff_eq_empty_iff
tff(fact_1798_subset__empty,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),bot_bot(set(A)))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% subset_empty
tff(fact_1799_empty__subsetI,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),bot_bot(set(A))),A3) ).

% empty_subsetI
tff(fact_1800_atLeastatMost__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or1337092689740270186AtMost(A,A2,B2) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% atLeastatMost_empty_iff2
tff(fact_1801_atLeastatMost__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% atLeastatMost_empty_iff
tff(fact_1802_atLeastatMost__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
         => ( set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastatMost_empty
tff(fact_1803_mod__pure,axiom,
    ! [B2: $o,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( rep_assn(pure_assn((B2)),H)
    <=> ( ( aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H) = bot_bot(set(nat)) )
        & (B2) ) ) ).

% mod_pure
tff(fact_1804_mod__h__bot__iff_I5_J,axiom,
    ! [P: assn,Q: assn,H: heap_ext(product_unit)] :
      ( rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q),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)),H),bot_bot(set(nat))))
        & rep_assn(Q,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(5)
tff(fact_1805_mod__h__bot__iff_I1_J,axiom,
    ! [B2: $o,H: heap_ext(product_unit)] :
      ( rep_assn(pure_assn((B2)),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))))
    <=> (B2) ) ).

% mod_h_bot_iff(1)
tff(fact_1806_mod__h__bot__iff_I4_J,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Q2: array(A),Y: list(A),H: heap_ext(product_unit)] : ~ rep_assn(aa(list(A),assn,snga_assn(A,Q2),Y),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(4)
tff(fact_1807_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_1808_Set_Oempty__def,axiom,
    ! [A: $tType] : bot_bot(set(A)) = collect(A,aTP_Lamp_bm(A,$o)) ).

% Set.empty_def
tff(fact_1809_set__notEmptyE,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( S2 != bot_bot(set(A)) )
     => ~ ! [X2: A] : ~ member(A,X2,S2) ) ).

% set_notEmptyE
tff(fact_1810_memb__imp__not__empty,axiom,
    ! [A: $tType,Xb: A,S2: set(A)] :
      ( member(A,Xb,S2)
     => ( S2 != bot_bot(set(A)) ) ) ).

% memb_imp_not_empty
tff(fact_1811_times__assn__raw_Ocases,axiom,
    ! [Xb: product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))] :
      ~ ! [P4: fun(product_prod(heap_ext(product_unit),set(nat)),$o),Q7: fun(product_prod(heap_ext(product_unit),set(nat)),$o),H2: heap_ext(product_unit),As: set(nat)] : Xb != aa(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),P4),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),aa(fun(product_prod(heap_ext(product_unit),set(nat)),$o),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(fun(product_prod(heap_ext(product_unit),set(nat)),$o),product_prod(heap_ext(product_unit),set(nat))),Q7),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)),H2),As))) ).

% times_assn_raw.cases
tff(fact_1812_sngr__assn__raw_Ocases,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xb: product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))] :
          ~ ! [R2: ref(A),X2: A,H2: heap_ext(product_unit),As: set(nat)] : Xb != aa(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),aa(ref(A),fun(product_prod(A,product_prod(heap_ext(product_unit),set(nat))),product_prod(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat))))),product_Pair(ref(A),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),R2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat))),aa(A,fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(A,product_prod(heap_ext(product_unit),set(nat)))),product_Pair(A,product_prod(heap_ext(product_unit),set(nat))),X2),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)),H2),As))) ) ).

% sngr_assn_raw.cases
tff(fact_1813_snga__assn__raw_Ocases,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xb: product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))] :
          ~ ! [R2: array(A),X2: list(A),H2: heap_ext(product_unit),As: set(nat)] : Xb != aa(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),aa(array(A),fun(product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),product_prod(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))))),product_Pair(array(A),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),R2),aa(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat))),aa(list(A),fun(product_prod(heap_ext(product_unit),set(nat)),product_prod(list(A),product_prod(heap_ext(product_unit),set(nat)))),product_Pair(list(A),product_prod(heap_ext(product_unit),set(nat))),X2),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)),H2),As))) ) ).

% snga_assn_raw.cases
tff(fact_1814_pure__assn__raw_Ocases,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod($o,product_prod(A,set(B)))] :
      ~ ! [B3: $o,H2: A,As: set(B)] : Xb != aa(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B))),aa($o,fun(product_prod(A,set(B)),product_prod($o,product_prod(A,set(B)))),product_Pair($o,product_prod(A,set(B))),(B3)),aa(set(B),product_prod(A,set(B)),aa(A,fun(set(B),product_prod(A,set(B))),product_Pair(A,set(B)),H2),As)) ).

% pure_assn_raw.cases
tff(fact_1815_one__assn__raw_Ocases,axiom,
    ! [Xb: product_prod(heap_ext(product_unit),set(nat))] :
      ~ ! [H2: heap_ext(product_unit),As: set(nat)] : Xb != 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)),H2),As) ).

% one_assn_raw.cases
tff(fact_1816_subset__minus__empty,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
     => ( aa(set(A),set(A),minus_minus(set(A),A3),B5) = bot_bot(set(A)) ) ) ).

% subset_minus_empty
tff(fact_1817_mod__h__bot__indep,axiom,
    ! [P: assn,H: heap_ext(product_unit),H3: heap_ext(product_unit)] :
      ( 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)),H3),bot_bot(set(nat)))) ) ).

% mod_h_bot_indep
tff(fact_1818_mod__emp,axiom,
    ! [H: product_prod(heap_ext(product_unit),set(nat))] :
      ( rep_assn(one_one(assn),H)
    <=> ( aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H) = bot_bot(set(nat)) ) ) ).

% mod_emp
tff(fact_1819_mod__emp__simp,axiom,
    ! [H: heap_ext(product_unit)] : rep_assn(one_one(assn),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_emp_simp
tff(fact_1820_verit__comp__simplify1_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),A2) ) ).

% verit_comp_simplify1(2)
tff(fact_1821_verit__la__disequality,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
          | ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% verit_la_disequality
tff(fact_1822_verit__comp__simplify1_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A] : ~ aa(A,$o,ord_less(A,A2),A2) ) ).

% verit_comp_simplify1(1)
tff(fact_1823_Collect__mono__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,P)),collect(A,Q))
    <=> ! [X: A] :
          ( aa(A,$o,P,X)
         => aa(A,$o,Q,X) ) ) ).

% Collect_mono_iff
tff(fact_1824_set__eq__subset,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( ( A3 = B5 )
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3) ) ) ).

% set_eq_subset
tff(fact_1825_subset__trans,axiom,
    ! [A: $tType,A3: set(A),B5: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5) ) ) ).

% subset_trans
tff(fact_1826_Collect__mono,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X2: A] :
          ( aa(A,$o,P,X2)
         => aa(A,$o,Q,X2) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,P)),collect(A,Q)) ) ).

% Collect_mono
tff(fact_1827_subset__refl,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),A3) ).

% subset_refl
tff(fact_1828_subset__iff,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
    <=> ! [T2: A] :
          ( member(A,T2,A3)
         => member(A,T2,B5) ) ) ).

% subset_iff
tff(fact_1829_Set_OequalityD2,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( ( A3 = B5 )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3) ) ).

% Set.equalityD2
tff(fact_1830_equalityD1,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( ( A3 = B5 )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5) ) ).

% equalityD1
tff(fact_1831_subset__eq,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
    <=> ! [X: A] :
          ( member(A,X,A3)
         => member(A,X,B5) ) ) ).

% subset_eq
tff(fact_1832_equalityE,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( ( A3 = B5 )
     => ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
         => ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3) ) ) ).

% equalityE
tff(fact_1833_subsetD,axiom,
    ! [A: $tType,A3: set(A),B5: set(A),C2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
     => ( member(A,C2,A3)
       => member(A,C2,B5) ) ) ).

% subsetD
tff(fact_1834_in__mono,axiom,
    ! [A: $tType,A3: set(A),B5: set(A),Xb: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
     => ( member(A,Xb,A3)
       => member(A,Xb,B5) ) ) ).

% in_mono
tff(fact_1835_double__diff,axiom,
    ! [A: $tType,A3: set(A),B5: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
       => ( aa(set(A),set(A),minus_minus(set(A),B5),aa(set(A),set(A),minus_minus(set(A),C5),A3)) = A3 ) ) ) ).

% double_diff
tff(fact_1836_Diff__subset,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B5)),A3) ).

% Diff_subset
tff(fact_1837_Diff__mono,axiom,
    ! [A: $tType,A3: set(A),C5: set(A),D6: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),D6),B5)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B5)),aa(set(A),set(A),minus_minus(set(A),C5),D6)) ) ) ).

% Diff_mono
tff(fact_1838_verit__la__generic,axiom,
    ! [A2: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),Xb)
      | ( A2 = Xb )
      | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),A2) ) ).

% verit_la_generic
tff(fact_1839_psubset__imp__ex__mem,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A3),B5)
     => ? [B3: A] : member(A,B3,aa(set(A),set(A),minus_minus(set(A),B5),A3)) ) ).

% psubset_imp_ex_mem
tff(fact_1840_less__eq__set__def,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
    <=> aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aTP_Lamp_av(set(A),fun(A,$o),A3)),aTP_Lamp_av(set(A),fun(A,$o),B5)) ) ).

% less_eq_set_def
tff(fact_1841_Collect__subset,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cv(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))),A3) ).

% Collect_subset
tff(fact_1842_verit__comp__simplify1_I3_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B4: A,A6: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B4),A6)
        <=> aa(A,$o,ord_less(A,A6),B4) ) ) ).

% verit_comp_simplify1(3)
tff(fact_1843_verit__sum__simplify,axiom,
    ! [A: $tType] :
      ( cancel1802427076303600483id_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),zero_zero(A)) = A2 ) ).

% verit_sum_simplify
tff(fact_1844_verit__eq__simplify_I10_J,axiom,
    ! [X23: num] : one2 != bit0(X23) ).

% verit_eq_simplify(10)
tff(fact_1845_verit__eq__simplify_I14_J,axiom,
    ! [X23: num,X33: num] : bit0(X23) != aa(num,num,bit1,X33) ).

% verit_eq_simplify(14)
tff(fact_1846_verit__eq__simplify_I12_J,axiom,
    ! [X33: num] : one2 != aa(num,num,bit1,X33) ).

% verit_eq_simplify(12)
tff(fact_1847_psubsetE,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A3),B5)
     => ~ ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
         => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3) ) ) ).

% psubsetE
tff(fact_1848_psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A3),B5)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
        & ( A3 != B5 ) ) ) ).

% psubset_eq
tff(fact_1849_psubset__imp__subset,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A3),B5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5) ) ).

% psubset_imp_subset
tff(fact_1850_psubset__subset__trans,axiom,
    ! [A: $tType,A3: set(A),B5: set(A),C5: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A3),B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
       => aa(set(A),$o,ord_less(set(A),A3),C5) ) ) ).

% psubset_subset_trans
tff(fact_1851_subset__not__subset__eq,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A3),B5)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
        & ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3) ) ) ).

% subset_not_subset_eq
tff(fact_1852_subset__psubset__trans,axiom,
    ! [A: $tType,A3: set(A),B5: set(A),C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
     => ( aa(set(A),$o,ord_less(set(A),B5),C5)
       => aa(set(A),$o,ord_less(set(A),A3),C5) ) ) ).

% subset_psubset_trans
tff(fact_1853_subset__iff__psubset__eq,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
    <=> ( aa(set(A),$o,ord_less(set(A),A3),B5)
        | ( A3 = B5 ) ) ) ).

% subset_iff_psubset_eq
tff(fact_1854_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_1855_TBOUND__mono,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),Ta: nat,T3: nat] :
      ( time_TBOUND(A,C2,Ta)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Ta),T3)
       => time_TBOUND(A,C2,T3) ) ) ).

% TBOUND_mono
tff(fact_1856_htt__htD,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Ta: nat] :
      ( time_htt(A,P,C2,Q,Ta)
     => hoare_hoare_triple(A,P,C2,Q) ) ).

% htt_htD
tff(fact_1857_max__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X3: A,Xa: A] :
          aa(A,A,ord_max(A,X3),Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa),Xa,X3) ) ).

% max_def_raw
tff(fact_1858_norm__pre__ex__rule__htt,axiom,
    ! [B: $tType,A: $tType,P: fun(A,assn),F: heap_Time_Heap(B),Q: fun(B,assn),Ta: nat] :
      ( ! [X2: A] : time_htt(B,aa(A,assn,P,X2),F,Q,Ta)
     => time_htt(B,ex_assn(A,P),F,Q,Ta) ) ).

% norm_pre_ex_rule_htt
tff(fact_1859_norm__post__ex__rule__htt,axiom,
    ! [B: $tType,A: $tType,P: assn,F: heap_Time_Heap(A),Q: fun(B,fun(A,assn)),Xb: B,Ta: nat] :
      ( time_htt(A,P,F,aa(B,fun(A,assn),Q,Xb),Ta)
     => time_htt(A,P,F,aTP_Lamp_cx(fun(B,fun(A,assn)),fun(A,assn),Q),Ta) ) ).

% norm_post_ex_rule_htt
tff(fact_1860_int__ops_I3_J,axiom,
    ! [Nb: num] : aa(nat,int,semiring_1_of_nat(int),aa(num,nat,numeral_numeral(nat),Nb)) = aa(num,int,numeral_numeral(int),Nb) ).

% int_ops(3)
tff(fact_1861_nat__int__comparison_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,ord_less(nat,A2),B2)
    <=> aa(int,$o,ord_less(int,aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ).

% nat_int_comparison(2)
tff(fact_1862_nat__int__comparison_I3_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ) ).

% nat_int_comparison(3)
tff(fact_1863_int__ops_I2_J,axiom,
    aa(nat,int,semiring_1_of_nat(int),one_one(nat)) = one_one(int) ).

% int_ops(2)
tff(fact_1864_int__ops_I5_J,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(5)
tff(fact_1865_int__plus,axiom,
    ! [Nb: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),aa(nat,int,semiring_1_of_nat(int),M)) ).

% int_plus
tff(fact_1866_int__ops_I7_J,axiom,
    ! [A2: nat,B2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),B2)) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)) ).

% int_ops(7)
tff(fact_1867_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_1868_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_1869_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_1870_TBOUNDD,axiom,
    ! [A: $tType,M: heap_Time_Heap(A),Ta: nat,H: heap_ext(product_unit)] :
      ( time_TBOUND(A,M,Ta)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,M,H)),Ta) ) ).

% TBOUNDD
tff(fact_1871_TBOUNDI,axiom,
    ! [A: $tType,M: heap_Time_Heap(A),Ta: nat] :
      ( ! [H2: heap_ext(product_unit)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,M,H2)),Ta)
     => time_TBOUND(A,M,Ta) ) ).

% TBOUNDI
tff(fact_1872_TBOUND__def,axiom,
    ! [A: $tType,M: heap_Time_Heap(A),Ta: nat] :
      ( time_TBOUND(A,M,Ta)
    <=> ! [H4: heap_ext(product_unit)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,M,H4)),Ta) ) ).

% TBOUND_def
tff(fact_1873_norm__pre__pure__iff__htt,axiom,
    ! [A: $tType,P: assn,B2: $o,F: heap_Time_Heap(A),Q: fun(A,assn),Ta: nat] :
      ( time_htt(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B2))),F,Q,Ta)
    <=> ( (B2)
       => time_htt(A,P,F,Q,Ta) ) ) ).

% norm_pre_pure_iff_htt
tff(fact_1874_norm__pre__pure__iff__htt_H,axiom,
    ! [A: $tType,B2: $o,P: assn,F: heap_Time_Heap(A),Q: fun(A,assn),Ta: nat] :
      ( time_htt(A,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),pure_assn((B2))),P),F,Q,Ta)
    <=> ( (B2)
       => time_htt(A,P,F,Q,Ta) ) ) ).

% norm_pre_pure_iff_htt'
tff(fact_1875_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_1876_nat__leq__as__int,axiom,
    ! [X3: nat,Xa: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X3),Xa)
    <=> aa(int,$o,aa(int,fun(int,$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_1877_httI__TBOUND,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Ta: nat] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( time_TBOUND(A,C2,Ta)
       => time_htt(A,P,C2,Q,Ta) ) ) ).

% httI_TBOUND
tff(fact_1878_TBOUND__assert_H__weak,axiom,
    ! [A: $tType,P: $o,M: heap_Time_Heap(A),Ta: nat] :
      ( ( (P)
       => time_TBOUND(A,M,Ta) )
     => time_TBOUND(A,heap_Time_bind(product_unit,A,refine_Imp_assert((P)),aTP_Lamp_cy(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),M)),Ta) ) ).

% TBOUND_assert'_weak
tff(fact_1879_int__Suc,axiom,
    ! [Nb: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,Nb)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),one_one(int)) ).

% int_Suc
tff(fact_1880_int__ops_I4_J,axiom,
    ! [A2: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,A2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),one_one(int)) ).

% int_ops(4)
tff(fact_1881_TBOUND__assert_H__bind__strong,axiom,
    ! [A: $tType,P: $o,M: heap_Time_Heap(A),Ta: nat] :
      ( ( (P)
       => time_TBOUND(A,M,Ta) )
     => time_TBOUND(A,heap_Time_bind(product_unit,A,refine_Imp_assert((P)),aTP_Lamp_cy(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),M)),
          $ite((P),Ta,zero_zero(nat))) ) ).

% TBOUND_assert'_bind_strong
tff(fact_1882_TBOUND__option__case,axiom,
    ! [B: $tType,A: $tType,Ta: option(A),F: heap_Time_Heap(B),Bnd: nat,F4: fun(A,heap_Time_Heap(B)),Bnd2: fun(A,nat)] :
      ( ( ( Ta = none(A) )
       => time_TBOUND(B,F,Bnd) )
     => ( ! [X2: A] :
            ( ( Ta = aa(A,option(A),some(A),X2) )
           => time_TBOUND(B,aa(A,heap_Time_Heap(B),F4,X2),aa(A,nat,Bnd2,X2)) )
       => time_TBOUND(B,case_option(heap_Time_Heap(B),A,F,F4,Ta),case_option(nat,A,Bnd,Bnd2,Ta)) ) ) ).

% TBOUND_option_case
tff(fact_1883_vebt__pred_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: option(nat)] :
      ( ( vEBT_vebt_pred(Xb,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = none(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),(Uv2))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((A4),(Uw2)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( Xb = vEBT_Leaf((A4),(B3)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                     => ( ( Y = $ite(
                              (B3),
                              aa(nat,option(nat),some(nat),one_one(nat)),
                              $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va) )
                   => ( ( Y = none(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_pred_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va)),Xaa)) ) )
               => ( ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = none(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,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)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                         => ( ( Y = $ite(
                                  aa(nat,$o,ord_less(nat,Ma),Xaa),
                                  aa(nat,option(nat),some(nat),Ma),
                                  $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)),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,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,TreeList),h),l)),
                                            $let(
                                              pr: option(nat),
                                              pr:= vEBT_vebt_pred(Summary,h),
                                              $ite(
                                                pr = none(nat),
                                                $ite(aa(nat,$o,ord_less(nat,Mi),Xaa),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,Va2))),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) ) ) ) ) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
tff(fact_1884_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 )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( Xb = vEBT_Leaf((A4),(B3)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( Xb = vEBT_Leaf((A4),(B3)) )
                 => ! [N2: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V6368547301243506412_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary)),Xaa)) ) )
               => ( ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TreeList,Summary) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),zero_zero(nat),TreeList,Summary)),Xaa)) ) )
                 => ( ! [Mi: nat,Ma: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary) )
                       => ( ( Y = one_one(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                         => ( ( Y = $ite(
                                  ( aa(nat,$o,ord_less(nat,Xaa),Mi)
                                  | aa(nat,$o,ord_less(nat,Ma),Xaa) ),
                                  one_one(nat),
                                  $ite(
                                    ( ( Xaa = Mi )
                                    & ( Xaa = Ma ) ),
                                    one_one(nat),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = 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(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,TreeList),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)),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)),
                                              $ite(vEBT_VEBT_minNull(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),vEBT_V1232361888498592333_e_t_e(Summary,h),one_one(nat))),
                                            one_one(nat) ) ) ) ) ) ) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e'.pelims
tff(fact_1885_vebt__delete_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_delete(Xb,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = vEBT_Leaf($false,(B3)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( Xb = vEBT_Leaf((A4),(B3)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = vEBT_Leaf((A4),$false) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( Xb = vEBT_Leaf((A4),(B3)) )
                 => ! [N2: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
                     => ( ( Y = vEBT_Leaf((A4),(B3)) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary) )
                   => ( ( Y = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_delete_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary)),Xaa)) ) )
               => ( ! [Mi: nat,Ma: nat,TrLst: list(vEBT_VEBT),Smry: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TrLst,Smry) )
                     => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TrLst,Smry) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),zero_zero(nat),TrLst,Smry)),Xaa)) ) )
                 => ( ! [Mi: nat,Ma: nat,Tr: list(vEBT_VEBT),Sm: 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),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) )
                       => ( ( 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),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),Tr,Sm)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                         => ( ( Y = $ite(
                                  ( aa(nat,$o,ord_less(nat,Xaa),Mi)
                                  | aa(nat,$o,ord_less(nat,Ma),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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),
                                  $ite(
                                    ( ( Xaa = Mi )
                                    & ( Xaa = Ma ) ),
                                    vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary),
                                    $let(
                                      xn: nat,
                                      xn:= 
                                        $ite(Xaa = 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(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,TreeList),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summary)))))),Xaa),
                                      $let(
                                        minn: nat,
                                        minn:= 
                                          $ite(Xaa = 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,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)),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,Va2))),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(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 = 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,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)))))) ),
                                                            Ma ))),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),
                                                        $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),h),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h)))),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),newlist,Summary) ) ) ),
                                            vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) ) ) ) ) ) ) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
tff(fact_1886_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 )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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,B3: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),(B3))),zero_zero(nat))) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N2: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((Uv2),(Uw2))),aa(nat,nat,suc,N2))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xaa)) ) )
             => ( ! [V3: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc,Vd2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Vc,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) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                       => ( ( Y = $ite(
                                aa(nat,$o,ord_less(nat,Xaa),Mi),
                                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)),TreeList)),
                                      $let(
                                        maxlow: option(nat),
                                        maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                        $ite(
                                          ( ( maxlow != none(nat) )
                                          & vEBT_VEBT_less(aa(nat,option(nat),some(nat),l),maxlow) ),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_s_u_c_c2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                          aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_s_u_c_c2(Summary,h)),one_one(nat)) ) ),
                                      one_one(nat) ) ) ) ) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c'.pelims
tff(fact_1887_vebt__insert_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_insert(Xb,Xaa) = Y )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = $ite(
                      Xaa = zero_zero(nat),
                      vEBT_Leaf($true,(B3)),
                      $ite(Xaa = one_one(nat),vEBT_Leaf((A4),$true),vEBT_Leaf((A4),(B3))) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero(nat),Ts2,S) )
               => ( ( Y = vEBT_Node(Info2,zero_zero(nat),Ts2,S) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,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(Info2,zero_zero(nat),Ts2,S)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S) )
                 => ( ( Y = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_insert_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S)),Xaa)) ) )
             => ( ! [V3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary) )
                   => ( ( Y = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Xaa),Xaa)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,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)),TreeList,Summary)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,ord_less(nat,Xaa),Mi),Mi,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)),TreeList))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = 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,Xaa),Mi),Xaa,Mi)),
                                        aa(nat,nat,ord_max(nat,xn),Ma))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),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,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_vebt_insert(Summary,h),Summary)),
                                  vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) ) ) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
tff(fact_1888_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 )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),(Uv2))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((A4),(Uw2)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( Xb = vEBT_Leaf((A4),(B3)) )
                 => ! [Va2: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,Va2)) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va)),Xaa)) ) )
               => ( ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,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)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                         => ( ( Y = $ite(
                                  aa(nat,$o,ord_less(nat,Ma),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)),TreeList)),
                                        $let(
                                          minlow: option(nat),
                                          minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),
                                          $ite(
                                            ( ( minlow != none(nat) )
                                            & vEBT_VEBT_greater(aa(nat,option(nat),some(nat),l),minlow) ),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),vEBT_T_p_r_e_d2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h),l)),
                                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_T_p_r_e_d2(Summary,h)),one_one(nat)) ) ),
                                        one_one(nat) ) ) ) ) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d'.pelims
tff(fact_1889_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 )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = one_one(nat) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero(nat),Ts2,S) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5076183648494686801_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts2,S)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,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(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S)),Xaa)) ) )
             => ( ! [V3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,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)),TreeList,Summary)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                     => ( ( Y = $let(
                              xn: nat,
                              xn:= 
                                $ite(aa(nat,$o,ord_less(nat,Xaa),Mi),Mi,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)),TreeList))
                                  & ~ ( ( Xaa = Mi )
                                      | ( Xaa = 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,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),
                                    $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t2(Summary,h),one_one(nat))),
                                  one_one(nat) ) ) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t'.pelims
tff(fact_1890_Diff__idemp,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A3),B5)),B5) = aa(set(A),set(A),minus_minus(set(A),A3),B5) ).

% Diff_idemp
tff(fact_1891_Diff__iff,axiom,
    ! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
      ( member(A,C2,aa(set(A),set(A),minus_minus(set(A),A3),B5))
    <=> ( member(A,C2,A3)
        & ~ member(A,C2,B5) ) ) ).

% Diff_iff
tff(fact_1892_DiffI,axiom,
    ! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
      ( member(A,C2,A3)
     => ( ~ member(A,C2,B5)
       => member(A,C2,aa(set(A),set(A),minus_minus(set(A),A3),B5)) ) ) ).

% DiffI
tff(fact_1893_minus__set__def,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),B5) = collect(A,aa(fun(A,$o),fun(A,$o),minus_minus(fun(A,$o),aTP_Lamp_av(set(A),fun(A,$o),A3)),aTP_Lamp_av(set(A),fun(A,$o),B5))) ).

% minus_set_def
tff(fact_1894_set__diff__eq,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),B5) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_cz(set(A),fun(set(A),fun(A,$o)),A3),B5)) ).

% set_diff_eq
tff(fact_1895_DiffD2,axiom,
    ! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
      ( member(A,C2,aa(set(A),set(A),minus_minus(set(A),A3),B5))
     => ~ member(A,C2,B5) ) ).

% DiffD2
tff(fact_1896_DiffD1,axiom,
    ! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
      ( member(A,C2,aa(set(A),set(A),minus_minus(set(A),A3),B5))
     => member(A,C2,A3) ) ).

% DiffD1
tff(fact_1897_DiffE,axiom,
    ! [A: $tType,C2: A,A3: set(A),B5: set(A)] :
      ( member(A,C2,aa(set(A),set(A),minus_minus(set(A),A3),B5))
     => ~ ( member(A,C2,A3)
         => member(A,C2,B5) ) ) ).

% DiffE
tff(fact_1898_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 )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                      $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T5837161174952499735_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2) )
                 => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Vc: 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,Vc) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,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,Vc)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                              $ite(
                                Xaa = Mi,
                                one_one(nat),
                                aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    Xaa = 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,Xaa),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),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)),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(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)) )) )) )) )) )) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r.pelims
tff(fact_1899_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 )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = one_one(nat) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8099345112685741742_r_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) ) )
           => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Vc: 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,Vc) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,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,Vc)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                     => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                Xaa = Mi,
                                zero_zero(nat),
                                $ite(
                                  Xaa = Ma,
                                  zero_zero(nat),
                                  $ite(
                                    aa(nat,$o,ord_less(nat,Xaa),Mi),
                                    zero_zero(nat),
                                    $ite(
                                      aa(nat,$o,ord_less(nat,Ma),Xaa),
                                      zero_zero(nat),
                                      $ite(
                                        ( aa(nat,$o,ord_less(nat,Mi),Xaa)
                                        & aa(nat,$o,ord_less(nat,Xaa),Ma) ),
                                        $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)),TreeList)),vEBT_T_m_e_m_b_e_r2(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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) ) ) ) ) )) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>e\<^sub>m\<^sub>b\<^sub>e\<^sub>r'.pelims
tff(fact_1900_vebt__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
      <=> (Y) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B3),$false) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),Xaa)) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,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,Uv2,Uw2)),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(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2)),Xaa)) ) )
             => ( ! [V3: product_prod(nat,nat),Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc) )
                   => ( ~ (Y)
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,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,Vc)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                     => ( ( (Y)
                        <=> $ite(
                              Xaa = Mi,
                              $true,
                              $ite(
                                Xaa = Ma,
                                $true,
                                $ite(
                                  aa(nat,$o,ord_less(nat,Xaa),Mi),
                                  $false,
                                  $ite(
                                    aa(nat,$o,ord_less(nat,Ma),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)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
tff(fact_1901_vebt__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)),Xaa)) )
           => ( ! [V3: product_prod(nat,nat),Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Uy2,Uz2) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Vc: 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,Vc) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_vebt_member_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),aa(nat,nat,suc,zero_zero(nat)),Vb2,Vc)),Xaa)) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                     => ( aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa))
                       => $ite(
                            Xaa = Mi,
                            $true,
                            $ite(
                              Xaa = Ma,
                              $true,
                              $ite(
                                aa(nat,$o,ord_less(nat,Xaa),Mi),
                                $false,
                                $ite(
                                  aa(nat,$o,ord_less(nat,Ma),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)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
tff(fact_1902_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_V5719532721284313246member(Xb,Xaa)
      <=> (Y) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( (Y)
                <=> $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B3),$false) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Leaf((A4),(B3))),Xaa)) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ( ~ (Y)
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Uv2,Uw2)),Xaa)) ) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S) )
                 => ( ( (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)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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))))),$false) ) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),TreeList,S)),Xaa)) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
tff(fact_1903_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_V5719532721284313246member(Xb,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,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),TreeList,S)),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)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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_1904_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_V5719532721284313246member(Xb,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),Xaa))
               => $ite(
                    Xaa = zero_zero(nat),
                    (A4),
                    $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ( ! [Uu2: option(product_prod(nat,nat)),Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V5765760719290551771er_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Uu2,zero_zero(nat),Uv2,Uw2)),Xaa)) )
           => ~ ! [Uy2: option(product_prod(nat,nat)),V3: nat,TreeList: list(vEBT_VEBT),S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Uy2,aa(nat,nat,suc,V3),TreeList,S) )
                 => ( aa(product_prod(vEBT_VEBT,nat),$o,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),TreeList,S)),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)),TreeList)),vEBT_V5719532721284313246member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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_1905_vebt__member_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( aa(nat,$o,vEBT_vebt_member(Xb),Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),Xaa))
               => ~ $ite(
                      Xaa = zero_zero(nat),
                      (A4),
                      $ite(Xaa = one_one(nat),(B3),$false) ) ) )
         => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa))
                 => ~ $ite(
                        Xaa = Mi,
                        $true,
                        $ite(
                          Xaa = Ma,
                          $true,
                          $ite(
                            aa(nat,$o,ord_less(nat,Xaa),Mi),
                            $false,
                            $ite(
                              aa(nat,$o,ord_less(nat,Ma),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)),TreeList)),aa(nat,$o,vEBT_vebt_member(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_VEBT_low(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))))),$false) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
tff(fact_1906_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_membermima(Xb,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),(Uv2))),Xaa)) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xaa)) )
           => ( ! [Mi: nat,Ma: nat,Va: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va,Vb2) )
                 => ( aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),zero_zero(nat),Va,Vb2)),Xaa))
                   => ( ( Xaa = Mi )
                      | ( Xaa = Ma ) ) ) )
             => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list(vEBT_VEBT),Vc: 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),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc) )
                   => ( aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc)),Xaa))
                     => ( ( Xaa = Mi )
                        | ( Xaa = Ma )
                        | $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)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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))))),$false) ) ) ) )
               => ~ ! [V3: nat,TreeList: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd2) )
                     => ( aa(product_prod(vEBT_VEBT,nat),$o,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),TreeList,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)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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_1907_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_membermima(Xb,Xaa)
      <=> (Y) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ~ (Y)
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),(Uv2))),Xaa)) ) )
         => ( ! [Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2) )
               => ( ~ (Y)
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),zero_zero(nat),Ux2,Uy2)),Xaa)) ) )
           => ( ! [Mi: nat,Ma: nat,Va: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va,Vb2) )
                 => ( ( (Y)
                    <=> ( ( Xaa = Mi )
                        | ( Xaa = Ma ) ) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),zero_zero(nat),Va,Vb2)),Xaa)) ) )
             => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list(vEBT_VEBT),Vc: 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),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc) )
                   => ( ( (Y)
                      <=> ( ( Xaa = Mi )
                          | ( Xaa = Ma )
                          | $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)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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))))),$false) ) ) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc)),Xaa)) ) )
               => ~ ! [V3: nat,TreeList: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,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)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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))))),$false) ) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),TreeList,Vd2)),Xaa)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
tff(fact_1908_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_membermima(Xb,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_V4351362008482014158ma_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),Xb),Xaa))
       => ( ! [Mi: nat,Ma: nat,Va: list(vEBT_VEBT),Vb2: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),Va,Vb2) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),zero_zero(nat),Va,Vb2)),Xaa))
               => ~ ( ( Xaa = Mi )
                    | ( Xaa = Ma ) ) ) )
         => ( ! [Mi: nat,Ma: nat,V3: nat,TreeList: list(vEBT_VEBT),Vc: 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),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,V3),TreeList,Vc)),Xaa))
                 => ~ ( ( Xaa = Mi )
                      | ( Xaa = Ma )
                      | $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)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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))))),$false) ) ) ) )
           => ~ ! [V3: nat,TreeList: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,V3),TreeList,Vd2) )
                 => ( aa(product_prod(vEBT_VEBT,nat),$o,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),TreeList,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)),TreeList)),vEBT_VEBT_membermima(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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_1909_minus__apply,axiom,
    ! [A: $tType,B: $tType] :
      ( minus(A)
     => ! [A3: fun(B,A),B5: fun(B,A),Xb: B] : aa(B,A,aa(fun(B,A),fun(B,A),minus_minus(fun(B,A),A3),B5),Xb) = aa(A,A,minus_minus(A,aa(B,A,A3,Xb)),aa(B,A,B5,Xb)) ) ).

% minus_apply
tff(fact_1910_pos__mult__pos__ge,axiom,
    ! [Xb: int,Nb: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),Nb),one_one(int))),aa(int,int,aa(int,fun(int,int),times_times(int),Nb),Xb)) ) ) ).

% pos_mult_pos_ge
tff(fact_1911_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_1912_max_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,A,ord_max(A,A2),B2) = B2 ) ) ) ).

% max.absorb4
tff(fact_1913_max_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,A,ord_max(A,A2),B2) = A2 ) ) ) ).

% max.absorb1
tff(fact_1914_max_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,A,ord_max(A,A2),B2) = B2 ) ) ) ).

% max.absorb2
tff(fact_1915_max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,ord_max(A,B2),C2)),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% max.bounded_iff
tff(fact_1916_max_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
         => ( aa(A,A,ord_max(A,A2),B2) = A2 ) ) ) ).

% max.absorb3
tff(fact_1917_max_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,ord_max(A,A2),B2)) ) ) ).

% max.coboundedI2
tff(fact_1918_max_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),aa(A,A,ord_max(A,A2),B2)) ) ) ).

% max.coboundedI1
tff(fact_1919_max_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,A,ord_max(A,A2),B2) = B2 ) ) ) ).

% max.absorb_iff2
tff(fact_1920_max_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,A,ord_max(A,A2),B2) = A2 ) ) ) ).

% max.absorb_iff1
tff(fact_1921_le__max__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Z: A,Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),aa(A,A,ord_max(A,Xb),Y))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xb)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y) ) ) ) ).

% le_max_iff_disj
tff(fact_1922_max_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,ord_max(A,A2),B2)) ) ).

% max.cobounded2
tff(fact_1923_max_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,ord_max(A,A2),B2)) ) ).

% max.cobounded1
tff(fact_1924_max_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( A2 = aa(A,A,ord_max(A,A2),B2) ) ) ) ).

% max.order_iff
tff(fact_1925_max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,ord_max(A,B2),C2)),A2) ) ) ) ).

% max.boundedI
tff(fact_1926_max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,ord_max(A,B2),C2)),A2)
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% max.boundedE
tff(fact_1927_max_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,ord_max(A,A2),B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% max.orderI
tff(fact_1928_max_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( A2 = aa(A,A,ord_max(A,A2),B2) ) ) ) ).

% max.orderE
tff(fact_1929_max_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,D2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),D2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,ord_max(A,C2),D2)),aa(A,A,ord_max(A,A2),B2)) ) ) ) ).

% max.mono
tff(fact_1930_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_1931_max_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,ord_max(A,B2),C2)),A2)
         => ~ ( aa(A,$o,ord_less(A,B2),A2)
             => ~ aa(A,$o,ord_less(A,C2),A2) ) ) ) ).

% max.strict_boundedE
tff(fact_1932_max_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
        <=> ( ( A2 = aa(A,A,ord_max(A,A2),B2) )
            & ( A2 != B2 ) ) ) ) ).

% max.strict_order_iff
tff(fact_1933_max_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,C2),A2)
         => aa(A,$o,ord_less(A,C2),aa(A,A,ord_max(A,A2),B2)) ) ) ).

% max.strict_coboundedI1
tff(fact_1934_max_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,C2),B2)
         => aa(A,$o,ord_less(A,C2),aa(A,A,ord_max(A,A2),B2)) ) ) ).

% max.strict_coboundedI2
tff(fact_1935_fun__diff__def,axiom,
    ! [B: $tType,A: $tType] :
      ( minus(B)
     => ! [A3: fun(A,B),B5: fun(A,B),X3: A] : aa(A,B,aa(fun(A,B),fun(A,B),minus_minus(fun(A,B),A3),B5),X3) = aa(B,B,minus_minus(B,aa(A,B,A3,X3)),aa(A,B,B5,X3)) ) ).

% fun_diff_def
tff(fact_1936_p2__eq__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: 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))),Nb) = one_one(word(A)) )
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% p2_eq_1
tff(fact_1937_word__unat__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: 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))),Nb) = 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))),Nb)) ) ).

% word_unat_power
tff(fact_1938_word__less__two__pow__divD,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: nat,M: nat] :
          ( aa(word(A),$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))),Nb)),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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
            & 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,Nb),M))) ) ) ) ).

% word_less_two_pow_divD
tff(fact_1939_less__1__helper,axiom,
    ! [Nb: int,M: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),M)
     => aa(int,$o,ord_less(int,aa(int,int,minus_minus(int,Nb),one_one(int))),M) ) ).

% less_1_helper
tff(fact_1940_enat__ord__number_I1_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Nb)) ) ).

% enat_ord_number(1)
tff(fact_1941_heigt__uplog__rel,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta)) = archimedean_ceiling(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),Nb))) ) ) ).

% heigt_uplog_rel
tff(fact_1942_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_1943_enat__ord__number_I2_J,axiom,
    ! [M: num,Nb: num] :
      ( aa(extended_enat,$o,ord_less(extended_enat,aa(num,extended_enat,numeral_numeral(extended_enat),M)),aa(num,extended_enat,numeral_numeral(extended_enat),Nb))
    <=> aa(nat,$o,ord_less(nat,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Nb)) ) ).

% enat_ord_number(2)
tff(fact_1944_set__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_se5668285175392031749et_bit(A,zero_zero(nat),A2) = 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),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% set_bit_0
tff(fact_1945_assert_H__bind__rule,axiom,
    ! [A: $tType,P: assn,Phi: $o,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( ! [H2: product_prod(heap_ext(product_unit),set(nat))] :
          ( rep_assn(P,H2)
         => (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_cy(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),C2)),Q) ) ) ).

% assert'_bind_rule
tff(fact_1946_succ__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_s_u_c_c(Ta,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)))))),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U))))) ) ) ).

% succ_bound_size_univ
tff(fact_1947_div__of__0__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),zero_zero(word(A))),Nb) = zero_zero(word(A)) ) ).

% div_of_0_id
tff(fact_1948_set__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se5668285175392031749et_bit(int,Nb,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% set_bit_nonnegative_int_iff
tff(fact_1949_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,aa(word(A),fun(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,aa(word(A),fun(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_1950_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_1951_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_1952_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_1953_ceiling__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_ceiling(A,one_one(A)) = one_one(int) ) ) ).

% ceiling_one
tff(fact_1954_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_1955_ceiling__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),zero_zero(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A)) ) ) ).

% ceiling_le_zero
tff(fact_1956_ceiling__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),aa(num,int,numeral_numeral(int),V))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(num,A,numeral_numeral(A),V)) ) ) ).

% ceiling_le_numeral
tff(fact_1957_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_1958_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_1959_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,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A)) ) ) ).

% ceiling_less_one
tff(fact_1960_one__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(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_1961_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_1962_ceiling__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),one_one(int))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),one_one(A)) ) ) ).

% ceiling_le_one
tff(fact_1963_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_1964_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_1965_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_1966_ceiling__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: num,Nb: nat] : archimedean_ceiling(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) ) ).

% ceiling_numeral_power
tff(fact_1967_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_1968_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,$o,aa(A,fun(A,$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_1969_numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(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_1970_word__subset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),R3: word(A),Y: word(A),S3: word(A)] :
          ( aa(set(word(A)),$o,aa(set(word(A)),fun(set(word(A)),$o),ord_less_eq(set(word(A))),set_or1337092689740270186AtMost(word(A),Xb,aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),R3)),one_one(word(A))))),set_or1337092689740270186AtMost(word(A),Y,aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),S3)),one_one(word(A)))))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),R3)),one_one(word(A))))
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),S3)),one_one(word(A))))
             => ( ( S3 != zero_zero(word(A)) )
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),R3),S3) ) ) ) ) ) ).

% word_subset_less
tff(fact_1971_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_1972_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_1973_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_1974_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_1975_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_1976_word__less__minus__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),C2: word(A),D2: word(A),B2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A2),C2)
         => ( aa(word(A),$o,ord_less(word(A),D2),B2)
           => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),A2),B2)),A2)
             => ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),C2),D2)),C2)
               => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),A2),B2)),aa(word(A),word(A),minus_minus(word(A),C2),D2)) ) ) ) ) ) ).

% word_less_minus_mono
tff(fact_1977_word__div__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_1978_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,aa(word(A),fun(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_1979_word__leq__le__minus__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_1980_word__less__nowrapI_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Z: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(A),word(A),minus_minus(word(A),Z),K))
         => ( aa(word(A),$o,aa(word(A),fun(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_1981_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),$o,aa(word(A),fun(word(A),$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_1982_word__diff__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),Nb)
         => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),M)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Nb),M)
             => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),M),Nb)),M) ) ) ) ) ).

% word_diff_less
tff(fact_1983_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),$o,aa(word(A),fun(word(A),$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_1984_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),$o,aa(word(A),fun(word(A),$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_1985_less__1__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),M: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Nb),one_one(word(A)))),M)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Nb),M)
            & ( Nb != zero_zero(word(A)) ) ) ) ) ).

% less_1_simp
tff(fact_1986_sub__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_1987_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),$o,aa(word(A),fun(word(A),$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_1988_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),$o,aa(word(A),fun(word(A),$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_1989_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),$o,aa(word(A),fun(word(A),$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_1990_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),$o,aa(word(A),fun(word(A),$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_1991_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),$o,aa(word(A),fun(word(A),$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_1992_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),$o,aa(word(A),fun(word(A),$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_1993_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),$o,aa(word(A),fun(word(A),$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_1994_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),$o,aa(word(A),fun(word(A),$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_1995_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),$o,aa(word(A),fun(word(A),$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_1996_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),$o,aa(word(A),fun(word(A),$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_1997_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),$o,aa(word(A),fun(word(A),$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_1998_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),$o,aa(word(A),fun(word(A),$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_1999_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),$o,aa(word(A),fun(word(A),$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_2000_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),$o,aa(word(A),fun(word(A),$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_2001_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),$o,aa(word(A),fun(word(A),$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_2002_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),$o,aa(word(A),fun(word(A),$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_2003_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),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),Ab)
           => aa(word(A),$o,aa(word(A),fun(word(A),$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_2004_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),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Z)
           => aa(word(A),$o,ord_less(word(A),Y),Z) ) ) ) ).

% word_less_minus_cancel
tff(fact_2005_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),$o,aa(word(A),fun(word(A),$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_2006_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),$o,aa(word(A),fun(word(A),$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_2007_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),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y) ) ) ).

% word_minus_one_le_leq
tff(fact_2008_word__less__imp__diff__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),Nb: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),K),Nb)
         => ( aa(word(A),$o,ord_less(word(A),Nb),M)
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),minus_minus(word(A),Nb),K)),M) ) ) ) ).

% word_less_imp_diff_less
tff(fact_2009_le__step__down__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),Nb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),Nb)
         => ( ( I != Nb )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),minus_minus(word(A),Nb),one_one(word(A)))) ) ) ) ).

% le_step_down_word
tff(fact_2010_le__step__down__word__2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y)
         => ( ( Xb != Y )
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(A),word(A),minus_minus(word(A),Y),one_one(word(A)))) ) ) ) ).

% le_step_down_word_2
tff(fact_2011_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,Xb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$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_2012_word__le__sub1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( Xb != zero_zero(word(A)) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),Xb)
          <=> aa(word(A),$o,aa(word(A),fun(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_2013_word__sub__1__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( ( Xb != zero_zero(word(A)) )
         => aa(word(A),$o,aa(word(A),fun(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_2014_word__must__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(A),word(A),minus_minus(word(A),Nb),one_one(word(A))))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Nb),Xb)
           => ( Nb = zero_zero(word(A)) ) ) ) ) ).

% word_must_wrap
tff(fact_2015_word__sub__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N3: word(A),Nb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),N3),Nb)
         => ( ( N3 != 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),Nb),N3)),one_one(word(A))) != zero_zero(word(A)) ) ) ) ) ).

% word_sub_plus_one_nonzero
tff(fact_2016_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_2017_More__Word_Oword__div__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C2: word(A),A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),C2)
         => ( aa(word(A),$o,ord_less(word(A),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),B2),C2))
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),C2)),B2) ) ) ) ).

% More_Word.word_div_mult
tff(fact_2018_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_2019_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_2020_div__less__dividend__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: word(A)] :
          ( ( Xb != zero_zero(word(A)) )
         => ( ( Nb != one_one(word(A)) )
           => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Xb),Nb)),Xb) ) ) ) ).

% div_less_dividend_word
tff(fact_2021_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_2022_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)
         => ? [N2: word(A)] : M = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),N2),one_one(word(A))) ) ) ).

% word_gr0_conv_Suc
tff(fact_2023_less__is__non__zero__p1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),K: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A2),K)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),one_one(word(A))) != zero_zero(word(A)) ) ) ) ).

% less_is_non_zero_p1
tff(fact_2024_word__div__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Nb),one_one(word(A))) = Nb ) ).

% word_div_1
tff(fact_2025_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_2026_word__plus__one__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_2027_lt1__neq0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),Xb)
        <=> ( Xb != zero_zero(word(A)) ) ) ) ).

% lt1_neq0
tff(fact_2028_div__to__mult__word__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Y),Z))
         => aa(word(A),$o,aa(word(A),fun(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)),Xb),Z)),Y) ) ) ).

% div_to_mult_word_lt
tff(fact_2029_word__le__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y)
         => ? [N2: nat] : Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),aa(nat,word(A),semiring_1_of_nat(word(A)),N2)) ) ) ).

% word_le_add
tff(fact_2030_word__le__plus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2))
         => ( aa(word(A),$o,ord_less(word(A),C2),B2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2)) ) ) ) ).

% word_le_plus
tff(fact_2031_word__le__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Nb: word(A),A2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Nb))
         => ( aa(word(A),$o,ord_less(word(A),A2),Nb)
           => aa(word(A),$o,aa(word(A),fun(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),A2)),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),A2)),one_one(word(A)))) ) ) ) ).

% word_le_plus_1
tff(fact_2032_plus__one__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Nb),one_one(word(A))))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Nb) ) ) ).

% plus_one_helper
tff(fact_2033_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),$o,aa(word(A),fun(word(A),$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_2034_word__le__not__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),B2),A2)
        <=> ~ aa(word(A),$o,ord_less(word(A),A2),B2) ) ) ).

% word_le_not_less
tff(fact_2035_word__not__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( ~ aa(word(A),$o,aa(word(A),fun(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_2036_plus__one__helper2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Nb)
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Nb),one_one(word(A))) != zero_zero(word(A)) )
           => aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Nb),one_one(word(A)))) ) ) ) ).

% plus_one_helper2
tff(fact_2037_word__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),Xb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),one_one(word(A)))),B2)
         => ( aa(word(A),$o,ord_less(word(A),A2),aa(nat,word(A),semiring_1_of_nat(word(A)),Xb))
           => aa(word(A),$o,ord_less(word(A),A2),B2) ) ) ) ).

% word_1_0
tff(fact_2038_div__le__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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),$o,aa(word(A),fun(word(A),$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_2039_word__le__less__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_2040_plus__le__left__cancel__nowrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y6: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y6))
         => ( aa(word(A),$o,aa(word(A),fun(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),Y6)),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),Y6),Y) ) ) ) ) ).

% plus_le_left_cancel_nowrap
tff(fact_2041_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),$o,aa(word(A),fun(word(A),$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_2042_inc__i,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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),$o,aa(word(A),fun(word(A),$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),$o,aa(word(A),fun(word(A),$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_2043_word__div__mult__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),$o,aa(word(A),fun(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)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),B2)),B2)),A2) ) ).

% word_div_mult_le
tff(fact_2044_i0__lb,axiom,
    ! [Nb: extended_enat] : aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),zero_zero(extended_enat)),Nb) ).

% i0_lb
tff(fact_2045_ile0__eq,axiom,
    ! [Nb: extended_enat] :
      ( aa(extended_enat,$o,aa(extended_enat,fun(extended_enat,$o),ord_less_eq(extended_enat),Nb),zero_zero(extended_enat))
    <=> ( Nb = zero_zero(extended_enat) ) ) ).

% ile0_eq
tff(fact_2046_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_2047_word__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),M: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [N2: word(A)] :
                ( aa(word(A),$o,P,N2)
               => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N2)) )
           => aa(word(A),$o,P,M) ) ) ) ).

% word_induct
tff(fact_2048_word__induct2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),Nb: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [N2: word(A)] :
                ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N2) != zero_zero(word(A)) )
               => ( aa(word(A),$o,P,N2)
                 => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N2)) ) )
           => aa(word(A),$o,P,Nb) ) ) ) ).

% word_induct2
tff(fact_2049_plus__le__left__cancel__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y6: word(A),Y: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y6)),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),Y6)),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),Y6),Y) ) ) ) ) ).

% plus_le_left_cancel_wrap
tff(fact_2050_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,Nb: extended_enat] :
      ( aa(extended_enat,$o,ord_less(extended_enat,zero_zero(extended_enat)),aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),Nb))
    <=> ( aa(extended_enat,$o,ord_less(extended_enat,zero_zero(extended_enat)),M)
        & aa(extended_enat,$o,ord_less(extended_enat,zero_zero(extended_enat)),Nb) ) ) ).

% enat_0_less_mult_iff
tff(fact_2051_word__induct__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),M: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [N2: word(A)] :
                ( aa(word(A),$o,ord_less(word(A),N2),M)
               => ( aa(word(A),$o,P,N2)
                 => aa(word(A),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),N2)) ) )
           => aa(word(A),$o,P,M) ) ) ) ).

% word_induct_less
tff(fact_2052_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_2053_set__bit__greater__eq,axiom,
    ! [K: int,Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),bit_se5668285175392031749et_bit(int,Nb,K)) ).

% set_bit_greater_eq
tff(fact_2054_ceiling__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Y)),archimedean_ceiling(A,Xb)) ) ) ).

% ceiling_mono
tff(fact_2055_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_2056_ceiling__add__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] : aa(int,$o,aa(int,fun(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_2057_le__minus,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & order(A) )
     => ! [Y: A,Xb: A,A2: word(B),C2: word(B),B2: word(B)] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),A2),C2)),B2)
           => ( aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),A2),aa(word(B),word(B),aa(word(B),fun(word(B),word(B)),plus_plus(word(B)),A2),C2))
             => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),C2),aa(word(B),word(B),minus_minus(word(B),B2),A2)) ) ) ) ) ).

% le_minus
tff(fact_2058_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_2059_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_2060_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_2061_Abs__fnat__hom__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: nat,B2: nat] : 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)),A2)),aa(nat,word(A),semiring_1_of_nat(word(A)),B2)) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),B2)) ) ).

% Abs_fnat_hom_mult
tff(fact_2062_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw: $o,Nb: nat] : vEBT_T_s_u_c_c(vEBT_Leaf((Uv),(Uw)),aa(nat,nat,suc,Nb)) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(2)
tff(fact_2063_mult__ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A2)),archimedean_ceiling(A,B2))) ) ) ) ).

% mult_ceiling_le
tff(fact_2064_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,Vaa: nat] : vEBT_T_s_u_c_c(vEBT_Node(none(product_prod(nat,nat)),Ux,Uy,Uz),Vaa) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(3)
tff(fact_2065_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I1_J,axiom,
    ! [Uu: $o,B2: $o] : vEBT_T_s_u_c_c(vEBT_Leaf((Uu),(B2)),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_2066_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),Vc2: 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),Vc2,Vd),Ve) = one_one(nat) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(4)
tff(fact_2067_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_2068_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_2069_assert_H__rule,axiom,
    ! [P: assn,Phi: $o] :
      ( ! [H2: product_prod(heap_ext(product_unit),set(nat))] :
          ( rep_assn(P,H2)
         => (Phi) )
     => hoare_hoare_triple(product_unit,P,refine_Imp_assert((Phi)),aTP_Lamp_da(assn,fun(product_unit,assn),P)) ) ).

% assert'_rule
tff(fact_2070_Abs__fnat__hom__add,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [A2: nat,B2: nat] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),A2)),aa(nat,A,semiring_1_of_nat(A),B2)) = aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)) ) ).

% Abs_fnat_hom_add
tff(fact_2071_ceiling__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
         => ( archimedean_ceiling(real,log(aa(nat,real,semiring_1_of_nat(real),B2),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),Nb)),one_one(int)) ) ) ) ) ).

% ceiling_log_nat_eq_if
tff(fact_2072_ceiling__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( archimedean_ceiling(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),Nb))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(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,Nb),one_one(nat))),aa(num,nat,numeral_numeral(nat),bit0(one2)))),one_one(nat)))))),one_one(int)) ) ) ).

% ceiling_log2_div2
tff(fact_2073_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => ( ( archimedean_ceiling(real,log(aa(nat,real,semiring_1_of_nat(real),B2),aa(nat,real,semiring_1_of_nat(real),K))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),Nb)),one_one(int)) )
        <=> ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
tff(fact_2074_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_2075_succ__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_s_u_c_c(Ta,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,Ta))),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_2076_htt__vebt__succi,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => time_htt(option(nat),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),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_az(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun(option(nat),assn))),Ta),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,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),Nb))))))) ) ).

% htt_vebt_succi
tff(fact_2077_htt__vebt__memberi__invar__vebt,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => time_htt($o,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),Tib),aa(nat,heap_Time_Heap($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($o,assn)),aTP_Lamp_be(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,fun($o,assn))),Ta),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,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),Nb))))))) ) ).

% htt_vebt_memberi_invar_vebt
tff(fact_2078_htt__vebt__inserti__invar__vebt,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Tib: vEBT_VEBTi,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => time_htt(vEBT_VEBTi,aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Ta),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(Ta,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,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),Nb))))))) ) ).

% htt_vebt_inserti_invar_vebt
tff(fact_2079_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,B3: $o] : Xb = vEBT_Leaf((Uu2),(B3))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) ) ) )
       => ( ( ? [Uv2: $o,Uw2: $o] : Xb = vEBT_Leaf((Uv2),(Uw2))
           => ( ? [N2: nat] : Xaa = aa(nat,nat,suc,N2)
             => ( 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),Vc: 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),Vc,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) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                     => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                            $ite(
                              aa(nat,$o,ord_less(nat,Xaa),Mi),
                              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)),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(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,TreeList),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                                      one_one(nat) )) ) ) )) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.elims
tff(fact_2080_log__ceil__idem,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => ( archimedean_ceiling(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),Xb)) = archimedean_ceiling(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_2081_T_092_060_094sub_062s_092_060_094sub_062u_092_060_094sub_062c_092_060_094sub_062c_Osimps_I6_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,ord_less(nat,Xb),Mia),
          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,Vaa))),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,Vaa))),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)),TreeLista)),
                  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,TreeLista),h)))),
                    $let(
                      maxlow: option(nat),
                      maxlow:= vEBT_vebt_maxt(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),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,TreeLista),h),l)),
                          $let(
                            sc: option(nat),
                            sc:= vEBT_vebt_succ(Summarya,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(Summarya,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,TreeLista),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                  one_one(nat) )) ) ) )) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.simps(6)
tff(fact_2082_insert__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_i_n_s_e_r_t(Ta,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)))))),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U))))) ) ) ).

% insert_bound_size_univ
tff(fact_2083_word__coorder_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),zero_zero(word(A)))
        <=> ( A2 = zero_zero(word(A)) ) ) ) ).

% word_coorder.extremum_unique
tff(fact_2084_word__le__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),zero_zero(word(A)))
        <=> ( Xb = zero_zero(word(A)) ) ) ) ).

% word_le_0_iff
tff(fact_2085_idiff__0,axiom,
    ! [Nb: extended_enat] : aa(extended_enat,extended_enat,minus_minus(extended_enat,zero_zero(extended_enat)),Nb) = zero_zero(extended_enat) ).

% idiff_0
tff(fact_2086_idiff__0__right,axiom,
    ! [Nb: extended_enat] : aa(extended_enat,extended_enat,minus_minus(extended_enat,Nb),zero_zero(extended_enat)) = Nb ).

% idiff_0_right
tff(fact_2087_of__int__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Z: int,Nb: num] :
          ( ( aa(int,A,ring_1_of_int(A),Z) = aa(num,A,numeral_numeral(A),Nb) )
        <=> ( Z = aa(num,int,numeral_numeral(int),Nb) ) ) ) ).

% of_int_eq_numeral_iff
tff(fact_2088_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_2089_of__int__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: int,Z: int] :
          ( aa(A,$o,aa(A,fun(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,aa(int,fun(int,$o),ord_less_eq(int),W),Z) ) ) ).

% of_int_le_iff
tff(fact_2090_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_2091_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_2092_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_2093_of__int__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),W),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_add
tff(fact_2094_of__int__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_mult
tff(fact_2095_nat__numeral,axiom,
    ! [K: num] : nat2(aa(num,int,numeral_numeral(int),K)) = aa(num,nat,numeral_numeral(nat),K) ).

% nat_numeral
tff(fact_2096_of__int__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [W: int,Z: int] : aa(int,A,ring_1_of_int(A),aa(int,int,minus_minus(int,W),Z)) = aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),W)),aa(int,A,ring_1_of_int(A),Z)) ) ).

% of_int_diff
tff(fact_2097_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: int,B2: 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),B2)),W) )
        <=> ( Xb = aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
tff(fact_2098_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [B2: int,W: nat,Xb: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W) = aa(int,A,ring_1_of_int(A),Xb) )
        <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W) = Xb ) ) ) ).

% of_int_eq_of_int_power_cancel_iff
tff(fact_2099_of__int__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int,Nb: nat] : aa(int,A,ring_1_of_int(A),aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),Z)),Nb) ) ).

% of_int_power
tff(fact_2100_nat__1,axiom,
    nat2(one_one(int)) = aa(nat,nat,suc,zero_zero(nat)) ).

% nat_1
tff(fact_2101_nat__le__0,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int))
     => ( nat2(Z) = zero_zero(nat) ) ) ).

% nat_le_0
tff(fact_2102_nat__0__iff,axiom,
    ! [I: int] :
      ( ( nat2(I) = zero_zero(nat) )
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int)) ) ).

% nat_0_iff
tff(fact_2103_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_2104_int__nat__eq,axiom,
    ! [Z: int] :
      aa(nat,int,semiring_1_of_nat(int),nat2(Z)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z),Z,zero_zero(int)) ).

% int_nat_eq
tff(fact_2105_ceiling__add__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archimedean_ceiling(A,Xb)),Z) ) ).

% ceiling_add_of_int
tff(fact_2106_ceiling__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] : archimedean_ceiling(A,aa(A,A,minus_minus(A,Xb),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,minus_minus(int,archimedean_ceiling(A,Xb)),Z) ) ).

% ceiling_diff_of_int
tff(fact_2107_of__int__le__0__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),zero_zero(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),zero_zero(int)) ) ) ).

% of_int_le_0_iff
tff(fact_2108_of__int__0__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) ) ) ).

% of_int_0_le_iff
tff(fact_2109_of__int__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),Nb)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),Nb)),Z) ) ) ).

% of_int_numeral_le_iff
tff(fact_2110_of__int__le__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),aa(num,int,numeral_numeral(int),Nb)) ) ) ).

% of_int_le_numeral_iff
tff(fact_2111_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_2112_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_2113_of__int__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num,Z: int] :
          ( aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),Nb)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,ord_less(int,aa(num,int,numeral_numeral(int),Nb)),Z) ) ) ).

% of_int_numeral_less_iff
tff(fact_2114_of__int__less__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int,Nb: num] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Z)),aa(num,A,numeral_numeral(A),Nb))
        <=> aa(int,$o,ord_less(int,Z),aa(num,int,numeral_numeral(int),Nb)) ) ) ).

% of_int_less_numeral_iff
tff(fact_2115_of__int__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),one_one(A))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),one_one(int)) ) ) ).

% of_int_le_1_iff
tff(fact_2116_of__int__1__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(int,A,ring_1_of_int(A),Z))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),Z) ) ) ).

% of_int_1_le_iff
tff(fact_2117_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_2118_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_2119_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,Xb: num,Nb: 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)),Nb) )
        <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
tff(fact_2120_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: num,Nb: nat,Y: int] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb) = 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)),Nb) = Y ) ) ) ).

% numeral_power_eq_of_int_cancel_iff
tff(fact_2121_of__int__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int,B2: int,W: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Xb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)) ) ) ).

% of_int_power_le_of_int_cancel_iff
tff(fact_2122_of__int__le__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,Xb: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),W)),aa(int,A,ring_1_of_int(A),Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)),Xb) ) ) ).

% of_int_le_of_int_power_cancel_iff
tff(fact_2123_of__int__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: int,B2: int,W: nat] :
          ( aa(A,$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),B2)),W))
        <=> aa(int,$o,ord_less(int,Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),B2),W)) ) ) ).

% of_int_power_less_of_int_cancel_iff
tff(fact_2124_of__int__less__of__int__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: int,W: nat,Xb: int] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(int,A,ring_1_of_int(A),B2)),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),B2),W)),Xb) ) ) ).

% of_int_less_of_int_power_cancel_iff
tff(fact_2125_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_2126_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_2127_of__nat__nat,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(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_2128_numeral__power__eq__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,Y: int] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb) = nat2(Y) )
    <=> ( aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
tff(fact_2129_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,Xb: num,Nb: nat] :
      ( ( nat2(Y) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb) )
    <=> ( Y = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) ) ) ).

% nat_eq_numeral_power_cancel_iff
tff(fact_2130_nat__ceiling__le__eq,axiom,
    ! [Xb: real,A2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),nat2(archimedean_ceiling(real,Xb))),A2)
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(nat,real,semiring_1_of_nat(real),A2)) ) ).

% nat_ceiling_le_eq
tff(fact_2131_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_2132_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb)) ) ) ).

% of_int_le_numeral_power_cancel_iff
tff(fact_2133_numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb)),A2) ) ) ).

% numeral_power_le_of_int_cancel_iff
tff(fact_2134_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xb: num,Nb: nat] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb))
        <=> aa(int,$o,ord_less(int,A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb)) ) ) ).

% of_int_less_numeral_power_cancel_iff
tff(fact_2135_numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A2: int] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,ord_less(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb)),A2) ) ) ).

% numeral_power_less_of_int_cancel_iff
tff(fact_2136_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_2137_numeral__power__less__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A2: int] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb)),nat2(A2))
    <=> aa(int,$o,ord_less(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb)),A2) ) ).

% numeral_power_less_nat_cancel_iff
tff(fact_2138_nat__less__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,nat2(A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb))
    <=> aa(int,$o,ord_less(int,A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb)) ) ).

% nat_less_numeral_power_cancel_iff
tff(fact_2139_numeral__power__le__nat__cancel__iff,axiom,
    ! [Xb: num,Nb: nat,A2: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb)),nat2(A2))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb)),A2) ) ).

% numeral_power_le_nat_cancel_iff
tff(fact_2140_nat__le__numeral__power__cancel__iff,axiom,
    ! [A2: int,Xb: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),nat2(A2)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),Xb)),Nb))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb)) ) ).

% nat_le_numeral_power_cancel_iff
tff(fact_2141_range__subset__card,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A),D2: word(A)] :
          ( aa(set(word(A)),$o,aa(set(word(A)),fun(set(word(A)),$o),ord_less_eq(set(word(A))),set_or1337092689740270186AtMost(word(A),A2,B2)),set_or1337092689740270186AtMost(word(A),C2,D2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),D2)
              & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),B2),A2)),aa(word(A),word(A),minus_minus(word(A),D2),C2)) ) ) ) ) ).

% range_subset_card
tff(fact_2142_word__diff__ls_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xaa: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(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(4)
tff(fact_2143_word__diff__ls_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A),W: word(A),Xaa: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xb)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Xb))
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(word(A),$o),ord_less_eq(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(3)
tff(fact_2144_Word_Oword__l__diffs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xaa: word(A),Z: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(word(A),$o),ord_less_eq(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(4)
tff(fact_2145_Word_Oword__l__diffs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xaa: word(A),Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Xb)),aa(word(A),word(A),minus_minus(word(A),Z),Xb))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Z)
           => aa(word(A),$o,aa(word(A),fun(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)),W),Xaa)),Z) ) ) ) ).

% Word.word_l_diffs(3)
tff(fact_2146_word__plus__mcs_I4_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa))
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(word(A),$o),ord_less_eq(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(4)
tff(fact_2147_word__plus__mcs_I3_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),Xc)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)),Xc))
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(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)),V),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xaa)) ) ) ) ).

% word_plus_mcs(3)
tff(fact_2148_More__Word_Oword__l__diffs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),aa(word(A),word(A),minus_minus(word(A),Z),Xb)) ) ) ) ).

% More_Word.word_l_diffs(4)
tff(fact_2149_More__Word_Oword__l__diffs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Z: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),aa(word(A),word(A),minus_minus(word(A),Z),Xb))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Z)
           => aa(word(A),$o,aa(word(A),fun(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)),W),Xb)),Z) ) ) ) ).

% More_Word.word_l_diffs(3)
tff(fact_2150_word__diff__ls_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xb)),W) ) ) ) ).

% word_diff_ls'(4)
tff(fact_2151_word__diff__ls_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xb)),W)
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(word(A),$o),ord_less_eq(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'(3)
tff(fact_2152_word__l__diffs_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb)),Z)
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(word(A),$o),ord_less_eq(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'(4)
tff(fact_2153_word__l__diffs_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb)),Xb)),aa(word(A),word(A),minus_minus(word(A),Z),Xb))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Z)
           => aa(word(A),$o,aa(word(A),fun(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)),W),Xb)),Z) ) ) ) ).

% word_l_diffs'(3)
tff(fact_2154_word__diff__ls_H_H_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),W: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(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''(4)
tff(fact_2155_word__diff__ls_H_H_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xb)),aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb)),Xb))
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(word(A),$o),ord_less_eq(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''(3)
tff(fact_2156_le__plus,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C2: word(A),B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),aa(word(A),word(A),minus_minus(word(A),B2),A2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
           => aa(word(A),$o,aa(word(A),fun(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)),A2),C2)),B2) ) ) ) ).

% le_plus
tff(fact_2157_le__plus_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),aa(word(A),word(A),minus_minus(word(A),B2),A2))
           => aa(word(A),$o,aa(word(A),fun(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)),A2),C2)),B2) ) ) ) ).

% le_plus'
tff(fact_2158_le__minus_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),C2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2)),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),aa(word(A),word(A),minus_minus(word(A),B2),A2)) ) ) ) ).

% le_minus'
tff(fact_2159_word__sub__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xb)
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Xb),Y)),Xb) ) ) ).

% word_sub_le
tff(fact_2160_word__sub__le__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Xb),Y)),Xb)
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xb) ) ) ).

% word_sub_le_iff
tff(fact_2161_word__le__minus__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),C2: word(A),D2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),C2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),D2),B2)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),A2),B2)),A2)
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),C2),D2)),C2)
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),A2),B2)),aa(word(A),word(A),minus_minus(word(A),C2),D2)) ) ) ) ) ) ).

% word_le_minus_mono
tff(fact_2162_word__le__minus__cancel,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A),Z: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Y),Xb)),aa(word(A),word(A),minus_minus(word(A),Z),Xb))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Z)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z) ) ) ) ).

% word_le_minus_cancel
tff(fact_2163_word__le__minus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(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_le_minus_mono_left
tff(fact_2164_word__le__imp__diff__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A),Nb: word(A),M: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),K),Nb)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Nb),M)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Nb),K)),M) ) ) ) ).

% word_le_imp_diff_le
tff(fact_2165_plus__minus__no__overflow__ab,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),Ab)
           => aa(word(A),$o,aa(word(A),fun(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_ab
tff(fact_2166_word__le__minus__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Z: word(A),Y: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Z),Y)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xb)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Z),Xb)
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Xb),Y)),aa(word(A),word(A),minus_minus(word(A),Xb),Z)) ) ) ) ) ).

% word_le_minus_mono_right
tff(fact_2167_plus__minus__not__NULL__ab,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Ab: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(A),word(A),minus_minus(word(A),Ab),C2))
         => ( aa(word(A),$o,aa(word(A),fun(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_2168_neq__0__no__wrap,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_2169_word__random,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),X4: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),X4))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),X4)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),Xb)) ) ) ) ).

% word_random
tff(fact_2170_word__sub__mono2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A),D2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),C2),D2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),A2)
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),B2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2))
             => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),D2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),C2),D2))
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),B2),D2) ) ) ) ) ) ).

% word_sub_mono2
tff(fact_2171_word__plus__mcs__3,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),W: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),V),W)
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(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)),V),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb)) ) ) ) ).

% word_plus_mcs_3
tff(fact_2172_word__plus__mcs__4,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [V: word(A),Xb: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),W),Xb))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),V),Xb))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),V),W) ) ) ) ).

% word_plus_mcs_4
tff(fact_2173_word__add__le__iff2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),I),K))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),J),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),J),K))
           => ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ).

% word_add_le_iff2
tff(fact_2174_word__plus__mcs__4_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),V: word(A),W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),V)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),W))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),V))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),V),W) ) ) ) ).

% word_plus_mcs_4'
tff(fact_2175_word__add__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),W: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),W)),Xb)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),W))
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),Xb) ) ) ) ).

% word_add_increasing
tff(fact_2176_word__plus__mono__left,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(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),Xb)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Z),Xb)) ) ) ) ).

% word_plus_mono_left
tff(fact_2177_constraint__expand,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Lower: word(A),Upper: word(A)] :
          ( member(word(A),Xb,collect(word(A),aa(word(A),fun(word(A),$o),aTP_Lamp_db(word(A),fun(word(A),fun(word(A),$o)),Lower),Upper)))
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Lower),Xb)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Upper) ) ) ) ).

% constraint_expand
tff(fact_2178_word__plus__mono__right2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),C2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),B2)
           => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),C2)) ) ) ) ).

% word_plus_mono_right2
tff(fact_2179_word__plus__mono__right,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Z: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Z)
         => ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(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),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Z)) ) ) ) ).

% word_plus_mono_right
tff(fact_2180_word__le__plus__either,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A),Z: word(A)] :
          ( ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y)
            | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Z) )
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Z))
           => aa(word(A),$o,aa(word(A),fun(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),Z)) ) ) ) ).

% word_le_plus_either
tff(fact_2181_le__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),B2)
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2))
           => aa(word(A),$o,aa(word(A),fun(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)),A2),B2)) ) ) ) ).

% le_no_overflow
tff(fact_2182_olen__add__eqv,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Y),Xb))
        <=> aa(word(A),$o,aa(word(A),fun(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)) ) ) ).

% olen_add_eqv
tff(fact_2183_word__coorder_Oextremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),zero_zero(word(A))),A2) ) ).

% word_coorder.extremum
tff(fact_2184_word__coorder_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),zero_zero(word(A)))
         => ( A2 = zero_zero(word(A)) ) ) ) ).

% word_coorder.extremum_uniqueI
tff(fact_2185_word__zero__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),zero_zero(word(A))),Y) ) ).

% word_zero_le
tff(fact_2186_iadd__is__0,axiom,
    ! [M: extended_enat,Nb: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),plus_plus(extended_enat),M),Nb) = zero_zero(extended_enat) )
    <=> ( ( M = zero_zero(extended_enat) )
        & ( Nb = zero_zero(extended_enat) ) ) ) ).

% iadd_is_0
tff(fact_2187_imult__is__0,axiom,
    ! [M: extended_enat,Nb: extended_enat] :
      ( ( aa(extended_enat,extended_enat,aa(extended_enat,fun(extended_enat,extended_enat),times_times(extended_enat),M),Nb) = zero_zero(extended_enat) )
    <=> ( ( M = zero_zero(extended_enat) )
        | ( Nb = zero_zero(extended_enat) ) ) ) ).

% imult_is_0
tff(fact_2188_zero__one__enat__neq_I1_J,axiom,
    zero_zero(extended_enat) != one_one(extended_enat) ).

% zero_one_enat_neq(1)
tff(fact_2189_ex__le__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z3: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),Z3)) ) ).

% ex_le_of_int
tff(fact_2190_ex__of__int__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z3: int] : aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Z3)),Xb) ) ).

% ex_of_int_less
tff(fact_2191_ex__less__of__int,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z3: int] : aa(A,$o,ord_less(A,Xb),aa(int,A,ring_1_of_int(A),Z3)) ) ).

% ex_less_of_int
tff(fact_2192_mult__of__int__commute,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Xb: int,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),Xb)),Y) = aa(A,A,aa(A,fun(A,A),times_times(A),Y),aa(int,A,ring_1_of_int(A),Xb)) ) ).

% mult_of_int_commute
tff(fact_2193_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_2194_nat__mono,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),Y)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),nat2(Xb)),nat2(Y)) ) ).

% nat_mono
tff(fact_2195_nat__one__as__int,axiom,
    one_one(nat) = nat2(one_one(int)) ).

% nat_one_as_int
tff(fact_2196_eq__nat__nat__iff,axiom,
    ! [Z: int,Z5: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z5)
       => ( ( nat2(Z) = nat2(Z5) )
        <=> ( Z = Z5 ) ) ) ) ).

% eq_nat_nat_iff
tff(fact_2197_all__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ! [X_1: nat] : aa(nat,$o,P,X_1)
    <=> ! [X: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
         => aa(nat,$o,P,nat2(X)) ) ) ).

% all_nat
tff(fact_2198_ex__nat,axiom,
    ! [P: fun(nat,$o)] :
      ( ? [X_1: nat] : aa(nat,$o,P,X_1)
    <=> ? [X: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),X)
          & aa(nat,$o,P,nat2(X)) ) ) ).

% ex_nat
tff(fact_2199_le__of__int__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(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_2200_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_2201_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_2202_nat__le__iff,axiom,
    ! [Xb: int,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),nat2(Xb)),Nb)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),aa(nat,int,semiring_1_of_nat(int),Nb)) ) ).

% nat_le_iff
tff(fact_2203_of__nat__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),R3),aa(nat,A,semiring_1_of_nat(A),nat2(archimedean_ceiling(A,R3)))) ) ).

% of_nat_ceiling
tff(fact_2204_nat__int__add,axiom,
    ! [A2: nat,B2: nat] : nat2(aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2) ).

% nat_int_add
tff(fact_2205_nat__0__le,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$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_2206_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( aa(nat,int,semiring_1_of_nat(int),M) = Z )
    <=> ( ( M = nat2(Z) )
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) ) ) ).

% int_eq_iff
tff(fact_2207_ceiling__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),Z)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% ceiling_le_iff
tff(fact_2208_ceiling__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),A2))
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_ceiling(A,Xb)),A2) ) ) ).

% ceiling_le
tff(fact_2209_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_2210_int__minus,axiom,
    ! [Nb: nat,M: nat] : aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,minus_minus(nat,Nb),M)) = aa(nat,int,semiring_1_of_nat(int),nat2(aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),Nb)),aa(nat,int,semiring_1_of_nat(int),M)))) ).

% int_minus
tff(fact_2211_real__of__int__div4,axiom,
    ! [Nb: int,Xb: int] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),Xb))),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Nb)),aa(int,real,ring_1_of_int(real),Xb))) ).

% real_of_int_div4
tff(fact_2212_real__nat__ceiling__ge,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(nat,real,semiring_1_of_nat(real),nat2(archimedean_ceiling(real,Xb)))) ).

% real_nat_ceiling_ge
tff(fact_2213_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_2214_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_2215_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_2216_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_2217_of__int__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Z: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(int,A,ring_1_of_int(A),Z)) ) ) ).

% of_int_nonneg
tff(fact_2218_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_2219_floor__exists1,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [X2: int] :
          ( aa(A,$o,aa(A,fun(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,$o,aa(A,fun(A,$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_2220_floor__exists,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
        ? [Z3: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z3)),Xb)
          & aa(A,$o,ord_less(A,Xb),aa(int,A,ring_1_of_int(A),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z3),one_one(int)))) ) ) ).

% floor_exists
tff(fact_2221_of__int__ceiling__le__add__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R3))),aa(A,A,aa(A,fun(A,A),plus_plus(A),R3),one_one(A))) ) ).

% of_int_ceiling_le_add_one
tff(fact_2222_of__int__ceiling__diff__one__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,R3))),one_one(A))),R3) ) ).

% of_int_ceiling_diff_one_le
tff(fact_2223_of__nat__less__of__int__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,Xb: int] :
          ( aa(A,$o,ord_less(A,aa(nat,A,semiring_1_of_nat(A),Nb)),aa(int,A,ring_1_of_int(A),Xb))
        <=> aa(int,$o,ord_less(int,aa(nat,int,semiring_1_of_nat(int),Nb)),Xb) ) ) ).

% of_nat_less_of_int_iff
tff(fact_2224_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$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_2225_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M = nat2(W) )
    <=> $ite(aa(int,$o,aa(int,fun(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_2226_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( nat2(W) = M )
    <=> $ite(aa(int,$o,aa(int,fun(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_2227_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( aa(int,$o,ord_less(int,zero_zero(int)),W)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),nat2(W)),nat2(Z))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),W),Z) ) ) ).

% nat_le_eq_zle
tff(fact_2228_nat__add__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( aa(int,$o,aa(int,fun(int,$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_2229_le__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),nat2(K))
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Nb)),K) ) ) ).

% le_nat_iff
tff(fact_2230_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_2231_nat__mult__distrib,axiom,
    ! [Z: int,Z5: int] :
      ( aa(int,$o,aa(int,fun(int,$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_2232_nat__diff__distrib,axiom,
    ! [Z5: int,Z: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z5)
     => ( aa(int,$o,aa(int,fun(int,$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_2233_nat__diff__distrib_H,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$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_2234_nat__div__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$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_2235_nat__div__distrib_H,axiom,
    ! [Y: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$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_2236_nat__power__eq,axiom,
    ! [Z: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Z)
     => ( nat2(aa(nat,int,aa(int,fun(nat,int),power_power(int),Z),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),nat2(Z)),Nb) ) ) ).

% nat_power_eq
tff(fact_2237_int__le__real__less,axiom,
    ! [Nb: int,M: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Nb),M)
    <=> aa(real,$o,ord_less(real,aa(int,real,ring_1_of_int(real),Nb)),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_2238_int__less__real__le,axiom,
    ! [Nb: int,M: int] :
      ( aa(int,$o,ord_less(int,Nb),M)
    <=> aa(real,$o,aa(real,fun(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),Nb)),one_one(real))),aa(int,real,ring_1_of_int(real),M)) ) ).

% int_less_real_le
tff(fact_2239_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_T_m_a_x_t(vEBT_Leaf((A2),(B2))) = 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)))) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(1)
tff(fact_2240_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : vEBT_T_m_a_x_t(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) = one_one(nat) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(2)
tff(fact_2241_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,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT,Xb: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,zero_zero(nat),Ts,S3),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(2)
tff(fact_2242_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list(vEBT_VEBT),Uw: vEBT_VEBT] : vEBT_T_m_i_n_t(vEBT_Node(none(product_prod(nat,nat)),Uu,Uv,Uw)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(2)
tff(fact_2243_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_2244_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,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),archimedean_ceiling(A,Xb))) ) ) ).

% ceiling_correct
tff(fact_2245_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,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),Z))
           => ( archimedean_ceiling(A,Xb) = Z ) ) ) ) ).

% ceiling_unique
tff(fact_2246_ceiling__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: int] :
          ( ( archimedean_ceiling(A,Xb) = A2 )
        <=> ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),A2)),one_one(A))),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(int,A,ring_1_of_int(A),A2)) ) ) ) ).

% ceiling_eq_iff
tff(fact_2247_ceiling__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archimedean_ceiling(A,Ta))
        <=> ! [I2: int] :
              ( ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,aa(int,A,ring_1_of_int(A),I2)),one_one(A))),Ta)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Ta),aa(int,A,ring_1_of_int(A),I2)) )
             => aa(int,$o,P,I2) ) ) ) ).

% ceiling_split
tff(fact_2248_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,$o,aa(A,fun(A,$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_2249_le__ceiling__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,aa(int,fun(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_2250_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( aa(int,$o,aa(int,fun(int,$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_2251_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$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_2252_real__of__int__div2,axiom,
    ! [Nb: int,Xb: int] : aa(real,$o,aa(real,fun(real,$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),Nb)),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),Nb),Xb)))) ).

% real_of_int_div2
tff(fact_2253_real__of__int__div3,axiom,
    ! [Nb: int,Xb: int] : aa(real,$o,aa(real,fun(real,$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),Nb)),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),Nb),Xb)))),one_one(real)) ).

% real_of_int_div3
tff(fact_2254_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_2255_maxt__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_a_x_t(Ta)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ).

% maxt_bound
tff(fact_2256_ceiling__divide__upper,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$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_2257_T_092_060_094sub_062m_092_060_094sub_062a_092_060_094sub_062x_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: 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),Mia),Maa)),Ux,Uy,Uz)) = one_one(nat) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.simps(3)
tff(fact_2258_mint__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_i_n_t(Ta)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ).

% mint_bound
tff(fact_2259_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] :
      vEBT_T_m_i_n_t(vEBT_Leaf((A2),(B2))) = 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)))) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(1)
tff(fact_2260_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,
    ! [Info: option(product_prod(nat,nat)),Ts: list(vEBT_VEBT),S3: vEBT_VEBT,Xb: nat] : vEBT_T_i_n_s_e_r_t(vEBT_Node(Info,aa(nat,nat,suc,zero_zero(nat)),Ts,S3),Xb) = one_one(nat) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(3)
tff(fact_2261_T_092_060_094sub_062m_092_060_094sub_062i_092_060_094sub_062n_092_060_094sub_062t_Osimps_I3_J,axiom,
    ! [Mia: nat,Maa: 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),Mia),Maa)),Ux,Uy,Uz)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.simps(3)
tff(fact_2262_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,
    ! [A2: $o,B2: $o,Xb: nat] :
      vEBT_T_i_n_s_e_r_t(vEBT_Leaf((A2),(B2)),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_2263_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_2264_ceiling__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: int,Xb: A] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),Nb)),Xb)
         => ( aa(A,$o,aa(A,fun(A,$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),Nb)),one_one(A)))
           => ( archimedean_ceiling(A,Xb) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),one_one(int)) ) ) ) ) ).

% ceiling_eq
tff(fact_2265_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,TreeLista: list(vEBT_VEBT),Summarya: 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)),TreeLista,Summarya),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_2266_insersimp,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Y: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ~ ? [X_13: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Ta),X_13)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(Ta,Y)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ) ) ).

% insersimp
tff(fact_2267_insertsimp,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,L: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_minNull(Ta)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(Ta,L)),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))) ) ) ).

% insertsimp
tff(fact_2268_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 )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ( Y != 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)))) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi: nat,Ma: 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),Mi),Ma)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.elims
tff(fact_2269_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 )
     => ( ! [A4: $o] :
            ( ? [B3: $o] : Xb = vEBT_Leaf((A4),(B3))
           => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                  $ite((A4),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) ) )
       => ( ( ? [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)
           => ( Y != one_one(nat) ) )
         => ~ ( ? [Mi: nat,Ma: 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),Mi),Ma)),Ux2,Uy2,Uz2)
             => ( Y != one_one(nat) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.elims
tff(fact_2270_insert__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_i_n_s_e_r_t(Ta,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,Ta))),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_2271_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 )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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,B3: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),(B3))),zero_zero(nat))) ) ) )
         => ( ! [Uv2: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((Uv2),(Uw2)) )
               => ! [N2: nat] :
                    ( ( Xaa = aa(nat,nat,suc,N2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((Uv2),(Uw2))),aa(nat,nat,suc,N2))) ) ) )
           => ( ! [Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_s_u_c_c_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Ux2,Uy2,Uz2)),Xaa)) ) )
             => ( ! [V3: product_prod(nat,nat),Vc: list(vEBT_VEBT),Vd2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vc,Vd2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Vc,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) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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)) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                       => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                $ite(
                                  aa(nat,$o,ord_less(nat,Xaa),Mi),
                                  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)),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(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))),
                                                      $ite(sc = none(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),sc)))))) ) )) )),
                                          one_one(nat) )) ) ) )) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>s\<^sub>u\<^sub>c\<^sub>c.pelims
tff(fact_2272_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 )
     => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
           => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
             => ( Y != one_one(nat) ) ) )
         => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
             => ( ? [N2: nat] : Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N2))
               => ( Y != one_one(nat) ) ) )
           => ( ( ? [Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary)
               => ( Y != one_one(nat) ) )
             => ( ( ? [Mi: nat,Ma: nat,TreeList: 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),Mi),Ma)),zero_zero(nat),TreeList,Summary)
                 => ( Y != one_one(nat) ) )
               => ( ( ? [Mi: nat,Ma: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary)
                   => ( Y != one_one(nat) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,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),Mi)
                                | aa(nat,$o,ord_less(nat,Ma),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 = Mi )
                                    & ( Xaa = 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(Xaa = 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(Summary)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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 = 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(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,TreeList),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)),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(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 = 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.elims
tff(fact_2273_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,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya),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),Mia)
          | aa(nat,$o,ord_less(nat,Maa),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 = Mia )
              & ( Xb = Maa ) ),
              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 = Mia,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(Summarya)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya)))))),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 = Mia,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(Summarya))),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,Vaa))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),aa(option(nat),nat,the2(nat),vEBT_vebt_mint(Summarya)))))),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,Vaa))),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,Vaa))),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)),TreeLista)),
                        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,TreeLista),h),l))),
                          $let(
                            newnode: vEBT_VEBT,
                            newnode:= vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h),l),
                            $let(
                              newlist: list(vEBT_VEBT),
                              newlist:= list_update(vEBT_VEBT,TreeLista,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(Summarya,h))),
                                    $let(
                                      sn: vEBT_VEBT,
                                      sn:= vEBT_vebt_delete(Summarya,h),
                                      aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                                        $ite(
                                          xn = Maa,
                                          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 = Maa,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_2274_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,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( ( Xaa = zero_zero(nat) )
           => ( Y != one_one(nat) ) ) )
       => ( ( ? [A4: $o,Uw2: $o] : Xb = vEBT_Leaf((A4),(Uw2))
           => ( ( 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)) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( Xb = vEBT_Leaf((A4),(B3)) )
               => ( ? [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((B3),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),Va: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va)
               => ( 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) ) )
                 => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                       => ( Y != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                              $ite(
                                aa(nat,$o,ord_less(nat,Ma),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)),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(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,TreeList),aa(option(nat),nat,the2(nat),pr)))))) ) )) ),
                                        one_one(nat) )) ) ) )) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.elims
tff(fact_2275_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I7_J,axiom,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
        $ite(
          aa(nat,$o,ord_less(nat,Maa),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,Vaa))),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,Vaa))),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)),TreeLista)),
                  $let(
                    minlow: option(nat),
                    minlow:= vEBT_vebt_mint(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),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,TreeLista),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,TreeLista),h),l)),
                        $let(
                          pr: option(nat),
                          pr:= vEBT_vebt_pred(Summarya,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(Summarya,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,TreeLista),aa(option(nat),nat,the2(nat),pr)))))) ) )) ),
                  one_one(nat) )) ) ) )) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(7)
tff(fact_2276_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 )
     => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
         => ( 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)))) ) )
       => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S: vEBT_VEBT] : Xb = vEBT_Node(Info2,zero_zero(nat),Ts2,S)
           => ( Y != one_one(nat) ) )
         => ( ( ? [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S: vEBT_VEBT] : Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S)
             => ( Y != one_one(nat) ) )
           => ( ( ? [V3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary)
               => ( Y != aa(num,nat,numeral_numeral(nat),bit0(one2)) ) )
             => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,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),Mi),Mi,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)),TreeList))
                                & ~ ( ( Xaa = Mi )
                                    | ( Xaa = 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,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,TreeList),h)))),
                                  $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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_2277_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 )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,B3: $o] :
                ( ( Xb = vEBT_Leaf((A4),(B3)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( Xb = vEBT_Leaf((A4),(B3)) )
                 => ! [N2: nat] :
                      ( ( Xaa = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,N2)))) ) ) )
             => ( ! [Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T8441311223069195367_e_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Deg2,TreeList,Summary)),Xaa)) ) )
               => ( ! [Mi: nat,Ma: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),zero_zero(nat),TreeList,Summary) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),zero_zero(nat),TreeList,Summary)),Xaa)) ) )
                 => ( ! [Mi: nat,Ma: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary) )
                       => ( ( Y = one_one(nat) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,zero_zero(nat)),TreeList,Summary)),Xaa)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,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),Mi)
                                    | aa(nat,$o,ord_less(nat,Ma),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 = Mi )
                                        & ( Xaa = 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(Xaa = 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(Summary)),vEBT_T_m_i_n_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),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 = 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(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,TreeList),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)),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(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 = 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))),
                                                              $ite(xn = Ma,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,newlist),h))),one_one(nat))) )) ) )),
                                                  one_one(nat) ) ) ) )) )) )) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>d\<^sub>e\<^sub>l\<^sub>e\<^sub>t\<^sub>e.pelims
tff(fact_2278_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_2279_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_2280_word__numeral__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(num,word(A),numeral_numeral(word(A)),B2) = aa(int,word(A),ring_1_of_int(word(A)),aa(num,int,numeral_numeral(int),B2)) ) ).

% word_numeral_alt
tff(fact_2281_wi__hom__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int,B2: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A2)),aa(int,word(A),ring_1_of_int(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),B2)) ) ).

% wi_hom_add
tff(fact_2282_wi__hom__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int,B2: int] : aa(word(A),word(A),minus_minus(word(A),aa(int,word(A),ring_1_of_int(word(A)),A2)),aa(int,word(A),ring_1_of_int(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,A2),B2)) ) ).

% wi_hom_sub
tff(fact_2283_word__of__int__power__hom,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int,Nb: 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)),A2)),Nb) = aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),Nb)) ) ).

% word_of_int_power_hom
tff(fact_2284_wi__hom__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int,B2: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(int,word(A),ring_1_of_int(word(A)),A2)),aa(int,word(A),ring_1_of_int(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2)) ) ).

% wi_hom_mult
tff(fact_2285_minNull__bound,axiom,
    ! [Ta: vEBT_VEBT] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_m_i_n_N_u_l_l(Ta)),one_one(nat)) ).

% minNull_bound
tff(fact_2286_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_2287_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,
    ! [Uv: $o] : vEBT_T_m_i_n_N_u_l_l(vEBT_Leaf($true,(Uv))) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(2)
tff(fact_2288_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_2289_word__of__int__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int] :
          ( aa(int,$o,aa(int,fun(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_2290_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] : vEBT_T_p_r_e_d(vEBT_Leaf((Uu),(Uv)),zero_zero(nat)) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(1)
tff(fact_2291_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),Vaa: vEBT_VEBT,Vb: nat] : vEBT_T_p_r_e_d(vEBT_Node(none(product_prod(nat,nat)),Uy,Uz,Vaa),Vb) = one_one(nat) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(4)
tff(fact_2292_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),Vaa: nat,Vb: list(vEBT_VEBT),Vc2: 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),Vaa,Vb,Vc2)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(5)
tff(fact_2293_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,
    ! [Uw: nat,Ux: list(vEBT_VEBT),Uy: vEBT_VEBT] : vEBT_T_m_i_n_N_u_l_l(vEBT_Node(none(product_prod(nat,nat)),Uw,Ux,Uy)) = one_one(nat) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.simps(4)
tff(fact_2294_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I3_J,axiom,
    ! [A2: $o,B2: $o,Vaa: nat] :
      vEBT_T_p_r_e_d(vEBT_Leaf((A2),(B2)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))) = 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)))) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.simps(3)
tff(fact_2295_word__of__int__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: 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))),Nb)) = 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))),Nb) ) ).

% word_of_int_2p
tff(fact_2296_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_2297_T_092_060_094sub_062p_092_060_094sub_062r_092_060_094sub_062e_092_060_094sub_062d_Osimps_I2_J,axiom,
    ! [A2: $o,Uw: $o] : vEBT_T_p_r_e_d(vEBT_Leaf((A2),(Uw)),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_2298_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_2299_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) ) )
       => ( ( ? [Uv2: $o] : Xb = vEBT_Leaf($true,(Uv2))
           => ( Y != one_one(nat) ) )
         => ( ( ? [Uu2: $o] : Xb = vEBT_Leaf((Uu2),$true)
             => ( Y != one_one(nat) ) )
           => ( ( ? [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] : Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)
               => ( Y != one_one(nat) ) )
             => ~ ( ? [Uz2: product_prod(nat,nat),Va: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] : Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va,Vb2,Vc)
                 => ( Y != one_one(nat) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.elims
tff(fact_2300_pred__bound__height,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),vEBT_T_p_r_e_d(Ta,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,Ta))),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_2301_pred__bound__size__univ,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,U: real,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( ( U = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),vEBT_T_p_r_e_d(Ta,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)))))),log(aa(num,real,numeral_numeral(real),bit0(one2)),log(aa(num,real,numeral_numeral(real),bit0(one2)),U))))) ) ) ).

% pred_bound_size_univ
tff(fact_2302_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,
    ! [Mia: nat,Maa: nat,Vaa: nat,TreeLista: list(vEBT_VEBT),Summarya: 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),Mia),Maa)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),TreeLista,Summarya),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),Mia),Mia,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,Vaa))),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)),TreeLista))
              & ~ ( ( Xb = Mia )
                  | ( Xb = Maa ) ) ),
              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,TreeLista),h),vEBT_VEBT_low(xn,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),vEBT_T_m_i_n_N_u_l_l(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)))),
                $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),h)),vEBT_T_i_n_s_e_r_t(Summarya,h),one_one(nat))),
              one_one(nat) ) ) )) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.simps(5)
tff(fact_2303_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 )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( Xaa = zero_zero(nat) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),(Uv2))),zero_zero(nat))) ) ) )
         => ( ! [A4: $o,Uw2: $o] :
                ( ( Xb = vEBT_Leaf((A4),(Uw2)) )
               => ( ( Xaa = aa(nat,nat,suc,zero_zero(nat)) )
                 => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(Uw2))),aa(nat,nat,suc,zero_zero(nat)))) ) ) )
           => ( ! [A4: $o,B3: $o] :
                  ( ( Xb = vEBT_Leaf((A4),(B3)) )
                 => ! [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)),
                              $ite((B3),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),aa(nat,nat,suc,aa(nat,nat,suc,Va2)))) ) ) )
             => ( ! [Uy2: nat,Uz2: list(vEBT_VEBT),Va: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T_p_r_e_d_rel2),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(none(product_prod(nat,nat)),Uy2,Uz2,Va)),Xaa)) ) )
               => ( ! [V3: product_prod(nat,nat),Vd2: list(vEBT_VEBT),Ve2: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),V3),zero_zero(nat),Vd2,Ve2) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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) )
                         => ~ aa(product_prod(vEBT_VEBT,nat),$o,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)) ) )
                   => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary) )
                         => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                                  $ite(
                                    aa(nat,$o,ord_less(nat,Ma),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)),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(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))),
                                                      $ite(pr = none(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),vEBT_T_m_a_x_t(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),aa(option(nat),nat,the2(nat),pr)))))) ) )) ),
                                            one_one(nat) )) ) ) )) )
                           => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ) ) ).

% T\<^sub>p\<^sub>r\<^sub>e\<^sub>d.pelims
tff(fact_2304_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 )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite(Xaa = zero_zero(nat),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,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((A4),(B3))),Xaa)) ) )
         => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,zero_zero(nat),Ts2,S) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,accp(product_prod(vEBT_VEBT,nat),vEBT_T9217963907923527482_t_rel),aa(nat,product_prod(vEBT_VEBT,nat),aa(vEBT_VEBT,fun(nat,product_prod(vEBT_VEBT,nat)),product_Pair(vEBT_VEBT,nat),vEBT_Node(Info2,zero_zero(nat),Ts2,S)),Xaa)) ) )
           => ( ! [Info2: option(product_prod(nat,nat)),Ts2: list(vEBT_VEBT),S: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(product_prod(vEBT_VEBT,nat),$o,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(Info2,aa(nat,nat,suc,zero_zero(nat)),Ts2,S)),Xaa)) ) )
             => ( ! [V3: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,V3)),TreeList,Summary) )
                   => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(one2)) )
                     => ~ aa(product_prod(vEBT_VEBT,nat),$o,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)),TreeList,Summary)),Xaa)) ) )
               => ~ ! [Mi: nat,Ma: nat,Va2: nat,TreeList: 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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,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),Mi),Mi,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)),TreeList))
                                    & ~ ( ( Xaa = Mi )
                                        | ( Xaa = 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,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,TreeList),h)))),
                                      $ite(vEBT_VEBT_minNull(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeList),h)),vEBT_T_i_n_s_e_r_t(Summary,h),one_one(nat))),
                                    one_one(nat) ) ) )) )
                       => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),Mi),Ma)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),TreeList,Summary)),Xaa)) ) ) ) ) ) ) ) ) ).

% T\<^sub>i\<^sub>n\<^sub>s\<^sub>e\<^sub>r\<^sub>t.pelims
tff(fact_2305_delete__correct,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(Ta,Xb)) = aa(set(nat),set(nat),minus_minus(set(nat),vEBT_set_vebt(Ta)),aa(set(nat),set(nat),insert(nat,Xb),bot_bot(set(nat)))) ) ) ).

% delete_correct
tff(fact_2306_lemma__termdiff3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [H: A,Z: A,K6: real,Nb: nat] :
          ( ( H != zero_zero(A) )
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),K6)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),H))),K6)
             => aa(real,$o,aa(real,fun(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)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,minus_minus(nat,Nb),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),Nb)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))))),aa(nat,real,aa(real,fun(nat,real),power_power(real),K6),aa(nat,nat,minus_minus(nat,Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),real_V7770717601297561774m_norm(A,H))) ) ) ) ) ).

% lemma_termdiff3
tff(fact_2307_lowi__def,axiom,
    ! [Xb: nat,Nb: nat] : vEBT_VEBT_lowi(Xb,Nb) = 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))),Nb))) ).

% lowi_def
tff(fact_2308_foldr__zero,axiom,
    ! [Xs: list(nat),D2: nat] :
      ( ! [I3: nat] :
          ( aa(nat,$o,ord_less(nat,I3),aa(list(nat),nat,size_size(list(nat)),Xs))
         => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,nth(nat,Xs),I3)) )
     => aa(nat,$o,aa(nat,fun(nat,$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,D2)),D2)) ) ).

% foldr_zero
tff(fact_2309_foldr__one,axiom,
    ! [D2: nat,Ys: list(nat)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),D2),foldr(nat,nat,plus_plus(nat),Ys,D2)) ).

% foldr_one
tff(fact_2310_mod__mod__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% mod_mod_trivial
tff(fact_2311_foldr__same__int,axiom,
    ! [Xs: list(nat),Y: nat] :
      ( ! [X2: nat,Y5: nat] :
          ( member(nat,X2,aa(list(nat),set(nat),set2(nat),Xs))
         => ( member(nat,Y5,aa(list(nat),set(nat),set2(nat),Xs))
           => ( X2 = Y5 ) ) )
     => ( ! [X2: nat] :
            ( member(nat,X2,aa(list(nat),set(nat),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_2312_low__def,axiom,
    ! [Xb: nat,Nb: nat] : vEBT_VEBT_low(Xb,Nb) = 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))),Nb)) ).

% low_def
tff(fact_2313_delete__correct_H,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat,Xb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => ( vEBT_VEBT_set_vebt(vEBT_vebt_delete(Ta,Xb)) = aa(set(nat),set(nat),minus_minus(set(nat),vEBT_VEBT_set_vebt(Ta)),aa(set(nat),set(nat),insert(nat,Xb),bot_bot(set(nat)))) ) ) ).

% delete_correct'
tff(fact_2314_foldr__mono,axiom,
    ! [Xs: list(nat),Ys: list(nat),C2: nat,D2: nat] :
      ( ( aa(list(nat),nat,size_size(list(nat)),Xs) = aa(list(nat),nat,size_size(list(nat)),Ys) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,ord_less(nat,I3),aa(list(nat),nat,size_size(list(nat)),Xs))
           => aa(nat,$o,ord_less(nat,aa(nat,nat,nth(nat,Xs),I3)),aa(nat,nat,nth(nat,Ys),I3)) )
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),C2),D2)
         => aa(nat,$o,aa(nat,fun(nat,$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,D2)) ) ) ) ).

% foldr_mono
tff(fact_2315_mod__self,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),A2) = zero_zero(A) ) ).

% mod_self
tff(fact_2316_mod__by__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),zero_zero(A)) = A2 ) ).

% mod_by_0
tff(fact_2317_mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),zero_zero(A)),A2) = zero_zero(A) ) ).

% mod_0
tff(fact_2318_mod__add__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% mod_add_self2
tff(fact_2319_mod__add__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% mod_add_self1
tff(fact_2320_minus__mod__self2,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,A2),B2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% minus_mod_self2
tff(fact_2321_insert__subset,axiom,
    ! [A: $tType,Xb: A,A3: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,Xb),A3)),B5)
    <=> ( member(A,Xb,B5)
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5) ) ) ).

% insert_subset
tff(fact_2322_insert__Diff1,axiom,
    ! [A: $tType,Xb: A,B5: set(A),A3: set(A)] :
      ( member(A,Xb,B5)
     => ( aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),insert(A,Xb),A3)),B5) = aa(set(A),set(A),minus_minus(set(A),A3),B5) ) ) ).

% insert_Diff1
tff(fact_2323_Diff__insert0,axiom,
    ! [A: $tType,Xb: A,A3: set(A),B5: set(A)] :
      ( ~ member(A,Xb,A3)
     => ( aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xb),B5)) = aa(set(A),set(A),minus_minus(set(A),A3),B5) ) ) ).

% Diff_insert0
tff(fact_2324_mod__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,M),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb) = M ) ) ).

% mod_less
tff(fact_2325_nat__mod__eq_H,axiom,
    ! [A2: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,A2),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),Nb) = A2 ) ) ).

% nat_mod_eq'
tff(fact_2326_singleton__conv,axiom,
    ! [A: $tType,A2: A] : collect(A,aTP_Lamp_dc(A,fun(A,$o),A2)) = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) ).

% singleton_conv
tff(fact_2327_singleton__conv2,axiom,
    ! [A: $tType,A2: A] : collect(A,aa(A,fun(A,$o),fequal(A),A2)) = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) ).

% singleton_conv2
tff(fact_2328_bits__mod__by__1,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),one_one(A)) = zero_zero(A) ) ).

% bits_mod_by_1
tff(fact_2329_mod__by__1,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),one_one(A)) = zero_zero(A) ) ).

% mod_by_1
tff(fact_2330_mod__mult__self1__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),B2) = zero_zero(A) ) ).

% mod_mult_self1_is_0
tff(fact_2331_mod__mult__self2__is__0,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),B2) = zero_zero(A) ) ).

% mod_mult_self2_is_0
tff(fact_2332_bits__mod__div__trivial,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),B2) = zero_zero(A) ) ).

% bits_mod_div_trivial
tff(fact_2333_mod__div__trivial,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),B2) = zero_zero(A) ) ).

% mod_div_trivial
tff(fact_2334_mod__mult__self1,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% mod_mult_self1
tff(fact_2335_mod__mult__self2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: 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),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% mod_mult_self2
tff(fact_2336_mod__mult__self3,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: 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),B2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% mod_mult_self3
tff(fact_2337_mod__mult__self4,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [B2: A,C2: A,A2: 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),B2),C2)),A2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% mod_mult_self4
tff(fact_2338_singleton__insert__inj__eq,axiom,
    ! [A: $tType,B2: A,A2: A,A3: set(A)] :
      ( ( aa(set(A),set(A),insert(A,B2),bot_bot(set(A))) = aa(set(A),set(A),insert(A,A2),A3) )
    <=> ( ( A2 = B2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq
tff(fact_2339_singleton__insert__inj__eq_H,axiom,
    ! [A: $tType,A2: A,A3: set(A),B2: A] :
      ( ( aa(set(A),set(A),insert(A,A2),A3) = aa(set(A),set(A),insert(A,B2),bot_bot(set(A))) )
    <=> ( ( A2 = B2 )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))) ) ) ).

% singleton_insert_inj_eq'
tff(fact_2340_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_2341_insert__Diff__single,axiom,
    ! [A: $tType,A2: A,A3: set(A)] : aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = aa(set(A),set(A),insert(A,A2),A3) ).

% insert_Diff_single
tff(fact_2342_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,Nb: 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),Nb)))),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Nb) ).

% Suc_mod_mult_self1
tff(fact_2343_Suc__mod__mult__self2,axiom,
    ! [M: nat,Nb: 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),Nb),K)))),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Nb) ).

% Suc_mod_mult_self2
tff(fact_2344_Suc__mod__mult__self3,axiom,
    ! [K: nat,Nb: 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),Nb)),M))),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Nb) ).

% Suc_mod_mult_self3
tff(fact_2345_Suc__mod__mult__self4,axiom,
    ! [Nb: 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),Nb),K)),M))),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Nb) ).

% Suc_mod_mult_self4
tff(fact_2346_foldr__length,axiom,
    ! [A: $tType,L: list(A)] : foldr(A,nat,aTP_Lamp_dd(A,fun(nat,nat)),L,zero_zero(nat)) = aa(list(A),nat,size_size(list(A)),L) ).

% foldr_length
tff(fact_2347_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_2348_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_2349_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_2350_Suc__times__numeral__mod__eq,axiom,
    ! [K: num,Nb: 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)),Nb))),aa(num,nat,numeral_numeral(nat),K)) = one_one(nat) ) ) ).

% Suc_times_numeral_mod_eq
tff(fact_2351_not__mod__2__eq__1__eq__0,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% not_mod_2_eq_1_eq_0
tff(fact_2352_not__mod__2__eq__0__eq__1,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) != zero_zero(A) )
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% not_mod_2_eq_0_eq_1
tff(fact_2353_not__mod2__eq__Suc__0__eq__0,axiom,
    ! [Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Nb),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),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))) = zero_zero(nat) ) ) ).

% not_mod2_eq_Suc_0_eq_0
tff(fact_2354_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_2355_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_2356_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,Nb: 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,Nb)))) = 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))),Nb)) ).

% mod_Suc_eq_mod_add3
tff(fact_2357_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_2358_of__nat__mod,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb)) = 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),Nb)) ) ).

% of_nat_mod
tff(fact_2359_insert__Collect,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] : aa(set(A),set(A),insert(A,A2),collect(A,P)) = collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_de(A,fun(fun(A,$o),fun(A,$o)),A2),P)) ).

% insert_Collect
tff(fact_2360_insert__compr,axiom,
    ! [A: $tType,A2: A,B5: set(A)] : aa(set(A),set(A),insert(A,A2),B5) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_df(A,fun(set(A),fun(A,$o)),A2),B5)) ).

% insert_compr
tff(fact_2361_mod__add__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: 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),A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ).

% mod_add_right_eq
tff(fact_2362_mod__add__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: 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),A2),C2)),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ).

% mod_add_left_eq
tff(fact_2363_mod__add__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A6: A,B2: A,B4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A6),C2) )
         => ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A6),B4)),C2) ) ) ) ) ).

% mod_add_cong
tff(fact_2364_mod__add__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: 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),A2),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) ) ).

% mod_add_eq
tff(fact_2365_mod__mult__right__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: 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),A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ).

% mod_mult_right_eq
tff(fact_2366_mod__mult__left__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: 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),A2),C2)),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ).

% mod_mult_left_eq
tff(fact_2367_mult__mod__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ).

% mult_mod_right
tff(fact_2368_mod__mult__mult2,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),C2) ) ).

% mod_mult_mult2
tff(fact_2369_mod__mult__cong,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,A6: A,B2: A,B4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A6),C2) )
         => ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A6),B4)),C2) ) ) ) ) ).

% mod_mult_cong
tff(fact_2370_mod__mult__eq,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,C2: A,B2: 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),A2),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ).

% mod_mult_eq
tff(fact_2371_mod__diff__right__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,A2),B2)),C2) ) ).

% mod_diff_right_eq
tff(fact_2372_mod__diff__left__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: 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),A2),C2)),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,A2),B2)),C2) ) ).

% mod_diff_left_eq
tff(fact_2373_mod__diff__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,A6: A,B2: A,B4: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A6),C2) )
         => ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),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,A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,A6),B4)),C2) ) ) ) ) ).

% mod_diff_cong
tff(fact_2374_mod__diff__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: 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),A2),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,A2),B2)),C2) ) ).

% mod_diff_eq
tff(fact_2375_power__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [A2: A,B2: A,Nb: 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),A2),B2)),Nb)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),B2) ) ).

% power_mod
tff(fact_2376_mod__Suc__Suc__eq,axiom,
    ! [M: nat,Nb: 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),Nb)))),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(nat,nat,suc,M))),Nb) ).

% mod_Suc_Suc_eq
tff(fact_2377_mod__Suc__eq,axiom,
    ! [M: nat,Nb: 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),Nb))),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Nb) ).

% mod_Suc_eq
tff(fact_2378_nat__mod__eq,axiom,
    ! [B2: nat,Nb: nat,A2: nat] :
      ( aa(nat,$o,ord_less(nat,B2),Nb)
     => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B2),Nb) )
       => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),Nb) = B2 ) ) ) ).

% nat_mod_eq
tff(fact_2379_mod__less__eq__dividend,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb)),M) ).

% mod_less_eq_dividend
tff(fact_2380_mod__plus__right,axiom,
    ! [A2: nat,Xb: nat,M: nat,B2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),Xb)),M) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),Xb)),M) )
    <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),M) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B2),M) ) ) ).

% mod_plus_right
tff(fact_2381_insert__mono,axiom,
    ! [A: $tType,C5: set(A),D6: set(A),A2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),C5),D6)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,A2),C5)),aa(set(A),set(A),insert(A,A2),D6)) ) ).

% insert_mono
tff(fact_2382_subset__insert,axiom,
    ! [A: $tType,Xb: A,A3: set(A),B5: set(A)] :
      ( ~ member(A,Xb,A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,Xb),B5))
      <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5) ) ) ).

% subset_insert
tff(fact_2383_subset__insertI,axiom,
    ! [A: $tType,B5: set(A),A2: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(A),set(A),insert(A,A2),B5)) ).

% subset_insertI
tff(fact_2384_subset__insertI2,axiom,
    ! [A: $tType,A3: set(A),B5: set(A),B2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,B2),B5)) ) ).

% subset_insertI2
tff(fact_2385_insert__Diff__if,axiom,
    ! [A: $tType,Xb: A,A3: set(A),B5: set(A)] :
      aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),insert(A,Xb),A3)),B5) = $ite(member(A,Xb,B5),aa(set(A),set(A),minus_minus(set(A),A3),B5),aa(set(A),set(A),insert(A,Xb),aa(set(A),set(A),minus_minus(set(A),A3),B5))) ).

% insert_Diff_if
tff(fact_2386_foldr__cong,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: A,L: list(B),K: list(B),F: fun(B,fun(A,A)),G: fun(B,fun(A,A))] :
      ( ( A2 = B2 )
     => ( ( L = K )
       => ( ! [A4: A,X2: B] :
              ( member(B,X2,aa(list(B),set(B),set2(B),L))
             => ( aa(A,A,aa(B,fun(A,A),F,X2),A4) = aa(A,A,aa(B,fun(A,A),G,X2),A4) ) )
         => ( foldr(B,A,F,L,A2) = foldr(B,A,G,K,B2) ) ) ) ) ).

% foldr_cong
tff(fact_2387_Collect__conv__if,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dg(A,fun(fun(A,$o),fun(A,$o)),A2),P)) = $ite(aa(A,$o,P,A2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if
tff(fact_2388_Collect__conv__if2,axiom,
    ! [A: $tType,A2: A,P: fun(A,$o)] :
      collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dh(A,fun(fun(A,$o),fun(A,$o)),A2),P)) = $ite(aa(A,$o,P,A2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))),bot_bot(set(A))) ).

% Collect_conv_if2
tff(fact_2389_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),A2) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
tff(fact_2390_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
         => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),B2) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
tff(fact_2391_cong__exp__iff__simps_I9_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,Nb: 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(Nb))),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),Nb)),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(9)
tff(fact_2392_cong__exp__iff__simps_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Nb: 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),Nb)),aa(num,A,numeral_numeral(A),one2)) ) ).

% cong_exp_iff_simps(4)
tff(fact_2393_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = A2 )
        <=> ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) ) ) ) ).

% mod_eq_self_iff_div_eq_0
tff(fact_2394_mod__eqE,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2) )
         => ~ ! [D3: A] : B2 != aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D3)) ) ) ).

% mod_eqE
tff(fact_2395_div__add1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: 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),A2),B2)),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),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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),A2),C2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2)) ) ).

% div_add1_eq
tff(fact_2396_mod__Suc,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,M)),Nb) = $ite(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb)) = Nb,zero_zero(nat),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb))) ).

% mod_Suc
tff(fact_2397_mod__induct,axiom,
    ! [P: fun(nat,$o),Nb: nat,P2: nat,M: nat] :
      ( aa(nat,$o,P,Nb)
     => ( aa(nat,$o,ord_less(nat,Nb),P2)
       => ( aa(nat,$o,ord_less(nat,M),P2)
         => ( ! [N2: nat] :
                ( aa(nat,$o,ord_less(nat,N2),P2)
               => ( aa(nat,$o,P,N2)
                 => aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,N2)),P2)) ) )
           => aa(nat,$o,P,M) ) ) ) ) ).

% mod_induct
tff(fact_2398_nat__mod__lem,axiom,
    ! [Nb: nat,B2: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,ord_less(nat,B2),Nb)
      <=> ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B2),Nb) = B2 ) ) ) ).

% nat_mod_lem
tff(fact_2399_mod__less__divisor,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb)),Nb) ) ).

% mod_less_divisor
tff(fact_2400_mod__Suc__le__divisor,axiom,
    ! [M: nat,Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),aa(nat,nat,suc,Nb))),Nb) ).

% mod_Suc_le_divisor
tff(fact_2401_word__rot__lem,axiom,
    ! [L: nat,K: nat,D2: nat,Nb: 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),D2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),K),L)) )
     => ( aa(nat,$o,ord_less(nat,Nb),L)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),D2),Nb)),L) = Nb ) ) ) ).

% word_rot_lem
tff(fact_2402_nat__minus__mod,axiom,
    ! [Nb: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Nb),M))),M) = zero_zero(nat) ).

% nat_minus_mod
tff(fact_2403_mod__if,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb) = $ite(aa(nat,$o,ord_less(nat,M),Nb),M,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,minus_minus(nat,M),Nb)),Nb)) ).

% mod_if
tff(fact_2404_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_2405_mod__geq,axiom,
    ! [M: nat,Nb: nat] :
      ( ~ aa(nat,$o,ord_less(nat,M),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,minus_minus(nat,M),Nb)),Nb) ) ) ).

% mod_geq
tff(fact_2406_mod__eq__0D,axiom,
    ! [M: nat,D2: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),D2) = zero_zero(nat) )
     => ? [Q3: nat] : M = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D2),Q3) ) ).

% mod_eq_0D
tff(fact_2407_le__mod__geq,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,minus_minus(nat,M),Nb)),Nb) ) ) ).

% le_mod_geq
tff(fact_2408_nat__minus__mod__plus__right,axiom,
    ! [Nb: 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),Nb),Xb)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Nb),M))),M) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),M) ).

% nat_minus_mod_plus_right
tff(fact_2409_msrevs_I2_J,axiom,
    ! [K: nat,Nb: nat,M: 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),times_times(nat),K),Nb)),M)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb) ).

% msrevs(2)
tff(fact_2410_nat__mod__eq__iff,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Y),Nb) )
    <=> ? [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),Nb),Q1)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Y),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),Q22)) ) ).

% nat_mod_eq_iff
tff(fact_2411_subset__singletonD,axiom,
    ! [A: $tType,A3: set(A),Xb: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))
     => ( ( A3 = bot_bot(set(A)) )
        | ( A3 = aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))) ) ) ) ).

% subset_singletonD
tff(fact_2412_subset__singleton__iff,axiom,
    ! [A: $tType,X5: set(A),A2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))
    <=> ( ( X5 = bot_bot(set(A)) )
        | ( X5 = aa(set(A),set(A),insert(A,A2),bot_bot(set(A))) ) ) ) ).

% subset_singleton_iff
tff(fact_2413_Diff__insert,axiom,
    ! [A: $tType,A3: set(A),A2: A,B5: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),B5)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A3),B5)),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))) ).

% Diff_insert
tff(fact_2414_insert__Diff,axiom,
    ! [A: $tType,A2: A,A3: set(A)] :
      ( member(A,A2,A3)
     => ( aa(set(A),set(A),insert(A,A2),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = A3 ) ) ).

% insert_Diff
tff(fact_2415_Diff__insert2,axiom,
    ! [A: $tType,A3: set(A),A2: A,B5: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),B5)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))),B5) ).

% Diff_insert2
tff(fact_2416_insert__minus__eq,axiom,
    ! [A: $tType,Xb: A,Y: A,A3: set(A)] :
      ( ( Xb != Y )
     => ( aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),insert(A,Xb),A3)),aa(set(A),set(A),insert(A,Y),bot_bot(set(A)))) = aa(set(A),set(A),insert(A,Xb),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Y),bot_bot(set(A))))) ) ) ).

% insert_minus_eq
tff(fact_2417_Diff__insert__absorb,axiom,
    ! [A: $tType,Xb: A,A3: set(A)] :
      ( ~ member(A,Xb,A3)
     => ( aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),insert(A,Xb),A3)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) = A3 ) ) ).

% Diff_insert_absorb
tff(fact_2418_set__minus__singleton__eq,axiom,
    ! [A: $tType,Xb: A,X5: set(A)] :
      ( ~ member(A,Xb,X5)
     => ( aa(set(A),set(A),minus_minus(set(A),X5),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) = X5 ) ) ).

% set_minus_singleton_eq
tff(fact_2419_subset__Diff__insert,axiom,
    ! [A: $tType,A3: set(A),B5: set(A),Xb: A,C5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),minus_minus(set(A),B5),aa(set(A),set(A),insert(A,Xb),C5)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),minus_minus(set(A),B5),C5))
        & ~ member(A,Xb,A3) ) ) ).

% subset_Diff_insert
tff(fact_2420_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
tff(fact_2421_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,A2),B2)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = A2 ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
tff(fact_2422_cong__exp__iff__simps_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num,Q2: num] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),bit0(Nb))),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),Nb)),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(2)
tff(fact_2423_cong__exp__iff__simps_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),Nb)),aa(num,A,numeral_numeral(A),one2)) = zero_zero(A) ) ).

% cong_exp_iff_simps(1)
tff(fact_2424_cong__exp__iff__simps_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,Nb: 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(Nb))),aa(num,A,numeral_numeral(A),bit0(Q2))) ) ).

% cong_exp_iff_simps(6)
tff(fact_2425_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_2426_cong__exp__iff__simps_I10_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,Nb: 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,Nb))),aa(num,A,numeral_numeral(A),bit0(Q2))) ) ).

% cong_exp_iff_simps(10)
tff(fact_2427_cong__exp__iff__simps_I12_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,Nb: 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(Nb))),aa(num,A,numeral_numeral(A),bit0(Q2))) ) ).

% cong_exp_iff_simps(12)
tff(fact_2428_cong__exp__iff__simps_I13_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Q2: num,Nb: 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,Nb))),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),Nb)),aa(num,A,numeral_numeral(A),Q2)) ) ) ) ).

% cong_exp_iff_simps(13)
tff(fact_2429_mult__div__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) = A2 ) ).

% mult_div_mod_eq
tff(fact_2430_mod__mult__div__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))) = A2 ) ).

% mod_mult_div_eq
tff(fact_2431_mod__div__mult__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)) = A2 ) ).

% mod_div_mult_eq
tff(fact_2432_div__mult__mod__eq,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: 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),A2),B2)),B2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) = A2 ) ).

% div_mult_mod_eq
tff(fact_2433_mod__div__decomp,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : A2 = 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),A2),B2)),B2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) ) ).

% mod_div_decomp
tff(fact_2434_cancel__div__mod__rules_I1_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: 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),A2),B2)),B2)),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) ) ).

% cancel_div_mod_rules(1)
tff(fact_2435_cancel__div__mod__rules_I2_J,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: 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),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2))),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2) ) ).

% cancel_div_mod_rules(2)
tff(fact_2436_div__mult1__eq,axiom,
    ! [A: $tType] :
      ( euclid3128863361964157862miring(A)
     => ! [A2: A,B2: 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2))),C2)) ) ).

% div_mult1_eq
tff(fact_2437_minus__mult__div__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2))) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% minus_mult_div_eq_mod
tff(fact_2438_minus__mod__eq__mult__div,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ).

% minus_mod_eq_mult_div
tff(fact_2439_minus__mod__eq__div__mult,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2) ) ).

% minus_mod_eq_div_mult
tff(fact_2440_minus__div__mult__eq__mod,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),B2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ).

% minus_div_mult_eq_mod
tff(fact_2441_zmde,axiom,
    ! [A: $tType] :
      ( ( group_add(A)
        & semiring_modulo(A) )
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) ) ).

% zmde
tff(fact_2442_mod__le__divisor,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb)),Nb) ) ).

% mod_le_divisor
tff(fact_2443_div__less__mono,axiom,
    ! [A3: nat,B5: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,A3),B5)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A3),Nb) = zero_zero(nat) )
         => ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B5),Nb) = zero_zero(nat) )
           => aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),A3),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B5),Nb)) ) ) ) ) ).

% div_less_mono
tff(fact_2444_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_2445_nat__mod__eq__lemma,axiom,
    ! [Xb: nat,Nb: nat,Y: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Y),Nb) )
     => ( aa(nat,$o,aa(nat,fun(nat,$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),Nb),Q3)) ) ) ).

% nat_mod_eq_lemma
tff(fact_2446_mod__eq__nat2E,axiom,
    ! [M: nat,Q2: nat,Nb: 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),Nb),Q2) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
       => ~ ! [S: nat] : Nb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S)) ) ) ).

% mod_eq_nat2E
tff(fact_2447_mod__eq__nat1E,axiom,
    ! [M: nat,Q2: nat,Nb: 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),Nb),Q2) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
       => ~ ! [S: nat] : M != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Q2),S)) ) ) ).

% mod_eq_nat1E
tff(fact_2448_mod__mult2__eq,axiom,
    ! [M: nat,Nb: 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),Nb),Q2)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),Q2))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb)) ).

% mod_mult2_eq
tff(fact_2449_div__mod__decomp,axiom,
    ! [A3: nat,Nb: nat] : A3 = 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),A3),Nb)),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A3),Nb)) ).

% div_mod_decomp
tff(fact_2450_modulo__nat__def,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb) = 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),Nb)),Nb)) ).

% modulo_nat_def
tff(fact_2451_subset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),Xb: A,B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,Xb),B5))
    <=> $ite(member(A,Xb,A3),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),B5),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)) ) ).

% subset_insert_iff
tff(fact_2452_Diff__single__insert,axiom,
    ! [A: $tType,A3: set(A),Xb: A,B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),B5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,Xb),B5)) ) ).

% Diff_single_insert
tff(fact_2453_atLeast0__atMost__Suc,axiom,
    ! [Nb: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,Nb)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ).

% atLeast0_atMost_Suc
tff(fact_2454_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( set_or1337092689740270186AtMost(nat,M,Nb) = aa(set(nat),set(nat),insert(nat,M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Nb)) ) ) ).

% Icc_eq_insert_lb_nat
tff(fact_2455_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Nb))
     => ( set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,Nb)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ) ).

% atLeastAtMostSuc_conv
tff(fact_2456_atLeastAtMost__insertL,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(set(nat),set(nat),insert(nat,M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Nb)) = set_or1337092689740270186AtMost(nat,M,Nb) ) ) ).

% atLeastAtMost_insertL
tff(fact_2457_remove__subset,axiom,
    ! [A: $tType,Xb: A,S2: set(A)] :
      ( member(A,Xb,S2)
     => aa(set(A),$o,ord_less(set(A),aa(set(A),set(A),minus_minus(set(A),S2),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),S2) ) ).

% remove_subset
tff(fact_2458_set__update__subset__insert,axiom,
    ! [A: $tType,Xs: list(A),I: nat,Xb: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),list_update(A,Xs,I,Xb))),aa(set(A),set(A),insert(A,Xb),aa(list(A),set(A),set2(A),Xs))) ).

% set_update_subset_insert
tff(fact_2459_cong__exp__iff__simps_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num,Q2: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb))),aa(num,A,numeral_numeral(A),bit0(Q2))) != zero_zero(A) ) ).

% cong_exp_iff_simps(3)
tff(fact_2460_mod__mult2__eq_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,M: nat,Nb: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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),Nb))) = 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),A2),aa(nat,A,semiring_1_of_nat(A),M))),aa(nat,A,semiring_1_of_nat(A),Nb)))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(nat,A,semiring_1_of_nat(A),M))) ) ).

% mod_mult2_eq'
tff(fact_2461_field__char__0__class_Oof__nat__div,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,Nb: nat] : aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)) = 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),Nb)))),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% field_char_0_class.of_nat_div
tff(fact_2462_split__mod,axiom,
    ! [P: fun(nat,$o),M: nat,Nb: nat] :
      ( aa(nat,$o,P,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(nat,$o,P,M) )
        & ( ( Nb != zero_zero(nat) )
         => ! [I2: nat,J3: nat] :
              ( aa(nat,$o,ord_less(nat,J3),Nb)
             => ( ( M = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),I2)),J3) )
               => aa(nat,$o,P,J3) ) ) ) ) ) ).

% split_mod
tff(fact_2463_mod__lemma,axiom,
    ! [C2: nat,R3: nat,B2: nat,Q2: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),C2)
     => ( aa(nat,$o,ord_less(nat,R3),B2)
       => 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),B2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Q2),C2))),R3)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),C2)) ) ) ).

% mod_lemma
tff(fact_2464_foldr__length__aux,axiom,
    ! [A: $tType,L: list(A),A2: nat] : foldr(A,nat,aTP_Lamp_dd(A,fun(nat,nat)),L,A2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),aa(list(A),nat,size_size(list(A)),L)) ).

% foldr_length_aux
tff(fact_2465_psubset__insert__iff,axiom,
    ! [A: $tType,A3: set(A),Xb: A,B5: set(A)] :
      ( aa(set(A),$o,ord_less(set(A),A3),aa(set(A),set(A),insert(A,Xb),B5))
    <=> $ite(
          member(A,Xb,B5),
          aa(set(A),$o,ord_less(set(A),A3),B5),
          $ite(member(A,Xb,A3),aa(set(A),$o,ord_less(set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))),B5),aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)) ) ) ).

% psubset_insert_iff
tff(fact_2466_real__of__nat__div__aux,axiom,
    ! [Xb: nat,D2: 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),D2)) = 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),D2))),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),D2))),aa(nat,real,semiring_1_of_nat(real),D2))) ).

% real_of_nat_div_aux
tff(fact_2467_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
tff(fact_2468_cong__exp__iff__simps_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: num,Nb: 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,Nb))),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),Nb)),aa(num,A,numeral_numeral(A),Q2)) = zero_zero(A) ) ) ) ).

% cong_exp_iff_simps(7)
tff(fact_2469_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_2470_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,Nb: 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)))),Nb) = 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)),Nb) ).

% Suc_mod_eq_add3_mod
tff(fact_2471_Suc__times__mod__eq,axiom,
    ! [M: nat,Nb: 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),Nb))),M) = one_one(nat) ) ) ).

% Suc_times_mod_eq
tff(fact_2472_lemma__NBseq__def2,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K7: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K7)
              & ! [N6: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N6))),K7) )
        <=> ? [N7: nat] :
            ! [N6: A] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(B,aa(A,B,X5,N6))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7))) ) ) ).

% lemma_NBseq_def2
tff(fact_2473_lemma__NBseq__def,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [X5: fun(A,B)] :
          ( ? [K7: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),K7)
              & ! [N6: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N6))),K7) )
        <=> ? [N7: nat] :
            ! [N6: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,X5,N6))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N7))) ) ) ).

% lemma_NBseq_def
tff(fact_2474_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)),aa(list(A),set(A),set2(A),list_update(A,L,I,Xb))) = aa(set(A),set(A),insert(A,Xb),aa(list(A),set(A),set2(A),L)) ) ) ).

% insert_swap_set_eq
tff(fact_2475_divmod__digit__0_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ) ) ) ).

% divmod_digit_0(2)
tff(fact_2476_bits__stable__imp__add__self,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(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,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ) ).

% bits_stable_imp_add_self
tff(fact_2477_div__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: 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),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb))),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),A2),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),Nb),M)))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) ) ).

% div_exp_mod_exp_eq
tff(fact_2478_power__mod__div,axiom,
    ! [Xb: nat,Nb: 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))),Nb))),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,Nb),M))) ).

% power_mod_div
tff(fact_2479_verit__le__mono__div,axiom,
    ! [A3: nat,B5: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,A3),B5)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => aa(nat,$o,
            aa(nat,fun(nat,$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),A3),Nb)),
                $ite(aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),B5),Nb) = zero_zero(nat),one_one(nat),zero_zero(nat)))),
            aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),B5),Nb)) ) ) ).

% verit_le_mono_div
tff(fact_2480_divmod__digit__0_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
         => ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2)
           => ( 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),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% divmod_digit_0(1)
tff(fact_2481_mult__exp__mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),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))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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,Nb),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_2482_mod__double__modulus,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: A,Xb: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),M)
         => ( aa(A,$o,aa(A,fun(A,$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_2483_divmod__digit__1_I2_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)))
             => ( aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) ) ) ) ) ) ).

% divmod_digit_1(2)
tff(fact_2484_set__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se5668285175392031749et_bit(A,aa(nat,nat,suc,Nb),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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_se5668285175392031749et_bit(A,Nb,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% set_bit_Suc
tff(fact_2485_divmod__digit__1_I1_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)))
             => ( 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),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2)))),one_one(A)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ) ).

% divmod_digit_1(1)
tff(fact_2486_norm__divide__numeral,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A2)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_divide_numeral
tff(fact_2487_norm__mult__numeral2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,W: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),W))) = aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,A2)),aa(num,real,numeral_numeral(real),W)) ) ).

% norm_mult_numeral2
tff(fact_2488_norm__mult__numeral1,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [W: num,A2: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),W)),A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),W)),real_V7770717601297561774m_norm(A,A2)) ) ).

% norm_mult_numeral1
tff(fact_2489_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,aa(nat,fun(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_2490_norm__le__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Xb)),zero_zero(real))
        <=> ( Xb = zero_zero(A) ) ) ) ).

% norm_le_zero_iff
tff(fact_2491_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_2492_foldr0,axiom,
    ! [Xs: list(real),C2: real,D2: real] : foldr(real,real,plus_plus(real),Xs,aa(real,real,aa(real,fun(real,real),plus_plus(real),C2),D2)) = aa(real,real,aa(real,fun(real,real),plus_plus(real),foldr(real,real,plus_plus(real),Xs,D2)),C2) ).

% foldr0
tff(fact_2493_foldr__same,axiom,
    ! [Xs: list(real),Y: real] :
      ( ! [X2: real,Y5: real] :
          ( member(real,X2,aa(list(real),set(real),set2(real),Xs))
         => ( member(real,Y5,aa(list(real),set(real),set2(real),Xs))
           => ( X2 = Y5 ) ) )
     => ( ! [X2: real] :
            ( member(real,X2,aa(list(real),set(real),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_2494_norm__one,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ( real_V7770717601297561774m_norm(A,one_one(A)) = one_one(real) ) ) ).

% norm_one
tff(fact_2495_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$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_2496_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$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_2497_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_2498_one__mod__exp__eq__one,axiom,
    ! [Nb: 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))),Nb))) = one_one(int) ).

% one_mod_exp_eq_one
tff(fact_2499_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_2500_mod__plus__cong,axiom,
    ! [B2: int,B4: int,Xb: int,X4: int,Y: int,Y6: int,Z5: int] :
      ( ( B2 = B4 )
     => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xb),B4) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X4),B4) )
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Y),B4) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Y6),B4) )
         => ( ( aa(int,int,aa(int,fun(int,int),plus_plus(int),X4),Y6) = Z5 )
           => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Y)),B2) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Z5),B4) ) ) ) ) ) ).

% mod_plus_cong
tff(fact_2501_zmod__helper,axiom,
    ! [Nb: int,M: int,K: int,A2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Nb),M) = K )
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),A2)),M) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),A2)),M) ) ) ).

% zmod_helper
tff(fact_2502_Word_Omod__minus__cong,axiom,
    ! [B2: int,B4: int,Xb: int,X4: int,Y: int,Y6: int,Z5: int] :
      ( ( B2 = B4 )
     => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xb),B4) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),X4),B4) )
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Y),B4) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Y6),B4) )
         => ( ( aa(int,int,minus_minus(int,X4),Y6) = Z5 )
           => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,minus_minus(int,Xb),Y)),B2) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Z5),B4) ) ) ) ) ) ).

% Word.mod_minus_cong
tff(fact_2503_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),M)
     => aa(int,$o,aa(int,fun(int,$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_2504_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_2505_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_2506_zmod__eq__0D,axiom,
    ! [M: int,D2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),D2) = zero_zero(int) )
     => ? [Q3: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q3) ) ).

% zmod_eq_0D
tff(fact_2507_zmod__eq__0__iff,axiom,
    ! [M: int,D2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),M),D2) = zero_zero(int) )
    <=> ? [Q4: int] : M = aa(int,int,aa(int,fun(int,int),times_times(int),D2),Q4) ) ).

% zmod_eq_0_iff
tff(fact_2508_int__mod__ge,axiom,
    ! [A2: int,Nb: int] :
      ( aa(int,$o,ord_less(int,A2),Nb)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),Nb)) ) ) ).

% int_mod_ge
tff(fact_2509_neg__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,ord_less(int,B2),zero_zero(int))
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)),zero_zero(int))
        & aa(int,$o,ord_less(int,B2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)) ) ) ).

% neg_mod_conj
tff(fact_2510_pos__mod__conj,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2))
        & aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)),B2) ) ) ).

% pos_mod_conj
tff(fact_2511_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,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I)
          & aa(int,$o,ord_less(int,I),K) )
        | ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),zero_zero(int))
          & aa(int,$o,ord_less(int,K),I) ) ) ) ).

% zmod_trivial_iff
tff(fact_2512_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(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_2513_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,L),zero_zero(int))
     => aa(int,$o,aa(int,fun(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_2514_int__mod__eq,axiom,
    ! [B2: int,Nb: int,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( aa(int,$o,ord_less(int,B2),Nb)
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),Nb) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),Nb) )
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),Nb) = B2 ) ) ) ) ).

% int_mod_eq
tff(fact_2515_int__mod__lem,axiom,
    ! [Nb: int,B2: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),Nb)
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
          & aa(int,$o,ord_less(int,B2),Nb) )
      <=> ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),Nb) = B2 ) ) ) ).

% int_mod_lem
tff(fact_2516_int__mod__le_H,axiom,
    ! [B2: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,minus_minus(int,B2),Nb))
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),Nb)),aa(int,int,minus_minus(int,B2),Nb)) ) ).

% int_mod_le'
tff(fact_2517_nonneg__mod__div,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2))
          & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) ) ) ) ).

% nonneg_mod_div
tff(fact_2518_zdiv__mono__strict,axiom,
    ! [A3: int,B5: int,Nb: int] :
      ( aa(int,$o,ord_less(int,A3),B5)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),Nb)
       => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),Nb) = zero_zero(int) )
         => ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B5),Nb) = zero_zero(int) )
           => aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),Nb)),aa(int,int,aa(int,fun(int,int),divide_divide(int),B5),Nb)) ) ) ) ) ).

% zdiv_mono_strict
tff(fact_2519_div__mod__decomp__int,axiom,
    ! [A3: int,Nb: int] : A3 = 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),A3),Nb)),Nb)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A3),Nb)) ).

% div_mod_decomp_int
tff(fact_2520_mod__div__equality__div__eq,axiom,
    ! [A2: int,B2: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),B2) = aa(int,int,minus_minus(int,A2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)) ).

% mod_div_equality_div_eq
tff(fact_2521_word__mod__div__equality,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),B2: 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)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Nb),B2)),B2)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),Nb),B2)) = Nb ) ).

% word_mod_div_equality
tff(fact_2522_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_2523_pos__mod__bound2,axiom,
    ! [A2: int] : aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% pos_mod_bound2
tff(fact_2524_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(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_2525_int__mod__ge_H,axiom,
    ! [B2: int,Nb: int] :
      ( aa(int,$o,ord_less(int,B2),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),B2),Nb)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B2),Nb)) ) ) ).

% int_mod_ge'
tff(fact_2526_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),L)
     => ( aa(int,$o,aa(int,fun(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_2527_nat__mod__distrib,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$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_2528_real__of__int__div__aux,axiom,
    ! [Xb: int,D2: 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),D2)) = 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),D2))),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),D2))),aa(int,real,ring_1_of_int(real),D2))) ).

% real_of_int_div_aux
tff(fact_2529_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Nb))
     => ( set_or1337092689740270186AtMost(int,M,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Nb)) = aa(set(int),set(int),insert(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Nb)),set_or1337092689740270186AtMost(int,M,Nb)) ) ) ).

% atLeastAtMostPlus1_int_conv
tff(fact_2530_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_2531_pos__mod__sign2,axiom,
    ! [A2: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),aa(num,int,numeral_numeral(int),bit0(one2)))) ).

% pos_mod_sign2
tff(fact_2532_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_2533_nmod2,axiom,
    ! [Nb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Nb),aa(num,int,numeral_numeral(int),bit0(one2))) = zero_zero(int) )
      | ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Nb),aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) ) ) ).

% nmod2
tff(fact_2534_mod__exp__less__eq__exp,axiom,
    ! [A2: int,Nb: nat] : aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ).

% mod_exp_less_eq_exp
tff(fact_2535_mod__power__lem,axiom,
    ! [A2: int,Nb: nat,M: nat] :
      ( aa(int,$o,ord_less(int,one_one(int)),A2)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),Nb)),aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),M)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb),zero_zero(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),A2),Nb)) ) ) ).

% mod_power_lem
tff(fact_2536_split__zmod,axiom,
    ! [P: fun(int,$o),Nb: int,K: int] :
      ( aa(int,$o,P,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Nb),K))
    <=> ( ( ( K = zero_zero(int) )
         => aa(int,$o,P,Nb) )
        & ( aa(int,$o,ord_less(int,zero_zero(int)),K)
         => ! [I2: int,J3: int] :
              ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
                & aa(int,$o,ord_less(int,J3),K)
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),J3) ) )
             => aa(int,$o,P,J3) ) )
        & ( aa(int,$o,ord_less(int,K),zero_zero(int))
         => ! [I2: int,J3: int] :
              ( ( aa(int,$o,ord_less(int,K),J3)
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
                & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),J3) ) )
             => aa(int,$o,P,J3) ) ) ) ) ).

% split_zmod
tff(fact_2537_int__mod__neg__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R3: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R3) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R3),zero_zero(int))
       => ( aa(int,$o,ord_less(int,B2),R3)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = R3 ) ) ) ) ).

% int_mod_neg_eq
tff(fact_2538_int__mod__pos__eq,axiom,
    ! [A2: int,B2: int,Q2: int,R3: int] :
      ( ( A2 = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),Q2)),R3) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R3)
       => ( aa(int,$o,ord_less(int,R3),B2)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = R3 ) ) ) ) ).

% int_mod_pos_eq
tff(fact_2539_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,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
           => ( aa(int,$o,aa(int,fun(int,$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,aa(int,fun(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_2540_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,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
           => ( aa(int,$o,aa(int,fun(int,$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_2541_zmod__zmult2__eq,axiom,
    ! [C2: int,A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),C2)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),aa(int,int,aa(int,fun(int,int),times_times(int),B2),C2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),B2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),C2))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)) ) ) ).

% zmod_zmult2_eq
tff(fact_2542_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_2543_z1pmod2,axiom,
    ! [B2: 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))),B2)),one_one(int))),aa(num,int,numeral_numeral(int),bit0(one2))) = one_one(int) ).

% z1pmod2
tff(fact_2544_verit__le__mono__div__int,axiom,
    ! [A3: int,B5: int,Nb: int] :
      ( aa(int,$o,ord_less(int,A3),B5)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),Nb)
       => aa(int,$o,
            aa(int,fun(int,$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),A3),Nb)),
                $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),B5),Nb) = zero_zero(int),one_one(int),zero_zero(int)))),
            aa(int,int,aa(int,fun(int,int),divide_divide(int),B5),Nb)) ) ) ).

% verit_le_mono_div_int
tff(fact_2545_split__neg__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Nb: int] :
      ( aa(int,$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),Nb),K)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Nb),K))
      <=> ! [I2: int,J3: int] :
            ( ( aa(int,$o,ord_less(int,K),J3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J3),zero_zero(int))
              & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I2),J3) ) ) ) ).

% split_neg_lemma
tff(fact_2546_split__pos__lemma,axiom,
    ! [K: int,P: fun(int,fun(int,$o)),Nb: int] :
      ( aa(int,$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),Nb),K)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Nb),K))
      <=> ! [I2: int,J3: int] :
            ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),J3)
              & aa(int,$o,ord_less(int,J3),K)
              & ( Nb = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),K),I2)),J3) ) )
           => aa(int,$o,aa(int,fun(int,$o),P,I2),J3) ) ) ) ).

% split_pos_lemma
tff(fact_2547_norm__minus__commute,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),B2)) = real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,B2),A2)) ) ).

% norm_minus_commute
tff(fact_2548_p1mod22k_H,axiom,
    ! [B2: int,Nb: 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))),B2))),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))),Nb))) = 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),B2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)))) ).

% p1mod22k'
tff(fact_2549_p1mod22k,axiom,
    ! [B2: int,Nb: 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))),B2)),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))),Nb))) = 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),B2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)))),one_one(int)) ).

% p1mod22k
tff(fact_2550_pos__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( 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))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = 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),B2),A2))) ) ) ).

% pos_zmod_mult_2
tff(fact_2551_sb__inc__lem,axiom,
    ! [A2: int,K: nat] :
      ( aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K))),zero_zero(int))
     => aa(int,$o,aa(int,fun(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),A2),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),A2),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_2552_neg__zmod__mult__2,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),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))),B2))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),A2)) = 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),B2),one_one(int))),A2))),one_one(int)) ) ) ).

% neg_zmod_mult_2
tff(fact_2553_norm__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A,Y: A] : 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),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y)) ) ).

% norm_mult
tff(fact_2554_norm__ge__zero,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),real_V7770717601297561774m_norm(A,Xb)) ) ).

% norm_ge_zero
tff(fact_2555_norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [A2: A,B2: A] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)) ) ).

% norm_divide
tff(fact_2556_norm__power,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A,Nb: nat] : real_V7770717601297561774m_norm(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Xb)),Nb) ) ).

% norm_power
tff(fact_2557_power__eq__imp__eq__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,Nb: nat,Z: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb) )
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
           => ( real_V7770717601297561774m_norm(A,W) = real_V7770717601297561774m_norm(A,Z) ) ) ) ) ).

% power_eq_imp_eq_norm
tff(fact_2558_nonzero__norm__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)) ) ) ) ).

% nonzero_norm_divide
tff(fact_2559_norm__mult__less,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xb: A,R3: real,Y: A,S3: real] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xb)),R3)
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Y)),S3)
           => aa(real,$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),R3),S3)) ) ) ) ).

% norm_mult_less
tff(fact_2560_norm__mult__ineq,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Xb: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_mult_ineq
tff(fact_2561_norm__add__less,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,R3: real,Y: A,S3: real] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xb)),R3)
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Y)),S3)
           => aa(real,$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),R3),S3)) ) ) ) ).

% norm_add_less
tff(fact_2562_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_2563_norm__add__leD,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),C2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A2)),C2)) ) ) ).

% norm_add_leD
tff(fact_2564_norm__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))),E3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y))),E3) ) ) ).

% norm_triangle_le
tff(fact_2565_norm__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))) ) ).

% norm_triangle_ineq
tff(fact_2566_norm__triangle__mono,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,R3: real,B2: A,S3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,A2)),R3)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,B2)),S3)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),R3),S3)) ) ) ) ).

% norm_triangle_mono
tff(fact_2567_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_2568_norm__power__ineq,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xb: A,Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),real_V7770717601297561774m_norm(A,Xb)),Nb)) ) ).

% norm_power_ineq
tff(fact_2569_norm__triangle__le__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E3: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Xb)),real_V7770717601297561774m_norm(A,Y))),E3)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Xb),Y))),E3) ) ) ).

% norm_triangle_le_diff
tff(fact_2570_norm__diff__triangle__le,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A,E1: real,Z: A,E22: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Xb),Y))),E1)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Y),Z))),E22)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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_le
tff(fact_2571_norm__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),B2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))) ) ).

% norm_triangle_ineq4
tff(fact_2572_norm__triangle__sub,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Xb)),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,Y)),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,Xb),Y)))) ) ).

% norm_triangle_sub
tff(fact_2573_norm__diff__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))),real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))) ) ).

% norm_diff_ineq
tff(fact_2574_norm__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),B2))) ) ).

% norm_triangle_ineq2
tff(fact_2575_power__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [W: A,Nb: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),W),Nb) = one_one(A) )
         => ( ( real_V7770717601297561774m_norm(A,W) = one_one(real) )
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% power_eq_1_iff
tff(fact_2576_norm__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D2)))),aa(real,real,aa(real,fun(real,real),plus_plus(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),C2))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,B2),D2)))) ) ).

% norm_diff_triangle_ineq
tff(fact_2577_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_2578_norm__power__diff,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A,W: A,M: nat] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,Z)),one_one(real))
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,W)),one_one(real))
           => aa(real,$o,aa(real,fun(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_2579_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_2580_rel__of__empty,axiom,
    ! [B: $tType,A: $tType,P: fun(product_prod(A,B),$o)] : rel_of(A,B,aTP_Lamp_di(A,option(B)),P) = bot_bot(set(product_prod(A,B))) ).

% rel_of_empty
tff(fact_2581_vebt__inserti_Osimps,axiom,
    ! [Ta: vEBT_VEBTi,Xb: nat] : aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),vEBT_vebt_inserti,Ta),Xb) = vEBT_case_VEBTi(heap_Time_Heap(vEBT_VEBTi),aTP_Lamp_dx(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),Xb),aTP_Lamp_dy(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),Xb),Ta) ).

% vebt_inserti.simps
tff(fact_2582_VEBTi_Osize_I3_J,axiom,
    ! [X11a: option(product_prod(nat,nat)),X12a: nat,X13a: array(vEBT_VEBTi),X14a: vEBT_VEBTi] : aa(vEBT_VEBTi,nat,size_size(vEBT_VEBTi),vEBT_Nodei(X11a,X12a,X13a,X14a)) = 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),X13a)),aa(vEBT_VEBTi,nat,size_size(vEBT_VEBTi),X14a))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBTi.size(3)
tff(fact_2583_list__every__elemnt__bound__sum__bound,axiom,
    ! [A: $tType,Xs: list(A),F: fun(A,nat),Bound: nat,I: nat] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F,X2)),Bound) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),foldr(nat,nat,plus_plus(nat),aa(list(A),list(nat),map(A,nat,F),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_2584_map__ident,axiom,
    ! [A: $tType,X3: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_dz(A,A)),X3) = X3 ).

% map_ident
tff(fact_2585_length__map,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),aa(list(B),list(A),map(B,A,F),Xs)) = aa(list(B),nat,size_size(list(B)),Xs) ).

% length_map
tff(fact_2586_map__eq__conv,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B),G: fun(B,A)] :
      ( ( aa(list(B),list(A),map(B,A,F),Xs) = aa(list(B),list(A),map(B,A,G),Xs) )
    <=> ! [X: B] :
          ( member(B,X,aa(list(B),set(B),set2(B),Xs))
         => ( aa(B,A,F,X) = aa(B,A,G,X) ) ) ) ).

% map_eq_conv
tff(fact_2587_nth__map,axiom,
    ! [B: $tType,A: $tType,Nb: nat,Xs: list(A),F: fun(A,B)] :
      ( aa(nat,$o,ord_less(nat,Nb),aa(list(A),nat,size_size(list(A)),Xs))
     => ( aa(nat,B,nth(B,aa(list(A),list(B),map(A,B,F),Xs)),Nb) = aa(A,B,F,aa(nat,A,nth(A,Xs),Nb)) ) ) ).

% nth_map
tff(fact_2588_list_Omap__ident,axiom,
    ! [A: $tType,Ta: list(A)] : aa(list(A),list(A),map(A,A,aTP_Lamp_dz(A,A)),Ta) = Ta ).

% list.map_ident
tff(fact_2589_map__eq__imp__length__eq,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(B,A),Xs: list(B),G: fun(C,A),Ys: list(C)] :
      ( ( aa(list(B),list(A),map(B,A,F),Xs) = aa(list(C),list(A),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_2590_list_Omap__cong,axiom,
    ! [B: $tType,A: $tType,Xb: list(A),Ya: list(A),F: fun(A,B),G: fun(A,B)] :
      ( ( Xb = Ya )
     => ( ! [Z3: A] :
            ( member(A,Z3,aa(list(A),set(A),set2(A),Ya))
           => ( aa(A,B,F,Z3) = aa(A,B,G,Z3) ) )
       => ( aa(list(A),list(B),map(A,B,F),Xb) = aa(list(A),list(B),map(A,B,G),Ya) ) ) ) ).

% list.map_cong
tff(fact_2591_list_Omap__cong0,axiom,
    ! [B: $tType,A: $tType,Xb: list(A),F: fun(A,B),G: fun(A,B)] :
      ( ! [Z3: A] :
          ( member(A,Z3,aa(list(A),set(A),set2(A),Xb))
         => ( aa(A,B,F,Z3) = aa(A,B,G,Z3) ) )
     => ( aa(list(A),list(B),map(A,B,F),Xb) = aa(list(A),list(B),map(A,B,G),Xb) ) ) ).

% list.map_cong0
tff(fact_2592_list_Oinj__map__strong,axiom,
    ! [B: $tType,A: $tType,Xb: list(A),Xaa: list(A),F: fun(A,B),Fa: fun(A,B)] :
      ( ! [Z3: A,Za: A] :
          ( member(A,Z3,aa(list(A),set(A),set2(A),Xb))
         => ( member(A,Za,aa(list(A),set(A),set2(A),Xaa))
           => ( ( aa(A,B,F,Z3) = aa(A,B,Fa,Za) )
             => ( Z3 = Za ) ) ) )
     => ( ( aa(list(A),list(B),map(A,B,F),Xb) = aa(list(A),list(B),map(A,B,Fa),Xaa) )
       => ( Xb = Xaa ) ) ) ).

% list.inj_map_strong
tff(fact_2593_map__ext,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),F: fun(A,B),G: fun(A,B)] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => ( aa(A,B,F,X2) = aa(A,B,G,X2) ) )
     => ( aa(list(A),list(B),map(A,B,F),Xs) = aa(list(A),list(B),map(A,B,G),Xs) ) ) ).

% map_ext
tff(fact_2594_map__idI,axiom,
    ! [A: $tType,Xs: list(A),F: fun(A,A)] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => ( aa(A,A,F,X2) = X2 ) )
     => ( aa(list(A),list(A),map(A,A,F),Xs) = Xs ) ) ).

% map_idI
tff(fact_2595_map__cong,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(A),F: fun(A,B),G: fun(A,B)] :
      ( ( Xs = Ys )
     => ( ! [X2: A] :
            ( member(A,X2,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,B,F,X2) = aa(A,B,G,X2) ) )
       => ( aa(list(A),list(B),map(A,B,F),Xs) = aa(list(A),list(B),map(A,B,G),Ys) ) ) ) ).

% map_cong
tff(fact_2596_ex__map__conv,axiom,
    ! [B: $tType,A: $tType,Ys: list(B),F: fun(A,B)] :
      ( ? [Xs3: list(A)] : Ys = aa(list(A),list(B),map(A,B,F),Xs3)
    <=> ! [X: B] :
          ( member(B,X,aa(list(B),set(B),set2(B),Ys))
         => ? [Xa3: A] : X = aa(A,B,F,Xa3) ) ) ).

% ex_map_conv
tff(fact_2597_map__eq__nth__eq,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),L: list(B),L3: list(B),I: nat] :
      ( ( aa(list(B),list(A),map(B,A,F),L) = aa(list(B),list(A),map(B,A,F),L3) )
     => ( aa(B,A,F,aa(nat,B,nth(B,L),I)) = aa(B,A,F,aa(nat,B,nth(B,L3),I)) ) ) ).

% map_eq_nth_eq
tff(fact_2598_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_2599_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_2600_map__upd__eq,axiom,
    ! [B: $tType,A: $tType,I: nat,L: list(A),F: fun(A,B),Xb: A] :
      ( ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),L))
       => ( aa(A,B,F,aa(nat,A,nth(A,L),I)) = aa(A,B,F,Xb) ) )
     => ( aa(list(A),list(B),map(A,B,F),list_update(A,L,I,Xb)) = aa(list(A),list(B),map(A,B,F),L) ) ) ).

% map_upd_eq
tff(fact_2601_TBOUND__len,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: array(A)] : time_TBOUND(nat,array_len(A,Xs),one_one(nat)) ) ).

% TBOUND_len
tff(fact_2602_VEBT__internal_Ocnt_H_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,vEBT_Node(Info,Deg,TreeLista,Summarya)) = 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,Summarya))),foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2),TreeLista),zero_zero(nat))) ).

% VEBT_internal.cnt'.simps(2)
tff(fact_2603_VEBT__internal_Ocnt_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Xb) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
         => ( Y != one_one(nat) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,Deg2,TreeList,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),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2),TreeList),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.cnt'.elims
tff(fact_2604_eq__diff__eq_H,axiom,
    ! [Xb: real,Y: real,Z: real] :
      ( ( Xb = aa(real,real,minus_minus(real,Y),Z) )
    <=> ( Y = aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Z) ) ) ).

% eq_diff_eq'
tff(fact_2605_VEBT__internal_Ospace_H_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_space2,vEBT_Node(Info,Deg,TreeLista,Summarya)) = 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,Summarya))),foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2),TreeLista),zero_zero(nat))) ).

% VEBT_internal.space'.simps(2)
tff(fact_2606_VEBT__internal_Ospace_H_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Xb) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
         => ( Y != aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,Deg2,TreeList,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),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2),TreeList),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.space'.elims
tff(fact_2607_VEBT__internal_Ospace_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_space,vEBT_Node(Info,Deg,TreeLista,Summarya)) = 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,Summarya))),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))),foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space),TreeLista),zero_zero(nat))) ).

% VEBT_internal.space.simps(2)
tff(fact_2608_VEBT__internal_Ospace_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,Xb) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
         => ( Y != aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,Deg2,TreeList,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)),TreeList))),foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space),TreeList),zero_zero(nat))) ) ) ) ) ).

% VEBT_internal.space.elims
tff(fact_2609_upd__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xs: list(A),A2: array(A),Xb: A] :
          ( aa(nat,$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,A2),Xs),array_upd(A,I,Xb,A2),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_ea(nat,fun(list(A),fun(array(A),fun(A,fun(array(A),assn)))),I),Xs),A2),Xb)) ) ) ).

% upd_rule
tff(fact_2610_length__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A2: array(A),Xs: list(A)] : hoare_hoare_triple(nat,aa(list(A),assn,snga_assn(A,A2),Xs),array_len(A,A2),aa(list(A),fun(nat,assn),aTP_Lamp_eb(array(A),fun(list(A),fun(nat,assn)),A2),Xs)) ) ).

% length_rule
tff(fact_2611_VEBTi_Osize__gen_I1_J,axiom,
    ! [X11a: option(product_prod(nat,nat)),X12a: nat,X13a: array(vEBT_VEBTi),X14a: vEBT_VEBTi] : aa(vEBT_VEBTi,nat,vEBT_size_VEBTi,vEBT_Nodei(X11a,X12a,X13a,X14a)) = 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,X13a)),aa(vEBT_VEBTi,nat,vEBT_size_VEBTi,X14a))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBTi.size_gen(1)
tff(fact_2612_length__corresp,axiom,
    ! [B: $tType,A: $tType] :
      ( heap(A)
     => ! [Tree_array: array(A),Tree_is: list(B)] :
          ( ( ex_assn(list(A),snga_assn(A,Tree_array)) = 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_array) ) ) ) ).

% length_corresp
tff(fact_2613_list__every__elemnt__bound__sum__bound__real,axiom,
    ! [A: $tType,Xs: list(A),F: fun(A,real),Bound: real,I: real] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F,X2)),Bound) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,F),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_2614_real__nat__list,axiom,
    ! [A: $tType,F: fun(A,nat),Xs: list(A),C2: nat] : aa(nat,real,semiring_1_of_nat(real),foldr(nat,nat,plus_plus(nat),aa(list(A),list(nat),map(A,nat,F),Xs),C2)) = foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,aTP_Lamp_ec(fun(A,nat),fun(A,real),F)),Xs),aa(nat,real,semiring_1_of_nat(real),C2)) ).

% real_nat_list
tff(fact_2615_f__g__map__foldr__bound,axiom,
    ! [A: $tType,Xs: list(A),F: fun(A,real),C2: real,G: fun(A,real),D2: real] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(A,real,F,X2)),aa(real,real,aa(real,fun(real,real),times_times(real),C2),aa(A,real,G,X2))) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,F),Xs),D2)),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),aa(list(A),list(real),map(A,real,G),Xs),zero_zero(real)))),D2)) ) ).

% f_g_map_foldr_bound
tff(fact_2616_vebt__memberi_Osimps,axiom,
    ! [Ta: vEBT_VEBTi,Xb: nat] : aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),vEBT_vebt_memberi,Ta),Xb) = vEBT_case_VEBTi(heap_Time_Heap($o),aTP_Lamp_ei(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))),Xb),aTP_Lamp_ej(nat,fun($o,fun($o,heap_Time_Heap($o))),Xb),Ta) ).

% vebt_memberi.simps
tff(fact_2617_listsum__bound,axiom,
    ! [A: $tType,Xs: list(A),F: fun(A,real),Y: real] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(A,real,F,X2)) )
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),foldr(real,real,plus_plus(real),aa(list(A),list(real),map(A,real,F),Xs),Y)) ) ).

% listsum_bound
tff(fact_2618_merge__true__star,axiom,
    aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),top_top(assn)),top_top(assn)) = top_top(assn) ).

% merge_true_star
tff(fact_2619_assn__basic__inequalities_I1_J,axiom,
    top_top(assn) != one_one(assn) ).

% assn_basic_inequalities(1)
tff(fact_2620_case__prod__conv,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,fun(C,A)),A2: B,B2: C] : aa(product_prod(B,C),A,product_case_prod(B,C,A,F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)) = aa(C,A,aa(B,fun(C,A),F,A2),B2) ).

% case_prod_conv
tff(fact_2621_map__fst__mk__snd,axiom,
    ! [B: $tType,A: $tType,K: B,L: list(A)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),aa(list(A),list(product_prod(A,B)),map(A,product_prod(A,B),aa(B,fun(A,product_prod(A,B)),aTP_Lamp_ek(B,fun(A,product_prod(A,B))),K)),L)) = L ).

% map_fst_mk_snd
tff(fact_2622_map__snd__mk__fst,axiom,
    ! [B: $tType,A: $tType,K: B,L: list(A)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),aa(list(A),list(product_prod(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_2623_nested__case__prod__simp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F: fun(B,fun(C,fun(D,A))),Xb: product_prod(B,C),Y: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),product_case_prod(B,C,fun(D,A),F),Xb),Y) = aa(product_prod(B,C),A,product_case_prod(B,C,A,aa(D,fun(B,fun(C,A)),aTP_Lamp_el(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F),Y)),Xb) ).

% nested_case_prod_simp
tff(fact_2624_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_2625_prod_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,D: $tType,C: $tType,H: fun(B,A),F: fun(C,fun(D,B)),Prod: product_prod(C,D)] : aa(B,A,H,aa(product_prod(C,D),B,product_case_prod(C,D,B,F),Prod)) = aa(product_prod(C,D),A,product_case_prod(C,D,A,aa(fun(C,fun(D,B)),fun(C,fun(D,A)),aTP_Lamp_em(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),H),F)),Prod) ).

% prod.case_distrib
tff(fact_2626_old_Oprod_Ocase,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,fun(C,A)),X1: B,X23: C] : aa(product_prod(B,C),A,product_case_prod(B,C,A,F),aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X1),X23)) = aa(C,A,aa(B,fun(C,A),F,X1),X23) ).

% old.prod.case
tff(fact_2627_merge__true__star__ctx,axiom,
    ! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),top_top(assn)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),top_top(assn)),P)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),top_top(assn)),P) ).

% merge_true_star_ctx
tff(fact_2628_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,$o,Q,aa(product_prod(B,C),A,product_case_prod(B,C,A,P),Z))
     => ~ ! [X2: B,Y5: C] :
            ( ( Z = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X2),Y5) )
           => ~ aa(A,$o,Q,aa(C,A,aa(B,fun(C,A),P,X2),Y5)) ) ) ).

% case_prodE2
tff(fact_2629_case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : product_case_prod(A,B,C,aTP_Lamp_en(fun(product_prod(A,B),C),fun(A,fun(B,C)),F)) = F ).

% case_prod_eta
tff(fact_2630_cond__case__prod__eta,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C)),G: fun(product_prod(A,B),C)] :
      ( ! [X2: A,Y5: B] : aa(B,C,aa(A,fun(B,C),F,X2),Y5) = 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),Y5))
     => ( product_case_prod(A,B,C,F) = G ) ) ).

% cond_case_prod_eta
tff(fact_2631_fn__fst__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,C)] : aTP_Lamp_eo(fun(A,C),fun(product_prod(A,B),C),F) = product_case_prod(A,B,C,aTP_Lamp_ep(fun(A,C),fun(A,fun(B,C)),F)) ).

% fn_fst_conv
tff(fact_2632_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_eq(A,fun(B,A))),Prod) ).

% fst_def
tff(fact_2633_fn__snd__conv,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(B,C)] : aTP_Lamp_er(fun(B,C),fun(product_prod(A,B),C),F) = product_case_prod(A,B,C,aTP_Lamp_es(fun(B,C),fun(A,fun(B,C)),F)) ).

% fn_snd_conv
tff(fact_2634_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_et(B,fun(A,A))),Prod) ).

% snd_def
tff(fact_2635_pair__list__eqI,axiom,
    ! [B: $tType,A: $tType,Xs: list(product_prod(A,B)),Ys: list(product_prod(A,B))] :
      ( ( aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Xs) = aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),Ys) )
     => ( ( aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Xs) = aa(list(product_prod(A,B)),list(B),map(product_prod(A,B),B,product_snd(A,B)),Ys) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
tff(fact_2636_split__beta,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(B,fun(C,A)),Prod: product_prod(B,C)] : aa(product_prod(B,C),A,product_case_prod(B,C,A,F),Prod) = aa(C,A,aa(B,fun(C,A),F,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_2637_case__prod__beta,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(B,fun(C,A)),P2: product_prod(B,C)] : aa(product_prod(B,C),A,product_case_prod(B,C,A,F),P2) = aa(C,A,aa(B,fun(C,A),F,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_2638_mod__star__trueE,axiom,
    ! [P: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),top_top(assn)),H)
     => ~ ! [H5: product_prod(heap_ext(product_unit),set(nat))] : ~ rep_assn(P,H5) ) ).

% mod_star_trueE
tff(fact_2639_mod__star__trueI,axiom,
    ! [P: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( rep_assn(P,H)
     => rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),top_top(assn)),H) ) ).

% mod_star_trueI
tff(fact_2640_case__prod__rule,axiom,
    ! [A: $tType,B: $tType,C: $tType,Xb: product_prod(A,B),P: assn,F: fun(A,fun(B,heap_Time_Heap(C))),Q: fun(C,assn)] :
      ( ! [A4: A,B3: B] :
          ( ( Xb = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
         => hoare_hoare_triple(C,P,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F,A4),B3),Q) )
     => hoare_hoare_triple(C,P,aa(product_prod(A,B),heap_Time_Heap(C),product_case_prod(A,B,heap_Time_Heap(C),F),Xb),Q) ) ).

% case_prod_rule
tff(fact_2641_split__comp__eq,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,F: fun(D,fun(B,C)),G: fun(A,D)] : aa(fun(A,D),fun(product_prod(A,B),C),aTP_Lamp_eu(fun(D,fun(B,C)),fun(fun(A,D),fun(product_prod(A,B),C)),F),G) = product_case_prod(A,B,C,aa(fun(A,D),fun(A,fun(B,C)),aTP_Lamp_ev(fun(D,fun(B,C)),fun(fun(A,D),fun(A,fun(B,C))),F),G)) ).

% split_comp_eq
tff(fact_2642_case__prod__beta_H,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C)),X3: product_prod(A,B)] : aa(product_prod(A,B),C,product_case_prod(A,B,C,F),X3) = aa(B,C,aa(A,fun(B,C),F,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_2643_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_2644_prod_Osplit__sel,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,$o),F: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
      ( aa(A,$o,P,aa(product_prod(B,C),A,product_case_prod(B,C,A,F),Prod))
    <=> ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
       => aa(A,$o,P,aa(C,A,aa(B,fun(C,A),F,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_2645_prod_Osplit__sel__asm,axiom,
    ! [A: $tType,C: $tType,B: $tType,P: fun(A,$o),F: fun(B,fun(C,A)),Prod: product_prod(B,C)] :
      ( aa(A,$o,P,aa(product_prod(B,C),A,product_case_prod(B,C,A,F),Prod))
    <=> ~ ( ( Prod = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),aa(product_prod(B,C),B,product_fst(B,C),Prod)),aa(product_prod(B,C),C,product_snd(B,C),Prod)) )
          & ~ aa(A,$o,P,aa(C,A,aa(B,fun(C,A),F,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_2646_mod__h__bot__iff_I2_J,axiom,
    ! [H: heap_ext(product_unit)] : rep_assn(top_top(assn),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(2)
tff(fact_2647_mod__star__trueE_H,axiom,
    ! [P: assn,H: product_prod(heap_ext(product_unit),set(nat))] :
      ( rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),top_top(assn)),H)
     => ~ ! [H5: product_prod(heap_ext(product_unit),set(nat))] :
            ( ( aa(product_prod(heap_ext(product_unit),set(nat)),heap_ext(product_unit),product_fst(heap_ext(product_unit),set(nat)),H5) = aa(product_prod(heap_ext(product_unit),set(nat)),heap_ext(product_unit),product_fst(heap_ext(product_unit),set(nat)),H) )
           => ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H5)),aa(product_prod(heap_ext(product_unit),set(nat)),set(nat),product_snd(heap_ext(product_unit),set(nat)),H))
             => ~ rep_assn(P,H5) ) ) ) ).

% mod_star_trueE'
tff(fact_2648_VEBTi_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] : aa(vEBT_VEBTi,nat,vEBT_size_VEBTi,vEBT_Leafi((X21),(X22))) = zero_zero(nat) ).

% VEBTi.size_gen(2)
tff(fact_2649_VEBT__internal_Ocnt_Osimps_I2_J,axiom,
    ! [Info: option(product_prod(nat,nat)),Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] : aa(vEBT_VEBT,real,vEBT_VEBT_cnt,vEBT_Node(Info,Deg,TreeLista,Summarya)) = 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,Summarya))),foldr(real,real,plus_plus(real),aa(list(vEBT_VEBT),list(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt),TreeLista),zero_zero(real))) ).

% VEBT_internal.cnt.simps(2)
tff(fact_2650_VEBT__internal_Ocnt_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: real] :
      ( ( aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Xb) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
         => ( Y != one_one(real) ) )
       => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Info2,Deg2,TreeList,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),aa(list(vEBT_VEBT),list(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt),TreeList),zero_zero(real))) ) ) ) ) ).

% VEBT_internal.cnt.elims
tff(fact_2651_vebt__memberi__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xb: nat,Ta: 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,Ta),Tib),Xb)) ).

% vebt_memberi_refines
tff(fact_2652_of__nat__code__if,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          aa(nat,A,semiring_1_of_nat(A),Nb) = $ite(Nb = zero_zero(nat),zero_zero(A),aa(product_prod(nat,nat),A,product_case_prod(nat,nat,A,aTP_Lamp_ew(nat,fun(nat,A))),divmod_nat(Nb,aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ).

% of_nat_code_if
tff(fact_2653_Suc__if__eq,axiom,
    ! [A: $tType,F: fun(nat,A),H: fun(nat,A),G: A,Nb: nat] :
      ( ! [N2: nat] : aa(nat,A,F,aa(nat,nat,suc,N2)) = aa(nat,A,H,N2)
     => ( ( aa(nat,A,F,zero_zero(nat)) = G )
       => ( aa(nat,A,F,Nb) = $ite(Nb = zero_zero(nat),G,aa(nat,A,H,aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))) ) ) ) ).

% Suc_if_eq
tff(fact_2654_vebt__succi__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xb: nat,Ta: 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,Ta),Tib),Xb)) ).

% vebt_succi_refines
tff(fact_2655_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,$o,aa(A,fun(A,$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_2656_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,$o,aa(A,fun(A,$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,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% mult_le_cancel_iff1
tff(fact_2657_refines__replicate,axiom,
    ! [A: $tType,F: heap_Time_Heap(A),F4: heap_Time_Heap(A),Nb: nat] :
      ( refine_Imp_refines(A,F,F4)
     => refine_Imp_refines(list(A),vEBT_VEBT_replicatei(A,Nb,F),vEBT_VEBT_replicatei(A,Nb,F4)) ) ).

% refines_replicate
tff(fact_2658_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))),F22: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A))))),F23: fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(A)))))] :
      ( ( Tib = Ti )
     => ( ! [A4: $o,B3: $o] : refine_Imp_refines(A,aa($o,heap_Time_Heap(A),aa($o,fun($o,heap_Time_Heap(A)),F1,(A4)),(B3)),aa($o,heap_Time_Heap(A),aa($o,fun($o,heap_Time_Heap(A)),F12,(A4)),(B3)))
       => ( ! [Info2: option(product_prod(nat,nat)),Deg2: 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)))),F22,Info2),Deg2),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)))),F23,Info2),Deg2),TreeArray),Summary))
         => refine_Imp_refines(A,vEBT_case_VEBTi(heap_Time_Heap(A),F22,F1,Tib),vEBT_case_VEBTi(heap_Time_Heap(A),F23,F12,Ti)) ) ) ) ).

% refines_case_VEBTi
tff(fact_2659_case__prodI2,axiom,
    ! [B: $tType,A: $tType,P2: product_prod(A,B),C2: fun(A,fun(B,$o))] :
      ( ! [A4: A,B3: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
         => aa(B,$o,aa(A,fun(B,$o),C2,A4),B3) )
     => aa(product_prod(A,B),$o,product_case_prod(A,B,$o,C2),P2) ) ).

% case_prodI2
tff(fact_2660_case__prodI,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(B,$o)),A2: A,B2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),F,A2),B2)
     => aa(product_prod(A,B),$o,product_case_prod(A,B,$o,F),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)) ) ).

% case_prodI
tff(fact_2661_mem__case__prodI2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: product_prod(A,B),Z: C,C2: fun(A,fun(B,set(C)))] :
      ( ! [A4: A,B3: B] :
          ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
         => member(C,Z,aa(B,set(C),aa(A,fun(B,set(C)),C2,A4),B3)) )
     => member(C,Z,aa(product_prod(A,B),set(C),product_case_prod(A,B,set(C),C2),P2)) ) ).

% mem_case_prodI2
tff(fact_2662_mem__case__prodI,axiom,
    ! [A: $tType,B: $tType,C: $tType,Z: A,C2: fun(B,fun(C,set(A))),A2: B,B2: C] :
      ( member(A,Z,aa(C,set(A),aa(B,fun(C,set(A)),C2,A2),B2))
     => 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),A2),B2))) ) ).

% mem_case_prodI
tff(fact_2663_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] :
      ( ! [A4: A,B3: B] :
          ( ( aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) = P2 )
         => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,A4),B3),Xb) )
     => aa(C,$o,aa(product_prod(A,B),fun(C,$o),product_case_prod(A,B,fun(C,$o),C2),P2),Xb) ) ).

% case_prodI2'
tff(fact_2664_Collect__const,axiom,
    ! [A: $tType,P: $o] :
      collect(A,aTP_Lamp_ex($o,fun(A,$o),(P))) = $ite((P),top_top(set(A)),bot_bot(set(A))) ).

% Collect_const
tff(fact_2665_Collect__const__case__prod,axiom,
    ! [B: $tType,A: $tType,P: $o] :
      collect(product_prod(A,B),product_case_prod(A,B,$o,aTP_Lamp_ey($o,fun(A,fun(B,$o)),(P)))) = $ite((P),top_top(set(product_prod(A,B))),bot_bot(set(product_prod(A,B)))) ).

% Collect_const_case_prod
tff(fact_2666_Diff__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),minus_minus(set(A),A3),top_top(set(A))) = bot_bot(set(A)) ).

% Diff_UNIV
tff(fact_2667_round__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: num] : archimedean_round(A,aa(num,A,numeral_numeral(A),Nb)) = aa(num,int,numeral_numeral(int),Nb) ) ).

% round_numeral
tff(fact_2668_round__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archimedean_round(A,one_one(A)) = one_one(int) ) ) ).

% round_1
tff(fact_2669_fst__divmod__nat,axiom,
    ! [M: nat,Nb: nat] : aa(product_prod(nat,nat),nat,product_fst(nat,nat),divmod_nat(M,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) ).

% fst_divmod_nat
tff(fact_2670_snd__divmod__nat,axiom,
    ! [M: nat,Nb: nat] : aa(product_prod(nat,nat),nat,product_snd(nat,nat),divmod_nat(M,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb) ).

% snd_divmod_nat
tff(fact_2671_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,Y5: C] :
            ( ( P2 = aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),X2),Y5) )
           => ~ member(A,Z,aa(C,set(A),aa(B,fun(C,set(A)),C2,X2),Y5)) ) ) ).

% mem_case_prodE
tff(fact_2672_subset__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),top_top(set(A))) ).

% subset_UNIV
tff(fact_2673_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,Y5: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y5) )
           => ~ aa(B,$o,aa(A,fun(B,$o),C2,X2),Y5) ) ) ).

% case_prodE
tff(fact_2674_case__prodD,axiom,
    ! [A: $tType,B: $tType,F: fun(A,fun(B,$o)),A2: A,B2: B] :
      ( aa(product_prod(A,B),$o,product_case_prod(A,B,$o,F),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),F,A2),B2) ) ).

% case_prodD
tff(fact_2675_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,$o),C2),P2),Z)
     => ~ ! [X2: A,Y5: B] :
            ( ( P2 = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y5) )
           => ~ aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),C2,X2),Y5),Z) ) ) ).

% case_prodE'
tff(fact_2676_case__prodD_H,axiom,
    ! [B: $tType,A: $tType,C: $tType,R: fun(A,fun(B,fun(C,$o))),A2: A,B2: B,C2: C] :
      ( aa(C,$o,aa(product_prod(A,B),fun(C,$o),product_case_prod(A,B,fun(C,$o),R),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)),C2)
     => aa(C,$o,aa(B,fun(C,$o),aa(A,fun(B,fun(C,$o)),R,A2),B2),C2) ) ).

% case_prodD'
tff(fact_2677_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_2678_htt__refine,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Ta: nat,C6: heap_Time_Heap(A)] :
      ( time_htt(A,P,C2,Q,Ta)
     => ( refine_Imp_refines(A,C6,C2)
       => time_htt(A,P,C6,Q,Ta) ) ) ).

% htt_refine
tff(fact_2679_refines__bind,axiom,
    ! [B: $tType,A: $tType,M: heap_Time_Heap(A),M6: heap_Time_Heap(A),F: fun(A,heap_Time_Heap(B)),F4: fun(A,heap_Time_Heap(B))] :
      ( refine_Imp_refines(A,M,M6)
     => ( ! [X2: A] : refine_Imp_refines(B,aa(A,heap_Time_Heap(B),F,X2),aa(A,heap_Time_Heap(B),F4,X2))
       => refine_Imp_refines(B,heap_Time_bind(A,B,M,F),heap_Time_bind(A,B,M6,F4)) ) ) ).

% refines_bind
tff(fact_2680_Product__Type_OCollect__case__prodD,axiom,
    ! [B: $tType,A: $tType,Xb: product_prod(A,B),A3: fun(A,fun(B,$o))] :
      ( member(product_prod(A,B),Xb,collect(product_prod(A,B),product_case_prod(A,B,$o,A3)))
     => aa(B,$o,aa(A,fun(B,$o),A3,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_2681_refines__case__prod__right,axiom,
    ! [C: $tType,B: $tType,A: $tType,M: heap_Time_Heap(C),M6: fun(A,fun(B,heap_Time_Heap(C))),Ta: product_prod(A,B)] :
      ( ! [A4: A,B3: B] : refine_Imp_refines(C,M,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),M6,A4),B3))
     => refine_Imp_refines(C,M,aa(product_prod(A,B),heap_Time_Heap(C),product_case_prod(A,B,heap_Time_Heap(C),M6),Ta)) ) ).

% refines_case_prod_right
tff(fact_2682_prod__case__refines,axiom,
    ! [C: $tType,B: $tType,A: $tType,P2: product_prod(A,B),P5: product_prod(A,B),F: fun(A,fun(B,heap_Time_Heap(C))),F4: fun(A,fun(B,heap_Time_Heap(C)))] :
      ( ( P2 = P5 )
     => ( ! [A4: A,B3: B] : refine_Imp_refines(C,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F,A4),B3),aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F4,A4),B3))
       => refine_Imp_refines(C,aa(product_prod(A,B),heap_Time_Heap(C),product_case_prod(A,B,heap_Time_Heap(C),F),P2),aa(product_prod(A,B),heap_Time_Heap(C),product_case_prod(A,B,heap_Time_Heap(C),F4),P5)) ) ) ).

% prod_case_refines
tff(fact_2683_refines__option,axiom,
    ! [B: $tType,A: $tType,A2: option(A),A6: 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))] :
      ( ( A2 = A6 )
     => ( 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,A2),case_option(heap_Time_Heap(B),A,M12,M23,A6)) ) ) ) ).

% refines_option
tff(fact_2684_not__UNIV__le__Icc,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [L: A,H: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),top_top(set(A))),set_or1337092689740270186AtMost(A,L,H)) ) ).

% not_UNIV_le_Icc
tff(fact_2685_round__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,Xb)),archimedean_round(A,Y)) ) ) ).

% round_mono
tff(fact_2686_divmod__nat__if,axiom,
    ! [M: nat,Nb: nat] :
      divmod_nat(M,Nb) = $ite(
        ( ( Nb = zero_zero(nat) )
        | aa(nat,$o,ord_less(nat,M),Nb) ),
        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_ez(nat,fun(nat,product_prod(nat,nat)))),divmod_nat(aa(nat,nat,minus_minus(nat,M),Nb),Nb)) ) ).

% divmod_nat_if
tff(fact_2687_ceiling__ge__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archimedean_round(A,Xb)),archimedean_ceiling(A,Xb)) ) ).

% ceiling_ge_round
tff(fact_2688_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_cy(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),Q2))) ) ).

% refines_assert'_bind
tff(fact_2689_TBOUND__prod__case,axiom,
    ! [C: $tType,B: $tType,A: $tType,Ta: product_prod(A,B),F: fun(A,fun(B,heap_Time_Heap(C))),Bnd: fun(A,fun(B,nat))] :
      ( ! [A4: A,B3: B] :
          ( ( Ta = aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A4),B3) )
         => time_TBOUND(C,aa(B,heap_Time_Heap(C),aa(A,fun(B,heap_Time_Heap(C)),F,A4),B3),aa(B,nat,aa(A,fun(B,nat),Bnd,A4),B3)) )
     => time_TBOUND(C,aa(product_prod(A,B),heap_Time_Heap(C),product_case_prod(A,B,heap_Time_Heap(C),F),Ta),aa(product_prod(A,B),nat,product_case_prod(A,B,nat,Bnd),Ta)) ) ).

% TBOUND_prod_case
tff(fact_2690_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,$o)),aTP_Lamp_fa(fun(A,option(B)),fun(fun(product_prod(A,B),$o),fun(A,fun(B,$o))),M),P))) ).

% rel_of_def
tff(fact_2691_divmod__nat__def,axiom,
    ! [M: nat,Nb: nat] : divmod_nat(M,Nb) = 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),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb)) ).

% divmod_nat_def
tff(fact_2692_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_fb(num,fun(int,fun(int,product_prod(int,int))),L)),Qr) ).

% divmod_step_int_def
tff(fact_2693_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_fc(num,fun(nat,fun(nat,product_prod(nat,nat))),L)),Qr) ).

% divmod_step_nat_def
tff(fact_2694_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_fd(num,fun(A,fun(A,product_prod(A,A))),L)),Qr) ) ).

% divmod_step_def
tff(fact_2695_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_2696_of__int__round__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(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_2697_of__int__round__ge,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(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_2698_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_2699_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,$o,aa(A,fun(A,$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,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ).

% mult_le_cancel_iff2
tff(fact_2700_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,$o)))))),Xaa: product_prod(vEBT_VEBTi,nat),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: vEBT_VEBTi,Nb: nat] :
      ( ! [Vebt_inserti: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))] :
          ( ! [A7: vEBT_VEBTi,B7: nat,H6: heap_ext(product_unit),H7: heap_ext(product_unit),R5: vEBT_VEBTi,N8: nat] :
              ( heap_Time_effect(vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),Vebt_inserti,A7),B7),H6,H7,R5,N8)
             => aa(nat,$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,$o))))),P,A7),B7),H6),H7),R5),N8) )
         => ! [T4: vEBT_VEBTi,X2: nat,Ta2: heap_ext(product_unit),Xa2: heap_ext(product_unit),R2: vEBT_VEBTi,N2: nat] :
              ( heap_Time_effect(vEBT_VEBTi,vEBT_case_VEBTi(heap_Time_Heap(vEBT_VEBTi),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))),aTP_Lamp_fq(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),Vebt_inserti),X2),aTP_Lamp_dy(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),X2),T4),Ta2,Xa2,R2,N2)
             => aa(nat,$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,$o))))),P,T4),X2),Ta2),Xa2),R2),N2) ) )
     => ( heap_Time_effect(vEBT_VEBTi,aa(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi),product_case_prod(vEBT_VEBTi,nat,heap_Time_Heap(vEBT_VEBTi),vEBT_vebt_inserti),Xaa),H,H3,R3,Nb)
       => aa(nat,$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),H3),R3),Nb) ) ) ).

% vebt_inserti.raw_induct
tff(fact_2701_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,$o)))))),Xaa: product_prod(vEBT_VEBTi,nat),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: option(nat),Nb: nat] :
      ( ! [Vebt_predi: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))] :
          ( ! [A7: vEBT_VEBTi,B7: nat,H6: heap_ext(product_unit),H7: heap_ext(product_unit),R5: option(nat),N8: nat] :
              ( heap_Time_effect(option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),Vebt_predi,A7),B7),H6,H7,R5,N8)
             => aa(nat,$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,$o))))),P,A7),B7),H6),H7),R5),N8) )
         => ! [T4: vEBT_VEBTi,X2: nat,Ta2: heap_ext(product_unit),Xa2: heap_ext(product_unit),R2: option(nat),N2: nat] :
              ( heap_Time_effect(option(nat),vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_fw(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Vebt_predi),X2),aTP_Lamp_ch(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),X2),T4),Ta2,Xa2,R2,N2)
             => aa(nat,$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,$o))))),P,T4),X2),Ta2),Xa2),R2),N2) ) )
     => ( heap_Time_effect(option(nat),aa(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat)),product_case_prod(vEBT_VEBTi,nat,heap_Time_Heap(option(nat)),vEBT_vebt_predi),Xaa),H,H3,R3,Nb)
       => aa(nat,$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),H3),R3),Nb) ) ) ).

% vebt_predi.raw_induct
tff(fact_2702_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,$o)))))),Xaa: product_prod(vEBT_VEBTi,nat),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: option(nat),Nb: nat] :
      ( ! [Vebt_succi: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))] :
          ( ! [A7: vEBT_VEBTi,B7: nat,H6: heap_ext(product_unit),H7: heap_ext(product_unit),R5: option(nat),N8: nat] :
              ( heap_Time_effect(option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),Vebt_succi,A7),B7),H6,H7,R5,N8)
             => aa(nat,$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,$o))))),P,A7),B7),H6),H7),R5),N8) )
         => ! [T4: vEBT_VEBTi,X2: nat,Ta2: heap_ext(product_unit),Xa2: heap_ext(product_unit),R2: option(nat),N2: nat] :
              ( heap_Time_effect(option(nat),vEBT_case_VEBTi(heap_Time_Heap(option(nat)),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))))),aTP_Lamp_gc(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Vebt_succi),X2),aTP_Lamp_cq(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),X2),T4),Ta2,Xa2,R2,N2)
             => aa(nat,$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,$o))))),P,T4),X2),Ta2),Xa2),R2),N2) ) )
     => ( heap_Time_effect(option(nat),aa(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat)),product_case_prod(vEBT_VEBTi,nat,heap_Time_Heap(option(nat)),vEBT_vebt_succi),Xaa),H,H3,R3,Nb)
       => aa(nat,$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),H3),R3),Nb) ) ) ).

% vebt_succi.raw_induct
tff(fact_2703_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),H3: heap_ext(product_unit),R3: $o,Nb: nat] :
      ( ! [Vebt_memberi: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))] :
          ( ! [A7: vEBT_VEBTi,B7: nat,H6: heap_ext(product_unit),H7: heap_ext(product_unit),R5: $o,N8: nat] :
              ( heap_Time_effect($o,aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),Vebt_memberi,A7),B7),H6,H7,(R5),N8)
             => 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,A7),B7),H6),H7),(R5)),N8) )
         => ! [T4: vEBT_VEBTi,X2: nat,Ta2: heap_ext(product_unit),Xa2: heap_ext(product_unit),R2: $o,N2: nat] :
              ( heap_Time_effect($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_gi(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_memberi),X2),aTP_Lamp_ej(nat,fun($o,fun($o,heap_Time_Heap($o))),X2),T4),Ta2,Xa2,(R2),N2)
             => aa(nat,$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,T4),X2),Ta2),Xa2),(R2)),N2) ) )
     => ( heap_Time_effect($o,aa(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o),product_case_prod(vEBT_VEBTi,nat,heap_Time_Heap($o),vEBT_vebt_memberi),Xaa),H,H3,(R3),Nb)
       => aa(nat,$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),H3),(R3)),Nb) ) ) ).

% vebt_memberi.raw_induct
tff(fact_2704_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_gj(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),L)),Qr) ).

% divmod_step_integer_def
tff(fact_2705_VEBT__internal_Ovebt__succi_H_Osimps,axiom,
    ! [Ta: 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,Ta),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_hc(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Ta),Xb),aTP_Lamp_cq(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Xb),Tib) ).

% VEBT_internal.vebt_succi'.simps
tff(fact_2706_vebt__buildupi__refines,axiom,
    ! [Nb: nat] : refine_Imp_refines(vEBT_VEBTi,vEBT_vebt_buildupi(Nb),vEBT_V739175172307565963ildupi(Nb)) ).

% vebt_buildupi_refines
tff(fact_2707_vebt__inserti__refines,axiom,
    ! [Tib: vEBT_VEBTi,Xb: nat,Ta: 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,Ta),Tib),Xb)) ).

% vebt_inserti_refines
tff(fact_2708_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_hd($o,fun(fun(A,fun(B,$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_2709_Collect__case__prod__mono,axiom,
    ! [B: $tType,A: $tType,A3: fun(A,fun(B,$o)),B5: fun(A,fun(B,$o))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),A3),B5)
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(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,A3))),collect(product_prod(A,B),product_case_prod(A,B,$o,B5))) ) ).

% Collect_case_prod_mono
tff(fact_2710_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,$o,aTP_Lamp_he(A,fun(B,$o))),Prod) ).

% prod.disc_eq_case
tff(fact_2711_UNIV__def,axiom,
    ! [A: $tType] : top_top(set(A)) = collect(A,aTP_Lamp_bn(A,$o)) ).

% UNIV_def
tff(fact_2712_VEBT_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),F22: fun($o,fun($o,B)),VEBT: vEBT_VEBT] : aa(B,A,H,vEBT_case_VEBT(B,F1,F22,VEBT)) = vEBT_case_VEBT(A,aa(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),aTP_Lamp_hf(fun(B,A),fun(fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,B)))),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A))))),H),F1),aa(fun($o,fun($o,B)),fun($o,fun($o,A)),aTP_Lamp_cs(fun(B,A),fun(fun($o,fun($o,B)),fun($o,fun($o,A))),H),F22),VEBT) ).

% VEBT.case_distrib
tff(fact_2713_is__Node__def,axiom,
    ! [VEBT: vEBT_VEBT] :
      ( vEBT_is_Node(VEBT)
    <=> ? [X112: option(product_prod(nat,nat)),X122: nat,X132: list(vEBT_VEBT),X142: vEBT_VEBT] : VEBT = vEBT_Node(X112,X122,X132,X142) ) ).

% is_Node_def
tff(fact_2714_VEBT_OdiscI_I1_J,axiom,
    ! [VEBT: vEBT_VEBT,X11a: option(product_prod(nat,nat)),X12a: nat,X13a: list(vEBT_VEBT),X14a: vEBT_VEBT] :
      ( ( VEBT = vEBT_Node(X11a,X12a,X13a,X14a) )
     => vEBT_is_Node(VEBT) ) ).

% VEBT.discI(1)
tff(fact_2715_VEBT_Odisc_I1_J,axiom,
    ! [X11a: option(product_prod(nat,nat)),X12a: nat,X13a: list(vEBT_VEBT),X14a: vEBT_VEBT] : vEBT_is_Node(vEBT_Node(X11a,X12a,X13a,X14a)) ).

% VEBT.disc(1)
tff(fact_2716_VEBT_Osimps_I5_J,axiom,
    ! [A: $tType,F1: fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,A)))),F22: fun($o,fun($o,A)),X11a: option(product_prod(nat,nat)),X12a: nat,X13a: list(vEBT_VEBT),X14a: vEBT_VEBT] : vEBT_case_VEBT(A,F1,F22,vEBT_Node(X11a,X12a,X13a,X14a)) = 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,X11a),X12a),X13a),X14a) ).

% VEBT.simps(5)
tff(fact_2717_option_Othe__def,axiom,
    ! [A: $tType,Option: option(A)] : aa(option(A),A,the2(A),Option) = case_option(A,A,undefined(A),aTP_Lamp_dz(A,A),Option) ).

% option.the_def
tff(fact_2718_one__natural_Orsp,axiom,
    one_one(nat) = one_one(nat) ).

% one_natural.rsp
tff(fact_2719_one__integer_Orsp,axiom,
    one_one(int) = one_one(int) ).

% one_integer.rsp
tff(fact_2720_less__eq__integer__code_I1_J,axiom,
    aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),zero_zero(code_integer)) ).

% less_eq_integer_code(1)
tff(fact_2721_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_2722_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_2723_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_2724_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_2725_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),H3: heap_ext(product_unit),R3: $o,Nb: nat] :
      ( ! [Vebt_memberi2: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))))] :
          ( ! [A7: vEBT_VEBT,B7: vEBT_VEBTi,Ba: nat,H6: heap_ext(product_unit),H7: heap_ext(product_unit),R5: $o,N8: nat] :
              ( heap_Time_effect($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($o))),Vebt_memberi2,A7),B7),Ba),H6,H7,(R5),N8)
             => aa(nat,$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,A7),B7),Ba),H6),H7),(R5)),N8) )
         => ! [T4: vEBT_VEBT,Ti2: vEBT_VEBTi,X2: nat,Ta2: heap_ext(product_unit),Tia: heap_ext(product_unit),Xa2: $o,N2: nat] :
              ( heap_Time_effect($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($o)))))),aTP_Lamp_ht(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))))),Vebt_memberi2),T4),X2),aTP_Lamp_ej(nat,fun($o,fun($o,heap_Time_Heap($o))),X2),Ti2),Ta2,Tia,(Xa2),N2)
             => 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,$o)))))),P,T4),Ti2),X2),Ta2),Tia),(Xa2)),N2) ) )
     => ( heap_Time_effect($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($o)),vEBT_V854960066525838166emberi)),Xaa),H,H3,(R3),Nb)
       => aa(nat,$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),H3),(R3)),Nb) ) ) ).

% VEBT_internal.vebt_memberi'.raw_induct
tff(fact_2726_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_2727_VEBT__internal_Ovebt__memberi_H_Osimps,axiom,
    ! [Ta: 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($o))),vEBT_V854960066525838166emberi,Ta),Tib),Xb) = vEBT_case_VEBTi(heap_Time_Heap($o),aa(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))),aTP_Lamp_ih(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))))),Ta),Xb),aTP_Lamp_ej(nat,fun($o,fun($o,heap_Time_Heap($o))),Xb),Tib) ).

% VEBT_internal.vebt_memberi'.simps
tff(fact_2728_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,$o))))))),Xaa: product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: vEBT_VEBTi,Nb: nat] :
      ( ! [Vebt_inserti2: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))))] :
          ( ! [A7: vEBT_VEBT,B7: vEBT_VEBTi,Ba: nat,H6: heap_ext(product_unit),H7: heap_ext(product_unit),R5: vEBT_VEBTi,N8: nat] :
              ( heap_Time_effect(vEBT_VEBTi,aa(nat,heap_Time_Heap(vEBT_VEBTi),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Vebt_inserti2,A7),B7),Ba),H6,H7,R5,N8)
             => aa(nat,$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,A7),B7),Ba),H6),H7),R5),N8) )
         => ! [T4: vEBT_VEBT,Ti2: vEBT_VEBTi,X2: nat,Ta2: heap_ext(product_unit),Tia: heap_ext(product_unit),Xa2: vEBT_VEBTi,N2: 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_je(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),T4),X2),aTP_Lamp_dy(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),X2),Ti2),Ta2,Tia,Xa2,N2)
             => 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,$o)))))),P,T4),Ti2),X2),Ta2),Tia),Xa2),N2) ) )
     => ( heap_Time_effect(vEBT_VEBTi,aa(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi),product_case_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(vEBT_VEBTi),product_case_prod(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)),vEBT_V3964819847710782039nserti)),Xaa),H,H3,R3,Nb)
       => aa(nat,$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),H3),R3),Nb) ) ) ).

% VEBT_internal.vebt_inserti'.raw_induct
tff(fact_2729_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,$o))))))),Xaa: product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: option(nat),Nb: nat] :
      ( ! [Vebt_predi2: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))] :
          ( ! [A7: vEBT_VEBT,B7: vEBT_VEBTi,Ba: nat,H6: heap_ext(product_unit),H7: heap_ext(product_unit),R5: option(nat),N8: nat] :
              ( heap_Time_effect(option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Vebt_predi2,A7),B7),Ba),H6,H7,R5,N8)
             => aa(nat,$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,A7),B7),Ba),H6),H7),R5),N8) )
         => ! [T4: vEBT_VEBT,Ti2: vEBT_VEBTi,X2: nat,Ta2: heap_ext(product_unit),Tia: heap_ext(product_unit),Xa2: option(nat),N2: 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_jv(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_predi2),T4),X2),aTP_Lamp_ch(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),X2),Ti2),Ta2,Tia,Xa2,N2)
             => 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,$o)))))),P,T4),Ti2),X2),Ta2),Tia),Xa2),N2) ) )
     => ( heap_Time_effect(option(nat),aa(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat)),product_case_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(option(nat)),product_case_prod(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_VEBT_vebt_predi)),Xaa),H,H3,R3,Nb)
       => aa(nat,$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),H3),R3),Nb) ) ) ).

% VEBT_internal.vebt_predi'.raw_induct
tff(fact_2730_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,$o))))))),Xaa: product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: option(nat),Nb: nat] :
      ( ! [Vebt_succi2: fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))] :
          ( ! [A7: vEBT_VEBT,B7: vEBT_VEBTi,Ba: nat,H6: heap_ext(product_unit),H7: heap_ext(product_unit),R5: option(nat),N8: nat] :
              ( heap_Time_effect(option(nat),aa(nat,heap_Time_Heap(option(nat)),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Vebt_succi2,A7),B7),Ba),H6,H7,R5,N8)
             => aa(nat,$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,A7),B7),Ba),H6),H7),R5),N8) )
         => ! [T4: vEBT_VEBT,Ti2: vEBT_VEBTi,X2: nat,Ta2: heap_ext(product_unit),Tia: heap_ext(product_unit),Xa2: option(nat),N2: 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_kk(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_succi2),T4),X2),aTP_Lamp_cq(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),X2),Ti2),Ta2,Tia,Xa2,N2)
             => 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,$o)))))),P,T4),Ti2),X2),Ta2),Tia),Xa2),N2) ) )
     => ( heap_Time_effect(option(nat),aa(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat)),product_case_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat,heap_Time_Heap(option(nat)),product_case_prod(vEBT_VEBT,vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))),vEBT_VEBT_vebt_succi)),Xaa),H,H3,R3,Nb)
       => aa(nat,$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),H3),R3),Nb) ) ) ).

% VEBT_internal.vebt_succi'.raw_induct
tff(fact_2731_VEBT__internal_Ovebt__inserti_H_Osimps,axiom,
    ! [Ta: 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,Ta),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_lh(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))),Ta),Xb),aTP_Lamp_dy(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),Xb),Tib) ).

% VEBT_internal.vebt_inserti'.simps
tff(fact_2732_VEBT__internal_Ovebt__predi_H_Osimps,axiom,
    ! [Ta: 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,Ta),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_lw(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))))),Ta),Xb),aTP_Lamp_ch(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Xb),Tib) ).

% VEBT_internal.vebt_predi'.simps
tff(fact_2733_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_2734_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_2735_divmod__algorithm__code_I6_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Nb: num] : unique8689654367752047608divmod(A,aa(num,num,bit1,M),bit0(Nb)) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_lx(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,Nb)) ) ).

% divmod_algorithm_code(6)
tff(fact_2736_neg__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),zero_zero(int))
     => ( eucl_rel_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),one_one(int)),B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R3))
       => 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))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),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))),R3)),one_one(int)))) ) ) ).

% neg_eucl_rel_int_mult_2
tff(fact_2737_divmod__algorithm__code_I7_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Nb: num] :
          unique8689654367752047608divmod(A,bit0(M),aa(num,num,bit1,Nb)) = $ite(aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),M),Nb),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,Nb),unique8689654367752047608divmod(A,bit0(M),bit0(aa(num,num,bit1,Nb))))) ) ).

% divmod_algorithm_code(7)
tff(fact_2738_divmod__algorithm__code_I8_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Nb: num] :
          unique8689654367752047608divmod(A,aa(num,num,bit1,M),aa(num,num,bit1,Nb)) = $ite(aa(num,$o,ord_less(num,M),Nb),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,Nb),unique8689654367752047608divmod(A,aa(num,num,bit1,M),bit0(aa(num,num,bit1,Nb))))) ) ).

% divmod_algorithm_code(8)
tff(fact_2739_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_2740_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_2741_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_2742_divmod__algorithm__code_I3_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : unique8689654367752047608divmod(A,one2,bit0(Nb)) = 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_2743_divmod__algorithm__code_I4_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : unique8689654367752047608divmod(A,one2,aa(num,num,bit1,Nb)) = 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_2744_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_2745_one__div__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb)) = aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,one2,Nb)) ) ).

% one_div_numeral
tff(fact_2746_one__mod__numeral,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Nb: num] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(num,A,numeral_numeral(A),Nb)) = aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,one2,Nb)) ) ).

% one_mod_numeral
tff(fact_2747_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_2748_divmod__algorithm__code_I5_J,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Nb: num] : unique8689654367752047608divmod(A,bit0(M),bit0(Nb)) = aa(product_prod(A,A),product_prod(A,A),product_case_prod(A,A,product_prod(A,A),aTP_Lamp_ly(A,fun(A,product_prod(A,A)))),unique8689654367752047608divmod(A,M,Nb)) ) ).

% divmod_algorithm_code(5)
tff(fact_2749_unique__remainder,axiom,
    ! [A2: int,B2: int,Q2: int,R3: int,Q5: int,R4: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R3))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R4))
       => ( R3 = R4 ) ) ) ).

% unique_remainder
tff(fact_2750_unique__quotient,axiom,
    ! [A2: int,B2: int,Q2: int,R3: int,Q5: int,R4: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R3))
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q5),R4))
       => ( Q2 = Q5 ) ) ) ).

% unique_quotient
tff(fact_2751_VEBT_Odisc__eq__case_I1_J,axiom,
    ! [VEBT: vEBT_VEBT] :
      ( vEBT_is_Node(VEBT)
    <=> vEBT_case_VEBT($o,aTP_Lamp_lz(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,$o)))),aTP_Lamp_ma($o,fun($o,$o)),VEBT) ) ).

% VEBT.disc_eq_case(1)
tff(fact_2752_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_2753_mod__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R3: 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),R3))
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L) = R3 ) ) ).

% mod_int_unique
tff(fact_2754_div__int__unique,axiom,
    ! [K: int,L: int,Q2: int,R3: 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),R3))
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),K),L) = Q2 ) ) ).

% div_int_unique
tff(fact_2755_divmod__integer_H__def,axiom,
    ! [M: num,Nb: num] : unique8689654367752047608divmod(code_integer,M,Nb) = 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),divide_divide(code_integer),aa(num,code_integer,numeral_numeral(code_integer),M)),aa(num,code_integer,numeral_numeral(code_integer),Nb))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),aa(num,code_integer,numeral_numeral(code_integer),M)),aa(num,code_integer,numeral_numeral(code_integer),Nb))) ).

% divmod_integer'_def
tff(fact_2756_fst__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Nb: num] : aa(product_prod(A,A),A,product_fst(A,A),unique8689654367752047608divmod(A,M,Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% fst_divmod
tff(fact_2757_snd__divmod,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Nb: num] : aa(product_prod(A,A),A,product_snd(A,A),unique8689654367752047608divmod(A,M,Nb)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Nb)) ) ).

% snd_divmod
tff(fact_2758_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_2759_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_2760_divmod__int__def,axiom,
    ! [M: num,Nb: num] : unique8689654367752047608divmod(int,M,Nb) = 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),Nb))),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Nb))) ).

% divmod_int_def
tff(fact_2761_divmod__def,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Nb: num] : unique8689654367752047608divmod(A,M,Nb) = 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),Nb))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Nb))) ) ).

% divmod_def
tff(fact_2762_divmod_H__nat__def,axiom,
    ! [M: num,Nb: num] : unique8689654367752047608divmod(nat,M,Nb) = 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),Nb))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Nb))) ).

% divmod'_nat_def
tff(fact_2763_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q2: int,R3: 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),R3))
    <=> ( ( 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)),R3) )
        & $ite(
            aa(int,$o,ord_less(int,zero_zero(int)),L),
            ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R3)
            & aa(int,$o,ord_less(int,R3),L) ),
            $ite(
              aa(int,$o,ord_less(int,L),zero_zero(int)),
              ( aa(int,$o,ord_less(int,L),R3)
              & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),R3),zero_zero(int)) ),
              Q2 = zero_zero(int) ) ) ) ) ).

% eucl_rel_int_iff
tff(fact_2764_divmod__divmod__step,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Nb: num] :
          unique8689654367752047608divmod(A,M,Nb) = $ite(aa(num,$o,ord_less(num,M),Nb),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,Nb,unique8689654367752047608divmod(A,M,bit0(Nb)))) ) ).

% divmod_divmod_step
tff(fact_2765_pos__eucl__rel__int__mult__2,axiom,
    ! [B2: int,A2: int,Q2: int,R3: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R3))
       => 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))),A2)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),B2),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))),R3)))) ) ) ).

% pos_eucl_rel_int_mult_2
tff(fact_2766_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_mz(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_2767_divides__aux__eq,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [Q2: A,R3: A] :
          ( unique5940410009612947441es_aux(A,aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Q2),R3))
        <=> ( R3 = zero_zero(A) ) ) ) ).

% divides_aux_eq
tff(fact_2768_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) ) )
         => ~ ! [N2: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
               => ( Y != $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2),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),N2),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),N2),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),N2),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),N2),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),N2),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),N2),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),N2),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),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.elims
tff(fact_2769_flip__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se8732182000553998342ip_bit(A,aa(nat,nat,suc,Nb),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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,Nb,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% flip_bit_Suc
tff(fact_2770_unset__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_se2638667681897837118et_bit(A,zero_zero(nat),A2) = 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),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% unset_bit_0
tff(fact_2771_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_nq(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_2772_dvd__0__right,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : aa(A,$o,dvd_dvd(A,A2),zero_zero(A)) ) ).

% dvd_0_right
tff(fact_2773_dvd__0__left__iff,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,zero_zero(A)),A2)
        <=> ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left_iff
tff(fact_2774_dvd__add__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2))
        <=> aa(A,$o,dvd_dvd(A,A2),B2) ) ) ).

% dvd_add_triv_right_iff
tff(fact_2775_dvd__add__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> aa(A,$o,dvd_dvd(A,A2),B2) ) ) ).

% dvd_add_triv_left_iff
tff(fact_2776_div__dvd__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,$o,dvd_dvd(A,A2),C2)
           => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2))
            <=> aa(A,$o,dvd_dvd(A,B2),C2) ) ) ) ) ).

% div_dvd_div
tff(fact_2777_unset__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se2638667681897837118et_bit(int,Nb,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% unset_bit_nonnegative_int_iff
tff(fact_2778_flip__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se8732182000553998342ip_bit(int,Nb,K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% flip_bit_nonnegative_int_iff
tff(fact_2779_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_2780_curry__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(product_prod(B,C),A),A2: B,B2: C] : aa(C,A,aa(B,fun(C,A),product_curry(B,C,A,F),A2),B2) = aa(product_prod(B,C),A,F,aa(C,product_prod(B,C),aa(B,fun(C,product_prod(B,C)),product_Pair(B,C),A2),B2)) ).

% curry_conv
tff(fact_2781_case__prod__curry,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(product_prod(A,B),C)] : product_case_prod(A,B,C,product_curry(A,B,C,F)) = F ).

% case_prod_curry
tff(fact_2782_curry__case__prod,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(A,fun(B,C))] : product_curry(A,B,C,product_case_prod(A,B,C,F)) = F ).

% curry_case_prod
tff(fact_2783_dvd__times__right__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2))
          <=> aa(A,$o,dvd_dvd(A,B2),C2) ) ) ) ).

% dvd_times_right_cancel_iff
tff(fact_2784_dvd__times__left__cancel__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2))
          <=> aa(A,$o,dvd_dvd(A,B2),C2) ) ) ) ).

% dvd_times_left_cancel_iff
tff(fact_2785_dvd__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
        <=> ( ( C2 = zero_zero(A) )
            | aa(A,$o,dvd_dvd(A,A2),B2) ) ) ) ).

% dvd_mult_cancel_right
tff(fact_2786_dvd__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
        <=> ( ( C2 = zero_zero(A) )
            | aa(A,$o,dvd_dvd(A,A2),B2) ) ) ) ).

% dvd_mult_cancel_left
tff(fact_2787_dvd__add__times__triv__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)),B2))
        <=> aa(A,$o,dvd_dvd(A,A2),B2) ) ) ).

% dvd_add_times_triv_left_iff
tff(fact_2788_dvd__add__times__triv__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)))
        <=> aa(A,$o,dvd_dvd(A,A2),B2) ) ) ).

% dvd_add_times_triv_right_iff
tff(fact_2789_unit__prod,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
           => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A)) ) ) ) ).

% unit_prod
tff(fact_2790_div__add,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A2)
         => ( aa(A,$o,dvd_dvd(A,C2),B2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).

% div_add
tff(fact_2791_unit__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
           => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),one_one(A)) ) ) ) ).

% unit_div
tff(fact_2792_unit__div__1__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)),one_one(A)) ) ) ).

% unit_div_1_unit
tff(fact_2793_unit__div__1__div__1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),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)),A2)) = A2 ) ) ) ).

% unit_div_1_div_1
tff(fact_2794_dvd__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)),A2) = B2 ) ) ) ).

% dvd_div_mult_self
tff(fact_2795_dvd__mult__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2)) = B2 ) ) ) ).

% dvd_mult_div_cancel
tff(fact_2796_div__diff,axiom,
    ! [A: $tType] :
      ( idom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A2)
         => ( aa(A,$o,dvd_dvd(A,C2),B2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,A2),B2)),C2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ) ).

% div_diff
tff(fact_2797_dvd__imp__mod__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),A2) = zero_zero(A) ) ) ) ).

% dvd_imp_mod_0
tff(fact_2798_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_2799_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_2800_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
    <=> ( ( K = zero_zero(nat) )
        | aa(nat,$o,dvd_dvd(nat,M),Nb) ) ) ).

% nat_mult_dvd_cancel_disj
tff(fact_2801_unit__mult__div__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) ) ) ) ).

% unit_mult_div_div
tff(fact_2802_unit__div__mult__self,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),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),B2),A2)),A2) = B2 ) ) ) ).

% unit_div_mult_self
tff(fact_2803_odd__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> ~ ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ) ).

% odd_add
tff(fact_2804_even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
          <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ) ).

% even_add
tff(fact_2805_even__mult__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ) ).

% even_mult_iff
tff(fact_2806_even__mod__2__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% even_mod_2_iff
tff(fact_2807_even__Suc,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,Nb))
    <=> ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ) ).

% even_Suc
tff(fact_2808_even__Suc__Suc__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Nb)))
    <=> aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ) ).

% even_Suc_Suc_iff
tff(fact_2809_dvd__numeral__simp,axiom,
    ! [A: $tType] :
      ( unique1627219031080169319umeral(A)
     => ! [M: num,Nb: num] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),M)),aa(num,A,numeral_numeral(A),Nb))
        <=> unique5940410009612947441es_aux(A,unique8689654367752047608divmod(A,Nb,M)) ) ) ).

% dvd_numeral_simp
tff(fact_2810_even__plus__one__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)))
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% even_plus_one_iff
tff(fact_2811_even__diff,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,minus_minus(A,A2),B2))
        <=> 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),A2),B2)) ) ) ).

% even_diff
tff(fact_2812_odd__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% odd_Suc_div_two
tff(fact_2813_even__Suc__div__two,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ) ).

% even_Suc_div_two
tff(fact_2814_even__succ__div__2,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( 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)),A2)),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_2
tff(fact_2815_odd__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),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),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ) ) ).

% odd_succ_div_two
tff(fact_2816_even__succ__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ).

% even_succ_div_two
tff(fact_2817_even__power,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            & aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ) ) ).

% even_power
tff(fact_2818_zero__le__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( aa(nat,$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,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq_numeral
tff(fact_2819_power__less__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),zero_zero(A))
        <=> ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
            & aa(A,$o,ord_less(A,A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq
tff(fact_2820_power__less__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,ord_less(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),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,A2),zero_zero(A)) ) ) ) ).

% power_less_zero_eq_numeral
tff(fact_2821_even__of__nat,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ) ) ).

% even_of_nat
tff(fact_2822_odd__Suc__minus__one,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(nat,nat,suc,aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))) = Nb ) ) ).

% odd_Suc_minus_one
tff(fact_2823_even__diff__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,minus_minus(nat,M),Nb))
    <=> ( aa(nat,$o,ord_less(nat,M),Nb)
        | 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),Nb)) ) ) ).

% even_diff_nat
tff(fact_2824_odd__two__times__div__two__succ,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( 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),A2),aa(num,A,numeral_numeral(A),bit0(one2))))),one_one(A)) = A2 ) ) ) ).

% odd_two_times_div_two_succ
tff(fact_2825_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] :
          ( aa(A,$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))),Nb)),one_one(A)))
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% semiring_parity_class.even_mask_iff
tff(fact_2826_zero__less__power__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W)))
        <=> ( ( aa(num,nat,numeral_numeral(nat),W) = zero_zero(nat) )
            | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
              & ( A2 != zero_zero(A) ) )
            | ( ~ aa(nat,$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)),A2) ) ) ) ) ).

% zero_less_power_eq_numeral
tff(fact_2827_odd__two__times__div__two__nat,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( 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),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,nat,minus_minus(nat,Nb),one_one(nat)) ) ) ).

% odd_two_times_div_two_nat
tff(fact_2828_power__le__zero__eq__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W))),zero_zero(A))
        <=> ( aa(nat,$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,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
              | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(num,nat,numeral_numeral(nat),W))
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq_numeral
tff(fact_2829_even__succ__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
           => ( 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)),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) ) ) ) ) ).

% even_succ_div_exp
tff(fact_2830_even__succ__mod__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
           => ( 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)),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = 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),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb))) ) ) ) ) ).

% even_succ_mod_exp
tff(fact_2831_dvd__refl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,$o,dvd_dvd(A,A2),A2) ) ).

% dvd_refl
tff(fact_2832_dvd__trans,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,$o,dvd_dvd(A,B2),C2)
           => aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% dvd_trans
tff(fact_2833_dvd__antisym,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,M),Nb)
     => ( aa(nat,$o,dvd_dvd(nat,Nb),M)
       => ( M = Nb ) ) ) ).

% dvd_antisym
tff(fact_2834_of__nat__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Nb: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,semiring_1_of_nat(A),M)),aa(nat,A,semiring_1_of_nat(A),Nb))
        <=> aa(nat,$o,dvd_dvd(nat,M),Nb) ) ) ).

% of_nat_dvd_iff
tff(fact_2835_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_nr(C,fun(product_prod(A,B),C),C2)),X3),Xa) = C2 ).

% curry_K
tff(fact_2836_dvd__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,zero_zero(A)),A2)
         => ( A2 = zero_zero(A) ) ) ) ).

% dvd_0_left
tff(fact_2837_dvd__add__right__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
          <=> aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% dvd_add_right_iff
tff(fact_2838_dvd__add__left__iff,axiom,
    ! [A: $tType] :
      ( comm_s4317794764714335236cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),C2)
         => ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2))
          <=> aa(A,$o,dvd_dvd(A,A2),B2) ) ) ) ).

% dvd_add_left_iff
tff(fact_2839_dvd__add,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,$o,dvd_dvd(A,A2),C2)
           => aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ) ) ).

% dvd_add
tff(fact_2840_one__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A] : aa(A,$o,dvd_dvd(A,one_one(A)),A2) ) ).

% one_dvd
tff(fact_2841_unit__imp__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => aa(A,$o,dvd_dvd(A,B2),A2) ) ) ).

% unit_imp_dvd
tff(fact_2842_dvd__unit__imp__unit,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
           => aa(A,$o,dvd_dvd(A,A2),one_one(A)) ) ) ) ).

% dvd_unit_imp_unit
tff(fact_2843_dvdE,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ~ ! [K3: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),B2),K3) ) ) ).

% dvdE
tff(fact_2844_dvdI,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [A2: A,B2: A,K: A] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K) )
         => aa(A,$o,dvd_dvd(A,B2),A2) ) ) ).

% dvdI
tff(fact_2845_dvd__def,axiom,
    ! [A: $tType] :
      ( dvd(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
        <=> ? [K4: A] : A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B2),K4) ) ) ).

% dvd_def
tff(fact_2846_dvd__mult,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),C2)
         => aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ).

% dvd_mult
tff(fact_2847_dvd__mult2,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ).

% dvd_mult2
tff(fact_2848_dvd__mult__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)
         => aa(A,$o,dvd_dvd(A,A2),C2) ) ) ).

% dvd_mult_left
tff(fact_2849_dvd__triv__left,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% dvd_triv_left
tff(fact_2850_mult__dvd__mono,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
         => ( aa(A,$o,dvd_dvd(A,C2),D2)
           => aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ).

% mult_dvd_mono
tff(fact_2851_dvd__mult__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)
         => aa(A,$o,dvd_dvd(A,B2),C2) ) ) ).

% dvd_mult_right
tff(fact_2852_dvd__triv__right,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] : aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) ) ).

% dvd_triv_right
tff(fact_2853_dvd__diff__commute,axiom,
    ! [A: $tType] :
      ( euclid5891614535332579305n_ring(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,minus_minus(A,C2),B2))
        <=> aa(A,$o,dvd_dvd(A,A2),aa(A,A,minus_minus(A,B2),C2)) ) ) ).

% dvd_diff_commute
tff(fact_2854_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_2855_dvd__div__eq__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A2)
         => ( aa(A,$o,dvd_dvd(A,C2),B2)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
            <=> ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_iff
tff(fact_2856_dvd__div__eq__cancel,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) )
         => ( aa(A,$o,dvd_dvd(A,C2),A2)
           => ( aa(A,$o,dvd_dvd(A,C2),B2)
             => ( A2 = B2 ) ) ) ) ) ).

% dvd_div_eq_cancel
tff(fact_2857_div__div__div__same,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [D2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,D2),B2)
         => ( aa(A,$o,dvd_dvd(A,B2),A2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),D2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ) ).

% div_div_div_same
tff(fact_2858_dvd__power__same,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xb: A,Y: A,Nb: nat] :
          ( aa(A,$o,dvd_dvd(A,Xb),Y)
         => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Nb)) ) ) ).

% dvd_power_same
tff(fact_2859_dvd__mod__iff,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B2)
         => ( aa(A,$o,dvd_dvd(A,C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2))
          <=> aa(A,$o,dvd_dvd(A,C2),A2) ) ) ) ).

% dvd_mod_iff
tff(fact_2860_dvd__mod__imp__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2))
         => ( aa(A,$o,dvd_dvd(A,C2),B2)
           => aa(A,$o,dvd_dvd(A,C2),A2) ) ) ) ).

% dvd_mod_imp_dvd
tff(fact_2861_dvd__mod,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [K: A,M: A,Nb: A] :
          ( aa(A,$o,dvd_dvd(A,K),M)
         => ( aa(A,$o,dvd_dvd(A,K),Nb)
           => aa(A,$o,dvd_dvd(A,K),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),M),Nb)) ) ) ) ).

% dvd_mod
tff(fact_2862_mod__mod__cancel,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B2)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),C2) ) ) ) ).

% mod_mod_cancel
tff(fact_2863_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),M)
     => ( aa(nat,$o,dvd_dvd(nat,K),Nb)
       => aa(nat,$o,dvd_dvd(nat,K),aa(nat,nat,minus_minus(nat,M),Nb)) ) ) ).

% dvd_diff_nat
tff(fact_2864_unset__bit__less__eq,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),bit_se2638667681897837118et_bit(int,Nb,K)),K) ).

% unset_bit_less_eq
tff(fact_2865_subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aTP_Lamp_ns(A,fun(A,$o),A2))),collect(A,aTP_Lamp_ns(A,fun(A,$o),B2)))
        <=> aa(A,$o,dvd_dvd(A,A2),B2) ) ) ).

% subset_divisors_dvd
tff(fact_2866_strict__subset__divisors__dvd,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,ord_less(set(A),collect(A,aTP_Lamp_ns(A,fun(A,$o),A2))),collect(A,aTP_Lamp_ns(A,fun(A,$o),B2)))
        <=> ( aa(A,$o,dvd_dvd(A,A2),B2)
            & ~ aa(A,$o,dvd_dvd(A,B2),A2) ) ) ) ).

% strict_subset_divisors_dvd
tff(fact_2867_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_2868_minf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S3: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z3)
         => ( ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S3))
          <=> ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S3)) ) ) ) ).

% minf(10)
tff(fact_2869_minf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S3: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,X3),Z3)
         => ( aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S3))
          <=> aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S3)) ) ) ) ).

% minf(9)
tff(fact_2870_pinf_I10_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S3: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z3),X3)
         => ( ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S3))
          <=> ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S3)) ) ) ) ).

% pinf(10)
tff(fact_2871_pinf_I9_J,axiom,
    ! [A: $tType] :
      ( ( plus(A)
        & linorder(A)
        & dvd(A) )
     => ! [D2: A,S3: A] :
        ? [Z3: A] :
        ! [X3: A] :
          ( aa(A,$o,ord_less(A,Z3),X3)
         => ( aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S3))
          <=> aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),S3)) ) ) ) ).

% pinf(9)
tff(fact_2872_dvd__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% dvd_div_eq_0_iff
tff(fact_2873_is__unit__mult__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),one_one(A))
        <=> ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
            & aa(A,$o,dvd_dvd(A,B2),one_one(A)) ) ) ) ).

% is_unit_mult_iff
tff(fact_2874_dvd__mult__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2))
          <=> aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% dvd_mult_unit_iff
tff(fact_2875_mult__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)
          <=> aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% mult_unit_dvd_iff
tff(fact_2876_dvd__mult__unit__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
          <=> aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% dvd_mult_unit_iff'
tff(fact_2877_mult__unit__dvd__iff_H,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2)
          <=> aa(A,$o,dvd_dvd(A,B2),C2) ) ) ) ).

% mult_unit_dvd_iff'
tff(fact_2878_unit__mult__left__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_left_cancel
tff(fact_2879_unit__mult__right__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_mult_right_cancel
tff(fact_2880_div__plus__div__distrib__dvd__right,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_right
tff(fact_2881_div__plus__div__distrib__dvd__left,axiom,
    ! [A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ) ).

% div_plus_div_distrib_dvd_left
tff(fact_2882_unit__div__cancel,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) )
          <=> ( B2 = C2 ) ) ) ) ).

% unit_div_cancel
tff(fact_2883_div__unit__dvd__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2)
          <=> aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% div_unit_dvd_iff
tff(fact_2884_dvd__div__unit__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2))
          <=> aa(A,$o,dvd_dvd(A,A2),C2) ) ) ) ).

% dvd_div_unit_iff
tff(fact_2885_dvd__div__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)),C2) ) ) ) ).

% dvd_div_mult
tff(fact_2886_div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ) ) ).

% div_mult_swap
tff(fact_2887_div__div__eq__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),B2)
         => ( aa(A,$o,dvd_dvd(A,B2),A2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% div_div_eq_right
tff(fact_2888_dvd__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)),A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ).

% dvd_div_mult2_eq
tff(fact_2889_dvd__mult__imp__div,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2)
         => aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) ) ) ).

% dvd_mult_imp_div
tff(fact_2890_div__mult__div__if__dvd,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,D2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( aa(A,$o,dvd_dvd(A,D2),C2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),D2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) ) ) ) ) ).

% div_mult_div_if_dvd
tff(fact_2891_div__power,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,Nb: nat] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),Nb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ).

% div_power
tff(fact_2892_mod__0__imp__dvd,axiom,
    ! [A: $tType] :
      ( semiring_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = zero_zero(A) )
         => aa(A,$o,dvd_dvd(A,B2),A2) ) ) ).

% mod_0_imp_dvd
tff(fact_2893_dvd__eq__mod__eq__0,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),B2)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),A2) = zero_zero(A) ) ) ) ).

% dvd_eq_mod_eq_0
tff(fact_2894_mod__eq__0__iff__dvd,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = zero_zero(A) )
        <=> aa(A,$o,dvd_dvd(A,B2),A2) ) ) ).

% mod_eq_0_iff_dvd
tff(fact_2895_dvd__power__le,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Xb: A,Y: A,Nb: nat,M: nat] :
          ( aa(A,$o,dvd_dvd(A,Xb),Y)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
           => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),M)) ) ) ) ).

% dvd_power_le
tff(fact_2896_power__le__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat,B2: A,M: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),B2)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
           => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),B2) ) ) ) ).

% power_le_dvd
tff(fact_2897_le__imp__power__dvd,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% le_imp_power_dvd
tff(fact_2898_mod__eq__dvd__iff,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,C2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),C2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),B2),C2) )
        <=> aa(A,$o,dvd_dvd(A,C2),aa(A,A,minus_minus(A,A2),B2)) ) ) ).

% mod_eq_dvd_iff
tff(fact_2899_dvd__minus__mod,axiom,
    ! [A: $tType] :
      ( semidom_modulo(A)
     => ! [B2: A,A2: A] : aa(A,$o,dvd_dvd(A,B2),aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2))) ) ).

% dvd_minus_mod
tff(fact_2900_nat__dvd__not__less,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(nat,$o,ord_less(nat,M),Nb)
       => ~ aa(nat,$o,dvd_dvd(nat,Nb),M) ) ) ).

% nat_dvd_not_less
tff(fact_2901_dvd__minus__self,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,minus_minus(nat,Nb),M))
    <=> ( aa(nat,$o,ord_less(nat,Nb),M)
        | aa(nat,$o,dvd_dvd(nat,M),Nb) ) ) ).

% dvd_minus_self
tff(fact_2902_less__eq__dvd__minus,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(nat,$o,dvd_dvd(nat,M),Nb)
      <=> aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,minus_minus(nat,Nb),M)) ) ) ).

% less_eq_dvd_minus
tff(fact_2903_dvd__diffD1,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),aa(nat,nat,minus_minus(nat,M),Nb))
     => ( aa(nat,$o,dvd_dvd(nat,K),M)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
         => aa(nat,$o,dvd_dvd(nat,K),Nb) ) ) ) ).

% dvd_diffD1
tff(fact_2904_dvd__diffD,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),aa(nat,nat,minus_minus(nat,M),Nb))
     => ( aa(nat,$o,dvd_dvd(nat,K),Nb)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
         => aa(nat,$o,dvd_dvd(nat,K),M) ) ) ) ).

% dvd_diffD
tff(fact_2905_even__unset__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2638667681897837118et_bit(A,M,A2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            | ( M = zero_zero(nat) ) ) ) ) ).

% even_unset_bit_iff
tff(fact_2906_even__flip__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se8732182000553998342ip_bit(A,M,A2))
        <=> ~ ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            <=> ( M = zero_zero(nat) ) ) ) ) ).

% even_flip_bit_iff
tff(fact_2907_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_2908_curry__def,axiom,
    ! [C: $tType,A: $tType,B: $tType,X3: fun(product_prod(A,B),C),Xa: A,Xb2: B] : aa(B,C,aa(A,fun(B,C),product_curry(A,B,C,X3),Xa),Xb2) = 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),Xb2)) ).

% curry_def
tff(fact_2909_even__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: num] : aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,A,numeral_numeral(A),bit0(Nb))) ) ).

% even_numeral
tff(fact_2910_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_2911_unit__dvdE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [C3: A] : B2 != aa(A,A,aa(A,fun(A,A),times_times(A),A2),C3) ) ) ) ).

% unit_dvdE
tff(fact_2912_unit__div__eq__0__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% unit_div_eq_0_iff
tff(fact_2913_dvd__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,A2),B2)
           => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = C2 )
            <=> ( B2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2) ) ) ) ) ) ).

% dvd_div_eq_mult
tff(fact_2914_div__dvd__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,B2),A2)
           => ( aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2)
            <=> aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2)) ) ) ) ) ).

% div_dvd_iff_mult
tff(fact_2915_dvd__div__iff__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,C2),B2)
           => ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))
            <=> aa(A,$o,dvd_dvd(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ) ).

% dvd_div_iff_mult
tff(fact_2916_dvd__div__div__eq__mult,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( C2 != zero_zero(A) )
           => ( aa(A,$o,dvd_dvd(A,A2),B2)
             => ( aa(A,$o,dvd_dvd(A,C2),D2)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),D2),C2) )
                <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),D2) ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
tff(fact_2917_inf__period_I4_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D6: A,Ta: A] :
          ( aa(A,$o,dvd_dvd(A,D2),D6)
         => ! [X3: A,K5: A] :
              ( ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Ta))
            <=> ~ aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,X3),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D6))),Ta)) ) ) ) ).

% inf_period(4)
tff(fact_2918_inf__period_I3_J,axiom,
    ! [A: $tType] :
      ( ( comm_ring(A)
        & dvd(A) )
     => ! [D2: A,D6: A,Ta: A] :
          ( aa(A,$o,dvd_dvd(A,D2),D6)
         => ! [X3: A,K5: A] :
              ( aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),X3),Ta))
            <=> aa(A,$o,dvd_dvd(A,D2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,X3),aa(A,A,aa(A,fun(A,A),times_times(A),K5),D6))),Ta)) ) ) ) ).

% inf_period(3)
tff(fact_2919_unit__eq__div1,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = C2 )
          <=> ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% unit_eq_div1
tff(fact_2920_unit__eq__div2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),B2) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = C2 ) ) ) ) ).

% unit_eq_div2
tff(fact_2921_div__mult__unit2,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,B2),A2)
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% div_mult_unit2
tff(fact_2922_unit__div__commute,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ) ) ).

% unit_div_commute
tff(fact_2923_unit__div__mult__swap,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) ) ) ) ).

% unit_div_mult_swap
tff(fact_2924_is__unit__div__mult2__eq,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,$o,dvd_dvd(A,C2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)),C2) ) ) ) ) ).

% is_unit_div_mult2_eq
tff(fact_2925_unit__imp__mod__eq__0,axiom,
    ! [A: $tType] :
      ( euclid3725896446679973847miring(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = zero_zero(A) ) ) ) ).

% unit_imp_mod_eq_0
tff(fact_2926_is__unit__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),one_one(A))
        <=> ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% is_unit_power_iff
tff(fact_2927_dvd__imp__le,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,K),Nb)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ) ).

% dvd_imp_le
tff(fact_2928_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),Nb))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => aa(nat,$o,dvd_dvd(nat,M),Nb) ) ) ).

% dvd_mult_cancel
tff(fact_2929_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,Nb: 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),Nb))
      <=> aa(nat,$o,dvd_dvd(nat,M),Nb) ) ) ).

% nat_mult_dvd_cancel1
tff(fact_2930_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb))
    <=> ~ aa(nat,$o,dvd_dvd(nat,Nb),M) ) ).

% mod_greater_zero_iff_not_dvd
tff(fact_2931_mod__eq__dvd__iff__nat,axiom,
    ! [Nb: nat,M: nat,Q2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),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),Nb),Q2) )
      <=> aa(nat,$o,dvd_dvd(nat,Q2),aa(nat,nat,minus_minus(nat,M),Nb)) ) ) ).

% mod_eq_dvd_iff_nat
tff(fact_2932_real__of__nat__div,axiom,
    ! [D2: nat,Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,D2),Nb)
     => ( aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),D2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,semiring_1_of_nat(real),D2)) ) ) ).

% real_of_nat_div
tff(fact_2933_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_2934_odd__even__add,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2)
           => aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% odd_even_add
tff(fact_2935_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_2936_evenE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ~ ! [B3: A] : A2 != aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B3) ) ) ).

% evenE
tff(fact_2937_bit__eq__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2) )
            & ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ) ) ) ).

% bit_eq_rec
tff(fact_2938_is__unitE,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ~ ( ( A2 != zero_zero(A) )
             => ! [B3: A] :
                  ( ( B3 != zero_zero(A) )
                 => ( aa(A,$o,dvd_dvd(A,B3),one_one(A))
                   => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) = B3 )
                     => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B3) = A2 )
                       => ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B3) = one_one(A) )
                         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2) != aa(A,A,aa(A,fun(A,A),times_times(A),C2),B3) ) ) ) ) ) ) ) ) ) ).

% is_unitE
tff(fact_2939_is__unit__div__mult__cancel__left,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_left
tff(fact_2940_is__unit__div__mult__cancel__right,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,$o,dvd_dvd(A,B2),one_one(A))
           => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),B2) ) ) ) ) ).

% is_unit_div_mult_cancel_right
tff(fact_2941_odd__numeral,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: num] : ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb))) ) ).

% odd_numeral
tff(fact_2942_dvd__power__iff,axiom,
    ! [A: $tType] :
      ( algebraic_semidom(A)
     => ! [Xb: A,M: nat,Nb: nat] :
          ( ( Xb != zero_zero(A) )
         => ( aa(A,$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),Nb))
          <=> ( aa(A,$o,dvd_dvd(A,Xb),one_one(A))
              | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ) ) ).

% dvd_power_iff
tff(fact_2943_dvd__power,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: nat,Xb: A] :
          ( ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
            | ( Xb = one_one(A) ) )
         => aa(A,$o,dvd_dvd(A,Xb),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)) ) ) ).

% dvd_power
tff(fact_2944_even__even__mod__4__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
    <=> aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))))) ) ).

% even_even_mod_4_iff
tff(fact_2945_dvd__mult__cancel1,axiom,
    ! [M: nat,Nb: 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),Nb)),M)
      <=> ( Nb = one_one(nat) ) ) ) ).

% dvd_mult_cancel1
tff(fact_2946_dvd__mult__cancel2,axiom,
    ! [M: nat,Nb: 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),Nb),M)),M)
      <=> ( Nb = one_one(nat) ) ) ) ).

% dvd_mult_cancel2
tff(fact_2947_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),I),Nb))
     => ( aa(nat,$o,ord_less(nat,one_one(nat)),I)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ).

% power_dvd_imp_le
tff(fact_2948_dvd__minus__add,axiom,
    ! [Q2: nat,Nb: nat,R3: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q2),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Q2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R3),M))
       => ( aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,minus_minus(nat,Nb),Q2))
        <=> aa(nat,$o,dvd_dvd(nat,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),R3),M)),Q2))) ) ) ) ).

% dvd_minus_add
tff(fact_2949_diff__mod__le,axiom,
    ! [A2: nat,D2: nat,B2: nat] :
      ( aa(nat,$o,ord_less(nat,A2),D2)
     => ( aa(nat,$o,dvd_dvd(nat,B2),D2)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,A2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),A2),B2))),aa(nat,nat,minus_minus(nat,D2),B2)) ) ) ).

% diff_mod_le
tff(fact_2950_mod__nat__eqI,axiom,
    ! [R3: nat,Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,R3),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R3),M)
       => ( aa(nat,$o,dvd_dvd(nat,Nb),aa(nat,nat,minus_minus(nat,M),R3))
         => ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb) = R3 ) ) ) ) ).

% mod_nat_eqI
tff(fact_2951_even__two__times__div__two,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( 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),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) = A2 ) ) ) ).

% even_two_times_div_two
tff(fact_2952_even__iff__mod__2__eq__zero,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ) ) ).

% even_iff_mod_2_eq_zero
tff(fact_2953_odd__iff__mod__2__eq__one,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
        <=> ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = one_one(A) ) ) ) ).

% odd_iff_mod_2_eq_one
tff(fact_2954_power__mono__odd,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ).

% power_mono_odd
tff(fact_2955_odd__pos,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ).

% odd_pos
tff(fact_2956_even__set__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se5668285175392031749et_bit(A,M,A2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            & ( M != zero_zero(nat) ) ) ) ) ).

% even_set_bit_iff
tff(fact_2957_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)
     => ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),K),Nb))
      <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ).

% dvd_power_iff_le
tff(fact_2958_oddE,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ~ ! [B3: A] : A2 != 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))),B3)),one_one(A)) ) ) ).

% oddE
tff(fact_2959_mod2__eq__if,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = $ite(aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2),zero_zero(A),one_one(A)) ) ).

% mod2_eq_if
tff(fact_2960_parity__cases,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] :
          ( ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
           => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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))),A2)
             => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) != one_one(A) ) ) ) ) ).

% parity_cases
tff(fact_2961_zero__le__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
        <=> ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
            | ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ) ).

% zero_le_power_eq
tff(fact_2962_zero__le__odd__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ) ).

% zero_le_odd_power
tff(fact_2963_zero__le__even__power,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ).

% zero_le_even_power
tff(fact_2964_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_2965_zero__less__power__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))
        <=> ( ( Nb = zero_zero(nat) )
            | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
              & ( A2 != zero_zero(A) ) )
            | ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
              & aa(A,$o,ord_less(A,zero_zero(A)),A2) ) ) ) ) ).

% zero_less_power_eq
tff(fact_2966_even__mask__div__iff_H,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Nb: nat] :
          ( aa(A,$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))),Nb)))
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ).

% even_mask_div_iff'
tff(fact_2967_power__le__zero__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),zero_zero(A))
        <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
            & ( ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
              | ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
                & ( A2 = zero_zero(A) ) ) ) ) ) ) ).

% power_le_zero_eq
tff(fact_2968_even__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Nb),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,Nb),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% even_mod_4_div_2
tff(fact_2969_even__mask__div__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Nb: nat] :
          ( aa(A,$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))),Nb)))
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) = zero_zero(A) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ) ).

% even_mask_div_iff
tff(fact_2970_odd__mod__4__div__2,axiom,
    ! [Nb: nat] :
      ( ( aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Nb),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,Nb),aa(nat,nat,suc,zero_zero(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ).

% odd_mod_4_div_2
tff(fact_2971_Bernoulli__inequality__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => aa(real,$o,aa(real,fun(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),Nb)),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)),Nb)) ) ).

% Bernoulli_inequality_even
tff(fact_2972_VEBT__internal_OT__vebt__buildupi_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      vEBT_V441764108873111860ildupi(aa(nat,nat,suc,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb),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),Nb),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),Nb),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),Nb),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),Nb),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),Nb),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),Nb),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),Nb),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),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))) ).

% VEBT_internal.T_vebt_buildupi.simps(3)
tff(fact_2973_even__mult__exp__div__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,Nb: nat] :
          ( aa(A,$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),A2),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))),Nb)))
        <=> ( aa(nat,$o,ord_less(nat,Nb),M)
            | ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) = zero_zero(A) )
            | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
              & aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),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,Nb),M)))) ) ) ) ) ).

% even_mult_exp_div_exp_iff
tff(fact_2974_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)) ) )
         => ~ ! [N2: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
               => ( Y != $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2),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),N2),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),N2),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),N2),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),N2),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),N2),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),N2),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),N2),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),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.elims
tff(fact_2975_VEBT__internal_OTb_H_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      vEBT_VEBT_Tb2(aa(nat,nat,suc,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb),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),Nb),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),Nb),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),Nb),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),Nb),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),Nb),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),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% VEBT_internal.Tb'.simps(3)
tff(fact_2976_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)) ) )
         => ~ ! [N2: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
               => ( Y != $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2),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),N2),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),N2),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),N2),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),N2),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),N2),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),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb'.elims
tff(fact_2977_VEBT__internal_OTb_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      vEBT_VEBT_Tb(aa(nat,nat,suc,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb),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),Nb),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),Nb),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),Nb),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),Nb),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),Nb),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),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ).

% VEBT_internal.Tb.simps(3)
tff(fact_2978_VEBT__internal_OT__vebt__buildupi_H_Osimps_I3_J,axiom,
    ! [Nb: nat] :
      vEBT_V9176841429113362141ildupi(aa(nat,nat,suc,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb),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),Nb),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),Nb),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),Nb),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),Nb),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),Nb),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),Nb),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),Nb),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),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))) ).

% VEBT_internal.T_vebt_buildupi'.simps(3)
tff(fact_2979_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,
    ! [Vaa: nat] :
      vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,aa(nat,nat,suc,Vaa))) = $ite(
        aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),
        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,Vaa))),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,Vaa))),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_2980_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,
    ! [Vaa: nat] :
      vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,aa(nat,nat,suc,Vaa))) = $ite(
        aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),
        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,Vaa))),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,Vaa))),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_2981_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)) ) )
         => ~ ! [N2: nat] :
                ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
               => ( Y != $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2),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),N2),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),N2),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),N2),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),N2),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),N2),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),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) ) ) ) ) ) ).

% VEBT_internal.Tb.elims
tff(fact_2982_unset__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se2638667681897837118et_bit(A,aa(nat,nat,suc,Nb),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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_se2638667681897837118et_bit(A,Nb,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% unset_bit_Suc
tff(fact_2983_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_2984_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_2985_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_oj(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_2986_pow__divides__pow__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
         => ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb))
          <=> aa(A,$o,dvd_dvd(A,A2),B2) ) ) ) ).

% pow_divides_pow_iff
tff(fact_2987_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_on(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_oq(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va2),half)) ) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.elims
tff(fact_2988_VEBT__internal_Ovebt__buildupi_H_Osimps_I3_J,axiom,
    ! [Vaa: nat] :
      vEBT_V739175172307565963ildupi(aa(nat,nat,suc,aa(nat,nat,suc,Vaa))) = $ite(
        aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),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_on(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Vaa),half)) ),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),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_oq(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Vaa),half)) ) ) ).

% VEBT_internal.vebt_buildupi'.simps(3)
tff(fact_2989_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_os(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_ou(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va2),half)) ) ) ) ) ) ) ) ).

% vebt_buildupi.elims
tff(fact_2990_Heap__Time__Monad_Obind__mono,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,B5: fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(C)),C5: fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D)))] :
      ( 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),B5)
     => ( ! [Y5: C] : comple7038119648293358887notone(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),heap_Time_Heap_ord(D),aa(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D)),C5,Y5))
       => comple7038119648293358887notone(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),heap_Time_Heap_ord(D),aa(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D)),aTP_Lamp_ow(fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(C)),fun(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),B5),C5)) ) ) ).

% Heap_Time_Monad.bind_mono
tff(fact_2991_VEBT__internal_Ovebt__inserti_H_Ofixp__induct,axiom,
    ! [P: fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),$o)] :
      ( comple1908693960933563346ssible(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),partial_fun_lub(heap_Time_Heap(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_lub(vEBT_VEBTi)),partial_fun_ord(heap_Time_Heap(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_ord(vEBT_VEBTi)),aTP_Lamp_ox(fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),$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_oz(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))))
       => ( ! [F2: 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)))),$o,P,F2)
             => aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),$o,P,aTP_Lamp_pa(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)))),F2)) )
         => 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_2992_curryI,axiom,
    ! [A: $tType,B: $tType,F: fun(product_prod(A,B),$o),A2: A,B2: B] :
      ( aa(product_prod(A,B),$o,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2))
     => aa(B,$o,aa(A,fun(B,$o),product_curry(A,B,$o,F),A2),B2) ) ).

% curryI
tff(fact_2993_Heap__Time__Monad_Obind__bind,axiom,
    ! [C: $tType,A: $tType,B: $tType,F: 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,F,G),K) = heap_Time_bind(C,A,F,aa(fun(B,heap_Time_Heap(A)),fun(C,heap_Time_Heap(A)),aTP_Lamp_pb(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_2994_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)))) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),aTP_Lamp_pc(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ).

% Heap_lub_empty
tff(fact_2995_timeFrame_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))] :
      ( ! [N2: nat,R2: A,H2: heap_ext(product_unit),N5: nat] : Xb != aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),N2),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),N5))))
     => ~ ! [N2: nat] : Xb != aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),N2),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))) ) ).

% timeFrame.cases
tff(fact_2996_zdvd__zdiffD,axiom,
    ! [K: int,M: int,Nb: int] :
      ( aa(int,$o,dvd_dvd(int,K),aa(int,int,minus_minus(int,M),Nb))
     => ( aa(int,$o,dvd_dvd(int,K),Nb)
       => aa(int,$o,dvd_dvd(int,K),M) ) ) ).

% zdvd_zdiffD
tff(fact_2997_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)),aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),Xb)) = aa(nat,nat,suc,zero_zero(nat)) ).

% Heap.size(2)
tff(fact_2998_curryE,axiom,
    ! [A: $tType,B: $tType,F: fun(product_prod(A,B),$o),A2: A,B2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),product_curry(A,B,$o,F),A2),B2)
     => aa(product_prod(A,B),$o,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)) ) ).

% curryE
tff(fact_2999_curryD,axiom,
    ! [A: $tType,B: $tType,F: fun(product_prod(A,B),$o),A2: A,B2: B] :
      ( aa(B,$o,aa(A,fun(B,$o),product_curry(A,B,$o,F),A2),B2)
     => aa(product_prod(A,B),$o,F,aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),A2),B2)) ) ).

% curryD
tff(fact_3000_TBOUND__empty,axiom,
    ! [A: $tType,Ta: nat] : time_TBOUND(A,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),aTP_Lamp_pc(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),Ta) ).

% TBOUND_empty
tff(fact_3001_refines__empty,axiom,
    ! [A: $tType,M: heap_Time_Heap(A)] : refine_Imp_refines(A,M,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),aTP_Lamp_pc(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))))) ).

% refines_empty
tff(fact_3002_zdvd__antisym__nonneg,axiom,
    ! [M: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),M)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb)
       => ( aa(int,$o,dvd_dvd(int,M),Nb)
         => ( aa(int,$o,dvd_dvd(int,Nb),M)
           => ( M = Nb ) ) ) ) ) ).

% zdvd_antisym_nonneg
tff(fact_3003_zdvd__mono,axiom,
    ! [K: int,M: int,Ta: int] :
      ( ( K != zero_zero(int) )
     => ( aa(int,$o,dvd_dvd(int,M),Ta)
      <=> aa(int,$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),Ta)) ) ) ).

% zdvd_mono
tff(fact_3004_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,Nb: int] :
      ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),K),M)),aa(int,int,aa(int,fun(int,int),times_times(int),K),Nb))
     => ( ( K != zero_zero(int) )
       => aa(int,$o,dvd_dvd(int,M),Nb) ) ) ).

% zdvd_mult_cancel
tff(fact_3005_zdvd__reduce,axiom,
    ! [K: int,Nb: int,M: int] :
      ( aa(int,$o,dvd_dvd(int,K),aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),aa(int,int,aa(int,fun(int,int),times_times(int),K),M)))
    <=> aa(int,$o,dvd_dvd(int,K),Nb) ) ).

% zdvd_reduce
tff(fact_3006_zdvd__period,axiom,
    ! [A2: int,D2: int,Xb: int,Ta: int,C2: int] :
      ( aa(int,$o,dvd_dvd(int,A2),D2)
     => ( aa(int,$o,dvd_dvd(int,A2),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Ta))
      <=> aa(int,$o,dvd_dvd(int,A2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),aa(int,int,aa(int,fun(int,int),times_times(int),C2),D2))),Ta)) ) ) ).

% zdvd_period
tff(fact_3007_zdvd__imp__le,axiom,
    ! [Z: int,Nb: int] :
      ( aa(int,$o,dvd_dvd(int,Z),Nb)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),Nb) ) ) ).

% zdvd_imp_le
tff(fact_3008_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_3009_real__of__int__div,axiom,
    ! [D2: int,Nb: int] :
      ( aa(int,$o,dvd_dvd(int,D2),Nb)
     => ( aa(int,real,ring_1_of_int(real),aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),D2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(int,real,ring_1_of_int(real),Nb)),aa(int,real,ring_1_of_int(real),D2)) ) ) ).

% real_of_int_div
tff(fact_3010_fold__if__return,axiom,
    ! [A: $tType,B2: $o,C2: A,D2: A] :
      $ite((B2),aa(A,heap_Time_Heap(A),heap_Time_return(A),C2),aa(A,heap_Time_Heap(A),heap_Time_return(A),D2)) = aa(A,heap_Time_Heap(A),heap_Time_return(A),
        $ite((B2),C2,D2)) ).

% fold_if_return
tff(fact_3011_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$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,aa(int,fun(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_3012_int__div__sub__1,axiom,
    ! [M: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$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,Nb),one_one(int))),M) = $ite(aa(int,$o,dvd_dvd(int,M),Nb),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),M)),one_one(int)),aa(int,int,aa(int,fun(int,int),divide_divide(int),Nb),M)) ) ) ).

% int_div_sub_1
tff(fact_3013_aset_I10_J,axiom,
    ! [D2: int,D6: int,A3: set(int),Ta: int] :
      ( aa(int,$o,dvd_dvd(int,D2),D6)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A3)
                 => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa2) ) ) )
         => ( ~ aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),Ta))
           => ~ aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6)),Ta)) ) ) ) ).

% aset(10)
tff(fact_3014_aset_I9_J,axiom,
    ! [D2: int,D6: int,A3: set(int),Ta: int] :
      ( aa(int,$o,dvd_dvd(int,D2),D6)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
             => ! [Xb3: int] :
                  ( member(int,Xb3,A3)
                 => ( X3 != aa(int,int,minus_minus(int,Xb3),Xa2) ) ) )
         => ( aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),Ta))
           => aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),D6)),Ta)) ) ) ) ).

% aset(9)
tff(fact_3015_bset_I10_J,axiom,
    ! [D2: int,D6: int,B5: set(int),Ta: int] :
      ( aa(int,$o,dvd_dvd(int,D2),D6)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B5)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa2) ) ) )
         => ( ~ aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),Ta))
           => ~ aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,X3),D6)),Ta)) ) ) ) ).

% bset(10)
tff(fact_3016_bset_I9_J,axiom,
    ! [D2: int,D6: int,B5: set(int),Ta: int] :
      ( aa(int,$o,dvd_dvd(int,D2),D6)
     => ! [X3: int] :
          ( ! [Xa2: int] :
              ( member(int,Xa2,set_or1337092689740270186AtMost(int,one_one(int),D6))
             => ! [Xb3: int] :
                  ( member(int,Xb3,B5)
                 => ( X3 != aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb3),Xa2) ) ) )
         => ( aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),X3),Ta))
           => aa(int,$o,dvd_dvd(int,D2),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,minus_minus(int,X3),D6)),Ta)) ) ) ) ).

% bset(9)
tff(fact_3017_distrib__if__bind,axiom,
    ! [A: $tType,B: $tType,B2: $o,C2: heap_Time_Heap(B),D2: heap_Time_Heap(B),F: fun(B,heap_Time_Heap(A))] :
      heap_Time_bind(B,A,
        $ite((B2),C2,D2),
        F) = $ite((B2),heap_Time_bind(B,A,C2,F),heap_Time_bind(B,A,D2,F)) ).

% distrib_if_bind
tff(fact_3018_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_3019_effect__LetI,axiom,
    ! [B: $tType,A: $tType,Xb: A,Ta: A,F: fun(A,heap_Time_Heap(B)),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: B,Nb: nat] :
      ( ( Xb = Ta )
     => ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),F,Xb),H,H3,R3,Nb)
       => heap_Time_effect(B,aa(A,heap_Time_Heap(B),F,Ta),H,H3,R3,Nb) ) ) ).

% effect_LetI
tff(fact_3020_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_3021_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( aa(nat,$o,dvd_dvd(nat,nat2(Z)),M)
    <=> $ite(aa(int,$o,aa(int,fun(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_3022_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_pk(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_3023_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_3024_dvd__productE,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [P2: A,A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,P2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
         => ~ ! [X2: A,Y5: A] :
                ( ( P2 = aa(A,A,aa(A,fun(A,A),times_times(A),X2),Y5) )
               => ( aa(A,$o,dvd_dvd(A,X2),A2)
                 => ~ aa(A,$o,dvd_dvd(A,Y5),B2) ) ) ) ) ).

% dvd_productE
tff(fact_3025_division__decomp,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2))
         => ? [B8: A,C7: A] :
              ( ( A2 = aa(A,A,aa(A,fun(A,A),times_times(A),B8),C7) )
              & aa(A,$o,dvd_dvd(A,B8),B2)
              & aa(A,$o,dvd_dvd(A,C7),C2) ) ) ) ).

% division_decomp
tff(fact_3026_Euclid__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),A2: nat,B2: nat] :
      ( ! [A4: nat,B3: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),P,A4),B3)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),P,B3),A4) )
     => ( ! [A4: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,A4),zero_zero(nat))
       => ( ! [A4: nat,B3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),P,A4),B3)
             => aa(nat,$o,aa(nat,fun(nat,$o),P,A4),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),B3)) )
         => aa(nat,$o,aa(nat,fun(nat,$o),P,A2),B2) ) ) ) ).

% Euclid_induct
tff(fact_3027_even__nat__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$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_3028_emep1,axiom,
    ! [Nb: int,D2: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)
     => ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),D2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),D2)
         => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),one_one(int))),D2) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Nb),D2)),one_one(int)) ) ) ) ) ).

% emep1
tff(fact_3029_eme1p,axiom,
    ! [Nb: int,D2: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb)
     => ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),D2)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),D2)
         => ( 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)),Nb)),D2) = 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),Nb),D2)) ) ) ) ) ).

% eme1p
tff(fact_3030_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_pl(list(A),fun(array(A),assn),Xs)) ) ).

% of_list_rule
tff(fact_3031_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_pm(heap_Time_Heap(B),fun(A,heap_Time_Heap(B)),C2)) ).

% heap.const_mono
tff(fact_3032_dvd__pos__nat,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,dvd_dvd(nat,M),Nb)
       => aa(nat,$o,ord_less(nat,zero_zero(nat)),M) ) ) ).

% dvd_pos_nat
tff(fact_3033_gcd__nat__induct,axiom,
    ! [P: fun(nat,fun(nat,$o)),M: nat,Nb: nat] :
      ( ! [M2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),P,M2),zero_zero(nat))
     => ( ! [M2: nat,N2: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N2)
           => ( aa(nat,$o,aa(nat,fun(nat,$o),P,N2),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M2),N2))
             => aa(nat,$o,aa(nat,fun(nat,$o),P,M2),N2) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),P,M),Nb) ) ) ).

% gcd_nat_induct
tff(fact_3034_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_qc(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_3035_bezout__lemma__nat,axiom,
    ! [D2: nat,A2: nat,B2: nat,Xb: nat,Y: nat] :
      ( aa(nat,$o,dvd_dvd(nat,D2),A2)
     => ( aa(nat,$o,dvd_dvd(nat,D2),B2)
       => ( ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y)),D2) )
            | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Xb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y)),D2) ) )
         => ? [X2: nat,Y5: nat] :
              ( aa(nat,$o,dvd_dvd(nat,D2),A2)
              & aa(nat,$o,dvd_dvd(nat,D2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))
              & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2) = 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),A2),B2)),Y5)),D2) )
                | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y5)),D2) ) ) ) ) ) ) ).

% bezout_lemma_nat
tff(fact_3036_bezout__add__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D3: nat,X2: nat,Y5: nat] :
      ( aa(nat,$o,dvd_dvd(nat,D3),A2)
      & aa(nat,$o,dvd_dvd(nat,D3),B2)
      & ( ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y5)),D3) )
        | ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y5)),D3) ) ) ) ).

% bezout_add_nat
tff(fact_3037_bezout1__nat,axiom,
    ! [A2: nat,B2: nat] :
    ? [D3: nat,X2: nat,Y5: nat] :
      ( aa(nat,$o,dvd_dvd(nat,D3),A2)
      & aa(nat,$o,dvd_dvd(nat,D3),B2)
      & ( ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y5)) = D3 )
        | ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y5)) = D3 ) ) ) ).

% bezout1_nat
tff(fact_3038_effect__returnI,axiom,
    ! [A: $tType,H: heap_ext(product_unit),H3: heap_ext(product_unit),Xb: A] :
      ( ( H = H3 )
     => heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),Xb),H,H3,Xb,one_one(nat)) ) ).

% effect_returnI
tff(fact_3039_effect__returnE,axiom,
    ! [A: $tType,Xb: A,H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: A,Nb: nat] :
      ( heap_Time_effect(A,aa(A,heap_Time_Heap(A),heap_Time_return(A),Xb),H,H3,R3,Nb)
     => ~ ( ( R3 = Xb )
         => ( ( H3 = H )
           => ( Nb != one_one(nat) ) ) ) ) ).

% effect_returnE
tff(fact_3040_effect__bindE,axiom,
    ! [B: $tType,A: $tType,F: heap_Time_Heap(B),G: fun(B,heap_Time_Heap(A)),H: heap_ext(product_unit),H8: heap_ext(product_unit),R4: A,Nb: nat] :
      ( heap_Time_effect(A,heap_Time_bind(B,A,F,G),H,H8,R4,Nb)
     => ~ ! [H5: heap_ext(product_unit),R2: B,N1: nat] :
            ( heap_Time_effect(B,F,H,H5,R2,N1)
           => ! [N22: nat] :
                ( heap_Time_effect(A,aa(B,heap_Time_Heap(A),G,R2),H5,H8,R4,N22)
               => ( Nb != aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N1),N22) ) ) ) ) ).

% effect_bindE
tff(fact_3041_effect__bindI,axiom,
    ! [B: $tType,A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: A,Nb: nat,G: fun(A,heap_Time_Heap(B)),H8: heap_ext(product_unit),R4: B,N3: nat] :
      ( heap_Time_effect(A,F,H,H3,R3,Nb)
     => ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),G,R3),H3,H8,R4,N3)
       => heap_Time_effect(B,heap_Time_bind(A,B,F,G),H,H8,R4,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),N3)) ) ) ).

% effect_bindI
tff(fact_3042_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_qk(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_3043_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_qs(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_3044_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_rh(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_3045_bezout__add__strong__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [D3: nat,X2: nat,Y5: nat] :
          ( aa(nat,$o,dvd_dvd(nat,D3),A2)
          & aa(nat,$o,dvd_dvd(nat,D3),B2)
          & ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y5)),D3) ) ) ) ).

% bezout_add_strong_nat
tff(fact_3046_vebt__buildupi_Osimps_I3_J,axiom,
    ! [Vaa: nat] :
      vEBT_vebt_buildupi(aa(nat,nat,suc,aa(nat,nat,suc,Vaa))) = $ite(
        aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),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_os(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Vaa),half)) ),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),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_ou(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Vaa),half)) ) ) ).

% vebt_buildupi.simps(3)
tff(fact_3047_vebt__inserti_Ofixp__induct,axiom,
    ! [P: fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),$o)] :
      ( comple1908693960933563346ssible(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),partial_fun_lub(heap_Time_Heap(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_lub(vEBT_VEBTi)),partial_fun_ord(heap_Time_Heap(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_ord(vEBT_VEBTi)),aTP_Lamp_ri(fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),$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_rj(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))))
       => ( ! [F2: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))] :
              ( aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),$o,P,F2)
             => aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),$o,P,aTP_Lamp_rk(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),F2)) )
         => aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),$o,P,vEBT_vebt_inserti) ) ) ) ).

% vebt_inserti.fixp_induct
tff(fact_3048_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_3049_VEBT__internal_Ovebt__succi_H_Ofixp__induct,axiom,
    ! [P: fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o)] :
      ( comple1908693960933563346ssible(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),partial_fun_lub(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_lub(option(nat))),partial_fun_ord(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_ord(option(nat))),aTP_Lamp_rl(fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$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_rn(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))))
       => ( ! [F2: 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))))),$o,P,F2)
             => aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o,P,aTP_Lamp_ro(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))))),F2)) )
         => 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_3050_VEBT__internal_Ovebt__predi_H_Ofixp__induct,axiom,
    ! [P: fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o)] :
      ( comple1908693960933563346ssible(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(option(nat))),partial_fun_lub(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_lub(option(nat))),partial_fun_ord(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap_ord(option(nat))),aTP_Lamp_rl(fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$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_rn(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))))
       => ( ! [F2: 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))))),$o,P,F2)
             => aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),$o,P,aTP_Lamp_rp(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))))),F2)) )
         => 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_3051_VEBT__internal_Ovebt__buildupi_H_Opelims,axiom,
    ! [Xb: nat,Y: heap_Time_Heap(vEBT_VEBTi)] :
      ( ( vEBT_V739175172307565963ildupi(Xb) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_V254170901696579886pi_rel),Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi($false,$false)) )
             => ~ aa(nat,$o,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)) )
               => ~ aa(nat,$o,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_on(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va2),half)) ),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            heap_Time_bind(list(vEBT_VEBTi),vEBT_VEBTi,vEBT_VEBT_replicatei(vEBT_VEBTi,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_V739175172307565963ildupi(half)),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_oq(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va2),half)) ) ) )
                   => ~ aa(nat,$o,accp(nat,vEBT_V254170901696579886pi_rel),aa(nat,nat,suc,aa(nat,nat,suc,Va2))) ) ) ) ) ) ) ).

% VEBT_internal.vebt_buildupi'.pelims
tff(fact_3052_prod__decode__aux_Ocases,axiom,
    ! [Xb: product_prod(nat,nat)] :
      ~ ! [K3: nat,M2: nat] : Xb != aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),K3),M2) ).

% prod_decode_aux.cases
tff(fact_3053_list__decode_Ocases,axiom,
    ! [Xb: nat] :
      ( ( Xb != zero_zero(nat) )
     => ~ ! [N2: nat] : Xb != aa(nat,nat,suc,N2) ) ).

% list_decode.cases
tff(fact_3054_vebt__buildupi_Opelims,axiom,
    ! [Xb: nat,Y: heap_Time_Heap(vEBT_VEBTi)] :
      ( ( vEBT_vebt_buildupi(Xb) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_v1230518104690509829pi_rel),Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = aa(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi),heap_Time_return(vEBT_VEBTi),vEBT_Leafi($false,$false)) )
             => ~ aa(nat,$o,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)) )
               => ~ aa(nat,$o,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_os(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va2),half)) ),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            heap_Time_bind(list(vEBT_VEBTi),vEBT_VEBTi,vEBT_VEBT_replicatei(vEBT_VEBTi,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildupi(half)),aa(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)),aTP_Lamp_ou(nat,fun(nat,fun(list(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Va2),half)) ) ) )
                   => ~ aa(nat,$o,accp(nat,vEBT_v1230518104690509829pi_rel),aa(nat,nat,suc,aa(nat,nat,suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildupi.pelims
tff(fact_3055_VEBT__internal_OT__vebt__buildupi_H_Opelims,axiom,
    ! [Xb: nat,Y: int] :
      ( ( vEBT_V9176841429113362141ildupi(Xb) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_V3352910403632780892pi_rel),Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = one_one(int) )
             => ~ aa(nat,$o,accp(nat,vEBT_V3352910403632780892pi_rel),zero_zero(nat)) ) )
         => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = one_one(int) )
               => ~ aa(nat,$o,accp(nat,vEBT_V3352910403632780892pi_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N2: nat] :
                  ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
                 => ( ( Y = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2),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),N2),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),N2),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),N2),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),N2),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),N2),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),N2),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),N2),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),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))) )
                   => ~ aa(nat,$o,accp(nat,vEBT_V3352910403632780892pi_rel),aa(nat,nat,suc,aa(nat,nat,suc,N2))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi'.pelims
tff(fact_3056_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 )
     => ( aa(nat,$o,accp(nat,vEBT_V5144397997797733112_d_rel),Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
             => ~ aa(nat,$o,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))) )
               => ~ aa(nat,$o,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))) )),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(aa(num,num,bit1,one2))))),vEBT_V8646137997579335489_i_l_d(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),vEBT_V8646137997579335489_i_l_d(half))) ) ) )
                   => ~ aa(nat,$o,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_3057_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 )
     => ( aa(nat,$o,accp(nat,vEBT_V1247956027447740395_p_rel),Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
             => ~ aa(nat,$o,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)) )
               => ~ aa(nat,$o,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)))) )),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,bit0(one2))))),vEBT_V8346862874174094_d_u_p(aa(nat,nat,suc,half)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),vEBT_V8346862874174094_d_u_p(half)),one_one(nat)))) ) ) )
                   => ~ aa(nat,$o,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_3058_VEBT__internal_OTb_Opelims,axiom,
    ! [Xb: nat,Y: int] :
      ( ( vEBT_VEBT_Tb(Xb) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel2),Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = aa(num,int,numeral_numeral(int),aa(num,num,bit1,one2)) )
             => ~ aa(nat,$o,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)) )
               => ~ aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel2),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N2: nat] :
                  ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
                 => ( ( Y = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2),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),N2),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),N2),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),N2),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),N2),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),N2),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),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) )
                   => ~ aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel2),aa(nat,nat,suc,aa(nat,nat,suc,N2))) ) ) ) ) ) ) ).

% VEBT_internal.Tb.pelims
tff(fact_3059_VEBT__internal_OTb_H_Opelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( vEBT_VEBT_Tb2(Xb) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel),Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
             => ~ aa(nat,$o,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)) )
               => ~ aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N2: nat] :
                  ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
                 => ( ( Y = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2),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),N2),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),N2),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),N2),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),N2),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),N2),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),N2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))))) )
                   => ~ aa(nat,$o,accp(nat,vEBT_VEBT_Tb_rel),aa(nat,nat,suc,aa(nat,nat,suc,N2))) ) ) ) ) ) ) ).

% VEBT_internal.Tb'.pelims
tff(fact_3060_VEBT__internal_OT__vebt__buildupi_Opelims,axiom,
    ! [Xb: nat,Y: nat] :
      ( ( vEBT_V441764108873111860ildupi(Xb) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_V2957053500504383685pi_rel),Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = aa(nat,nat,suc,zero_zero(nat)) )
             => ~ aa(nat,$o,accp(nat,vEBT_V2957053500504383685pi_rel),zero_zero(nat)) ) )
         => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = aa(nat,nat,suc,zero_zero(nat)) )
               => ~ aa(nat,$o,accp(nat,vEBT_V2957053500504383685pi_rel),aa(nat,nat,suc,zero_zero(nat))) ) )
           => ~ ! [N2: nat] :
                  ( ( Xb = aa(nat,nat,suc,aa(nat,nat,suc,N2)) )
                 => ( ( Y = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2),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),N2),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),N2),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),N2),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),N2),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),N2),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),N2),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),N2),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),N2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))))))))) )
                   => ~ aa(nat,$o,accp(nat,vEBT_V2957053500504383685pi_rel),aa(nat,nat,suc,aa(nat,nat,suc,N2))) ) ) ) ) ) ) ).

% VEBT_internal.T_vebt_buildupi.pelims
tff(fact_3061_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($o)),aTP_Lamp_rq(fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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)))),$o,P,aTP_Lamp_rs(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))))
       => ( ! [F2: 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)))),$o,P,F2)
             => aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),$o,P,aTP_Lamp_rt(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),F2)) )
         => aa(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),$o,P,vEBT_V854960066525838166emberi) ) ) ) ).

% VEBT_internal.vebt_memberi'.fixp_induct
tff(fact_3062_vebt__predi_Ofixp__induct,axiom,
    ! [P: fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o)] :
      ( comple1908693960933563346ssible(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),partial_fun_lub(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_lub(option(nat))),partial_fun_ord(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_ord(option(nat))),aTP_Lamp_ru(fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$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_rv(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))
       => ( ! [F2: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))] :
              ( aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o,P,F2)
             => aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o,P,aTP_Lamp_rw(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),F2)) )
         => aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o,P,vEBT_vebt_predi) ) ) ) ).

% vebt_predi.fixp_induct
tff(fact_3063_vebt__succi_Ofixp__induct,axiom,
    ! [P: fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o)] :
      ( comple1908693960933563346ssible(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(option(nat))),partial_fun_lub(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_lub(option(nat))),partial_fun_ord(heap_Time_Heap(option(nat)),heap_Time_Heap(option(nat)),product_prod(vEBT_VEBTi,nat),heap_Time_Heap_ord(option(nat))),aTP_Lamp_ru(fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$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_rv(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))
       => ( ! [F2: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))] :
              ( aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o,P,F2)
             => aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o,P,aTP_Lamp_rx(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),F2)) )
         => aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),$o,P,vEBT_vebt_succi) ) ) ) ).

% vebt_succi.fixp_induct
tff(fact_3064_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($o)),aTP_Lamp_ry(fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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))),$o,P,aTP_Lamp_rz(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))))
       => ( ! [F2: fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))] :
              ( aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),$o,P,F2)
             => aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),$o,P,aTP_Lamp_sa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),F2)) )
         => aa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),$o,P,vEBT_vebt_memberi) ) ) ) ).

% vebt_memberi.fixp_induct
tff(fact_3065_heap_Ofixp__induct__uc,axiom,
    ! [A: $tType,B: $tType,C: $tType,U2: fun(C,fun(A,heap_Time_Heap(B))),F5: fun(C,C),C5: fun(fun(A,heap_Time_Heap(B)),C),F: C,P: fun(fun(A,heap_Time_Heap(B)),$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_sb(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))
     => ( ( F = 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_sc(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))) )
       => ( ! [F2: fun(A,heap_Time_Heap(B))] : aa(C,fun(A,heap_Time_Heap(B)),U2,aa(fun(A,heap_Time_Heap(B)),C,C5,F2)) = F2
         => ( 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)),$o,P,aTP_Lamp_se(A,heap_Time_Heap(B)))
             => ( ! [F2: C] :
                    ( aa(fun(A,heap_Time_Heap(B)),$o,P,aa(C,fun(A,heap_Time_Heap(B)),U2,F2))
                   => aa(fun(A,heap_Time_Heap(B)),$o,P,aa(C,fun(A,heap_Time_Heap(B)),U2,aa(C,C,F5,F2))) )
               => aa(fun(A,heap_Time_Heap(B)),$o,P,aa(C,fun(A,heap_Time_Heap(B)),U2,F)) ) ) ) ) ) ) ).

% heap.fixp_induct_uc
tff(fact_3066_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_3067_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_3068_tap__def,axiom,
    ! [A: $tType,F: fun(heap_ext(product_unit),A)] : heap_Time_tap(A,F) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),aTP_Lamp_sf(fun(heap_ext(product_unit),A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),F)) ).

% tap_def
tff(fact_3069_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_3070_effect__tapI,axiom,
    ! [A: $tType,H3: heap_ext(product_unit),H: heap_ext(product_unit),R3: A,F: fun(heap_ext(product_unit),A)] :
      ( ( H3 = H )
     => ( ( R3 = aa(heap_ext(product_unit),A,F,H) )
       => heap_Time_effect(A,heap_Time_tap(A,F),H,H3,R3,one_one(nat)) ) ) ).

% effect_tapI
tff(fact_3071_effect__tapE,axiom,
    ! [A: $tType,F: fun(heap_ext(product_unit),A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: A,Nb: nat] :
      ( heap_Time_effect(A,heap_Time_tap(A,F),H,H3,R3,Nb)
     => ~ ( ( H3 = H )
         => ( ( R3 = aa(heap_ext(product_unit),A,F,H) )
           => ( Nb != one_one(nat) ) ) ) ) ).

% effect_tapE
tff(fact_3072_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),F: 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_sb(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))
     => ( ( F = 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_sc(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))) )
       => ( ! [F2: C] : aa(fun(A,heap_Time_Heap(B)),C,C5,aa(C,fun(A,heap_Time_Heap(B)),U2,F2)) = F2
         => ( F = aa(C,C,F5,F) ) ) ) ) ).

% heap.fixp_rule_uc
tff(fact_3073_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_sg(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_3074_fixp__induct__heap,axiom,
    ! [C: $tType,B: $tType,A: $tType,U2: fun(C,fun(A,heap_Time_Heap(B))),F5: fun(C,C),C5: fun(fun(A,heap_Time_Heap(B)),C),F: C,P: fun(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,$o))))),Xb: A,H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: B,Nb: 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_sb(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))
     => ( ( F = 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_sc(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))) )
       => ( ! [F2: fun(A,heap_Time_Heap(B))] : aa(C,fun(A,heap_Time_Heap(B)),U2,aa(fun(A,heap_Time_Heap(B)),C,C5,F2)) = F2
         => ( ! [F2: C,X2: A,H2: heap_ext(product_unit),H5: heap_ext(product_unit),R2: B,N2: 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,F2),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,$o)))),P,Xa),Ha),H_a),Ra),Na) )
               => ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),aa(C,fun(A,heap_Time_Heap(B)),U2,aa(C,C,F5,F2)),X2),H2,H5,R2,N2)
                 => aa(nat,$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),H2),H5),R2),N2) ) )
           => ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),aa(C,fun(A,heap_Time_Heap(B)),U2,F),Xb),H,H3,R3,Nb)
             => 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),H3),R3),Nb) ) ) ) ) ) ).

% fixp_induct_heap
tff(fact_3075_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_3076_product__nth,axiom,
    ! [A: $tType,B: $tType,Nb: nat,Xs: list(A),Ys: list(B)] :
      ( aa(nat,$o,ord_less(nat,Nb),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)),Nb) = 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),Nb),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),Nb),aa(list(B),nat,size_size(list(B)),Ys)))) ) ) ).

% product_nth
tff(fact_3077_pred__subset__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S2: set(product_prod(A,B))] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),aTP_Lamp_sh(set(product_prod(A,B)),fun(A,fun(B,$o)),R)),aTP_Lamp_sh(set(product_prod(A,B)),fun(A,fun(B,$o)),S2))
    <=> aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R),S2) ) ).

% pred_subset_eq2
tff(fact_3078_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_3079_triangle__def,axiom,
    ! [Nb: nat] : nat_triangle(Nb) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Nb))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% triangle_def
tff(fact_3080_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_3081_triangle__Suc,axiom,
    ! [Nb: nat] : nat_triangle(aa(nat,nat,suc,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),nat_triangle(Nb)),aa(nat,nat,suc,Nb)) ).

% triangle_Suc
tff(fact_3082_exhaustive__int_H_Ocases,axiom,
    ! [Xb: product_prod(fun(int,option(product_prod($o,list(code_term)))),product_prod(int,int))] :
      ~ ! [F2: fun(int,option(product_prod($o,list(code_term)))),D3: int,I3: int] : Xb != aa(product_prod(int,int),product_prod(fun(int,option(product_prod($o,list(code_term)))),product_prod(int,int)),aa(fun(int,option(product_prod($o,list(code_term)))),fun(product_prod(int,int),product_prod(fun(int,option(product_prod($o,list(code_term)))),product_prod(int,int))),product_Pair(fun(int,option(product_prod($o,list(code_term)))),product_prod(int,int)),F2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D3),I3)) ).

% exhaustive_int'.cases
tff(fact_3083_full__exhaustive__int_H_Ocases,axiom,
    ! [Xb: product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(int,int))] :
      ~ ! [F2: fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),D3: int,I3: int] : Xb != aa(product_prod(int,int),product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(int,int)),aa(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),fun(product_prod(int,int),product_prod(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(int,int))),product_Pair(fun(product_prod(int,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(int,int)),F2),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D3),I3)) ).

% full_exhaustive_int'.cases
tff(fact_3084_exhaustive__integer_H_Ocases,axiom,
    ! [Xb: product_prod(fun(code_integer,option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer))] :
      ~ ! [F2: fun(code_integer,option(product_prod($o,list(code_term)))),D3: code_integer,I3: code_integer] : Xb != aa(product_prod(code_integer,code_integer),product_prod(fun(code_integer,option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer)),aa(fun(code_integer,option(product_prod($o,list(code_term)))),fun(product_prod(code_integer,code_integer),product_prod(fun(code_integer,option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer))),product_Pair(fun(code_integer,option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer)),F2),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),D3),I3)) ).

% exhaustive_integer'.cases
tff(fact_3085_full__exhaustive__integer_H_Ocases,axiom,
    ! [Xb: product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer))] :
      ~ ! [F2: fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),D3: code_integer,I3: code_integer] : Xb != aa(product_prod(code_integer,code_integer),product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer)),aa(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),fun(product_prod(code_integer,code_integer),product_prod(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer))),product_Pair(fun(product_prod(code_integer,fun(product_unit,code_term)),option(product_prod($o,list(code_term)))),product_prod(code_integer,code_integer)),F2),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),D3),I3)) ).

% full_exhaustive_integer'.cases
tff(fact_3086_pred__equals__eq2,axiom,
    ! [B: $tType,A: $tType,R: set(product_prod(A,B)),S2: 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),S2) )
    <=> ( R = S2 ) ) ).

% pred_equals_eq2
tff(fact_3087_subrelI,axiom,
    ! [B: $tType,A: $tType,R3: set(product_prod(A,B)),S3: set(product_prod(A,B))] :
      ( ! [X2: A,Y5: B] :
          ( member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y5),R3)
         => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),X2),Y5),S3) )
     => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),R3),S3) ) ).

% subrelI
tff(fact_3088_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_3089_pred__subset__eq,axiom,
    ! [A: $tType,R: set(A),S2: set(A)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),aTP_Lamp_av(set(A),fun(A,$o),R)),aTP_Lamp_av(set(A),fun(A,$o),S2))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),R),S2) ) ).

% pred_subset_eq
tff(fact_3090_exE__realizer,axiom,
    ! [C: $tType,A: $tType,B: $tType,P: fun(A,fun(B,$o)),P2: product_prod(B,A),Q: fun(C,$o),F: fun(B,fun(A,C))] :
      ( aa(B,$o,aa(A,fun(B,$o),P,aa(product_prod(B,A),A,product_snd(B,A),P2)),aa(product_prod(B,A),B,product_fst(B,A),P2))
     => ( ! [X2: B,Y5: A] :
            ( aa(B,$o,aa(A,fun(B,$o),P,Y5),X2)
           => aa(C,$o,Q,aa(A,C,aa(B,fun(A,C),F,X2),Y5)) )
       => aa(C,$o,Q,aa(product_prod(B,A),C,product_case_prod(B,A,C,F),P2)) ) ) ).

% exE_realizer
tff(fact_3091_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 )
     => ( ! [A4: $o,B3: $o] :
            ( ( Xb = vEBT_Leaf((A4),(B3)) )
           => ! [Ai: $o,Bi: $o] :
                ( ( Xaa = vEBT_Leafi((Ai),(Bi)) )
               => ( Y != pure_assn(( ( (Ai) = (A4) )
                      & ( (Bi) = (B3) ) )) ) ) )
       => ( ! [Mmo2: option(product_prod(nat,nat)),Deg2: nat,Tree_list2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mmo2,Deg2,Tree_list2,Summary) )
             => ! [Mmoi2: option(product_prod(nat,nat)),Degi2: nat,Tree_array2: array(vEBT_VEBTi),Summaryi2: vEBT_VEBTi] :
                  ( ( Xaa = vEBT_Nodei(Mmoi2,Degi2,Tree_array2,Summaryi2) )
                 => ( 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(( ( Mmoi2 = Mmo2 )
                                & ( Degi2 = Deg2 ) ))),
                            aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),Summaryi2))),
                        ex_assn(list(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn),aTP_Lamp_ba(list(vEBT_VEBT),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn)),Tree_list2),Tree_array2))) ) ) )
         => ( ( ? [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_3092_in__set__enumerate__eq,axiom,
    ! [A: $tType,P2: product_prod(nat,A),Nb: nat,Xs: list(A)] :
      ( member(product_prod(nat,A),P2,aa(list(product_prod(nat,A)),set(product_prod(nat,A)),set2(product_prod(nat,A)),enumerate(A,Nb,Xs)))
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(product_prod(nat,A),nat,product_fst(nat,A),P2))
        & aa(nat,$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)),Nb))
        & ( aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,aa(product_prod(nat,A),nat,product_fst(nat,A),P2)),Nb)) = aa(product_prod(nat,A),A,product_snd(nat,A),P2) ) ) ) ).

% in_set_enumerate_eq
tff(fact_3093_obtain__set__pred,axiom,
    ! [Z: nat,Xb: nat,A3: set(nat)] :
      ( aa(nat,$o,ord_less(nat,Z),Xb)
     => ( vEBT_VEBT_min_in_set(A3,Z)
       => ( finite_finite2(nat,A3)
         => ? [X_13: nat] : vEBT_is_pred_in_set(A3,Xb,X_13) ) ) ) ).

% obtain_set_pred
tff(fact_3094_obtain__set__succ,axiom,
    ! [Xb: nat,Z: nat,A3: set(nat),B5: set(nat)] :
      ( aa(nat,$o,ord_less(nat,Xb),Z)
     => ( vEBT_VEBT_max_in_set(A3,Z)
       => ( finite_finite2(nat,B5)
         => ( ( A3 = B5 )
           => ? [X_13: nat] : vEBT_is_succ_in_set(A3,Xb,X_13) ) ) ) ) ).

% obtain_set_succ
tff(fact_3095_set__vebt__finite,axiom,
    ! [Ta: vEBT_VEBT,Nb: nat] :
      ( vEBT_invar_vebt(Ta,Nb)
     => finite_finite2(nat,vEBT_VEBT_set_vebt(Ta)) ) ).

% set_vebt_finite
tff(fact_3096_succ__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_13: nat] : vEBT_is_succ_in_set(Xs,A2,X_13)
     => ( finite_finite2(nat,Xs)
       => ~ ? [X3: nat] :
              ( member(nat,X3,Xs)
              & aa(nat,$o,ord_less(nat,A2),X3) ) ) ) ).

% succ_none_empty
tff(fact_3097_pred__none__empty,axiom,
    ! [Xs: set(nat),A2: nat] :
      ( ~ ? [X_13: nat] : vEBT_is_pred_in_set(Xs,A2,X_13)
     => ( finite_finite2(nat,Xs)
       => ~ ? [X3: nat] :
              ( member(nat,X3,Xs)
              & aa(nat,$o,ord_less(nat,X3),A2) ) ) ) ).

% pred_none_empty
tff(fact_3098_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_3099_List_Ofinite__set,axiom,
    ! [A: $tType,Xs: list(A)] : finite_finite2(A,aa(list(A),set(A),set2(A),Xs)) ).

% List.finite_set
tff(fact_3100_star__false__right,axiom,
    ! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),bot_bot(assn)) = bot_bot(assn) ).

% star_false_right
tff(fact_3101_star__false__left,axiom,
    ! [P: assn] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),bot_bot(assn)),P) = bot_bot(assn) ).

% star_false_left
tff(fact_3102_false__rule,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),Q: fun(A,assn)] : hoare_hoare_triple(A,bot_bot(assn),C2,Q) ).

% false_rule
tff(fact_3103_length__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),nat,size_size(list(product_prod(nat,A))),enumerate(A,Nb,Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ).

% length_enumerate
tff(fact_3104_pure__assn__eq__false__iff,axiom,
    ! [P: $o] :
      ( ( pure_assn((P)) = bot_bot(assn) )
    <=> ~ (P) ) ).

% pure_assn_eq_false_iff
tff(fact_3105_pure__false,axiom,
    pure_assn($false) = bot_bot(assn) ).

% pure_false
tff(fact_3106_assn__basic__inequalities_I3_J,axiom,
    bot_bot(assn) != one_one(assn) ).

% assn_basic_inequalities(3)
tff(fact_3107_map__snd__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),list(A),map(product_prod(nat,A),A,product_snd(nat,A)),enumerate(A,Nb,Xs)) = Xs ).

% map_snd_enumerate
tff(fact_3108_infinite__Icc__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ finite_finite2(A,set_or1337092689740270186AtMost(A,A2,B2))
        <=> aa(A,$o,ord_less(A,A2),B2) ) ) ).

% infinite_Icc_iff
tff(fact_3109_snga__same__false,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [P2: array(A),Xb: list(A),Y: list(A)] : aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(A),assn,snga_assn(A,P2),Xb)),aa(list(A),assn,snga_assn(A,P2),Y)) = bot_bot(assn) ) ).

% snga_same_false
tff(fact_3110_finite__if__eq__beyond__finite,axiom,
    ! [A: $tType,S2: set(A),S4: set(A)] :
      ( finite_finite2(A,S2)
     => finite_finite2(set(A),collect(set(A),aa(set(A),fun(set(A),$o),aTP_Lamp_si(set(A),fun(set(A),fun(set(A),$o)),S2),S4))) ) ).

% finite_if_eq_beyond_finite
tff(fact_3111_finite__nat__set__iff__bounded,axiom,
    ! [N: set(nat)] :
      ( finite_finite2(nat,N)
    <=> ? [M7: nat] :
        ! [X: nat] :
          ( member(nat,X,N)
         => aa(nat,$o,ord_less(nat,X),M7) ) ) ).

% finite_nat_set_iff_bounded
tff(fact_3112_bounded__nat__set__is__finite,axiom,
    ! [N: set(nat),Nb: nat] :
      ( ! [X2: nat] :
          ( member(nat,X2,N)
         => aa(nat,$o,ord_less(nat,X2),Nb) )
     => finite_finite2(nat,N) ) ).

% bounded_nat_set_is_finite
tff(fact_3113_finite__nat__set__iff__bounded__le,axiom,
    ! [N: set(nat)] :
      ( finite_finite2(nat,N)
    <=> ? [M7: nat] :
        ! [X: nat] :
          ( member(nat,X,N)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),M7) ) ) ).

% finite_nat_set_iff_bounded_le
tff(fact_3114_finite__list,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => ? [Xs2: list(A)] : aa(list(A),set(A),set2(A),Xs2) = A3 ) ).

% finite_list
tff(fact_3115_finite__M__bounded__by__nat,axiom,
    ! [P: fun(nat,$o),I: nat] : finite_finite2(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_sj(fun(nat,$o),fun(nat,fun(nat,$o)),P),I))) ).

% finite_M_bounded_by_nat
tff(fact_3116_finite__less__ub,axiom,
    ! [F: fun(nat,nat),U: nat] :
      ( ! [N2: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),aa(nat,nat,F,N2))
     => finite_finite2(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_sk(fun(nat,nat),fun(nat,fun(nat,$o)),F),U))) ) ).

% finite_less_ub
tff(fact_3117_finite__lists__length__eq,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( finite_finite2(A,A3)
     => finite_finite2(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_sl(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_length_eq
tff(fact_3118_infinite__Icc,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ~ finite_finite2(A,set_or1337092689740270186AtMost(A,A2,B2)) ) ) ).

% infinite_Icc
tff(fact_3119_finite__lists__length__le,axiom,
    ! [A: $tType,A3: set(A),Nb: nat] :
      ( finite_finite2(A,A3)
     => finite_finite2(list(A),collect(list(A),aa(nat,fun(list(A),$o),aTP_Lamp_sm(set(A),fun(nat,fun(list(A),$o)),A3),Nb))) ) ).

% finite_lists_length_le
tff(fact_3120_vebt__assn__raw_Osimps_I3_J,axiom,
    ! [V: option(product_prod(nat,nat)),Vaa: nat,Vb: list(vEBT_VEBT),Vc2: vEBT_VEBT,Vd: $o,Ve: $o] : aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_Node(V,Vaa,Vb,Vc2)),vEBT_Leafi((Vd),(Ve))) = bot_bot(assn) ).

% vebt_assn_raw.simps(3)
tff(fact_3121_vebt__assn__raw_Osimps_I4_J,axiom,
    ! [Vd: $o,Ve: $o,V: option(product_prod(nat,nat)),Vaa: nat,Vb: array(vEBT_VEBTi),Vc2: vEBT_VEBTi] : aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_Leaf((Vd),(Ve))),vEBT_Nodei(V,Vaa,Vb,Vc2)) = bot_bot(assn) ).

% vebt_assn_raw.simps(4)
tff(fact_3122_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => finite_finite2(nat,collect(nat,aTP_Lamp_sn(nat,fun(nat,$o),M))) ) ).

% finite_divisors_nat
tff(fact_3123_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N: set(nat),Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))
     => finite_finite2(nat,N) ) ).

% subset_eq_atLeast0_atMost_finite
tff(fact_3124_finite__roots__unity,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => finite_finite2(A,collect(A,aTP_Lamp_so(nat,fun(A,$o),Nb))) ) ) ).

% finite_roots_unity
tff(fact_3125_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,Y5: A] : ~ aa(B,$o,aa(A,fun(B,$o),P,Y5),X2) ) ).

% exE_realizer'
tff(fact_3126_nth__enumerate__eq,axiom,
    ! [A: $tType,M: nat,Xs: list(A),Nb: nat] :
      ( aa(nat,$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,Nb,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),Nb),M)),aa(nat,A,nth(A,Xs),M)) ) ) ).

% nth_enumerate_eq
tff(fact_3127_disjE__realizer2,axiom,
    ! [B: $tType,A: $tType,P: $o,Q: fun(A,$o),Xb: option(A),R: fun(B,$o),F: B,G: fun(A,B)] :
      ( case_option($o,A,(P),Q,Xb)
     => ( ( (P)
         => aa(B,$o,R,F) )
       => ( ! [Q3: A] :
              ( aa(A,$o,Q,Q3)
             => aa(B,$o,R,aa(A,B,G,Q3)) )
         => aa(B,$o,R,case_option(B,A,F,G,Xb)) ) ) ) ).

% disjE_realizer2
tff(fact_3128_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_3129_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_3130_finite__Diff__insert,axiom,
    ! [A: $tType,A3: set(A),A2: A,B5: set(A)] :
      ( finite_finite2(A,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),B5)))
    <=> finite_finite2(A,aa(set(A),set(A),minus_minus(set(A),A3),B5)) ) ).

% finite_Diff_insert
tff(fact_3131_finite__Collect__le__nat,axiom,
    ! [K: nat] : finite_finite2(nat,collect(nat,aTP_Lamp_sp(nat,fun(nat,$o),K))) ).

% finite_Collect_le_nat
tff(fact_3132_finite__Collect__less__nat,axiom,
    ! [K: nat] : finite_finite2(nat,collect(nat,aTP_Lamp_sq(nat,fun(nat,$o),K))) ).

% finite_Collect_less_nat
tff(fact_3133_finite__Collect__subsets,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => finite_finite2(set(A),collect(set(A),aTP_Lamp_sr(set(A),fun(set(A),$o),A3))) ) ).

% finite_Collect_subsets
tff(fact_3134_finite__Collect__not,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( finite_finite2(A,collect(A,P))
     => ( finite_finite2(A,collect(A,aTP_Lamp_ss(fun(A,$o),fun(A,$o),P)))
      <=> finite_finite2(A,top_top(set(A))) ) ) ).

% finite_Collect_not
tff(fact_3135_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,$o),aTP_Lamp_st(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q))) ) ).

% finite_Collect_conjI
tff(fact_3136_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,$o),aTP_Lamp_su(fun(A,$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_3137_finite__interval__int1,axiom,
    ! [A2: int,B2: int] : finite_finite2(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_sv(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int1
tff(fact_3138_finite__interval__int4,axiom,
    ! [A2: int,B2: int] : finite_finite2(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_sw(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int4
tff(fact_3139_finite__Diff2,axiom,
    ! [A: $tType,B5: set(A),A3: set(A)] :
      ( finite_finite2(A,B5)
     => ( finite_finite2(A,aa(set(A),set(A),minus_minus(set(A),A3),B5))
      <=> finite_finite2(A,A3) ) ) ).

% finite_Diff2
tff(fact_3140_finite__Diff,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( finite_finite2(A,A3)
     => finite_finite2(A,aa(set(A),set(A),minus_minus(set(A),A3),B5)) ) ).

% finite_Diff
tff(fact_3141_finite__interval__int2,axiom,
    ! [A2: int,B2: int] : finite_finite2(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_sx(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int2
tff(fact_3142_finite__interval__int3,axiom,
    ! [A2: int,B2: int] : finite_finite2(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_sy(int,fun(int,fun(int,$o)),A2),B2))) ).

% finite_interval_int3
tff(fact_3143_finite__maxlen,axiom,
    ! [A: $tType,M8: set(list(A))] :
      ( finite_finite2(list(A),M8)
     => ? [N2: nat] :
        ! [X3: list(A)] :
          ( member(list(A),X3,M8)
         => aa(nat,$o,ord_less(nat,aa(list(A),nat,size_size(list(A)),X3)),N2) ) ) ).

% finite_maxlen
tff(fact_3144_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero(int) )
     => finite_finite2(int,collect(int,aTP_Lamp_sz(int,fun(int,$o),I))) ) ).

% finite_divisors_int
tff(fact_3145_not__finite__existsD,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ~ finite_finite2(A,collect(A,P))
     => ? [X_13: A] : aa(A,$o,P,X_13) ) ).

% not_finite_existsD
tff(fact_3146_pigeonhole__infinite__rel,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B5: set(B),R: fun(A,fun(B,$o))] :
      ( ~ finite_finite2(A,A3)
     => ( finite_finite2(B,B5)
       => ( ! [X2: A] :
              ( member(A,X2,A3)
             => ? [Xa: B] :
                  ( member(B,Xa,B5)
                  & aa(B,$o,aa(A,fun(B,$o),R,X2),Xa) ) )
         => ? [X2: B] :
              ( member(B,X2,B5)
              & ~ finite_finite2(A,collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_ta(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),A3),R),X2))) ) ) ) ) ).

% pigeonhole_infinite_rel
tff(fact_3147_finite__has__maximal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( finite_finite2(A,A3)
         => ( member(A,A2,A3)
           => ? [X2: A] :
                ( member(A,X2,A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),X2)
                & ! [Xa: A] :
                    ( member(A,Xa,A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xa)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal2
tff(fact_3148_finite__has__minimal2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A),A2: A] :
          ( finite_finite2(A,A3)
         => ( member(A,A2,A3)
           => ? [X2: A] :
                ( member(A,X2,A3)
                & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),A2)
                & ! [Xa: A] :
                    ( member(A,Xa,A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X2)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal2
tff(fact_3149_finite__subset,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
     => ( finite_finite2(A,B5)
       => finite_finite2(A,A3) ) ) ).

% finite_subset
tff(fact_3150_infinite__super,axiom,
    ! [A: $tType,S2: set(A),T5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
     => ( ~ finite_finite2(A,S2)
       => ~ finite_finite2(A,T5) ) ) ).

% infinite_super
tff(fact_3151_rev__finite__subset,axiom,
    ! [A: $tType,B5: set(A),A3: set(A)] :
      ( finite_finite2(A,B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
       => finite_finite2(A,A3) ) ) ).

% rev_finite_subset
tff(fact_3152_Diff__infinite__finite,axiom,
    ! [A: $tType,T5: set(A),S2: set(A)] :
      ( finite_finite2(A,T5)
     => ( ~ finite_finite2(A,S2)
       => ~ finite_finite2(A,aa(set(A),set(A),minus_minus(set(A),S2),T5)) ) ) ).

% Diff_infinite_finite
tff(fact_3153_finite__has__minimal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X2: A] :
                ( member(A,X2,A3)
                & ! [Xa: A] :
                    ( member(A,Xa,A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xa),X2)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_minimal
tff(fact_3154_finite__has__maximal,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A3: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ? [X2: A] :
                ( member(A,X2,A3)
                & ! [Xa: A] :
                    ( member(A,Xa,A3)
                   => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xa)
                     => ( X2 = Xa ) ) ) ) ) ) ) ).

% finite_has_maximal
tff(fact_3155_finite__subset__induct_H,axiom,
    ! [A: $tType,F5: set(A),A3: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,F5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F5),A3)
       => ( aa(set(A),$o,P,bot_bot(set(A)))
         => ( ! [A4: A,F6: set(A)] :
                ( finite_finite2(A,F6)
               => ( member(A,A4,A3)
                 => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F6),A3)
                   => ( ~ member(A,A4,F6)
                     => ( aa(set(A),$o,P,F6)
                       => aa(set(A),$o,P,aa(set(A),set(A),insert(A,A4),F6)) ) ) ) ) )
           => aa(set(A),$o,P,F5) ) ) ) ) ).

% finite_subset_induct'
tff(fact_3156_finite__subset__induct,axiom,
    ! [A: $tType,F5: set(A),A3: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,F5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),F5),A3)
       => ( aa(set(A),$o,P,bot_bot(set(A)))
         => ( ! [A4: A,F6: set(A)] :
                ( finite_finite2(A,F6)
               => ( member(A,A4,A3)
                 => ( ~ member(A,A4,F6)
                   => ( aa(set(A),$o,P,F6)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,A4),F6)) ) ) ) )
           => aa(set(A),$o,P,F5) ) ) ) ) ).

% finite_subset_induct
tff(fact_3157_infinite__remove,axiom,
    ! [A: $tType,S2: set(A),A2: A] :
      ( ~ finite_finite2(A,S2)
     => ~ finite_finite2(A,aa(set(A),set(A),minus_minus(set(A),S2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) ) ).

% infinite_remove
tff(fact_3158_infinite__coinduct,axiom,
    ! [A: $tType,X5: fun(set(A),$o),A3: set(A)] :
      ( aa(set(A),$o,X5,A3)
     => ( ! [A8: set(A)] :
            ( aa(set(A),$o,X5,A8)
           => ? [X3: A] :
                ( member(A,X3,A8)
                & ( aa(set(A),$o,X5,aa(set(A),set(A),minus_minus(set(A),A8),aa(set(A),set(A),insert(A,X3),bot_bot(set(A)))))
                  | ~ finite_finite2(A,aa(set(A),set(A),minus_minus(set(A),A8),aa(set(A),set(A),insert(A,X3),bot_bot(set(A))))) ) ) )
       => ~ finite_finite2(A,A3) ) ) ).

% infinite_coinduct
tff(fact_3159_finite__empty__induct,axiom,
    ! [A: $tType,A3: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,A3)
     => ( aa(set(A),$o,P,A3)
       => ( ! [A4: A,A8: set(A)] :
              ( finite_finite2(A,A8)
             => ( member(A,A4,A8)
               => ( aa(set(A),$o,P,A8)
                 => aa(set(A),$o,P,aa(set(A),set(A),minus_minus(set(A),A8),aa(set(A),set(A),insert(A,A4),bot_bot(set(A))))) ) ) )
         => aa(set(A),$o,P,bot_bot(set(A))) ) ) ) ).

% finite_empty_induct
tff(fact_3160_remove__induct,axiom,
    ! [A: $tType,P: fun(set(A),$o),B5: set(A)] :
      ( aa(set(A),$o,P,bot_bot(set(A)))
     => ( ( ~ finite_finite2(A,B5)
         => aa(set(A),$o,P,B5) )
       => ( ! [A8: set(A)] :
              ( finite_finite2(A,A8)
             => ( ( A8 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A8),B5)
                 => ( ! [X3: A] :
                        ( member(A,X3,A8)
                       => aa(set(A),$o,P,aa(set(A),set(A),minus_minus(set(A),A8),aa(set(A),set(A),insert(A,X3),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A8) ) ) ) )
         => aa(set(A),$o,P,B5) ) ) ) ).

% remove_induct
tff(fact_3161_finite__remove__induct,axiom,
    ! [A: $tType,B5: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,B5)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [A8: set(A)] :
              ( finite_finite2(A,A8)
             => ( ( A8 != bot_bot(set(A)) )
               => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A8),B5)
                 => ( ! [X3: A] :
                        ( member(A,X3,A8)
                       => aa(set(A),$o,P,aa(set(A),set(A),minus_minus(set(A),A8),aa(set(A),set(A),insert(A,X3),bot_bot(set(A))))) )
                   => aa(set(A),$o,P,A8) ) ) ) )
         => aa(set(A),$o,P,B5) ) ) ) ).

% finite_remove_induct
tff(fact_3162_finite__induct__select,axiom,
    ! [A: $tType,S2: set(A),P: fun(set(A),$o)] :
      ( finite_finite2(A,S2)
     => ( aa(set(A),$o,P,bot_bot(set(A)))
       => ( ! [T6: set(A)] :
              ( aa(set(A),$o,ord_less(set(A),T6),S2)
             => ( aa(set(A),$o,P,T6)
               => ? [X3: A] :
                    ( member(A,X3,aa(set(A),set(A),minus_minus(set(A),S2),T6))
                    & aa(set(A),$o,P,aa(set(A),set(A),insert(A,X3),T6)) ) ) )
         => aa(set(A),$o,P,S2) ) ) ) ).

% finite_induct_select
tff(fact_3163_finite__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => finite_finite2(complex,collect(complex,aa(complex,fun(complex,$o),aTP_Lamp_tb(nat,fun(complex,fun(complex,$o)),Nb),C2))) ) ).

% finite_nth_roots
tff(fact_3164_set__encode__insert,axiom,
    ! [A3: set(nat),Nb: nat] :
      ( finite_finite2(nat,A3)
     => ( ~ member(nat,Nb,A3)
       => ( aa(set(nat),nat,nat_set_encode,aa(set(nat),set(nat),insert(nat,Nb),A3)) = 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))),Nb)),aa(set(nat),nat,nat_set_encode,A3)) ) ) ) ).

% set_encode_insert
tff(fact_3165_diff__preserves__multiset,axiom,
    ! [A: $tType,M8: fun(A,nat),N: fun(A,nat)] :
      ( finite_finite2(A,collect(A,aTP_Lamp_tc(fun(A,nat),fun(A,$o),M8)))
     => finite_finite2(A,collect(A,aa(fun(A,nat),fun(A,$o),aTP_Lamp_td(fun(A,nat),fun(fun(A,nat),fun(A,$o)),M8),N))) ) ).

% diff_preserves_multiset
tff(fact_3166_add__mset__in__multiset,axiom,
    ! [A: $tType,M8: fun(A,nat),A2: A] :
      ( finite_finite2(A,collect(A,aTP_Lamp_tc(fun(A,nat),fun(A,$o),M8)))
     => finite_finite2(A,collect(A,aa(A,fun(A,$o),aTP_Lamp_te(fun(A,nat),fun(A,fun(A,$o)),M8),A2))) ) ).

% add_mset_in_multiset
tff(fact_3167_finite__linorder__max__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),P: fun(set(A),$o)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B3: A,A8: set(A)] :
                  ( finite_finite2(A,A8)
                 => ( ! [X3: A] :
                        ( member(A,X3,A8)
                       => aa(A,$o,ord_less(A,X3),B3) )
                   => ( aa(set(A),$o,P,A8)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,B3),A8)) ) ) )
             => aa(set(A),$o,P,A3) ) ) ) ) ).

% finite_linorder_max_induct
tff(fact_3168_size__empty,axiom,
    ! [A: $tType] : aa(multiset(A),nat,size_size(multiset(A)),zero_zero(multiset(A))) = zero_zero(nat) ).

% size_empty
tff(fact_3169_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_3170_size__union,axiom,
    ! [A: $tType,M8: multiset(A),N: 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),N)) = 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)),N)) ).

% size_union
tff(fact_3171_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,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,M,X2)),aa(A,nat,M,Y4)) ) ) ) ).

% ex_has_least_nat
tff(fact_3172_nonempty__has__size,axiom,
    ! [A: $tType,S2: multiset(A)] :
      ( ( S2 != zero_zero(multiset(A)) )
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(multiset(A),nat,size_size(multiset(A)),S2)) ) ).

% nonempty_has_size
tff(fact_3173_Lattices__Big_Oex__has__greatest__nat,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F: fun(A,nat),B2: nat] :
      ( aa(A,$o,P,K)
     => ( ! [Y5: A] :
            ( aa(A,$o,P,Y5)
           => aa(nat,$o,ord_less(nat,aa(A,nat,F,Y5)),B2) )
       => ? [X2: A] :
            ( aa(A,$o,P,X2)
            & ! [Y4: A] :
                ( aa(A,$o,P,Y4)
               => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F,Y4)),aa(A,nat,F,X2)) ) ) ) ) ).

% Lattices_Big.ex_has_greatest_nat
tff(fact_3174_diff__size__le__size__Diff,axiom,
    ! [A: $tType,M8: multiset(A),M9: multiset(A)] : aa(nat,$o,aa(nat,fun(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_3175_complex__mod__triangle__ineq2,axiom,
    ! [B2: complex,A2: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),B2),A2))),real_V7770717601297561774m_norm(complex,B2))),real_V7770717601297561774m_norm(complex,A2)) ).

% complex_mod_triangle_ineq2
tff(fact_3176_even__set__encode__iff,axiom,
    ! [A3: set(nat)] :
      ( finite_finite2(nat,A3)
     => ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(set(nat),nat,nat_set_encode,A3))
      <=> ~ member(nat,zero_zero(nat),A3) ) ) ).

% even_set_encode_iff
tff(fact_3177_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_3178_ex__min__if__finite,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [S2: set(A)] :
          ( finite_finite2(A,S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ? [X2: A] :
                ( member(A,X2,S2)
                & ~ ? [Xa: A] :
                      ( member(A,Xa,S2)
                      & aa(A,$o,ord_less(A,Xa),X2) ) ) ) ) ) ).

% ex_min_if_finite
tff(fact_3179_ex__has__greatest__nat__lemma,axiom,
    ! [A: $tType,P: fun(A,$o),K: A,F: fun(A,nat),Nb: nat] :
      ( aa(A,$o,P,K)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ? [Y4: A] :
                ( aa(A,$o,P,Y4)
                & ~ aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F,Y4)),aa(A,nat,F,X2)) ) )
       => ? [Y5: A] :
            ( aa(A,$o,P,Y5)
            & ~ aa(nat,$o,ord_less(nat,aa(A,nat,F,Y5)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,F,K)),Nb)) ) ) ) ).

% ex_has_greatest_nat_lemma
tff(fact_3180_filter__preserves__multiset,axiom,
    ! [A: $tType,M8: fun(A,nat),P: fun(A,$o)] :
      ( finite_finite2(A,collect(A,aTP_Lamp_tc(fun(A,nat),fun(A,$o),M8)))
     => finite_finite2(A,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_tf(fun(A,nat),fun(fun(A,$o),fun(A,$o)),M8),P))) ) ).

% filter_preserves_multiset
tff(fact_3181_finite__ranking__induct,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [S2: set(A),P: fun(set(A),$o),F: fun(A,B)] :
          ( finite_finite2(A,S2)
         => ( 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,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Y4)),aa(A,B,F,X2)) )
                   => ( aa(set(A),$o,P,S5)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,X2),S5)) ) ) )
             => aa(set(A),$o,P,S2) ) ) ) ) ).

% finite_ranking_induct
tff(fact_3182_finite__linorder__min__induct,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),P: fun(set(A),$o)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,P,bot_bot(set(A)))
           => ( ! [B3: A,A8: set(A)] :
                  ( finite_finite2(A,A8)
                 => ( ! [X3: A] :
                        ( member(A,X3,A8)
                       => aa(A,$o,ord_less(A,B3),X3) )
                   => ( aa(set(A),$o,P,A8)
                     => aa(set(A),$o,P,aa(set(A),set(A),insert(A,B3),A8)) ) ) )
             => aa(set(A),$o,P,A3) ) ) ) ) ).

% finite_linorder_min_induct
tff(fact_3183_bezw_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: product_prod(int,int)] :
      ( ( bezw(Xb,Xaa) = Y )
     => ( aa(product_prod(nat,nat),$o,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 = $ite(Xaa = zero_zero(nat),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),one_one(int)),zero_zero(int)),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Xaa)))),aa(int,int,minus_minus(int,aa(product_prod(int,int),int,product_fst(int,int),bezw(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Xaa)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Xaa)))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Xb),Xaa)))))) )
           => ~ aa(product_prod(nat,nat),$o,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_3184_old_Oprod_Orec,axiom,
    ! [B: $tType,A: $tType,C: $tType,F1: fun(B,fun(C,A)),A2: B,B2: 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),A2),B2)) = aa(C,A,aa(B,fun(C,A),F1,A2),B2) ).

% old.prod.rec
tff(fact_3185_list__assn__aux__ineq__len,axiom,
    ! [B: $tType,A: $tType,L: list(A),Li2: list(B),A3: 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,A3,L),Li2) = bot_bot(assn) ) ) ).

% list_assn_aux_ineq_len
tff(fact_3186_accp__subset,axiom,
    ! [A: $tType,R1: fun(A,fun(A,$o)),R22: fun(A,fun(A,$o))] :
      ( aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),R1),R22)
     => aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),accp(A,R22)),accp(A,R1)) ) ).

% accp_subset
tff(fact_3187_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_3188_intind,axiom,
    ! [A: $tType,I: nat,Nb: nat,P: fun(A,$o),Xb: A] :
      ( aa(nat,$o,ord_less(nat,I),Nb)
     => ( aa(A,$o,P,Xb)
       => aa(A,$o,P,aa(nat,A,nth(A,replicate(A,Nb,Xb)),I)) ) ) ).

% intind
tff(fact_3189_repli__cons__repl,axiom,
    ! [B: $tType,A: $tType,Q: assn,Xb: heap_Time_Heap(A),A3: fun(B,fun(A,assn)),Y: B,Nb: 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_tg(assn,fun(fun(B,fun(A,assn)),fun(B,fun(A,assn))),Q),A3),Y))
     => hoare_hoare_triple(list(A),Q,vEBT_VEBT_replicatei(A,Nb,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_th(assn,fun(fun(B,fun(A,assn)),fun(B,fun(nat,fun(list(A),assn)))),Q),A3),Y),Nb)) ) ).

% repli_cons_repl
tff(fact_3190_repli__emp,axiom,
    ! [A: $tType,B: $tType,Xb: heap_Time_Heap(A),A3: fun(B,fun(A,assn)),Y: B,Nb: nat] :
      ( hoare_hoare_triple(A,one_one(assn),Xb,aa(B,fun(A,assn),A3,Y))
     => hoare_hoare_triple(list(A),one_one(assn),vEBT_VEBT_replicatei(A,Nb,Xb),vEBT_List_list_assn(B,A,A3,replicate(B,Nb,Y))) ) ).

% repli_emp
tff(fact_3191_union__eq__empty,axiom,
    ! [A: $tType,M8: multiset(A),N: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M8),N) = zero_zero(multiset(A)) )
    <=> ( ( M8 = zero_zero(multiset(A)) )
        & ( N = zero_zero(multiset(A)) ) ) ) ).

% union_eq_empty
tff(fact_3192_empty__eq__union,axiom,
    ! [A: $tType,M8: multiset(A),N: multiset(A)] :
      ( ( zero_zero(multiset(A)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M8),N) )
    <=> ( ( M8 = zero_zero(multiset(A)) )
        & ( N = zero_zero(multiset(A)) ) ) ) ).

% empty_eq_union
tff(fact_3193_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_3194_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_3195_length__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : aa(list(A),nat,size_size(list(A)),replicate(A,Nb,Xb)) = Nb ).

% length_replicate
tff(fact_3196_diff__diff__add__mset,axiom,
    ! [A: $tType,M8: multiset(A),N: multiset(A),P: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),minus_minus(multiset(A),M8),N)),P) = aa(multiset(A),multiset(A),minus_minus(multiset(A),M8),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N),P)) ).

% diff_diff_add_mset
tff(fact_3197_Ball__set__replicate,axiom,
    ! [A: $tType,Nb: nat,A2: A,P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),replicate(A,Nb,A2)))
         => aa(A,$o,P,X) )
    <=> ( aa(A,$o,P,A2)
        | ( Nb = zero_zero(nat) ) ) ) ).

% Ball_set_replicate
tff(fact_3198_Bex__set__replicate,axiom,
    ! [A: $tType,Nb: nat,A2: A,P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),replicate(A,Nb,A2)))
          & aa(A,$o,P,X) )
    <=> ( aa(A,$o,P,A2)
        & ( Nb != zero_zero(nat) ) ) ) ).

% Bex_set_replicate
tff(fact_3199_in__set__replicate,axiom,
    ! [A: $tType,Xb: A,Nb: nat,Y: A] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),replicate(A,Nb,Y)))
    <=> ( ( Xb = Y )
        & ( Nb != zero_zero(nat) ) ) ) ).

% in_set_replicate
tff(fact_3200_nth__replicate,axiom,
    ! [A: $tType,I: nat,Nb: nat,Xb: A] :
      ( aa(nat,$o,ord_less(nat,I),Nb)
     => ( aa(nat,A,nth(A,replicate(A,Nb,Xb)),I) = Xb ) ) ).

% nth_replicate
tff(fact_3201_set__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      ( ( Nb != zero_zero(nat) )
     => ( aa(list(A),set(A),set2(A),replicate(A,Nb,Xb)) = aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))) ) ) ).

% set_replicate
tff(fact_3202_map__fst__mk__fst,axiom,
    ! [B: $tType,A: $tType,K: A,L: list(B)] : aa(list(product_prod(A,B)),list(A),map(product_prod(A,B),A,product_fst(A,B)),aa(list(B),list(product_prod(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_3203_map__snd__mk__snd,axiom,
    ! [B: $tType,A: $tType,K: A,L: list(B)] : aa(list(product_prod(B,A)),list(A),map(product_prod(B,A),A,product_snd(B,A)),aa(list(B),list(product_prod(B,A)),map(B,product_prod(B,A),aa(A,fun(B,product_prod(B,A)),aTP_Lamp_ti(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_3204_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_3205_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_3206_union__le__mono1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B5: multiset(A),D6: multiset(A),C5: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),B5),D6)
         => 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)),B5),C5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),D6),C5)) ) ) ).

% union_le_mono1
tff(fact_3207_union__le__mono2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B5: multiset(A),D6: multiset(A),C5: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),B5),D6)
         => 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),B5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),D6)) ) ) ).

% union_le_mono2
tff(fact_3208_union__less__mono,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A3: multiset(A),C5: multiset(A),B5: multiset(A),D6: multiset(A)] :
          ( aa(multiset(A),$o,ord_less(multiset(A),A3),C5)
         => ( aa(multiset(A),$o,ord_less(multiset(A),B5),D6)
           => aa(multiset(A),$o,ord_less(multiset(A),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B5)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),C5),D6)) ) ) ) ).

% union_less_mono
tff(fact_3209_Multiset_Odiff__add,axiom,
    ! [A: $tType,M8: multiset(A),N: multiset(A),Q: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),M8),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N),Q)) = aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),minus_minus(multiset(A),M8),N)),Q) ).

% Multiset.diff_add
tff(fact_3210_diff__union__cancelL,axiom,
    ! [A: $tType,N: multiset(A),M8: 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)),N),M8)),N) = M8 ).

% diff_union_cancelL
tff(fact_3211_diff__union__cancelR,axiom,
    ! [A: $tType,M8: multiset(A),N: 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)),M8),N)),N) = M8 ).

% diff_union_cancelR
tff(fact_3212_multi__union__self__other__eq,axiom,
    ! [A: $tType,A3: multiset(A),X5: multiset(A),Y7: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),X5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),Y7) )
     => ( X5 = Y7 ) ) ).

% multi_union_self_other_eq
tff(fact_3213_union__right__cancel,axiom,
    ! [A: $tType,M8: multiset(A),K6: multiset(A),N: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M8),K6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N),K6) )
    <=> ( M8 = N ) ) ).

% union_right_cancel
tff(fact_3214_union__left__cancel,axiom,
    ! [A: $tType,K6: multiset(A),M8: multiset(A),N: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),K6),M8) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),K6),N) )
    <=> ( M8 = N ) ) ).

% union_left_cancel
tff(fact_3215_union__commute,axiom,
    ! [A: $tType,M8: multiset(A),N: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M8),N) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N),M8) ).

% union_commute
tff(fact_3216_union__lcomm,axiom,
    ! [A: $tType,M8: multiset(A),N: multiset(A),K6: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M8),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N),K6)) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M8),K6)) ).

% union_lcomm
tff(fact_3217_union__assoc,axiom,
    ! [A: $tType,M8: multiset(A),N: multiset(A),K6: multiset(A)] : aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M8),N)),K6) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M8),aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),N),K6)) ).

% union_assoc
tff(fact_3218_Multiset_Odiff__right__commute,axiom,
    ! [A: $tType,M8: multiset(A),N: multiset(A),Q: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),minus_minus(multiset(A),M8),N)),Q) = aa(multiset(A),multiset(A),minus_minus(multiset(A),aa(multiset(A),multiset(A),minus_minus(multiset(A),M8),Q)),N) ).

% Multiset.diff_right_commute
tff(fact_3219_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_3220_Multiset_Odiff__cancel,axiom,
    ! [A: $tType,A3: multiset(A)] : aa(multiset(A),multiset(A),minus_minus(multiset(A),A3),A3) = zero_zero(multiset(A)) ).

% Multiset.diff_cancel
tff(fact_3221_mset__distrib,axiom,
    ! [A: $tType,A3: multiset(A),B5: multiset(A),M8: multiset(A),N: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),B5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),M8),N) )
     => ~ ! [Am: multiset(A),An: multiset(A)] :
            ( ( A3 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),An) )
           => ! [Bm: multiset(A),Bn: multiset(A)] :
                ( ( B5 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Bm),Bn) )
               => ( ( M8 = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),Am),Bm) )
                 => ( N != aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),An),Bn) ) ) ) ) ) ).

% mset_distrib
tff(fact_3222_union__diff__assoc,axiom,
    ! [A: $tType,C5: multiset(A),B5: multiset(A),A3: multiset(A)] :
      ( ( aa(multiset(A),multiset(A),minus_minus(multiset(A),C5),B5) = 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)),A3),B5)),C5) = aa(multiset(A),multiset(A),aa(multiset(A),fun(multiset(A),multiset(A)),plus_plus(multiset(A)),A3),aa(multiset(A),multiset(A),minus_minus(multiset(A),B5),C5)) ) ) ).

% union_diff_assoc
tff(fact_3223_replicate__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,Xb: A] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = Nb )
     => ( ! [Y5: A] :
            ( member(A,Y5,aa(list(A),set(A),set2(A),Xs))
           => ( Y5 = Xb ) )
       => ( Xs = replicate(A,Nb,Xb) ) ) ) ).

% replicate_eqI
tff(fact_3224_replicate__length__same,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => ( X2 = Xb ) )
     => ( replicate(A,aa(list(A),nat,size_size(list(A)),Xs),Xb) = Xs ) ) ).

% replicate_length_same
tff(fact_3225_map__replicate__const,axiom,
    ! [B: $tType,A: $tType,K: A,Lst: list(B)] : aa(list(B),list(A),map(B,A,aa(A,fun(B,A),aTP_Lamp_eq(A,fun(B,A)),K)),Lst) = replicate(A,aa(list(B),nat,size_size(list(B)),Lst),K) ).

% map_replicate_const
tff(fact_3226_set__replicate__Suc,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : aa(list(A),set(A),set2(A),replicate(A,aa(nat,nat,suc,Nb),Xb)) = aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))) ).

% set_replicate_Suc
tff(fact_3227_set__replicate__conv__if,axiom,
    ! [A: $tType,Nb: nat,Xb: A] :
      aa(list(A),set(A),set2(A),replicate(A,Nb,Xb)) = $ite(Nb = 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_3228_list__assn__cong,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xs4: list(A),Xsi: list(B),Xsi2: list(B),A3: fun(A,fun(B,assn)),A9: fun(A,fun(B,assn))] :
      ( ( Xs = Xs4 )
     => ( ( Xsi = Xsi2 )
       => ( ! [X2: A,Xi: B] :
              ( member(A,X2,aa(list(A),set(A),set2(A),Xs4))
             => ( member(B,Xi,aa(list(B),set(B),set2(B),Xsi2))
               => ( aa(B,assn,aa(A,fun(B,assn),A3,X2),Xi) = aa(B,assn,aa(A,fun(B,assn),A9,X2),Xi) ) ) )
         => ( aa(list(B),assn,vEBT_List_list_assn(A,B,A3,Xs),Xsi) = aa(list(B),assn,vEBT_List_list_assn(A,B,A9,Xs4),Xsi2) ) ) ) ) ).

% list_assn_cong
tff(fact_3229_accp__subset__induct,axiom,
    ! [A: $tType,D6: fun(A,$o),R: fun(A,fun(A,$o)),Xb: A,P: fun(A,$o)] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),D6),accp(A,R))
     => ( ! [X2: A,Z3: A] :
            ( aa(A,$o,D6,X2)
           => ( aa(A,$o,aa(A,fun(A,$o),R,Z3),X2)
             => aa(A,$o,D6,Z3) ) )
       => ( aa(A,$o,D6,Xb)
         => ( ! [X2: A] :
                ( aa(A,$o,D6,X2)
               => ( ! [Z4: A] :
                      ( aa(A,$o,aa(A,fun(A,$o),R,Z4),X2)
                     => aa(A,$o,P,Z4) )
                 => aa(A,$o,P,X2) ) )
           => aa(A,$o,P,Xb) ) ) ) ) ).

% accp_subset_induct
tff(fact_3230_vebt__buildup_Osimps_I3_J,axiom,
    ! [Vaa: nat] :
      vEBT_vebt_buildup(aa(nat,nat,suc,aa(nat,nat,suc,Vaa))) = $ite(
        aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),
        $let(
          half: nat,
          half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Vaa))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),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,Vaa))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
          vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Vaa)),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_3231_extract__pre__list__assn__lengthD,axiom,
    ! [B: $tType,A: $tType,A3: 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,A3,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_3232_vebt__buildup_Opelims,axiom,
    ! [Xb: nat,Y: vEBT_VEBT] :
      ( ( vEBT_vebt_buildup(Xb) = Y )
     => ( aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),Xb)
       => ( ( ( Xb = zero_zero(nat) )
           => ( ( Y = vEBT_Leaf($false,$false) )
             => ~ aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),zero_zero(nat)) ) )
         => ( ( ( Xb = aa(nat,nat,suc,zero_zero(nat)) )
             => ( ( Y = vEBT_Leaf($false,$false) )
               => ~ aa(nat,$o,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)) ),
                          $let(
                            half: nat,
                            half:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,suc,aa(nat,nat,suc,Va2))),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                            vEBT_Node(none(product_prod(nat,nat)),aa(nat,nat,suc,aa(nat,nat,suc,Va2)),replicate(vEBT_VEBT,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,nat,suc,half)),vEBT_vebt_buildup(half)),vEBT_vebt_buildup(aa(nat,nat,suc,half))) ) ) )
                   => ~ aa(nat,$o,accp(nat,vEBT_v4011308405150292612up_rel),aa(nat,nat,suc,aa(nat,nat,suc,Va2))) ) ) ) ) ) ) ).

% vebt_buildup.pelims
tff(fact_3233_VEBT__internal_Ospace_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space,Xb) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel2),Xb)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2)) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel2),vEBT_Leaf((A4),(B3))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,Deg2,TreeList,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)),TreeList))),foldr(nat,nat,plus_plus(nat),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space),TreeList),zero_zero(nat))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel2),vEBT_Node(Info2,Deg2,TreeList,Summary)) ) ) ) ) ) ).

% VEBT_internal.space.pelims
tff(fact_3234_VEBT__internal_Ospace_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_space2,Xb) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel),Xb)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = aa(num,nat,numeral_numeral(nat),bit0(bit0(one2))) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel),vEBT_Leaf((A4),(B3))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,Deg2,TreeList,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),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_space2),TreeList),zero_zero(nat))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_space_rel),vEBT_Node(Info2,Deg2,TreeList,Summary)) ) ) ) ) ) ).

% VEBT_internal.space'.pelims
tff(fact_3235_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 )
     => ( aa(product_prod(vEBT_VEBT,vEBT_VEBTi),$o,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))
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ! [Ai: $o,Bi: $o] :
                  ( ( Xaa = vEBT_Leafi((Ai),(Bi)) )
                 => ( ( Y = pure_assn(( ( (Ai) = (A4) )
                          & ( (Bi) = (B3) ) )) )
                   => ~ aa(product_prod(vEBT_VEBT,vEBT_VEBTi),$o,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((A4),(B3))),vEBT_Leafi((Ai),(Bi)))) ) ) )
         => ( ! [Mmo2: option(product_prod(nat,nat)),Deg2: nat,Tree_list2: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mmo2,Deg2,Tree_list2,Summary) )
               => ! [Mmoi2: option(product_prod(nat,nat)),Degi2: nat,Tree_array2: array(vEBT_VEBTi),Summaryi2: vEBT_VEBTi] :
                    ( ( Xaa = vEBT_Nodei(Mmoi2,Degi2,Tree_array2,Summaryi2) )
                   => ( ( 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(( ( Mmoi2 = Mmo2 )
                                    & ( Degi2 = Deg2 ) ))),
                                aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summary),Summaryi2))),
                            ex_assn(list(vEBT_VEBTi),aa(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn),aTP_Lamp_ba(list(vEBT_VEBT),fun(array(vEBT_VEBTi),fun(list(vEBT_VEBTi),assn)),Tree_list2),Tree_array2))) )
                     => ~ aa(product_prod(vEBT_VEBT,vEBT_VEBTi),$o,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(Mmo2,Deg2,Tree_list2,Summary)),vEBT_Nodei(Mmoi2,Degi2,Tree_array2,Summaryi2))) ) ) )
           => ( ! [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: $o] :
                      ( ( Xaa = vEBT_Leafi((Vd3),(Ve3)) )
                     => ( ( Y = bot_bot(assn) )
                       => ~ aa(product_prod(vEBT_VEBT,vEBT_VEBTi),$o,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: $o] :
                    ( ( Xb = vEBT_Leaf((Vd3),(Ve3)) )
                   => ! [V3: option(product_prod(nat,nat)),Va2: nat,Vb3: array(vEBT_VEBTi),Vc3: vEBT_VEBTi] :
                        ( ( Xaa = vEBT_Nodei(V3,Va2,Vb3,Vc3) )
                       => ( ( Y = bot_bot(assn) )
                         => ~ aa(product_prod(vEBT_VEBT,vEBT_VEBTi),$o,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_3236_vebt__maxti_Opelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_Time_Heap(option(nat))] :
      ( ( vEBT_vebt_maxti(Xb) = Y )
     => ( aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_vebt_maxti_rel),Xb)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leafi((A4),(B3)) )
             => ( ( Y = $ite(
                      (B3),
                      aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),one_one(nat))),
                      $ite((A4),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),zero_zero(nat))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))) ) )
               => ~ aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_vebt_maxti_rel),vEBT_Leafi((A4),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: array(vEBT_VEBTi),Uw2: vEBT_VEBTi] :
                ( ( Xb = vEBT_Nodei(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)) )
                 => ~ aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_vebt_maxti_rel),vEBT_Nodei(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array(vEBT_VEBTi),Uz2: vEBT_VEBTi] :
                  ( ( Xb = vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),Ma)) )
                   => ~ aa(vEBT_VEBTi,$o,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),Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_maxti.pelims
tff(fact_3237_less__eq__multiset__def,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [M8: multiset(A),N: multiset(A)] :
          ( aa(multiset(A),$o,aa(multiset(A),fun(multiset(A),$o),ord_less_eq(multiset(A)),M8),N)
        <=> ( aa(multiset(A),$o,ord_less(multiset(A),M8),N)
            | ( M8 = N ) ) ) ) ).

% less_eq_multiset_def
tff(fact_3238_vebt__minti_Opelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_Time_Heap(option(nat))] :
      ( ( vEBT_vebt_minti(Xb) = Y )
     => ( aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_vebt_minti_rel),Xb)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leafi((A4),(B3)) )
             => ( ( Y = $ite(
                      (A4),
                      aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),zero_zero(nat))),
                      $ite((B3),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),one_one(nat))),aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat))) ) )
               => ~ aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_vebt_minti_rel),vEBT_Leafi((A4),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: array(vEBT_VEBTi),Uw2: vEBT_VEBTi] :
                ( ( Xb = vEBT_Nodei(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),none(nat)) )
                 => ~ aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_vebt_minti_rel),vEBT_Nodei(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: array(vEBT_VEBTi),Uz2: vEBT_VEBTi] :
                  ( ( Xb = vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(option(nat),heap_Time_Heap(option(nat)),heap_Time_return(option(nat)),aa(nat,option(nat),some(nat),Mi)) )
                   => ~ aa(vEBT_VEBTi,$o,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),Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_minti.pelims
tff(fact_3239_VEBT__internal_Ocnt_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: real] :
      ( ( aa(vEBT_VEBT,real,vEBT_VEBT_cnt,Xb) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2),Xb)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = one_one(real) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2),vEBT_Leaf((A4),(B3))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,Deg2,TreeList,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),aa(list(vEBT_VEBT),list(real),map(vEBT_VEBT,real,vEBT_VEBT_cnt),TreeList),zero_zero(real))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel2),vEBT_Node(Info2,Deg2,TreeList,Summary)) ) ) ) ) ) ).

% VEBT_internal.cnt.pelims
tff(fact_3240_VEBT__internal_Ocnt_H_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_cnt2,Xb) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel),Xb)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = one_one(nat) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel),vEBT_Leaf((A4),(B3))) ) )
         => ~ ! [Info2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Info2,Deg2,TreeList,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),aa(list(vEBT_VEBT),list(nat),map(vEBT_VEBT,nat,vEBT_VEBT_cnt2),TreeList),zero_zero(nat))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_cnt_rel),vEBT_Node(Info2,Deg2,TreeList,Summary)) ) ) ) ) ) ).

% VEBT_internal.cnt'.pelims
tff(fact_3241_vebt__maxt_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_maxt(Xb) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_maxt_rel),Xb)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = $ite(
                      (B3),
                      aa(nat,option(nat),some(nat),one_one(nat)),
                      $ite((A4),aa(nat,option(nat),some(nat),zero_zero(nat)),none(nat)) ) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_maxt_rel),vEBT_Leaf((A4),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = none(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_maxt_rel),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Ma) )
                   => ~ aa(vEBT_VEBT,$o,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),Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_maxt.pelims
tff(fact_3242_vebt__mint_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: option(nat)] :
      ( ( vEBT_vebt_mint(Xb) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_mint_rel),Xb)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = $ite(
                      (A4),
                      aa(nat,option(nat),some(nat),zero_zero(nat)),
                      $ite((B3),aa(nat,option(nat),some(nat),one_one(nat)),none(nat)) ) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_mint_rel),vEBT_Leaf((A4),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = none(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_vebt_mint_rel),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y = aa(nat,option(nat),some(nat),Mi) )
                   => ~ aa(vEBT_VEBT,$o,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),Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% vebt_mint.pelims
tff(fact_3243_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 )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel),Xb)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite((A4),zero_zero(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel),vEBT_Leaf((A4),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_i_n_t_rel),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(vEBT_VEBT,$o,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),Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>t.pelims
tff(fact_3244_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 )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel),Xb)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),
                      $ite((B3),one_one(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)))) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel),vEBT_Leaf((A4),(B3))) ) )
         => ( ! [Uu2: nat,Uv2: list(vEBT_VEBT),Uw2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T_m_a_x_t_rel),vEBT_Node(none(product_prod(nat,nat)),Uu2,Uv2,Uw2)) ) )
           => ~ ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list(vEBT_VEBT),Uz2: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),product_Pair(nat,nat),Mi),Ma)),Ux2,Uy2,Uz2) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(vEBT_VEBT,$o,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),Mi),Ma)),Ux2,Uy2,Uz2)) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>a\<^sub>x\<^sub>t.pelims
tff(fact_3245_VEBT__internal_OminNulli_Opelims,axiom,
    ! [Xb: vEBT_VEBTi,Y: heap_Time_Heap($o)] :
      ( ( vEBT_VEBT_minNulli(Xb) = Y )
     => ( aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_V5740978063120863272li_rel),Xb)
       => ( ( ( Xb = vEBT_Leafi($false,$false) )
           => ( ( Y = aa($o,heap_Time_Heap($o),heap_Time_return($o),$true) )
             => ~ aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_V5740978063120863272li_rel),vEBT_Leafi($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xb = vEBT_Leafi($true,(Uv2)) )
               => ( ( Y = aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) )
                 => ~ aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_V5740978063120863272li_rel),vEBT_Leafi($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leafi((Uu2),$true) )
                 => ( ( Y = aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) )
                   => ~ aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_V5740978063120863272li_rel),vEBT_Leafi((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: array(vEBT_VEBTi),Uy2: vEBT_VEBTi] :
                    ( ( Xb = vEBT_Nodei(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
                   => ( ( Y = aa($o,heap_Time_Heap($o),heap_Time_return($o),$true) )
                     => ~ aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_V5740978063120863272li_rel),vEBT_Nodei(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va: nat,Vb2: array(vEBT_VEBTi),Vc: vEBT_VEBTi] :
                      ( ( Xb = vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va,Vb2,Vc) )
                     => ( ( Y = aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) )
                       => ~ aa(vEBT_VEBTi,$o,accp(vEBT_VEBTi,vEBT_V5740978063120863272li_rel),vEBT_Nodei(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va,Vb2,Vc)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNulli.pelims
tff(fact_3246_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 )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( ( Y = one_one(nat) )
             => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv2)) )
               => ( ( Y = one_one(nat) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ( Y = one_one(nat) )
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
                   => ( ( Y = one_one(nat) )
                     => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_T5462971552011256508_l_rel),vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va,Vb2,Vc) )
                     => ( ( Y = one_one(nat) )
                       => ~ aa(vEBT_VEBT,$o,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),Va,Vb2,Vc)) ) ) ) ) ) ) ) ) ).

% T\<^sub>m\<^sub>i\<^sub>n\<^sub>N\<^sub>u\<^sub>l\<^sub>l.pelims
tff(fact_3247_fold__atLeastAtMost__nat_Opinduct,axiom,
    ! [A: $tType,A0: fun(nat,fun(A,A)),A1: nat,A22: nat,A32: A,P: fun(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))))] :
      ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),aa(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),aa(fun(nat,fun(A,A)),fun(product_prod(nat,product_prod(nat,A)),product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A)))),product_Pair(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),A0),aa(product_prod(nat,A),product_prod(nat,product_prod(nat,A)),aa(nat,fun(product_prod(nat,A),product_prod(nat,product_prod(nat,A))),product_Pair(nat,product_prod(nat,A)),A1),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),A22),A32))))
     => ( ! [F2: fun(nat,fun(A,A)),A4: nat,B3: nat,Acc: A] :
            ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),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))),F2),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)),A4),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),B3),Acc))))
           => ( ( ~ aa(nat,$o,ord_less(nat,B3),A4)
               => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A4),one_one(nat))),B3),aa(A,A,aa(nat,fun(A,A),F2,A4),Acc)) )
             => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,F2),A4),B3),Acc) ) )
       => aa(A,$o,aa(nat,fun(A,$o),aa(nat,fun(nat,fun(A,$o)),aa(fun(nat,fun(A,A)),fun(nat,fun(nat,fun(A,$o))),P,A0),A1),A22),A32) ) ) ).

% fold_atLeastAtMost_nat.pinduct
tff(fact_3248_fold__atLeastAtMost__nat_Opsimps,axiom,
    ! [A: $tType,F: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc2: A] :
      ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),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))),F),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),Acc2))))
     => ( set_fo6178422350223883121st_nat(A,F,A2,B2,Acc2) = $ite(aa(nat,$o,ord_less(nat,B2),A2),Acc2,set_fo6178422350223883121st_nat(A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F,A2),Acc2))) ) ) ).

% fold_atLeastAtMost_nat.psimps
tff(fact_3249_fold__atLeastAtMost__nat_Opelims,axiom,
    ! [A: $tType,Xb: fun(nat,fun(A,A)),Xaa: nat,Xba: nat,Xc: A,Y: A] :
      ( ( set_fo6178422350223883121st_nat(A,Xb,Xaa,Xba,Xc) = Y )
     => ( aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),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))),Xb),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)),Xaa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xba),Xc))))
       => ~ ( ( 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))) )
           => ~ aa(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),$o,accp(product_prod(fun(nat,fun(A,A)),product_prod(nat,product_prod(nat,A))),set_fo1817059534552279752at_rel(A)),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))),Xb),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)),Xaa),aa(A,product_prod(nat,A),aa(nat,fun(A,product_prod(nat,A)),product_Pair(nat,A),Xba),Xc)))) ) ) ) ).

% fold_atLeastAtMost_nat.pelims
tff(fact_3250_VEBT__internal_Ooption__shift_Opelims,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,A)),Xaa: option(A),Xba: option(A),Y: option(A)] :
      ( ( aa(option(A),option(A),aa(option(A),fun(option(A),option(A)),vEBT_V2048590022279873568_shift(A,Xb),Xaa),Xba) = Y )
     => ( aa(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),$o,accp(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),vEBT_V459564278314245337ft_rel(A)),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))),Xb),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)),Xaa),Xba)))
       => ( ( ( Xaa = none(A) )
           => ( ( Y = none(A) )
             => ~ aa(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),$o,accp(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),vEBT_V459564278314245337ft_rel(A)),aa(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),aa(fun(A,fun(A,A)),fun(product_prod(option(A),option(A)),product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A)))),product_Pair(fun(A,fun(A,A)),product_prod(option(A),option(A))),Xb),aa(option(A),product_prod(option(A),option(A)),aa(option(A),fun(option(A),product_prod(option(A),option(A))),product_Pair(option(A),option(A)),none(A)),Xba))) ) )
         => ( ! [V3: A] :
                ( ( Xaa = aa(A,option(A),some(A),V3) )
               => ( ( Xba = none(A) )
                 => ( ( Y = none(A) )
                   => ~ aa(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),$o,accp(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),vEBT_V459564278314245337ft_rel(A)),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))),Xb),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)))) ) ) )
           => ~ ! [A4: A] :
                  ( ( Xaa = aa(A,option(A),some(A),A4) )
                 => ! [B3: A] :
                      ( ( Xba = aa(A,option(A),some(A),B3) )
                     => ( ( Y = aa(A,option(A),some(A),aa(A,A,aa(A,fun(A,A),Xb,A4),B3)) )
                       => ~ aa(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),$o,accp(product_prod(fun(A,fun(A,A)),product_prod(option(A),option(A))),vEBT_V459564278314245337ft_rel(A)),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))),Xb),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),A4)),aa(A,option(A),some(A),B3)))) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.pelims
tff(fact_3251_quickcheck__narrowing__samples_Onarrowing__samples_Opinduct,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),A0: code_integer,P: fun(code_integer,$o)] :
          ( aa(code_integer,$o,accp(code_integer,code_T1710151556404007877es_rel(A,A_of_integer)),A0)
         => ( ! [I3: code_integer] :
                ( aa(code_integer,$o,accp(code_integer,code_T1710151556404007877es_rel(A,A_of_integer)),I3)
               => ( ! [X3: product_prod(A,A),Xa: A,Y4: A] :
                      ( aa(code_integer,$o,ord_less(code_integer,zero_zero(code_integer)),I3)
                     => ( ( X3 = aa(code_integer,product_prod(A,A),A_of_integer,I3) )
                       => ( ( aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xa),Y4) = X3 )
                         => aa(code_integer,$o,P,aa(code_integer,code_integer,minus_minus(code_integer,I3),one_one(code_integer))) ) ) )
                 => aa(code_integer,$o,P,I3) ) )
           => aa(code_integer,$o,P,A0) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.pinduct
tff(fact_3252_in__measure,axiom,
    ! [A: $tType,Xb: A,Y: A,F: fun(A,nat)] :
      ( member(product_prod(A,A),aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),Xb),Y),measure(A,F))
    <=> aa(nat,$o,ord_less(nat,aa(A,nat,F,Xb)),aa(A,nat,F,Y)) ) ).

% in_measure
tff(fact_3253_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_3254_fold__atLeastAtMost__nat_Osimps,axiom,
    ! [A: $tType,F: fun(nat,fun(A,A)),A2: nat,B2: nat,Acc2: A] :
      set_fo6178422350223883121st_nat(A,F,A2,B2,Acc2) = $ite(aa(nat,$o,ord_less(nat,B2),A2),Acc2,set_fo6178422350223883121st_nat(A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),one_one(nat)),B2,aa(A,A,aa(nat,fun(A,A),F,A2),Acc2))) ).

% fold_atLeastAtMost_nat.simps
tff(fact_3255_freeze__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A2: array(A),Xs: list(A)] : hoare_hoare_triple(list(A),aa(list(A),assn,snga_assn(A,A2),Xs),array_freeze(A,A2),aa(list(A),fun(list(A),assn),aTP_Lamp_tj(array(A),fun(list(A),fun(list(A),assn)),A2),Xs)) ) ).

% freeze_rule
tff(fact_3256_in__finite__psubset,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( member(product_prod(set(A),set(A)),aa(set(A),product_prod(set(A),set(A)),aa(set(A),fun(set(A),product_prod(set(A),set(A))),product_Pair(set(A),set(A)),A3),B5),finite_psubset(A))
    <=> ( aa(set(A),$o,ord_less(set(A),A3),B5)
        & finite_finite2(A,B5) ) ) ).

% in_finite_psubset
tff(fact_3257_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_tk(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_tk(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,$o)),aTP_Lamp_tl(set(A),fun(fun(A,B),fun(fun(A,B),fun(A,$o))),I5),Xb),Y))) ) ) ) ).

% prod.finite_Collect_op
tff(fact_3258_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_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))) ) ) ) ).

% sum.finite_Collect_op
tff(fact_3259_ceiling__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,one_one(real)),B2)
       => ( ( archimedean_ceiling(real,log(B2,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,$o,ord_less(real,powr(real,B2,aa(nat,real,semiring_1_of_nat(real),K))),Xb)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),powr(real,B2,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_3260_powr__one__eq__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [A2: A] : powr(A,one_one(A),A2) = one_one(A) ) ).

% powr_one_eq_one
tff(fact_3261_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_3262_powr__nonneg__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,A2,Xb)),zero_zero(real))
    <=> ( A2 = zero_zero(real) ) ) ).

% powr_nonneg_iff
tff(fact_3263_powr__less__cancel__iff,axiom,
    ! [Xb: real,A2: real,B2: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xb)
     => ( aa(real,$o,ord_less(real,powr(real,Xb,A2)),powr(real,Xb,B2))
      <=> aa(real,$o,ord_less(real,A2),B2) ) ) ).

% powr_less_cancel_iff
tff(fact_3264_powr__eq__one__iff,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),A2)
     => ( ( powr(real,A2,Xb) = one_one(real) )
      <=> ( Xb = zero_zero(real) ) ) ) ).

% powr_eq_one_iff
tff(fact_3265_powr__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,one_one(real)) = Xb ) ) ).

% powr_one
tff(fact_3266_powr__one__gt__zero__iff,axiom,
    ! [Xb: real] :
      ( ( powr(real,Xb,one_one(real)) = Xb )
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb) ) ).

% powr_one_gt_zero_iff
tff(fact_3267_powr__le__cancel__iff,axiom,
    ! [Xb: real,A2: real,B2: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xb,A2)),powr(real,Xb,B2))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2) ) ) ).

% powr_le_cancel_iff
tff(fact_3268_numeral__powr__numeral__real,axiom,
    ! [M: num,Nb: num] : powr(real,aa(num,real,numeral_numeral(real),M),aa(num,real,numeral_numeral(real),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),M)),aa(num,nat,numeral_numeral(nat),Nb)) ).

% numeral_powr_numeral_real
tff(fact_3269_log__powr__cancel,axiom,
    ! [A2: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( log(A2,powr(real,A2,Y)) = Y ) ) ) ).

% log_powr_cancel
tff(fact_3270_powr__log__cancel,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( powr(real,A2,log(A2,Xb)) = Xb ) ) ) ) ).

% powr_log_cancel
tff(fact_3271_powr__numeral,axiom,
    ! [Xb: real,Nb: num] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(num,real,numeral_numeral(real),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(num,nat,numeral_numeral(nat),Nb)) ) ) ).

% powr_numeral
tff(fact_3272_powr__powr,axiom,
    ! [Xb: real,A2: real,B2: real] : powr(real,powr(real,Xb,A2),B2) = powr(real,Xb,aa(real,real,aa(real,fun(real,real),times_times(real),A2),B2)) ).

% powr_powr
tff(fact_3273_powr__ge__pzero,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),powr(real,Xb,Y)) ).

% powr_ge_pzero
tff(fact_3274_powr__mono2,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xb,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_mono2
tff(fact_3275_powr__less__mono,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( aa(real,$o,ord_less(real,A2),B2)
     => ( aa(real,$o,ord_less(real,one_one(real)),Xb)
       => aa(real,$o,ord_less(real,powr(real,Xb,A2)),powr(real,Xb,B2)) ) ) ).

% powr_less_mono
tff(fact_3276_powr__less__cancel,axiom,
    ! [Xb: real,A2: real,B2: real] :
      ( aa(real,$o,ord_less(real,powr(real,Xb,A2)),powr(real,Xb,B2))
     => ( aa(real,$o,ord_less(real,one_one(real)),Xb)
       => aa(real,$o,ord_less(real,A2),B2) ) ) ).

% powr_less_cancel
tff(fact_3277_powr__mono,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xb,A2)),powr(real,Xb,B2)) ) ) ).

% powr_mono
tff(fact_3278_powr__mono2_H,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Y,A2)),powr(real,Xb,A2)) ) ) ) ).

% powr_mono2'
tff(fact_3279_powr__less__mono2,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(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,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_less_mono2
tff(fact_3280_powr__inj,axiom,
    ! [A2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( ( powr(real,A2,Xb) = powr(real,A2,Y) )
        <=> ( Xb = Y ) ) ) ) ).

% powr_inj
tff(fact_3281_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_3282_powr__le1,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xb,A2)),one_one(real)) ) ) ) ).

% powr_le1
tff(fact_3283_powr__mono__both,axiom,
    ! [A2: real,B2: real,Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,Xb,A2)),powr(real,Y,B2)) ) ) ) ) ).

% powr_mono_both
tff(fact_3284_ge__one__powr__ge__zero,axiom,
    ! [Xb: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),powr(real,Xb,A2)) ) ) ).

% ge_one_powr_ge_zero
tff(fact_3285_powr__divide,axiom,
    ! [Xb: real,Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y),A2) = aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,Xb,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_divide
tff(fact_3286_powr__mult,axiom,
    ! [Xb: real,Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( powr(real,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y),A2) = aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,Xb,A2)),powr(real,Y,A2)) ) ) ) ).

% powr_mult
tff(fact_3287_log__base__powr,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( ( A2 != zero_zero(real) )
     => ( log(powr(real,A2,B2),Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),log(A2,Xb)),B2) ) ) ).

% log_base_powr
tff(fact_3288_log__powr,axiom,
    ! [Xb: real,B2: real,Y: real] :
      ( ( Xb != zero_zero(real) )
     => ( log(B2,powr(real,Xb,Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),Y),log(B2,Xb)) ) ) ).

% log_powr
tff(fact_3289_powr__add,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A,A2: A,B2: A] : powr(A,Xb,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),powr(A,Xb,A2)),powr(A,Xb,B2)) ) ).

% powr_add
tff(fact_3290_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_3291_powr__realpow,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(nat,real,semiring_1_of_nat(real),Nb)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb) ) ) ).

% powr_realpow
tff(fact_3292_less__log__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,Y),log(B2,Xb))
        <=> aa(real,$o,ord_less(real,powr(real,B2,Y)),Xb) ) ) ) ).

% less_log_iff
tff(fact_3293_log__less__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,log(B2,Xb)),Y)
        <=> aa(real,$o,ord_less(real,Xb),powr(real,B2,Y)) ) ) ) ).

% log_less_iff
tff(fact_3294_less__powr__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,Xb),powr(real,B2,Y))
        <=> aa(real,$o,ord_less(real,log(B2,Xb)),Y) ) ) ) ).

% less_powr_iff
tff(fact_3295_powr__less__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,ord_less(real,powr(real,B2,Y)),Xb)
        <=> aa(real,$o,ord_less(real,Y),log(B2,Xb)) ) ) ) ).

% powr_less_iff
tff(fact_3296_powr__mult__base,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$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_3297_le__log__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),log(B2,Xb))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Y)),Xb) ) ) ) ).

% le_log_iff
tff(fact_3298_log__le__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),log(B2,Xb)),Y)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),powr(real,B2,Y)) ) ) ) ).

% log_le_iff
tff(fact_3299_le__powr__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),powr(real,B2,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),log(B2,Xb)),Y) ) ) ) ).

% le_powr_iff
tff(fact_3300_powr__le__iff,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),B2)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,Y)),Xb)
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),log(B2,Xb)) ) ) ) ).

% powr_le_iff
tff(fact_3301_log__add__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),log(B2,Xb)),Y) = log(B2,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),powr(real,B2,Y))) ) ) ) ) ).

% log_add_eq_powr
tff(fact_3302_add__log__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Y),log(B2,Xb)) = log(B2,aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,B2,Y)),Xb)) ) ) ) ) ).

% add_log_eq_powr
tff(fact_3303_minus__log__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( aa(real,real,minus_minus(real,Y),log(B2,Xb)) = log(B2,aa(real,real,aa(real,fun(real,real),divide_divide(real),powr(real,B2,Y)),Xb)) ) ) ) ) ).

% minus_log_eq_powr
tff(fact_3304_finite__psubset__def,axiom,
    ! [A: $tType] : finite_psubset(A) = collect(product_prod(set(A),set(A)),product_case_prod(set(A),set(A),$o,aTP_Lamp_to(set(A),fun(set(A),$o)))) ).

% finite_psubset_def
tff(fact_3305_sum__gp,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,M: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_or1337092689740270186AtMost(nat,M,Nb)) = $ite(
            aa(nat,$o,ord_less(nat,Nb),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),Nb),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,Nb)))),aa(A,A,minus_minus(A,one_one(A)),Xb))) ) ) ).

% sum_gp
tff(fact_3306_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_3307_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_tp(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_3308_VEBT__internal_Oheight_Osimps_I1_J,axiom,
    ! [A2: $o,B2: $o] : aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Leaf((A2),(B2))) = zero_zero(nat) ).

% VEBT_internal.height.simps(1)
tff(fact_3309_sum_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_tq(B,A)),A3) = zero_zero(A) ) ).

% sum.neutral_const
tff(fact_3310_of__nat__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [F: fun(B,nat),A3: set(B)] : aa(nat,A,semiring_1_of_nat(A),aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,F),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_tr(fun(B,nat),fun(B,A),F)),A3) ) ).

% of_nat_sum
tff(fact_3311_of__int__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [F: fun(B,int),A3: set(B)] : aa(int,A,ring_1_of_int(A),aa(set(B),int,groups7311177749621191930dd_sum(B,int,F),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_ts(fun(B,int),fun(B,A),F)),A3) ) ).

% of_int_sum
tff(fact_3312_sum_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B)] :
          ( finite_finite2(A,S2)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tt(A,fun(fun(A,B),fun(A,B)),A2),B2)),S2) = $ite(member(A,A2,S2),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta'
tff(fact_3313_sum_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B)] :
          ( finite_finite2(A,S2)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_tu(A,fun(fun(A,B),fun(A,B)),A2),B2)),S2) = $ite(member(A,A2,S2),aa(A,B,B2,A2),zero_zero(B)) ) ) ) ).

% sum.delta
tff(fact_3314_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_3315_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_3316_sum_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),Xb: A,G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ~ member(A,Xb,A3)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),insert(A,Xb),A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)) ) ) ) ) ).

% sum.insert
tff(fact_3317_powser__sums__zero__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A),Xb: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_tv(fun(nat,A),fun(nat,A),A2)),Xb)
        <=> ( aa(nat,A,A2,zero_zero(nat)) = Xb ) ) ) ).

% powser_sums_zero_iff
tff(fact_3318_sum_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,ord_less(nat,aa(nat,nat,suc,Nb)),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,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb)))) ) ).

% sum.cl_ivl_Suc
tff(fact_3319_sum__zero__power,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [C2: fun(nat,A),A3: set(nat)] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_tw(fun(nat,A),fun(nat,A),C2)),A3) = $ite(
            ( finite_finite2(nat,A3)
            & member(nat,zero_zero(nat),A3) ),
            aa(nat,A,C2,zero_zero(nat)),
            zero_zero(A) ) ) ).

% sum_zero_power
tff(fact_3320_sum__zero__power_H,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [C2: fun(nat,A),D2: fun(nat,A),A3: set(nat)] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_tx(fun(nat,A),fun(fun(nat,A),fun(nat,A)),C2),D2)),A3) = $ite(
            ( finite_finite2(nat,A3)
            & member(nat,zero_zero(nat),A3) ),
            aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,C2,zero_zero(nat))),aa(nat,A,D2,zero_zero(nat))),
            zero_zero(A) ) ) ).

% sum_zero_power'
tff(fact_3321_sum_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,fun(C,A)),B5: set(C),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(set(C),fun(B,A),aTP_Lamp_ty(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B5)),A3) = aa(set(C),A,groups7311177749621191930dd_sum(C,A,aa(set(B),fun(C,A),aTP_Lamp_ua(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A3)),B5) ) ).

% sum.swap
tff(fact_3322_norm__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(B,A),A3: set(B)] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F),A3))),aa(set(B),real,groups7311177749621191930dd_sum(B,real,aTP_Lamp_ub(fun(B,A),fun(B,real),F)),A3)) ) ).

% norm_sum
tff(fact_3323_sum__norm__le,axiom,
    ! [B: $tType,A: $tType] :
      ( real_V822414075346904944vector(B)
     => ! [S2: set(A),F: fun(A,B),G: fun(A,real)] :
          ( ! [X2: A] :
              ( member(A,X2,S2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,F,X2))),aa(A,real,G,X2)) )
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),S2))),aa(set(A),real,groups7311177749621191930dd_sum(A,real,G),S2)) ) ) ).

% sum_norm_le
tff(fact_3324_sum__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [K6: set(A),F: fun(A,B),G: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,K6)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I3)),aa(A,B,G,I3)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),K6)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),K6)) ) ) ).

% sum_mono
tff(fact_3325_sum_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),H: fun(B,A),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_uc(fun(B,A),fun(fun(B,A),fun(B,A)),G),H)),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,H),A3)) ) ).

% sum.distrib
tff(fact_3326_sum__distrib__left,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [R3: A,F: fun(B,A),A3: set(B)] : aa(A,A,aa(A,fun(A,A),times_times(A),R3),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_ud(A,fun(fun(B,A),fun(B,A)),R3),F)),A3) ) ).

% sum_distrib_left
tff(fact_3327_sum__distrib__right,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F: fun(B,A),A3: set(B),R3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F),A3)),R3) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_ue(fun(B,A),fun(A,fun(B,A)),F),R3)),A3) ) ).

% sum_distrib_right
tff(fact_3328_sum__product,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( semiring_0(A)
     => ! [F: fun(B,A),A3: set(B),G: fun(C,A),B5: set(C)] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F),A3)),aa(set(C),A,groups7311177749621191930dd_sum(C,A,G),B5)) = 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_ug(fun(B,A),fun(fun(C,A),fun(set(C),fun(B,A))),F),G),B5)),A3) ) ).

% sum_product
tff(fact_3329_sum__subtractf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(B,A),G: fun(B,A),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_uh(fun(B,A),fun(fun(B,A),fun(B,A)),F),G)),A3) = aa(A,A,minus_minus(A,aa(set(B),A,groups7311177749621191930dd_sum(B,A,F),A3)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)) ) ).

% sum_subtractf
tff(fact_3330_sum__divide__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F: fun(B,A),A3: set(B),R3: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F),A3)),R3) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_ui(fun(B,A),fun(A,fun(B,A)),F),R3)),A3) ) ).

% sum_divide_distrib
tff(fact_3331_sum_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [A3: set(A),B5: set(B),G: fun(A,fun(B,C)),R: fun(A,fun(B,$o))] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(B,B5)
           => ( 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_uk(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B5),G),R)),A3) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_um(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A3),G),R)),B5) ) ) ) ) ).

% sum.swap_restrict
tff(fact_3332_mod__sum__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F: fun(B,A),A2: A,A3: 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_un(fun(B,A),fun(A,fun(B,A)),F),A2)),A3)),A2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F),A3)),A2) ) ).

% mod_sum_eq
tff(fact_3333_sum__nonpos,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),zero_zero(B)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),A3)),zero_zero(B)) ) ) ).

% sum_nonpos
tff(fact_3334_sum__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),A3)) ) ) ).

% sum_nonneg
tff(fact_3335_sum__mono__inv,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere8940638589300402666id_add(A)
     => ! [F: fun(B,A),I5: set(B),G: fun(B,A),I: B] :
          ( ( aa(set(B),A,groups7311177749621191930dd_sum(B,A,F),I5) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),I5) )
         => ( ! [I3: B] :
                ( member(B,I3,I5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F,I3)),aa(B,A,G,I3)) )
           => ( member(B,I,I5)
             => ( finite_finite2(B,I5)
               => ( aa(B,A,F,I) = aa(B,A,G,I) ) ) ) ) ) ) ).

% sum_mono_inv
tff(fact_3336_sum__cong__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A3: set(nat),F: fun(nat,A),G: fun(nat,A)] :
          ( ~ member(nat,zero_zero(nat),A3)
         => ( ! [X2: nat] :
                ( member(nat,aa(nat,nat,suc,X2),A3)
               => ( aa(nat,A,F,aa(nat,nat,suc,X2)) = aa(nat,A,G,aa(nat,nat,suc,X2)) ) )
           => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),A3) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),A3) ) ) ) ) ).

% sum_cong_Suc
tff(fact_3337_sum_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cv(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_uo(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P)),A3) ) ) ) ).

% sum.inter_filter
tff(fact_3338_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_up(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% sum.shift_bounds_cl_Suc_ivl
tff(fact_3339_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Nb: 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),Nb),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_uq(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% sum.shift_bounds_cl_nat_ivl
tff(fact_3340_sum__le__included,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(C)
     => ! [S3: set(A),Ta: set(B),G: fun(B,C),I: fun(B,A),F: fun(A,C)] :
          ( finite_finite2(A,S3)
         => ( finite_finite2(B,Ta)
           => ( ! [X2: B] :
                  ( member(B,X2,Ta)
                 => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),zero_zero(C)),aa(B,C,G,X2)) )
             => ( ! [X2: A] :
                    ( member(A,X2,S3)
                   => ? [Xa: B] :
                        ( member(B,Xa,Ta)
                        & ( aa(B,A,I,Xa) = X2 )
                        & aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(A,C,F,X2)),aa(B,C,G,Xa)) ) )
               => aa(C,$o,aa(C,fun(C,$o),ord_less_eq(C),aa(set(A),C,groups7311177749621191930dd_sum(A,C,F),S3)),aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),Ta)) ) ) ) ) ) ).

% sum_le_included
tff(fact_3341_sum__nonneg__eq__0__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ! [X2: A] :
                ( member(A,X2,A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X2)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),A3) = zero_zero(B) )
            <=> ! [X: A] :
                  ( member(A,X,A3)
                 => ( aa(A,B,F,X) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
tff(fact_3342_sum__strict__mono__ex1,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [A3: set(A),F: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ! [X2: A] :
                ( member(A,X2,A3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,G,X2)) )
           => ( ? [X3: A] :
                  ( member(A,X3,A3)
                  & aa(B,$o,ord_less(B,aa(A,B,F,X3)),aa(A,B,G,X3)) )
             => aa(B,$o,ord_less(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)) ) ) ) ) ).

% sum_strict_mono_ex1
tff(fact_3343_sum_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [R: fun(A,fun(A,$o)),S2: 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,X24: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X15),X24)
                  & 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),X24),Y23)) )
           => ( finite_finite2(B,S2)
             => ( ! [X2: B] :
                    ( member(B,X2,S2)
                   => 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),S2)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),S2)) ) ) ) ) ) ).

% sum.related
tff(fact_3344_sum__strict__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( strict7427464778891057005id_add(B)
     => ! [A3: set(A),F: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [X2: A] :
                  ( member(A,X2,A3)
                 => aa(B,$o,ord_less(B,aa(A,B,F,X2)),aa(A,B,G,X2)) )
             => aa(B,$o,ord_less(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3)) ) ) ) ) ).

% sum_strict_mono
tff(fact_3345_sum_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),Xb: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),insert(A,Xb),A3)) = $ite(member(A,Xb,A3),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3))) ) ) ) ).

% sum.insert_if
tff(fact_3346_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S6: set(A),T7: set(B),S2: set(A),I: fun(B,A),J: fun(A,B),T5: set(B),G: fun(A,C),H: fun(B,C)] :
          ( finite_finite2(A,S6)
         => ( finite_finite2(B,T7)
           => ( ! [A4: A] :
                  ( member(A,A4,aa(set(A),set(A),minus_minus(set(A),S2),S6))
                 => ( aa(B,A,I,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),minus_minus(set(A),S2),S6))
                   => member(B,aa(A,B,J,A4),aa(set(B),set(B),minus_minus(set(B),T5),T7)) )
               => ( ! [B3: B] :
                      ( member(B,B3,aa(set(B),set(B),minus_minus(set(B),T5),T7))
                     => ( aa(A,B,J,aa(B,A,I,B3)) = B3 ) )
                 => ( ! [B3: B] :
                        ( member(B,B3,aa(set(B),set(B),minus_minus(set(B),T5),T7))
                       => member(A,aa(B,A,I,B3),aa(set(A),set(A),minus_minus(set(A),S2),S6)) )
                   => ( ! [A4: A] :
                          ( member(A,A4,S6)
                         => ( aa(A,C,G,A4) = zero_zero(C) ) )
                     => ( ! [B3: B] :
                            ( member(B,B3,T7)
                           => ( aa(B,C,H,B3) = zero_zero(C) ) )
                       => ( ! [A4: A] :
                              ( member(A,A4,S2)
                             => ( aa(B,C,H,aa(A,B,J,A4)) = aa(A,C,G,A4) ) )
                         => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,G),S2) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,H),T5) ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
tff(fact_3347_sum__nonneg__0,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S3: set(A),F: fun(A,B),I: A] :
          ( finite_finite2(A,S3)
         => ( ! [I3: A] :
                ( member(A,I3,S3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I3)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),S3) = zero_zero(B) )
             => ( member(A,I,S3)
               => ( aa(A,B,F,I) = zero_zero(B) ) ) ) ) ) ) ).

% sum_nonneg_0
tff(fact_3348_sum__nonneg__leq__bound,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [S3: set(A),F: fun(A,B),B5: B,I: A] :
          ( finite_finite2(A,S3)
         => ( ! [I3: A] :
                ( member(A,I3,S3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I3)) )
           => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),S3) = B5 )
             => ( member(A,I,S3)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I)),B5) ) ) ) ) ) ).

% sum_nonneg_leq_bound
tff(fact_3349_sum_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),minus_minus(set(A),A3),collect(A,aTP_Lamp_ur(fun(A,B),fun(A,$o),G)))) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) ) ) ) ).

% sum.setdiff_irrelevant
tff(fact_3350_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_us(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_3351_sum_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,Nb,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ut(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),M)),set_or1337092689740270186AtMost(nat,Nb,M)) ) ).

% sum.atLeastAtMost_rev
tff(fact_3352_sum__pos2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),I: A,F: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,ord_less(B,zero_zero(B)),aa(A,B,F,I))
             => ( ! [I3: A] :
                    ( member(A,I3,I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I3)) )
               => aa(B,$o,ord_less(B,zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),I5)) ) ) ) ) ) ).

% sum_pos2
tff(fact_3353_sum__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),F: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( member(A,I3,I5)
                 => aa(B,$o,ord_less(B,zero_zero(B)),aa(A,B,F,I3)) )
             => aa(B,$o,ord_less(B,zero_zero(B)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),I5)) ) ) ) ) ).

% sum_pos
tff(fact_3354_sum_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C5: set(A),A3: set(A),B5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),minus_minus(set(A),C5),A3))
                   => ( aa(A,B,G,A4) = zero_zero(B) ) )
               => ( ! [B3: A] :
                      ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),C5),B5))
                     => ( aa(A,B,H,B3) = zero_zero(B) ) )
                 => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),B5) )
                  <=> ( 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_3355_sum_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [C5: set(A),A3: set(A),B5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),minus_minus(set(A),C5),A3))
                   => ( aa(A,B,G,A4) = zero_zero(B) ) )
               => ( ! [B3: A] :
                      ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),C5),B5))
                     => ( aa(A,B,H,B3) = 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),A3) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),B5) ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
tff(fact_3356_sum_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T5: set(A),S2: set(A),G: fun(A,B)] :
          ( finite_finite2(A,T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,G,X2) = zero_zero(B) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),S2) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),T5) ) ) ) ) ) ).

% sum.mono_neutral_left
tff(fact_3357_sum_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T5: set(A),S2: set(A),G: fun(A,B)] :
          ( finite_finite2(A,T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,G,X2) = zero_zero(B) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),T5) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),S2) ) ) ) ) ) ).

% sum.mono_neutral_right
tff(fact_3358_sum_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T5: set(A),S2: set(A),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,H,X2) = zero_zero(B) ) )
             => ( ! [X2: A] :
                    ( member(A,X2,S2)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
               => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),S2) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),T5) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
tff(fact_3359_sum_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [T5: set(A),S2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,G,X2) = zero_zero(B) ) )
             => ( ! [X2: A] :
                    ( member(A,X2,S2)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
               => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),T5) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),S2) ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
tff(fact_3360_sum_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [B5: set(A),A3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
         => ( finite_finite2(A,A3)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),minus_minus(set(A),A3),B5))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),B5)) ) ) ) ) ).

% sum.subset_diff
tff(fact_3361_sum__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A3: set(A),B5: set(A),F: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),aa(set(A),set(A),minus_minus(set(A),A3),B5)) = aa(B,B,minus_minus(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),B5)) ) ) ) ) ).

% sum_diff
tff(fact_3362_powser__sums__zero,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [A2: fun(nat,A)] : aa(A,$o,sums(A,aTP_Lamp_tv(fun(nat,A),fun(nat,A),A2)),aa(nat,A,A2,zero_zero(nat))) ) ).

% powser_sums_zero
tff(fact_3363_sum__shift__lb__Suc0__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_or1337092689740270186AtMost(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0
tff(fact_3364_sum_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = 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),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% sum.atLeast0_atMost_Suc
tff(fact_3365_sum_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,aa(nat,nat,suc,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Nb))) ) ) ) ).

% sum.nat_ivl_Suc'
tff(fact_3366_sum_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Nb)) = 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),Nb))) ) ) ) ).

% sum.atLeast_Suc_atMost
tff(fact_3367_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_3368_sum_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) = 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_up(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Nb))) ) ) ) ).

% sum.Suc_reindex_ivl
tff(fact_3369_sum__Suc__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Nb: nat,F: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Nb))
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_uu(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,M,Nb)) = aa(A,A,minus_minus(A,aa(nat,A,F,aa(nat,nat,suc,Nb))),aa(nat,A,F,M)) ) ) ) ).

% sum_Suc_diff
tff(fact_3370_sum__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(nat,A),A2: nat,B2: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_uv(fun(nat,A),fun(nat,fun(A,A)),F),A2,B2,zero_zero(A)) ) ).

% sum_atLeastAtMost_code
tff(fact_3371_sum__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [B5: set(A),A3: set(A),F: fun(A,B)] :
          ( finite_finite2(A,B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
           => ( ! [B3: A] :
                  ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),B5),A3))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,B3)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),B5)) ) ) ) ) ).

% sum_mono2
tff(fact_3372_sum_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),G: fun(A,B),Xb: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),insert(A,Xb),A3)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ).

% sum.insert_remove
tff(fact_3373_sum_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [A3: set(A),Xb: A,G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( member(A,Xb,A3)
           => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,G,Xb)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ) ).

% sum.remove
tff(fact_3374_sum__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [A3: set(A),F: fun(A,B),A2: A] :
          ( finite_finite2(A,A3)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = $ite(member(A,A2,A3),aa(B,B,minus_minus(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),A3)),aa(A,B,F,A2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),A3)) ) ) ) ).

% sum_diff1
tff(fact_3375_sum_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A),P2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),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),Nb),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,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P2)))) ) ) ) ).

% sum.ub_add_nat
tff(fact_3376_sum_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B),C2: fun(A,B)] :
          ( finite_finite2(A,S2)
         => ( 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_uw(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C2)),S2) = $ite(member(A,A2,S2),aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(A,B,B2,A2)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,C2),aa(set(A),set(A),minus_minus(set(A),S2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,C2),aa(set(A),set(A),minus_minus(set(A),S2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))) ) ) ) ).

% sum.delta_remove
tff(fact_3377_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_3378_sum__strict__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( ordere8940638589300402666id_add(B)
     => ! [B5: set(A),A3: set(A),B2: A,F: fun(A,B)] :
          ( finite_finite2(A,B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
           => ( member(A,B2,aa(set(A),set(A),minus_minus(set(A),B5),A3))
             => ( aa(B,$o,ord_less(B,zero_zero(B)),aa(A,B,F,B2))
               => ( ! [X2: A] :
                      ( member(A,X2,B5)
                     => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X2)) )
                 => aa(B,$o,ord_less(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),A3)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),B5)) ) ) ) ) ) ) ).

% sum_strict_mono2
tff(fact_3379_member__le__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ordere6911136660526730532id_add(B)
        & semiring_1(B) )
     => ! [I: A,A3: set(A),F: fun(A,B)] :
          ( member(A,I,A3)
         => ( ! [X2: A] :
                ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,I),bot_bot(set(A)))))
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X2)) )
           => ( finite_finite2(A,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),A3)) ) ) ) ) ).

% member_le_sum
tff(fact_3380_sum__natinterval__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(nat,A),M: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_ux(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,M,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb),aa(A,A,minus_minus(A,aa(nat,A,F,M)),aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))),zero_zero(A)) ) ).

% sum_natinterval_diff
tff(fact_3381_sum__telescope_H_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Nb: nat,F: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_uy(fun(nat,A),fun(nat,A),F)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),Nb)) = aa(A,A,minus_minus(A,aa(nat,A,F,Nb)),aa(nat,A,F,M)) ) ) ) ).

% sum_telescope''
tff(fact_3382_mask__eq__sum__exp,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [Nb: nat] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),one_one(A)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2)))),collect(nat,aTP_Lamp_sq(nat,fun(nat,$o),Nb))) ) ).

% mask_eq_sum_exp
tff(fact_3383_sum__gp__multiplied,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Nb: nat,Xb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( 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,Nb))) = 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,Nb))) ) ) ) ).

% sum_gp_multiplied
tff(fact_3384_sum_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: 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))),Nb)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_uz(fun(nat,A),fun(nat,A),G)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% sum.in_pairs
tff(fact_3385_VEBT__internal_Oheight_Ocases,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ! [A4: $o,B3: $o] : Xb != vEBT_Leaf((A4),(B3))
     => ~ ! [Uu2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] : Xb != vEBT_Node(Uu2,Deg2,TreeList,Summary) ) ).

% VEBT_internal.height.cases
tff(fact_3386_sums__if_H,axiom,
    ! [G: fun(nat,real),Xb: real] :
      ( aa(real,$o,sums(real,G),Xb)
     => aa(real,$o,sums(real,aTP_Lamp_va(fun(nat,real),fun(nat,real),G)),Xb) ) ).

% sums_if'
tff(fact_3387_sums__if,axiom,
    ! [G: fun(nat,real),Xb: real,F: fun(nat,real),Y: real] :
      ( aa(real,$o,sums(real,G),Xb)
     => ( aa(real,$o,sums(real,F),Y)
       => aa(real,$o,sums(real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_vb(fun(nat,real),fun(fun(nat,real),fun(nat,real)),G),F)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)) ) ) ).

% sums_if
tff(fact_3388_mask__eq__sum__exp__nat,axiom,
    ! [Nb: 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))),Nb)),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_sq(nat,fun(nat,$o),Nb))) ).

% mask_eq_sum_exp_nat
tff(fact_3389_gauss__sum__nat,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_vc(nat,nat)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,suc,Nb))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% gauss_sum_nat
tff(fact_3390_double__gauss__sum,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: 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),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))) ) ).

% double_gauss_sum
tff(fact_3391_double__arith__series,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,D2: A,Nb: 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_vd(A,fun(A,fun(nat,A)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))) = 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),Nb)),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))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),D2))) ) ).

% double_arith_series
tff(fact_3392_arith__series__nat,axiom,
    ! [A2: nat,D2: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_ve(nat,fun(nat,fun(nat,nat)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = 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,Nb)),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))),A2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),D2)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% arith_series_nat
tff(fact_3393_Sum__Icc__nat,axiom,
    ! [M: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_vc(nat,nat)),set_or1337092689740270186AtMost(nat,M,Nb)) = 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),Nb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),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_3394_double__gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: 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)),Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))) ) ).

% double_gauss_sum_from_Suc_0
tff(fact_3395_gauss__sum,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = 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),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum
tff(fact_3396_arith__series,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [A2: A,D2: A,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_vf(A,fun(A,fun(nat,A)),A2),D2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = 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),Nb)),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))),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),D2)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% arith_series
tff(fact_3397_sum__gp__offset,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,M: nat,Nb: 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),Nb))) = $ite(Xb = one_one(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),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,Nb))))),aa(A,A,minus_minus(A,one_one(A)),Xb))) ) ).

% sum_gp_offset
tff(fact_3398_gauss__sum__from__Suc__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,semiring_1_of_nat(A)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = 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),Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% gauss_sum_from_Suc_0
tff(fact_3399_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_3400_power__half__series,axiom,
    aa(real,$o,sums(real,aTP_Lamp_vg(nat,real)),one_one(real)) ).

% power_half_series
tff(fact_3401_sums__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => aa(A,$o,sums(A,aTP_Lamp_vh(nat,A)),zero_zero(A)) ) ).

% sums_zero
tff(fact_3402_sums__If__finite__set_H,axiom,
    ! [A: $tType] :
      ( ( topolo1287966508704411220up_add(A)
        & topological_t2_space(A) )
     => ! [G: fun(nat,A),S2: A,A3: set(nat),S6: A,F: fun(nat,A)] :
          ( aa(A,$o,sums(A,G),S2)
         => ( finite_finite2(nat,A3)
           => ( ( S6 = aa(A,A,aa(A,fun(A,A),plus_plus(A),S2),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vi(fun(nat,A),fun(fun(nat,A),fun(nat,A)),G),F)),A3)) )
             => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aa(set(nat),fun(fun(nat,A),fun(nat,A)),aTP_Lamp_vj(fun(nat,A),fun(set(nat),fun(fun(nat,A),fun(nat,A))),G),A3),F)),S6) ) ) ) ) ).

% sums_If_finite_set'
tff(fact_3403_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_vk(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_3404_int__sum,axiom,
    ! [A: $tType,F: fun(A,nat),A3: set(A)] : aa(nat,int,semiring_1_of_nat(int),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F),A3)) = aa(set(A),int,groups7311177749621191930dd_sum(A,int,aTP_Lamp_vl(fun(A,nat),fun(A,int),F)),A3) ).

% int_sum
tff(fact_3405_sum__subtractf__nat,axiom,
    ! [A: $tType,A3: set(A),G: fun(A,nat),F: fun(A,nat)] :
      ( ! [X2: A] :
          ( member(A,X2,A3)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,G,X2)),aa(A,nat,F,X2)) )
     => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_vm(fun(A,nat),fun(fun(A,nat),fun(A,nat)),G),F)),A3) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F),A3)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,G),A3)) ) ) ).

% sum_subtractf_nat
tff(fact_3406_sum__SucD,axiom,
    ! [A: $tType,F: fun(A,nat),A3: set(A),Nb: nat] :
      ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F),A3) = aa(nat,nat,suc,Nb) )
     => ? [X2: A] :
          ( member(A,X2,A3)
          & aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(A,nat,F,X2)) ) ) ).

% sum_SucD
tff(fact_3407_sum__eq__Suc0__iff,axiom,
    ! [A: $tType,A3: set(A),F: fun(A,nat)] :
      ( finite_finite2(A,A3)
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F),A3) = aa(nat,nat,suc,zero_zero(nat)) )
      <=> ? [X: A] :
            ( member(A,X,A3)
            & ( aa(A,nat,F,X) = aa(nat,nat,suc,zero_zero(nat)) )
            & ! [Xa3: A] :
                ( member(A,Xa3,A3)
               => ( ( X != Xa3 )
                 => ( aa(A,nat,F,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
tff(fact_3408_sum__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F: fun(A,nat)] :
      ( finite_finite2(A,A3)
     => ( ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F),A3) = one_one(nat) )
      <=> ? [X: A] :
            ( member(A,X,A3)
            & ( aa(A,nat,F,X) = one_one(nat) )
            & ! [Xa3: A] :
                ( member(A,Xa3,A3)
               => ( ( X != Xa3 )
                 => ( aa(A,nat,F,Xa3) = zero_zero(nat) ) ) ) ) ) ) ).

% sum_eq_1_iff
tff(fact_3409_sum__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,one_one(nat)),Nb)
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_vn(complex,complex)),collect(complex,aTP_Lamp_vo(nat,fun(complex,$o),Nb))) = zero_zero(complex) ) ) ).

% sum_roots_unity
tff(fact_3410_sum__nth__roots,axiom,
    ! [Nb: nat,C2: complex] :
      ( aa(nat,$o,ord_less(nat,one_one(nat)),Nb)
     => ( aa(set(complex),complex,groups7311177749621191930dd_sum(complex,complex,aTP_Lamp_vn(complex,complex)),collect(complex,aa(complex,fun(complex,$o),aTP_Lamp_tb(nat,fun(complex,fun(complex,$o)),Nb),C2))) = zero_zero(complex) ) ) ).

% sum_nth_roots
tff(fact_3411_sum__diff__nat,axiom,
    ! [A: $tType,B5: set(A),A3: set(A),F: fun(A,nat)] :
      ( finite_finite2(A,B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
       => ( aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F),aa(set(A),set(A),minus_minus(set(A),A3),B5)) = aa(nat,nat,minus_minus(nat,aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F),A3)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F),B5)) ) ) ) ).

% sum_diff_nat
tff(fact_3412_sum__diff1__nat,axiom,
    ! [A: $tType,F: fun(A,nat),A3: set(A),A2: A] :
      aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = $ite(member(A,A2,A3),aa(nat,nat,minus_minus(nat,aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F),A3)),aa(A,nat,F,A2)),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,F),A3)) ).

% sum_diff1_nat
tff(fact_3413_sums__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A),G: fun(nat,A),S3: A,Ta: A] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,N2)),aa(nat,A,G,N2))
         => ( aa(A,$o,sums(A,F),S3)
           => ( aa(A,$o,sums(A,G),Ta)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),S3),Ta) ) ) ) ) ).

% sums_le
tff(fact_3414_sums__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F: fun(nat,A)] : aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vp(nat,fun(fun(nat,A),fun(nat,A)),I),F)),aa(nat,A,F,I)) ) ).

% sums_single
tff(fact_3415_sums__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( aa(A,$o,sums(A,F),A2)
         => ( aa(A,$o,sums(A,G),B2)
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F),G)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ) ) ).

% sums_add
tff(fact_3416_sums__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F: fun(nat,A),A2: A,C2: A] :
          ( aa(A,$o,sums(A,F),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_vr(fun(nat,A),fun(A,fun(nat,A)),F),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ).

% sums_mult2
tff(fact_3417_sums__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F: fun(nat,A),A2: A,C2: A] :
          ( aa(A,$o,sums(A,F),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_vs(fun(nat,A),fun(A,fun(nat,A)),F),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) ) ) ).

% sums_mult
tff(fact_3418_sums__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),A2: A,G: fun(nat,A),B2: A] :
          ( aa(A,$o,sums(A,F),A2)
         => ( aa(A,$o,sums(A,G),B2)
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vt(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F),G)),aa(A,A,minus_minus(A,A2),B2)) ) ) ) ).

% sums_diff
tff(fact_3419_sums__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(nat,A),A2: A,C2: A] :
          ( aa(A,$o,sums(A,F),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_vu(fun(nat,A),fun(A,fun(nat,A)),F),C2)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)) ) ) ).

% sums_divide
tff(fact_3420_sums__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F: fun(A,fun(nat,B)),Xb: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => aa(B,$o,sums(B,aa(A,fun(nat,B),F,I3)),aa(A,B,Xb,I3)) )
         => aa(B,$o,sums(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_vw(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F)),aa(set(A),B,groups7311177749621191930dd_sum(A,B,Xb),I5)) ) ) ).

% sums_sum
tff(fact_3421_Sum__Icc__int,axiom,
    ! [M: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M),Nb)
     => ( aa(set(int),int,groups7311177749621191930dd_sum(int,int,aTP_Lamp_vx(int,int)),set_or1337092689740270186AtMost(int,M,Nb)) = 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),Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),Nb),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_3422_sums__mult2__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F: fun(nat,A),D2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vy(A,fun(fun(nat,A),fun(nat,A)),C2),F)),aa(A,A,aa(A,fun(A,A),times_times(A),D2),C2))
          <=> aa(A,$o,sums(A,F),D2) ) ) ) ).

% sums_mult2_iff
tff(fact_3423_sums__mult__iff,axiom,
    ! [A: $tType] :
      ( ( field(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [C2: A,F: fun(nat,A),D2: A] :
          ( ( C2 != zero_zero(A) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vz(A,fun(fun(nat,A),fun(nat,A)),C2),F)),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2))
          <=> aa(A,$o,sums(A,F),D2) ) ) ) ).

% sums_mult_iff
tff(fact_3424_sums__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F: fun(nat,A),A2: A] :
          ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wa(A,fun(fun(nat,A),fun(nat,A)),C2),F)),A2)
         => ( ( C2 != zero_zero(A) )
           => aa(A,$o,sums(A,F),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),C2)) ) ) ) ).

% sums_mult_D
tff(fact_3425_sums__Suc__imp,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),S3: A] :
          ( ( aa(nat,A,F,zero_zero(nat)) = zero_zero(A) )
         => ( aa(A,$o,sums(A,aTP_Lamp_wb(fun(nat,A),fun(nat,A),F)),S3)
           => aa(A,$o,sums(A,F),S3) ) ) ) ).

% sums_Suc_imp
tff(fact_3426_sums__Suc,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F: fun(nat,A),L: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_wc(fun(nat,A),fun(nat,A),F)),L)
         => aa(A,$o,sums(A,F),aa(A,A,aa(A,fun(A,A),plus_plus(A),L),aa(nat,A,F,zero_zero(nat)))) ) ) ).

% sums_Suc
tff(fact_3427_sums__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),S3: A] :
          ( aa(A,$o,sums(A,aTP_Lamp_wb(fun(nat,A),fun(nat,A),F)),S3)
        <=> aa(A,$o,sums(A,F),aa(A,A,aa(A,fun(A,A),plus_plus(A),S3),aa(nat,A,F,zero_zero(nat)))) ) ) ).

% sums_Suc_iff
tff(fact_3428_sums__zero__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Nb: nat,F: fun(nat,A),S3: A] :
          ( ! [I3: nat] :
              ( aa(nat,$o,ord_less(nat,I3),Nb)
             => ( aa(nat,A,F,I3) = zero_zero(A) ) )
         => ( aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wd(nat,fun(fun(nat,A),fun(nat,A)),Nb),F)),S3)
          <=> aa(A,$o,sums(A,F),S3) ) ) ) ).

% sums_zero_iff_shift
tff(fact_3429_sums__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [N: set(nat),F: fun(nat,A)] :
          ( finite_finite2(nat,N)
         => ( ! [N2: nat] :
                ( ~ member(nat,N2,N)
               => ( aa(nat,A,F,N2) = zero_zero(A) ) )
           => aa(A,$o,sums(A,F),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),N)) ) ) ) ).

% sums_finite
tff(fact_3430_sums__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F: fun(nat,A)] :
          ( finite_finite2(nat,collect(nat,P))
         => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_we(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),collect(nat,P))) ) ) ).

% sums_If_finite
tff(fact_3431_sums__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F: fun(nat,A)] :
          ( finite_finite2(nat,A3)
         => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wf(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),A3)) ) ) ).

% sums_If_finite_set
tff(fact_3432_lemma__termdiff2,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [H: A,Z: A,Nb: 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)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb))),H)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),aa(nat,nat,minus_minus(nat,Nb),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_wh(A,fun(A,fun(nat,fun(nat,A))),H),Z),Nb)),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))))) ) ) ) ).

% lemma_termdiff2
tff(fact_3433_TBOUND__fi_H__adm,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,Foo: fun(A,fun(B,fun(C,nat)))] : comple1908693960933563346ssible(fun(product_prod(product_prod(A,B),C),heap_Time_Heap(D)),partial_fun_lub(heap_Time_Heap(D),heap_Time_Heap(D),product_prod(product_prod(A,B),C),heap_Time_Heap_lub(D)),partial_fun_ord(heap_Time_Heap(D),heap_Time_Heap(D),product_prod(product_prod(A,B),C),heap_Time_Heap_ord(D)),aTP_Lamp_wi(fun(A,fun(B,fun(C,nat))),fun(fun(product_prod(product_prod(A,B),C),heap_Time_Heap(D)),$o),Foo)) ).

% TBOUND_fi'_adm
tff(fact_3434_pochhammer__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,Nb: 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))),Nb)) = 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))),Nb)))),comm_s3205402744901411588hammer(A,Z,Nb))),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)))),Nb)) ) ).

% pochhammer_double
tff(fact_3435_of__nat__code,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] : aa(nat,A,semiring_1_of_nat(A),Nb) = semiri8178284476397505188at_aux(A,aTP_Lamp_wj(A,A),Nb,zero_zero(A)) ) ).

% of_nat_code
tff(fact_3436_floor__log__nat__eq__powr__iff,axiom,
    ! [B2: nat,K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => ( ( archim6421214686448440834_floor(real,log(aa(nat,real,semiring_1_of_nat(real),B2),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),Nb) )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
            & aa(nat,$o,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
tff(fact_3437_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_3438_pochhammer__1,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,one_one(nat)) = A2 ) ).

% pochhammer_1
tff(fact_3439_lessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_lessThan(A,Xb)),set_ord_lessThan(A,Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% lessThan_subset_iff
tff(fact_3440_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_3441_floor__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ( archim6421214686448440834_floor(A,one_one(A)) = one_one(int) ) ) ).

% floor_one
tff(fact_3442_pochhammer__0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,zero_zero(nat)) = one_one(A) ) ).

% pochhammer_0
tff(fact_3443_pochhammer__Suc0,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% pochhammer_Suc0
tff(fact_3444_sum_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,Nb))),aa(nat,A,G,Nb)) ) ).

% sum.lessThan_Suc
tff(fact_3445_single__Diff__lessThan,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [K: A] : aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),insert(A,K),bot_bot(set(A)))),set_ord_lessThan(A,K)) = aa(set(A),set(A),insert(A,K),bot_bot(set(A))) ) ).

% single_Diff_lessThan
tff(fact_3446_floor__diff__of__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] : archim6421214686448440834_floor(A,aa(A,A,minus_minus(A,Xb),aa(int,A,ring_1_of_int(A),Z))) = aa(int,int,minus_minus(int,archim6421214686448440834_floor(A,Xb)),Z) ) ).

% floor_diff_of_int
tff(fact_3447_zero__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb) ) ) ).

% zero_le_floor
tff(fact_3448_numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),V)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),V)),Xb) ) ) ).

% numeral_le_floor
tff(fact_3449_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_3450_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_3451_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,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ).

% zero_less_floor
tff(fact_3452_floor__le__zero,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(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_3453_one__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),one_one(int)),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ).

% one_le_floor
tff(fact_3454_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_3455_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_3456_floor__numeral__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: num,Nb: nat] : archim6421214686448440834_floor(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),Xb)),Nb)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),Xb)),Nb) ) ).

% floor_numeral_power
tff(fact_3457_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_3458_floor__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A2)),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),A2)),aa(num,int,numeral_numeral(int),B2)) ).

% floor_divide_eq_div_numeral
tff(fact_3459_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,$o,aa(A,fun(A,$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_3460_floor__le__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(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_3461_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,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(num,A,numeral_numeral(A),bit0(one2))),Xb) ) ) ).

% one_less_floor
tff(fact_3462_floor__le__one,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(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_3463_floor__one__divide__eq__div__numeral,axiom,
    ! [B2: num] : archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),aa(num,real,numeral_numeral(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),one_one(int)),aa(num,int,numeral_numeral(int),B2)) ).

% floor_one_divide_eq_div_numeral
tff(fact_3464_lessThan__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_lessThan(A,U) = collect(A,aTP_Lamp_wk(A,fun(A,$o),U)) ) ).

% lessThan_def
tff(fact_3465_finite__set__of__finite__funs,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B5: set(B),D2: B] :
      ( finite_finite2(A,A3)
     => ( finite_finite2(B,B5)
       => finite_finite2(fun(A,B),collect(fun(A,B),aa(B,fun(fun(A,B),$o),aa(set(B),fun(B,fun(fun(A,B),$o)),aTP_Lamp_wl(set(A),fun(set(B),fun(B,fun(fun(A,B),$o))),A3),B5),D2))) ) ) ).

% finite_set_of_finite_funs
tff(fact_3466_of__int__floor__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(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_3467_floor__mono,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Y)) ) ) ).

% floor_mono
tff(fact_3468_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_3469_floor__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),archimedean_ceiling(A,Xb)) ) ).

% floor_le_ceiling
tff(fact_3470_lessThan__strict__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Nb: A] :
          ( aa(set(A),$o,ord_less(set(A),set_ord_lessThan(A,M)),set_ord_lessThan(A,Nb))
        <=> aa(A,$o,ord_less(A,M),Nb) ) ) ).

% lessThan_strict_subset_iff
tff(fact_3471_pochhammer__pos,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Nb: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xb)
         => aa(A,$o,ord_less(A,zero_zero(A)),comm_s3205402744901411588hammer(A,Xb,Nb)) ) ) ).

% pochhammer_pos
tff(fact_3472_lessThan__Suc,axiom,
    ! [K: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),insert(nat,K),set_ord_lessThan(nat,K)) ).

% lessThan_Suc
tff(fact_3473_pochhammer__eq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat,M: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Nb) = zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
           => ( comm_s3205402744901411588hammer(A,A2,M) = zero_zero(A) ) ) ) ) ).

% pochhammer_eq_0_mono
tff(fact_3474_pochhammer__neq__0__mono,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat,Nb: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,M) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
           => ( comm_s3205402744901411588hammer(A,A2,Nb) != zero_zero(A) ) ) ) ) ).

% pochhammer_neq_0_mono
tff(fact_3475_floor__le__round,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),archim6421214686448440834_floor(A,Xb)),archimedean_round(A,Xb)) ) ).

% floor_le_round
tff(fact_3476_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_wm(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_3477_le__floor__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Z),archim6421214686448440834_floor(A,Xb))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),Z)),Xb) ) ) ).

% le_floor_iff
tff(fact_3478_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_3479_floor__add__int,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(A,Xb)),Z) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),aa(int,A,ring_1_of_int(A),Z))) ) ).

% floor_add_int
tff(fact_3480_int__add__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] : aa(int,int,aa(int,fun(int,int),plus_plus(int),Z),archim6421214686448440834_floor(A,Xb)) = archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),Z)),Xb)) ) ).

% int_add_floor
tff(fact_3481_le__floor__add,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] : aa(int,$o,aa(int,fun(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_3482_floor__power,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Nb: 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),Nb)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),archim6421214686448440834_floor(A,Xb)),Nb) ) ) ) ).

% floor_power
tff(fact_3483_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_3484_pochhammer__nonneg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Xb: A,Nb: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),comm_s3205402744901411588hammer(A,Xb,Nb)) ) ) ).

% pochhammer_nonneg
tff(fact_3485_of__nat__aux_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Inc: fun(A,A),Nb: nat,I: A] : semiri8178284476397505188at_aux(A,Inc,aa(nat,nat,suc,Nb),I) = semiri8178284476397505188at_aux(A,Inc,Nb,aa(A,A,Inc,I)) ) ).

% of_nat_aux.simps(2)
tff(fact_3486_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_3487_pochhammer__0__left,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Nb: nat] :
          comm_s3205402744901411588hammer(A,zero_zero(A),Nb) = $ite(Nb = zero_zero(nat),one_one(A),zero_zero(A)) ) ).

% pochhammer_0_left
tff(fact_3488_sum_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wn(fun(nat,A),fun(nat,fun(nat,A)),G),Nb)),set_ord_lessThan(nat,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,Nb)) ) ).

% sum.nat_diff_reindex
tff(fact_3489_sum__diff__distrib,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Q: fun(A,nat),P: fun(A,nat),Nb: A] :
          ( ! [X2: A] : aa(nat,$o,aa(nat,fun(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,Nb))),aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,Q),set_ord_lessThan(A,Nb))) = aa(set(A),nat,groups7311177749621191930dd_sum(A,nat,aa(fun(A,nat),fun(A,nat),aTP_Lamp_wo(fun(A,nat),fun(fun(A,nat),fun(A,nat)),Q),P)),set_ord_lessThan(A,Nb)) ) ) ) ).

% sum_diff_distrib
tff(fact_3490_of__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [R3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),R3)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),nat2(archim6421214686448440834_floor(A,R3)))),R3) ) ) ).

% of_nat_floor
tff(fact_3491_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_3492_floor__divide__of__nat__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [M: nat,Nb: 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),Nb))) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)) ) ).

% floor_divide_of_nat_eq
tff(fact_3493_le__mult__nat__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(archim6421214686448440834_floor(A,A2))),nat2(archim6421214686448440834_floor(A,B2)))),nat2(archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))) ) ).

% le_mult_nat_floor
tff(fact_3494_nat__floor__neg,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
     => ( nat2(archim6421214686448440834_floor(real,Xb)) = zero_zero(nat) ) ) ).

% nat_floor_neg
tff(fact_3495_floor__eq3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(real,$o,ord_less(real,aa(nat,real,semiring_1_of_nat(real),Nb)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb)))
       => ( nat2(archim6421214686448440834_floor(real,Xb)) = Nb ) ) ) ).

% floor_eq3
tff(fact_3496_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_3497_le__nat__floor,axiom,
    ! [Xb: nat,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Xb)),A2)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Xb),nat2(archim6421214686448440834_floor(real,A2))) ) ).

% le_nat_floor
tff(fact_3498_ceiling__diff__floor__le__1,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(int,$o,aa(int,fun(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_3499_floor__eq,axiom,
    ! [Nb: int,Xb: real] :
      ( aa(real,$o,ord_less(real,aa(int,real,ring_1_of_int(real),Nb)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),Nb)),one_one(real)))
       => ( archim6421214686448440834_floor(real,Xb) = Nb ) ) ) ).

% floor_eq
tff(fact_3500_real__of__int__floor__add__one__gt,axiom,
    ! [R3: real] : aa(real,$o,ord_less(real,R3),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R3))),one_one(real))) ).

% real_of_int_floor_add_one_gt
tff(fact_3501_real__of__int__floor__add__one__ge,axiom,
    ! [R3: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R3),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R3))),one_one(real))) ).

% real_of_int_floor_add_one_ge
tff(fact_3502_real__of__int__floor__gt__diff__one,axiom,
    ! [R3: real] : aa(real,$o,ord_less(real,aa(real,real,minus_minus(real,R3),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R3))) ).

% real_of_int_floor_gt_diff_one
tff(fact_3503_real__of__int__floor__ge__diff__one,axiom,
    ! [R3: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,minus_minus(real,R3),one_one(real))),aa(int,real,ring_1_of_int(real),archim6421214686448440834_floor(real,R3))) ).

% real_of_int_floor_ge_diff_one
tff(fact_3504_pochhammer__rec,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A)),Nb)) ) ).

% pochhammer_rec
tff(fact_3505_pochhammer__Suc,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,A2,Nb)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(nat,A,semiring_1_of_nat(A),Nb))) ) ).

% pochhammer_Suc
tff(fact_3506_pochhammer__rec_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,Nb: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),Nb))),comm_s3205402744901411588hammer(A,Z,Nb)) ) ).

% pochhammer_rec'
tff(fact_3507_pochhammer__product_H,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Z: A,Nb: nat,M: nat] : comm_s3205402744901411588hammer(A,Z,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,Nb)),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Z),aa(nat,A,semiring_1_of_nat(A),Nb)),M)) ) ).

% pochhammer_product'
tff(fact_3508_sum_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = 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_up(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb))) ) ).

% sum.lessThan_Suc_shift
tff(fact_3509_sum__lessThan__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(nat,A),M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_uu(fun(nat,A),fun(nat,A),F)),set_ord_lessThan(nat,M)) = aa(A,A,minus_minus(A,aa(nat,A,F,M)),aa(nat,A,F,zero_zero(nat))) ) ).

% sum_lessThan_telescope
tff(fact_3510_sum__lessThan__telescope_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(nat,A),M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_wp(fun(nat,A),fun(nat,A),F)),set_ord_lessThan(nat,M)) = aa(A,A,minus_minus(A,aa(nat,A,F,zero_zero(nat))),aa(nat,A,F,M)) ) ).

% sum_lessThan_telescope'
tff(fact_3511_sumr__diff__mult__const2,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [F: fun(nat,A),Nb: nat,R3: A] : aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_ord_lessThan(nat,Nb))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),R3)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_wq(fun(nat,A),fun(A,fun(nat,A)),F),R3)),set_ord_lessThan(nat,Nb)) ) ).

% sumr_diff_mult_const2
tff(fact_3512_sum_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_up(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb)) ) ).

% sum.atLeast1_atMost_eq
tff(fact_3513_sums__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),Nb: nat,S3: A] :
          ( aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_wr(fun(nat,A),fun(nat,fun(nat,A)),F),Nb)),S3)
        <=> aa(A,$o,sums(A,F),aa(A,A,aa(A,fun(A,A),plus_plus(A),S3),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_ord_lessThan(nat,Nb)))) ) ) ).

% sums_iff_shift
tff(fact_3514_sums__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),S3: A,Nb: nat] :
          ( aa(A,$o,sums(A,F),S3)
         => aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_wr(fun(nat,A),fun(nat,fun(nat,A)),F),Nb)),aa(A,A,minus_minus(A,S3),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_ord_lessThan(nat,Nb)))) ) ) ).

% sums_split_initial_segment
tff(fact_3515_sums__iff__shift_H,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),Nb: nat,S3: A] :
          ( aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_wr(fun(nat,A),fun(nat,fun(nat,A)),F),Nb)),aa(A,A,minus_minus(A,S3),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_ord_lessThan(nat,Nb))))
        <=> aa(A,$o,sums(A,F),S3) ) ) ).

% sums_iff_shift'
tff(fact_3516_floor__split,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [P: fun(int,$o),Ta: A] :
          ( aa(int,$o,P,archim6421214686448440834_floor(A,Ta))
        <=> ! [I2: int] :
              ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),I2)),Ta)
                & aa(A,$o,ord_less(A,Ta),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),I2)),one_one(A))) )
             => aa(int,$o,P,I2) ) ) ) ).

% floor_split
tff(fact_3517_floor__eq__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: int] :
          ( ( archim6421214686448440834_floor(A,Xb) = A2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),Xb)
            & aa(A,$o,ord_less(A,Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(int,A,ring_1_of_int(A),A2)),one_one(A))) ) ) ) ).

% floor_eq_iff
tff(fact_3518_floor__unique,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(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_3519_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,$o,aa(A,fun(A,$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_3520_floor__le__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Z: int] :
          ( aa(int,$o,aa(int,fun(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_3521_le__mult__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),times_times(int),archim6421214686448440834_floor(A,A2)),archim6421214686448440834_floor(A,B2))),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))) ) ) ) ).

% le_mult_floor
tff(fact_3522_floor__correct,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(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_3523_floor__eq4,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,semiring_1_of_nat(real),Nb)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb)))
       => ( nat2(archim6421214686448440834_floor(real,Xb)) = Nb ) ) ) ).

% floor_eq4
tff(fact_3524_floor__eq2,axiom,
    ! [Nb: int,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(int,real,ring_1_of_int(real),Nb)),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),Nb)),one_one(real)))
       => ( archim6421214686448440834_floor(real,Xb) = Nb ) ) ) ).

% floor_eq2
tff(fact_3525_floor__divide__real__eq__div,axiom,
    ! [B2: int,A2: real] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
     => ( archim6421214686448440834_floor(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),aa(int,real,ring_1_of_int(real),B2))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),archim6421214686448440834_floor(real,A2)),B2) ) ) ).

% floor_divide_real_eq_div
tff(fact_3526_power__diff__1__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)),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,Nb))) ) ).

% power_diff_1_eq
tff(fact_3527_one__diff__power__eq,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] : aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)) = 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,Nb))) ) ).

% one_diff_power_eq
tff(fact_3528_geometric__sum,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Xb: A,Nb: 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,Nb)) = 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),Nb)),one_one(A))),aa(A,A,minus_minus(A,Xb),one_one(A))) ) ) ) ).

% geometric_sum
tff(fact_3529_pochhammer__product,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [M: nat,Nb: nat,Z: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( comm_s3205402744901411588hammer(A,Z,Nb) = 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,Nb),M))) ) ) ) ).

% pochhammer_product
tff(fact_3530_floor__divide__lower,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Q2: A,P2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Q2)
         => aa(A,$o,aa(A,fun(A,$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_3531_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_ws(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_wt(A,fun(A,fun(nat,fun(nat,A))),Z),H),M)),set_ord_lessThan(nat,M)) ) ).

% lemma_termdiff1
tff(fact_3532_sum__gp__strict,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_ord_lessThan(nat,Nb)) = $ite(Xb = one_one(A),aa(nat,A,semiring_1_of_nat(A),Nb),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),Nb))),aa(A,A,minus_minus(A,one_one(A)),Xb))) ) ).

% sum_gp_strict
tff(fact_3533_power__diff__sumr2,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat,Y: A] : aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Nb)) = 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_wu(A,fun(nat,fun(A,fun(nat,A))),Xb),Nb),Y)),set_ord_lessThan(nat,Nb))) ) ).

% power_diff_sumr2
tff(fact_3534_diff__power__eq__sum,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: 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,Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),aa(nat,nat,suc,Nb))) = 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_wv(A,fun(nat,fun(A,fun(nat,A))),Xb),Nb),Y)),set_ord_lessThan(nat,aa(nat,nat,suc,Nb)))) ) ).

% diff_power_eq_sum
tff(fact_3535_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_3536_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_3537_real__sum__nat__ivl__bounded2,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,F: fun(nat,A),K6: A,K: nat] :
          ( ! [P6: nat] :
              ( aa(nat,$o,ord_less(nat,P6),Nb)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,P6)),K6) )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),K6)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_ord_lessThan(nat,aa(nat,nat,minus_minus(nat,Nb),K)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K6)) ) ) ) ).

% real_sum_nat_ivl_bounded2
tff(fact_3538_one__diff__power__eq_H,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] : aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)) = 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_ww(A,fun(nat,fun(nat,A)),Xb),Nb)),set_ord_lessThan(nat,Nb))) ) ).

% one_diff_power_eq'
tff(fact_3539_TBOUND__adm,axiom,
    ! [B: $tType,A: $tType,Ta: fun(A,nat)] : comple1908693960933563346ssible(fun(A,heap_Time_Heap(B)),partial_fun_lub(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_lub(B)),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),aTP_Lamp_wx(fun(A,nat),fun(fun(A,heap_Time_Heap(B)),$o),Ta)) ).

% TBOUND_adm
tff(fact_3540_refines__adm,axiom,
    ! [B: $tType,A: $tType,Ta: fun(A,heap_Time_Heap(B))] : comple1908693960933563346ssible(fun(A,heap_Time_Heap(B)),partial_fun_lub(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_lub(B)),partial_fun_ord(heap_Time_Heap(B),heap_Time_Heap(B),A,heap_Time_Heap_ord(B)),aTP_Lamp_wy(fun(A,heap_Time_Heap(B)),fun(fun(A,heap_Time_Heap(B)),$o),Ta)) ).

% refines_adm
tff(fact_3541_sum__split__even__odd,axiom,
    ! [F: fun(nat,real),G: fun(nat,real),Nb: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_wz(fun(nat,real),fun(fun(nat,real),fun(nat,real)),F),G)),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_xa(fun(nat,real),fun(nat,real),F)),set_ord_lessThan(nat,Nb))),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_xb(fun(nat,real),fun(nat,real),G)),set_ord_lessThan(nat,Nb))) ).

% sum_split_even_odd
tff(fact_3542_floor__log__eq__powr__iff,axiom,
    ! [Xb: real,B2: real,K: int] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,one_one(real)),B2)
       => ( ( archim6421214686448440834_floor(real,log(B2,Xb)) = K )
        <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,B2,aa(int,real,ring_1_of_int(real),K))),Xb)
            & aa(real,$o,ord_less(real,Xb),powr(real,B2,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_3543_pochhammer__code,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] :
          comm_s3205402744901411588hammer(A,A2,Nb) = $ite(Nb = zero_zero(nat),one_one(A),set_fo6178422350223883121st_nat(A,aTP_Lamp_xc(A,fun(nat,fun(A,A)),A2),zero_zero(nat),aa(nat,nat,minus_minus(nat,Nb),one_one(nat)),one_one(A))) ) ).

% pochhammer_code
tff(fact_3544_TBOUND__fi__adm,axiom,
    ! [C: $tType,B: $tType,A: $tType,Foo: fun(A,fun(B,nat))] : comple1908693960933563346ssible(fun(product_prod(A,B),heap_Time_Heap(C)),partial_fun_lub(heap_Time_Heap(C),heap_Time_Heap(C),product_prod(A,B),heap_Time_Heap_lub(C)),partial_fun_ord(heap_Time_Heap(C),heap_Time_Heap(C),product_prod(A,B),heap_Time_Heap_ord(C)),aTP_Lamp_xd(fun(A,fun(B,nat)),fun(fun(product_prod(A,B),heap_Time_Heap(C)),$o),Foo)) ).

% TBOUND_fi_adm
tff(fact_3545_floor__log2__div2,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( archim6421214686448440834_floor(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),Nb))) = aa(int,int,aa(int,fun(int,int),plus_plus(int),archim6421214686448440834_floor(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),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))),one_one(int)) ) ) ).

% floor_log2_div2
tff(fact_3546_floor__log__nat__eq__if,axiom,
    ! [B2: nat,Nb: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),Nb)),K)
     => ( aa(nat,$o,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),B2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),B2)
         => ( archim6421214686448440834_floor(real,log(aa(nat,real,semiring_1_of_nat(real),B2),aa(nat,real,semiring_1_of_nat(real),K))) = aa(nat,int,semiring_1_of_nat(int),Nb) ) ) ) ) ).

% floor_log_nat_eq_if
tff(fact_3547_sum__bounds__lt__plus1,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(nat,A),Mm: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_up(fun(nat,A),fun(nat,A),F)),set_ord_lessThan(nat,Mm)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_or1337092689740270186AtMost(nat,one_one(nat),Mm)) ) ).

% sum_bounds_lt_plus1
tff(fact_3548_pochhammer__times__pochhammer__half,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Z: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),comm_s3205402744901411588hammer(A,Z,aa(nat,nat,suc,Nb))),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,Nb))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_xe(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))),Nb)),one_one(nat)))) ) ).

% pochhammer_times_pochhammer_half
tff(fact_3549_sumr__cos__zero__one,axiom,
    ! [Nb: nat] : aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_xf(nat,real)),set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = one_one(real) ).

% sumr_cos_zero_one
tff(fact_3550_finite__nat__bounded,axiom,
    ! [S2: set(nat)] :
      ( finite_finite2(nat,S2)
     => ? [K3: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S2),set_ord_lessThan(nat,K3)) ) ).

% finite_nat_bounded
tff(fact_3551_finite__nat__iff__bounded,axiom,
    ! [S2: set(nat)] :
      ( finite_finite2(nat,S2)
    <=> ? [K4: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S2),set_ord_lessThan(nat,K4)) ) ).

% finite_nat_iff_bounded
tff(fact_3552_prod_Oneutral__const,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A3: set(B)] : groups7121269368397514597t_prod(B,A,aTP_Lamp_xg(B,A),A3) = one_one(A) ) ).

% prod.neutral_const
tff(fact_3553_of__nat__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F: fun(B,nat),A3: set(B)] : aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(B,nat,F,A3)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_xh(fun(B,nat),fun(B,A),F),A3) ) ).

% of_nat_prod
tff(fact_3554_of__int__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [F: fun(B,int),A3: set(B)] : aa(int,A,ring_1_of_int(A),groups7121269368397514597t_prod(B,int,F,A3)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_xi(fun(B,int),fun(B,A),F),A3) ) ).

% of_int_prod
tff(fact_3555_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_3556_prod_Oinfinite,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ~ finite_finite2(A,A3)
         => ( groups7121269368397514597t_prod(A,B,G,A3) = one_one(B) ) ) ) ).

% prod.infinite
tff(fact_3557_cos__coeff__0,axiom,
    cos_coeff(zero_zero(nat)) = one_one(real) ).

% cos_coeff_0
tff(fact_3558_prod_Odelta_H,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B)] :
          ( finite_finite2(A,S2)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xj(A,fun(fun(A,B),fun(A,B)),A2),B2),S2) = $ite(member(A,A2,S2),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta'
tff(fact_3559_prod_Odelta,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B)] :
          ( finite_finite2(A,S2)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_xk(A,fun(fun(A,B),fun(A,B)),A2),B2),S2) = $ite(member(A,A2,S2),aa(A,B,B2,A2),one_one(B)) ) ) ) ).

% prod.delta
tff(fact_3560_prod_Oinsert,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),Xb: A,G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ~ member(A,Xb,A3)
           => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),insert(A,Xb),A3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xb)),groups7121269368397514597t_prod(A,B,G,A3)) ) ) ) ) ).

% prod.insert
tff(fact_3561_prod_OlessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,Nb))),aa(nat,A,G,Nb)) ) ).

% prod.lessThan_Suc
tff(fact_3562_prod_Ocl__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] :
          groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,ord_less(nat,aa(nat,nat,suc,Nb)),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb)))) ) ).

% prod.cl_ivl_Suc
tff(fact_3563_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A3: set(B)] :
          ( ( groups7121269368397514597t_prod(B,A,G,A3) != one_one(A) )
         => ~ ! [A4: B] :
                ( member(B,A4,A3)
               => ( aa(B,A,G,A4) = one_one(A) ) ) ) ) ).

% prod.not_neutral_contains_not_neutral
tff(fact_3564_prod_Oneutral,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A3)
             => ( aa(A,B,G,X2) = one_one(B) ) )
         => ( groups7121269368397514597t_prod(A,B,G,A3) = one_one(B) ) ) ) ).

% prod.neutral
tff(fact_3565_norm__prod__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [F: fun(B,A),A3: set(B)] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,groups7121269368397514597t_prod(B,A,F,A3))),groups7121269368397514597t_prod(B,real,aTP_Lamp_xl(fun(B,A),fun(B,real),F),A3)) ) ).

% norm_prod_le
tff(fact_3566_prod__norm,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [F: fun(A,B),A3: set(A)] : groups7121269368397514597t_prod(A,real,aTP_Lamp_xm(fun(A,B),fun(A,real),F),A3) = real_V7770717601297561774m_norm(B,groups7121269368397514597t_prod(A,B,F,A3)) ) ).

% prod_norm
tff(fact_3567_prod_Oswap,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,fun(C,A)),B5: set(C),A3: set(B)] : groups7121269368397514597t_prod(B,A,aa(set(C),fun(B,A),aTP_Lamp_xn(fun(B,fun(C,A)),fun(set(C),fun(B,A)),G),B5),A3) = groups7121269368397514597t_prod(C,A,aa(set(B),fun(C,A),aTP_Lamp_xp(fun(B,fun(C,A)),fun(set(B),fun(C,A)),G),A3),B5) ) ).

% prod.swap
tff(fact_3568_prod_Odistrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),H: fun(B,A),A3: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xq(fun(B,A),fun(fun(B,A),fun(B,A)),G),H),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(B,A,G,A3)),groups7121269368397514597t_prod(B,A,H,A3)) ) ).

% prod.distrib
tff(fact_3569_prod__dividef,axiom,
    ! [A: $tType,B: $tType] :
      ( field(A)
     => ! [F: fun(B,A),G: fun(B,A),A3: set(B)] : groups7121269368397514597t_prod(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_xr(fun(B,A),fun(fun(B,A),fun(B,A)),F),G),A3) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(B,A,F,A3)),groups7121269368397514597t_prod(B,A,G,A3)) ) ).

% prod_dividef
tff(fact_3570_prod__power__distrib,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F: fun(B,A),A3: set(B),Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),groups7121269368397514597t_prod(B,A,F,A3)),Nb) = groups7121269368397514597t_prod(B,A,aa(nat,fun(B,A),aTP_Lamp_xs(fun(B,A),fun(nat,fun(B,A)),F),Nb),A3) ) ).

% prod_power_distrib
tff(fact_3571_prod_Oswap__restrict,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [A3: set(A),B5: set(B),G: fun(A,fun(B,C)),R: fun(A,fun(B,$o))] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(B,B5)
           => ( 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_xt(set(B),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(A,C))),B5),G),R),A3) = groups7121269368397514597t_prod(B,C,aa(fun(A,fun(B,$o)),fun(B,C),aa(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C)),aTP_Lamp_xv(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),A3),G),R),B5) ) ) ) ) ).

% prod.swap_restrict
tff(fact_3572_mod__prod__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( euclid4440199948858584721cancel(A)
     => ! [F: fun(B,A),A2: A,A3: set(B)] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),groups7121269368397514597t_prod(B,A,aa(A,fun(B,A),aTP_Lamp_un(fun(B,A),fun(A,fun(B,A)),F),A2),A3)),A2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),groups7121269368397514597t_prod(B,A,F,A3)),A2) ) ).

% mod_prod_eq
tff(fact_3573_prod__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F: fun(A,B),G: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,A3)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I3))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,I3)),aa(A,B,G,I3)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups7121269368397514597t_prod(A,B,F,A3)),groups7121269368397514597t_prod(A,B,G,A3)) ) ) ).

% prod_mono
tff(fact_3574_prod__nonneg,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),groups7121269368397514597t_prod(A,B,F,A3)) ) ) ).

% prod_nonneg
tff(fact_3575_prod__pos,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A3)
             => aa(B,$o,ord_less(B,zero_zero(B)),aa(A,B,F,X2)) )
         => aa(B,$o,ord_less(B,zero_zero(B)),groups7121269368397514597t_prod(A,B,F,A3)) ) ) ).

% prod_pos
tff(fact_3576_prod__ge__1,axiom,
    ! [B: $tType,A: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A3)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F,X2)) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),groups7121269368397514597t_prod(A,B,F,A3)) ) ) ).

% prod_ge_1
tff(fact_3577_prod_Ointer__filter,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),P: fun(A,$o)] :
          ( finite_finite2(A,A3)
         => ( groups7121269368397514597t_prod(A,B,G,collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cv(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) = groups7121269368397514597t_prod(A,B,aa(fun(A,$o),fun(A,B),aTP_Lamp_xw(fun(A,B),fun(fun(A,$o),fun(A,B)),G),P),A3) ) ) ) ).

% prod.inter_filter
tff(fact_3578_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_xx(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% prod.shift_bounds_cl_Suc_ivl
tff(fact_3579_power__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [C2: A,F: fun(B,nat),A3: set(B)] : aa(nat,A,aa(A,fun(nat,A),power_power(A),C2),aa(set(B),nat,groups7311177749621191930dd_sum(B,nat,F),A3)) = groups7121269368397514597t_prod(B,A,aa(fun(B,nat),fun(B,A),aTP_Lamp_xy(A,fun(fun(B,nat),fun(B,A)),C2),F),A3) ) ).

% power_sum
tff(fact_3580_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Nb: 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),Nb),K))) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_xz(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% prod.shift_bounds_cl_nat_ivl
tff(fact_3581_prod__le__1,axiom,
    ! [A: $tType,B: $tType] :
      ( linord181362715937106298miring(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( ! [X2: A] :
              ( member(A,X2,A3)
             => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,X2))
                & aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),one_one(B)) ) )
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups7121269368397514597t_prod(A,B,F,A3)),one_one(B)) ) ) ).

% prod_le_1
tff(fact_3582_prod_Orelated,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [R: fun(A,fun(A,$o)),S2: 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,X24: A,Y23: A] :
                ( ( aa(A,$o,aa(A,fun(A,$o),R,X15),X24)
                  & 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),X24),Y23)) )
           => ( finite_finite2(B,S2)
             => ( ! [X2: B] :
                    ( member(B,X2,S2)
                   => 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,S2)),groups7121269368397514597t_prod(B,A,G,S2)) ) ) ) ) ) ).

% prod.related
tff(fact_3583_prod_Oinsert__if,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),Xb: A] :
          ( finite_finite2(A,A3)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),insert(A,Xb),A3)) = $ite(member(A,Xb,A3),groups7121269368397514597t_prod(A,B,G,A3),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xb)),groups7121269368397514597t_prod(A,B,G,A3))) ) ) ) ).

% prod.insert_if
tff(fact_3584_prod_Oreindex__bij__witness__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S6: set(A),T7: set(B),S2: set(A),I: fun(B,A),J: fun(A,B),T5: set(B),G: fun(A,C),H: fun(B,C)] :
          ( finite_finite2(A,S6)
         => ( finite_finite2(B,T7)
           => ( ! [A4: A] :
                  ( member(A,A4,aa(set(A),set(A),minus_minus(set(A),S2),S6))
                 => ( aa(B,A,I,aa(A,B,J,A4)) = A4 ) )
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),minus_minus(set(A),S2),S6))
                   => member(B,aa(A,B,J,A4),aa(set(B),set(B),minus_minus(set(B),T5),T7)) )
               => ( ! [B3: B] :
                      ( member(B,B3,aa(set(B),set(B),minus_minus(set(B),T5),T7))
                     => ( aa(A,B,J,aa(B,A,I,B3)) = B3 ) )
                 => ( ! [B3: B] :
                        ( member(B,B3,aa(set(B),set(B),minus_minus(set(B),T5),T7))
                       => member(A,aa(B,A,I,B3),aa(set(A),set(A),minus_minus(set(A),S2),S6)) )
                   => ( ! [A4: A] :
                          ( member(A,A4,S6)
                         => ( aa(A,C,G,A4) = one_one(C) ) )
                     => ( ! [B3: B] :
                            ( member(B,B3,T7)
                           => ( aa(B,C,H,B3) = one_one(C) ) )
                       => ( ! [A4: A] :
                              ( member(A,A4,S2)
                             => ( aa(B,C,H,aa(A,B,J,A4)) = aa(A,C,G,A4) ) )
                         => ( groups7121269368397514597t_prod(A,C,G,S2) = groups7121269368397514597t_prod(B,C,H,T5) ) ) ) ) ) ) ) ) ) ) ) ).

% prod.reindex_bij_witness_not_neutral
tff(fact_3585_prod__dvd__prod__subset,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B5: set(A),A3: set(A),F: fun(A,B)] :
          ( finite_finite2(A,B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
           => aa(B,$o,dvd_dvd(B,groups7121269368397514597t_prod(A,B,F,A3)),groups7121269368397514597t_prod(A,B,F,B5)) ) ) ) ).

% prod_dvd_prod_subset
tff(fact_3586_prod__dvd__prod__subset2,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_1(B)
     => ! [B5: set(A),A3: set(A),F: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
           => ( ! [A4: A] :
                  ( member(A,A4,A3)
                 => aa(B,$o,dvd_dvd(B,aa(A,B,F,A4)),aa(A,B,G,A4)) )
             => aa(B,$o,dvd_dvd(B,groups7121269368397514597t_prod(A,B,F,A3)),groups7121269368397514597t_prod(A,B,G,B5)) ) ) ) ) ).

% prod_dvd_prod_subset2
tff(fact_3587_prod_Osetdiff__irrelevant,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A3),collect(A,aTP_Lamp_ya(fun(A,B),fun(A,$o),G)))) = groups7121269368397514597t_prod(A,B,G,A3) ) ) ) ).

% prod.setdiff_irrelevant
tff(fact_3588_prod_Onat__diff__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_yb(fun(nat,A),fun(nat,fun(nat,A)),G),Nb),set_ord_lessThan(nat,Nb)) = groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,Nb)) ) ).

% prod.nat_diff_reindex
tff(fact_3589_prod_OatLeastAtMost__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,Nb,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_yc(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),M),set_or1337092689740270186AtMost(nat,Nb,M)) ) ).

% prod.atLeastAtMost_rev
tff(fact_3590_less__1__prod2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),I: A,F: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( member(A,I,I5)
           => ( aa(B,$o,ord_less(B,one_one(B)),aa(A,B,F,I))
             => ( ! [I3: A] :
                    ( member(A,I3,I5)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F,I3)) )
               => aa(B,$o,ord_less(B,one_one(B)),groups7121269368397514597t_prod(A,B,F,I5)) ) ) ) ) ) ).

% less_1_prod2
tff(fact_3591_less__1__prod,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),F: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( ( I5 != bot_bot(set(A)) )
           => ( ! [I3: A] :
                  ( member(A,I3,I5)
                 => aa(B,$o,ord_less(B,one_one(B)),aa(A,B,F,I3)) )
             => aa(B,$o,ord_less(B,one_one(B)),groups7121269368397514597t_prod(A,B,F,I5)) ) ) ) ) ).

% less_1_prod
tff(fact_3592_prod_Osame__carrier,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(A),A3: set(A),B5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),minus_minus(set(A),C5),A3))
                   => ( aa(A,B,G,A4) = one_one(B) ) )
               => ( ! [B3: A] :
                      ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),C5),B5))
                     => ( aa(A,B,H,B3) = one_one(B) ) )
                 => ( ( groups7121269368397514597t_prod(A,B,G,A3) = groups7121269368397514597t_prod(A,B,H,B5) )
                  <=> ( groups7121269368397514597t_prod(A,B,G,C5) = groups7121269368397514597t_prod(A,B,H,C5) ) ) ) ) ) ) ) ) ).

% prod.same_carrier
tff(fact_3593_prod_Osame__carrierI,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [C5: set(A),A3: set(A),B5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,C5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
             => ( ! [A4: A] :
                    ( member(A,A4,aa(set(A),set(A),minus_minus(set(A),C5),A3))
                   => ( aa(A,B,G,A4) = one_one(B) ) )
               => ( ! [B3: A] :
                      ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),C5),B5))
                     => ( aa(A,B,H,B3) = one_one(B) ) )
                 => ( ( groups7121269368397514597t_prod(A,B,G,C5) = groups7121269368397514597t_prod(A,B,H,C5) )
                   => ( groups7121269368397514597t_prod(A,B,G,A3) = groups7121269368397514597t_prod(A,B,H,B5) ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
tff(fact_3594_prod_Omono__neutral__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T5: set(A),S2: set(A),G: fun(A,B)] :
          ( finite_finite2(A,T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( groups7121269368397514597t_prod(A,B,G,S2) = groups7121269368397514597t_prod(A,B,G,T5) ) ) ) ) ) ).

% prod.mono_neutral_left
tff(fact_3595_prod_Omono__neutral__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T5: set(A),S2: set(A),G: fun(A,B)] :
          ( finite_finite2(A,T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( groups7121269368397514597t_prod(A,B,G,T5) = groups7121269368397514597t_prod(A,B,G,S2) ) ) ) ) ) ).

% prod.mono_neutral_right
tff(fact_3596_prod_Omono__neutral__cong__left,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T5: set(A),S2: set(A),H: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,H,X2) = one_one(B) ) )
             => ( ! [X2: A] :
                    ( member(A,X2,S2)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
               => ( groups7121269368397514597t_prod(A,B,G,S2) = groups7121269368397514597t_prod(A,B,H,T5) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
tff(fact_3597_prod_Omono__neutral__cong__right,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [T5: set(A),S2: set(A),G: fun(A,B),H: fun(A,B)] :
          ( finite_finite2(A,T5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
           => ( ! [X2: A] :
                  ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T5),S2))
                 => ( aa(A,B,G,X2) = one_one(B) ) )
             => ( ! [X2: A] :
                    ( member(A,X2,S2)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
               => ( groups7121269368397514597t_prod(A,B,G,T5) = groups7121269368397514597t_prod(A,B,H,S2) ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
tff(fact_3598_prod_Osubset__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [B5: set(A),A3: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
         => ( finite_finite2(A,A3)
           => ( groups7121269368397514597t_prod(A,B,G,A3) = aa(B,B,aa(B,fun(B,B),times_times(B),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A3),B5))),groups7121269368397514597t_prod(A,B,G,B5)) ) ) ) ) ).

% prod.subset_diff
tff(fact_3599_prod_OatLeast0__atMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% prod.atLeast0_atMost_Suc
tff(fact_3600_prod_OatLeast__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Nb)) = 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),Nb))) ) ) ) ).

% prod.atLeast_Suc_atMost
tff(fact_3601_prod_Onat__ivl__Suc_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Nb))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,aa(nat,nat,suc,Nb))),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Nb))) ) ) ) ).

% prod.nat_ivl_Suc'
tff(fact_3602_prod_OlessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_xx(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,Nb))) ) ).

% prod.lessThan_Suc_shift
tff(fact_3603_prod_OSuc__reindex__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,M)),groups7121269368397514597t_prod(nat,A,aTP_Lamp_xx(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Nb))) ) ) ) ).

% prod.Suc_reindex_ivl
tff(fact_3604_prod_OatLeast1__atMost__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_xx(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,Nb)) ) ).

% prod.atLeast1_atMost_eq
tff(fact_3605_prod__mono__strict,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_semidom(B)
     => ! [A3: set(A),F: fun(A,B),G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( ! [I3: A] :
                ( member(A,I3,A3)
               => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,I3))
                  & aa(B,$o,ord_less(B,aa(A,B,F,I3)),aa(A,B,G,I3)) ) )
           => ( ( A3 != bot_bot(set(A)) )
             => aa(B,$o,ord_less(B,groups7121269368397514597t_prod(A,B,F,A3)),groups7121269368397514597t_prod(A,B,G,A3)) ) ) ) ) ).

% prod_mono_strict
tff(fact_3606_prod__atLeastAtMost__code,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [F: fun(nat,A),A2: nat,B2: nat] : groups7121269368397514597t_prod(nat,A,F,set_or1337092689740270186AtMost(nat,A2,B2)) = set_fo6178422350223883121st_nat(A,aTP_Lamp_yd(fun(nat,A),fun(nat,fun(A,A)),F),A2,B2,one_one(A)) ) ).

% prod_atLeastAtMost_code
tff(fact_3607_even__prod__iff,axiom,
    ! [B: $tType,A: $tType] :
      ( semiring_parity(B)
     => ! [A3: set(A),F: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),groups7121269368397514597t_prod(A,B,F,A3))
          <=> ? [X: A] :
                ( member(A,X,A3)
                & aa(B,$o,dvd_dvd(B,aa(num,B,numeral_numeral(B),bit0(one2))),aa(A,B,F,X)) ) ) ) ) ).

% even_prod_iff
tff(fact_3608_prod_Oinsert__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),G: fun(A,B),Xb: A] :
          ( finite_finite2(A,A3)
         => ( groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),insert(A,Xb),A3)) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xb)),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ).

% prod.insert_remove
tff(fact_3609_prod_Oremove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [A3: set(A),Xb: A,G: fun(A,B)] :
          ( finite_finite2(A,A3)
         => ( member(A,Xb,A3)
           => ( groups7121269368397514597t_prod(A,B,G,A3) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,G,Xb)),groups7121269368397514597t_prod(A,B,G,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))))) ) ) ) ) ).

% prod.remove
tff(fact_3610_prod_Oub__add__nat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A),P2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)))
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P2))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Nb))),groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),P2)))) ) ) ) ).

% prod.ub_add_nat
tff(fact_3611_prod_Odelta__remove,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),A2: A,B2: fun(A,B),C2: fun(A,B)] :
          ( finite_finite2(A,S2)
         => ( groups7121269368397514597t_prod(A,B,aa(fun(A,B),fun(A,B),aa(fun(A,B),fun(fun(A,B),fun(A,B)),aTP_Lamp_ye(A,fun(fun(A,B),fun(fun(A,B),fun(A,B))),A2),B2),C2),S2) = $ite(member(A,A2,S2),aa(B,B,aa(B,fun(B,B),times_times(B),aa(A,B,B2,A2)),groups7121269368397514597t_prod(A,B,C2,aa(set(A),set(A),minus_minus(set(A),S2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))),groups7121269368397514597t_prod(A,B,C2,aa(set(A),set(A),minus_minus(set(A),S2),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))) ) ) ) ).

% prod.delta_remove
tff(fact_3612_norm__prod__diff,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),Z: fun(A,B),W: fun(A,B)] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,Z,I3))),one_one(real)) )
         => ( ! [I3: A] :
                ( member(A,I3,I5)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(B,aa(A,B,W,I3))),one_one(real)) )
           => aa(real,$o,aa(real,fun(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_yf(fun(A,B),fun(fun(A,B),fun(A,real)),Z),W)),I5)) ) ) ) ).

% norm_prod_diff
tff(fact_3613_prod__mono2,axiom,
    ! [B: $tType,A: $tType] :
      ( linordered_idom(B)
     => ! [B5: set(A),A3: set(A),F: fun(A,B)] :
          ( finite_finite2(A,B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
           => ( ! [B3: A] :
                  ( member(A,B3,aa(set(A),set(A),minus_minus(set(A),B5),A3))
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),one_one(B)),aa(A,B,F,B3)) )
             => ( ! [A4: A] :
                    ( member(A,A4,A3)
                   => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,F,A4)) )
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),groups7121269368397514597t_prod(A,B,F,A3)),groups7121269368397514597t_prod(A,B,F,B5)) ) ) ) ) ) ).

% prod_mono2
tff(fact_3614_prod__diff1,axiom,
    ! [B: $tType,A: $tType] :
      ( semidom_divide(B)
     => ! [A3: set(A),F: fun(A,B),A2: A] :
          ( finite_finite2(A,A3)
         => ( ( aa(A,B,F,A2) != zero_zero(B) )
           => ( groups7121269368397514597t_prod(A,B,F,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) = $ite(member(A,A2,A3),aa(B,B,aa(B,fun(B,B),divide_divide(B),groups7121269368397514597t_prod(A,B,F,A3)),aa(A,B,F,A2)),groups7121269368397514597t_prod(A,B,F,A3)) ) ) ) ) ).

% prod_diff1
tff(fact_3615_pochhammer__Suc__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_yg(A,fun(nat,A),A2),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_Suc_prod
tff(fact_3616_pochhammer__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,Nb) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_yh(A,fun(nat,fun(nat,A)),A2),Nb),set_or1337092689740270186AtMost(nat,one_one(nat),Nb)) ) ).

% pochhammer_prod_rev
tff(fact_3617_prod_Oin__pairs,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: 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))),Nb)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_yi(fun(nat,A),fun(nat,A),G),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% prod.in_pairs
tff(fact_3618_pochhammer__Suc__prod__rev,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,aa(nat,nat,suc,Nb)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_yh(A,fun(nat,fun(nat,A)),A2),Nb),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_Suc_prod_rev
tff(fact_3619_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set(nat)] :
      ( ! [M2: nat] :
          ( aa(nat,$o,ord_less(nat,K),M2)
         => ? [N8: nat] :
              ( aa(nat,$o,ord_less(nat,M2),N8)
              & member(nat,N8,S2) ) )
     => ~ finite_finite2(nat,S2) ) ).

% unbounded_k_infinite
tff(fact_3620_infinite__nat__iff__unbounded,axiom,
    ! [S2: set(nat)] :
      ( ~ finite_finite2(nat,S2)
    <=> ! [M7: nat] :
        ? [N6: nat] :
          ( aa(nat,$o,ord_less(nat,M7),N6)
          & member(nat,N6,S2) ) ) ).

% infinite_nat_iff_unbounded
tff(fact_3621_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set(nat)] :
      ( ~ finite_finite2(nat,S2)
    <=> ! [M7: nat] :
        ? [N6: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),N6)
          & member(nat,N6,S2) ) ) ).

% infinite_nat_iff_unbounded_le
tff(fact_3622_round__altdef,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          archimedean_round(A,Xb) = $ite(aa(A,$o,aa(A,fun(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_3623_gchoose__row__sum__weighted,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [R3: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_yj(A,fun(nat,A),R3)),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,R3),aa(nat,nat,suc,M))) ) ).

% gchoose_row_sum_weighted
tff(fact_3624_central__binomial__lower__bound,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => aa(real,$o,aa(real,fun(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)))),Nb)),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),Nb)))),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))),Nb)),Nb))) ) ).

% central_binomial_lower_bound
tff(fact_3625_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Xb: A,Y: B,A2: product_prod(A,B)] :
      ( aa(B,$o,aa(A,fun(B,$o),P,Xb),Y)
     => ( ( A2 = 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),A2)),aa(product_prod(A,B),B,product_snd(A,B),A2)) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
tff(fact_3626_binomial__Suc__n,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),Nb) = aa(nat,nat,suc,Nb) ).

% binomial_Suc_n
tff(fact_3627_binomial__n__n,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),Nb) = one_one(nat) ).

% binomial_n_n
tff(fact_3628_gbinomial__1,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : aa(nat,A,gbinomial(A,A2),one_one(nat)) = A2 ) ).

% gbinomial_1
tff(fact_3629_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_3630_binomial__1,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),aa(nat,nat,suc,zero_zero(nat))) = Nb ).

% binomial_1
tff(fact_3631_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_3632_binomial__eq__0__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( ( aa(nat,nat,binomial(Nb),K) = zero_zero(nat) )
    <=> aa(nat,$o,ord_less(nat,Nb),K) ) ).

% binomial_eq_0_iff
tff(fact_3633_binomial__Suc__Suc,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K))) ).

% binomial_Suc_Suc
tff(fact_3634_gbinomial__0_I1_J,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : aa(nat,A,gbinomial(A,A2),zero_zero(nat)) = one_one(A) ) ).

% gbinomial_0(1)
tff(fact_3635_binomial__n__0,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),zero_zero(nat)) = one_one(nat) ).

% binomial_n_0
tff(fact_3636_gbinomial__Suc0,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,zero_zero(nat))) = A2 ) ).

% gbinomial_Suc0
tff(fact_3637_prod__eq__1__iff,axiom,
    ! [A: $tType,A3: set(A),F: fun(A,nat)] :
      ( finite_finite2(A,A3)
     => ( ( groups7121269368397514597t_prod(A,nat,F,A3) = one_one(nat) )
      <=> ! [X: A] :
            ( member(A,X,A3)
           => ( aa(A,nat,F,X) = one_one(nat) ) ) ) ) ).

% prod_eq_1_iff
tff(fact_3638_zero__less__binomial__iff,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,binomial(Nb),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb) ) ).

% zero_less_binomial_iff
tff(fact_3639_prod__pos__nat__iff,axiom,
    ! [A: $tType,A3: set(A),F: fun(A,nat)] :
      ( finite_finite2(A,A3)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),groups7121269368397514597t_prod(A,nat,F,A3))
      <=> ! [X: A] :
            ( member(A,X,A3)
           => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(A,nat,F,X)) ) ) ) ).

% prod_pos_nat_iff
tff(fact_3640_choose__one,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),one_one(nat)) = Nb ).

% choose_one
tff(fact_3641_int__prod,axiom,
    ! [A: $tType,F: fun(A,nat),A3: set(A)] : aa(nat,int,semiring_1_of_nat(int),groups7121269368397514597t_prod(A,nat,F,A3)) = groups7121269368397514597t_prod(A,int,aTP_Lamp_vl(fun(A,nat),fun(A,int),F),A3) ).

% int_prod
tff(fact_3642_binomial__eq__0,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),K)
     => ( aa(nat,nat,binomial(Nb),K) = zero_zero(nat) ) ) ).

% binomial_eq_0
tff(fact_3643_Suc__times__binomial,axiom,
    ! [K: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,binomial(Nb),K)) ).

% Suc_times_binomial
tff(fact_3644_Suc__times__binomial__eq,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,Nb)),aa(nat,nat,binomial(Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,K))),aa(nat,nat,suc,K)) ).

% Suc_times_binomial_eq
tff(fact_3645_binomial__symmetric,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( aa(nat,nat,binomial(Nb),K) = aa(nat,nat,binomial(Nb),aa(nat,nat,minus_minus(nat,Nb),K)) ) ) ).

% binomial_symmetric
tff(fact_3646_choose__mult__lemma,axiom,
    ! [M: nat,R3: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),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),M),R3)),K)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K))),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),K)),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),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),M),R3)),K)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),R3)),M)) ).

% choose_mult_lemma
tff(fact_3647_binomial__le__pow,axiom,
    ! [R3: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R3),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),R3)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),R3)) ) ).

% binomial_le_pow
tff(fact_3648_frac__ge__0,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),archimedean_frac(A,Xb)) ) ).

% frac_ge_0
tff(fact_3649_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_3650_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_3651_zero__less__binomial,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(nat,nat,binomial(Nb),K)) ) ).

% zero_less_binomial
tff(fact_3652_Suc__times__binomial__add,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,A2)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),aa(nat,nat,suc,A2))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,B2)),aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2))),A2)) ).

% Suc_times_binomial_add
tff(fact_3653_binomial__Suc__Suc__eq__times,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),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,Nb)),aa(nat,nat,binomial(Nb),K))),aa(nat,nat,suc,K)) ).

% binomial_Suc_Suc_eq_times
tff(fact_3654_choose__mult,axiom,
    ! [K: nat,M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
       => ( aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Nb),M)),aa(nat,nat,binomial(M),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Nb),K)),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% choose_mult
tff(fact_3655_binomial__absorb__comp,axiom,
    ! [Nb: nat,K: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,Nb),K)),aa(nat,nat,binomial(Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Nb),one_one(nat))),K)) ).

% binomial_absorb_comp
tff(fact_3656_gbinomial__Suc__Suc,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,A2),K)),aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc_Suc
tff(fact_3657_gbinomial__of__nat__symmetric,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Nb)),K) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Nb)),aa(nat,nat,minus_minus(nat,Nb),K)) ) ) ) ).

% gbinomial_of_nat_symmetric
tff(fact_3658_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_vx(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_3659_small__lazy_H_Ocases,axiom,
    ! [Xb: product_prod(int,int)] :
      ~ ! [D3: int,I3: int] : Xb != aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),D3),I3) ).

% small_lazy'.cases
tff(fact_3660_binomial__absorption,axiom,
    ! [K: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,K)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Nb),one_one(nat))),K)) ).

% binomial_absorption
tff(fact_3661_gbinomial__addition__formula,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),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,A2),one_one(A))),aa(nat,nat,suc,K))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),K)) ) ).

% gbinomial_addition_formula
tff(fact_3662_gbinomial__absorb__comp,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,A2),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),K)) ) ).

% gbinomial_absorb_comp
tff(fact_3663_gbinomial__mult__1,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,A2),K)) = 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),K)),aa(nat,A,gbinomial(A,A2),K))),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,A2),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1
tff(fact_3664_gbinomial__mult__1_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),A2) = 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),K)),aa(nat,A,gbinomial(A,A2),K))),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,A2),aa(nat,nat,suc,K)))) ) ).

% gbinomial_mult_1'
tff(fact_3665_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,semiring_1_of_nat(A),K)),A2)
         => aa(A,$o,aa(A,fun(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),A2),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,gbinomial(A,A2),K)) ) ) ).

% gbinomial_ge_n_over_k_pow_k
tff(fact_3666_frac__def,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : archimedean_frac(A,Xb) = aa(A,A,minus_minus(A,Xb),aa(int,A,ring_1_of_int(A),archim6421214686448440834_floor(A,Xb))) ) ).

% frac_def
tff(fact_3667_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_vx(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_3668_binomial__ge__n__over__k__pow__k,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => aa(A,$o,aa(A,fun(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),Nb)),aa(nat,A,semiring_1_of_nat(A),K))),K)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K))) ) ) ).

% binomial_ge_n_over_k_pow_k
tff(fact_3669_binomial__maximum,axiom,
    ! [Nb: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ).

% binomial_maximum
tff(fact_3670_binomial__antimono,axiom,
    ! [K: nat,K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))),K)
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K2)),aa(nat,nat,binomial(Nb),K)) ) ) ) ).

% binomial_antimono
tff(fact_3671_binomial__mono,axiom,
    ! [K: nat,K2: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2)),Nb)
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),K2)) ) ) ).

% binomial_mono
tff(fact_3672_binomial__maximum_H,axiom,
    ! [Nb: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$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))),Nb)),K)),aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),Nb)) ).

% binomial_maximum'
tff(fact_3673_binomial__le__pow2,axiom,
    ! [Nb: nat,K: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,binomial(Nb),K)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% binomial_le_pow2
tff(fact_3674_choose__reduce__nat,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
       => ( aa(nat,nat,binomial(Nb),K) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Nb),one_one(nat))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Nb),one_one(nat))),K)) ) ) ) ).

% choose_reduce_nat
tff(fact_3675_times__binomial__minus1__eq,axiom,
    ! [K: nat,Nb: 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(Nb),K)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Nb),one_one(nat))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))) ) ) ).

% times_binomial_minus1_eq
tff(fact_3676_subset__Collect__iff,axiom,
    ! [A: $tType,B5: set(A),A3: set(A),P: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cv(set(A),fun(fun(A,$o),fun(A,$o)),A3),P)))
      <=> ! [X: A] :
            ( member(A,X,B5)
           => aa(A,$o,P,X) ) ) ) ).

% subset_Collect_iff
tff(fact_3677_subset__CollectI,axiom,
    ! [A: $tType,B5: set(A),A3: set(A),Q: fun(A,$o),P: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
     => ( ! [X2: A] :
            ( member(A,X2,B5)
           => ( aa(A,$o,Q,X2)
             => aa(A,$o,P,X2) ) )
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cv(set(A),fun(fun(A,$o),fun(A,$o)),B5),Q))),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cv(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) ) ) ).

% subset_CollectI
tff(fact_3678_Suc__times__gbinomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: 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),A2),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),A2),one_one(A))),aa(nat,A,gbinomial(A,A2),K)) ) ).

% Suc_times_gbinomial
tff(fact_3679_gbinomial__absorption,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: 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,A2),aa(nat,nat,suc,K))) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),K)) ) ).

% gbinomial_absorption
tff(fact_3680_gbinomial__trinomial__revision,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,M: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),M)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),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,A2),K)),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,nat,minus_minus(nat,M),K))) ) ) ) ).

% gbinomial_trinomial_revision
tff(fact_3681_frac__eq,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( ( archimedean_frac(A,Xb) = Xb )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
            & aa(A,$o,ord_less(A,Xb),one_one(A)) ) ) ) ).

% frac_eq
tff(fact_3682_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_3683_binomial__less__binomial__Suc,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,K),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))
     => aa(nat,$o,ord_less(nat,aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,suc,K))) ) ).

% binomial_less_binomial_Suc
tff(fact_3684_binomial__strict__mono,axiom,
    ! [K: nat,K2: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,K),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K2)),Nb)
       => aa(nat,$o,ord_less(nat,aa(nat,nat,binomial(Nb),K)),aa(nat,nat,binomial(Nb),K2)) ) ) ).

% binomial_strict_mono
tff(fact_3685_binomial__strict__antimono,axiom,
    ! [K: nat,K2: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,K),K2)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K))
       => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K2),Nb)
         => aa(nat,$o,ord_less(nat,aa(nat,nat,binomial(Nb),K2)),aa(nat,nat,binomial(Nb),K)) ) ) ) ).

% binomial_strict_antimono
tff(fact_3686_central__binomial__odd,axiom,
    ! [Nb: nat] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(nat,nat,binomial(Nb),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))) = aa(nat,nat,binomial(Nb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% central_binomial_odd
tff(fact_3687_binomial__addition__formula,axiom,
    ! [Nb: nat,K: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,nat,binomial(Nb),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,Nb),one_one(nat))),aa(nat,nat,suc,K))),aa(nat,nat,binomial(aa(nat,nat,minus_minus(nat,Nb),one_one(nat))),K)) ) ) ).

% binomial_addition_formula
tff(fact_3688_gbinomial__rec,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))) ) ).

% gbinomial_rec
tff(fact_3689_gbinomial__factors,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),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),A2),one_one(A))),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,K)))),aa(nat,A,gbinomial(A,A2),K)) ) ).

% gbinomial_factors
tff(fact_3690_choose__two,axiom,
    ! [Nb: nat] : aa(nat,nat,binomial(Nb),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),Nb),aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% choose_two
tff(fact_3691_gbinomial__reduce__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),K)) ) ) ) ).

% gbinomial_reduce_nat
tff(fact_3692_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_3693_gbinomial__sum__up__index,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_yk(nat,fun(nat,A),K)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(nat,A,gbinomial(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),one_one(A))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ).

% gbinomial_sum_up_index
tff(fact_3694_gbinomial__absorption_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
         => ( aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,semiring_1_of_nat(A),K))),aa(nat,A,gbinomial(A,aa(A,A,minus_minus(A,A2),one_one(A))),aa(nat,nat,minus_minus(nat,K),one_one(nat)))) ) ) ) ).

% gbinomial_absorption'
tff(fact_3695_choose__odd__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
         => ( 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_yl(nat,fun(nat,A),Nb)),set_ord_atMost(nat,Nb))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) ) ) ) ).

% choose_odd_sum
tff(fact_3696_choose__even__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
         => ( 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_ym(nat,fun(nat,A),Nb)),set_ord_atMost(nat,Nb))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) ) ) ) ).

% choose_even_sum
tff(fact_3697_gbinomial__partial__row__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_yj(A,fun(nat,A),A2)),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,A2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),one_one(nat)))) ) ).

% gbinomial_partial_row_sum
tff(fact_3698_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_3699_atMost__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,K: A] :
          ( member(A,I,set_ord_atMost(A,K))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),K) ) ) ).

% atMost_iff
tff(fact_3700_atMost__subset__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atMost(A,Xb)),set_ord_atMost(A,Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% atMost_subset_iff
tff(fact_3701_Icc__subset__Iic__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [L: A,H: A,H3: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,L,H)),set_ord_atMost(A,H3))
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),H)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),H),H3) ) ) ) ).

% Icc_subset_Iic_iff
tff(fact_3702_sum_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% sum.atMost_Suc
tff(fact_3703_prod_OatMost__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,Nb))),aa(nat,A,G,aa(nat,nat,suc,Nb))) ) ).

% prod.atMost_Suc
tff(fact_3704_atMost__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [U: A] : set_ord_atMost(A,U) = collect(A,aTP_Lamp_yn(A,fun(A,$o),U)) ) ).

% atMost_def
tff(fact_3705_lessThan__Suc__atMost,axiom,
    ! [K: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,K)) = set_ord_atMost(nat,K) ).

% lessThan_Suc_atMost
tff(fact_3706_atMost__Suc,axiom,
    ! [K: nat] : set_ord_atMost(nat,aa(nat,nat,suc,K)) = aa(set(nat),set(nat),insert(nat,aa(nat,nat,suc,K)),set_ord_atMost(nat,K)) ).

% atMost_Suc
tff(fact_3707_not__UNIV__le__Iic,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [H: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),top_top(set(A))),set_ord_atMost(A,H)) ) ).

% not_UNIV_le_Iic
tff(fact_3708_not__Iic__le__Icc,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [H: A,L3: A,H3: A] : ~ aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atMost(A,H)),set_or1337092689740270186AtMost(A,L3,H3)) ) ).

% not_Iic_le_Icc
tff(fact_3709_finite__nat__iff__bounded__le,axiom,
    ! [S2: set(nat)] :
      ( finite_finite2(nat,S2)
    <=> ? [K4: nat] : aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),S2),set_ord_atMost(nat,K4)) ) ).

% finite_nat_iff_bounded_le
tff(fact_3710_Iic__subset__Iio__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_ord_atMost(A,A2)),set_ord_lessThan(A,B2))
        <=> aa(A,$o,ord_less(A,A2),B2) ) ) ).

% Iic_subset_Iio_iff
tff(fact_3711_sum__choose__upper,axiom,
    ! [M: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_yo(nat,fun(nat,nat),M)),set_ord_atMost(nat,Nb)) = aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),aa(nat,nat,suc,M)) ).

% sum_choose_upper
tff(fact_3712_sum_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,aa(nat,nat,suc,Nb))) = 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_up(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Nb))) ) ).

% sum.atMost_Suc_shift
tff(fact_3713_sum__telescope,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(nat,A),I: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_wp(fun(nat,A),fun(nat,A),F)),set_ord_atMost(nat,I)) = aa(A,A,minus_minus(A,aa(nat,A,F,zero_zero(nat))),aa(nat,A,F,aa(nat,nat,suc,I))) ) ).

% sum_telescope
tff(fact_3714_polyfun__eq__coeffs,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat,D2: fun(nat,A)] :
          ( ! [X: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_yp(fun(nat,A),fun(A,fun(nat,A)),C2),X)),set_ord_atMost(nat,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_yp(fun(nat,A),fun(A,fun(nat,A)),D2),X)),set_ord_atMost(nat,Nb))
        <=> ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
             => ( aa(nat,A,C2,I2) = aa(nat,A,D2,I2) ) ) ) ) ).

% polyfun_eq_coeffs
tff(fact_3715_prod_OatMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_xx(fun(nat,A),fun(nat,A),G),set_ord_atMost(nat,Nb))) ) ).

% prod.atMost_Suc_shift
tff(fact_3716_sum_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_yq(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_ord_atMost(nat,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ys(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_ord_lessThan(nat,Nb)) ) ).

% sum.nested_swap'
tff(fact_3717_prod_Onested__swap_H,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_yt(fun(nat,fun(nat,A)),fun(nat,A),A2),set_ord_atMost(nat,Nb)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_yv(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb),set_ord_lessThan(nat,Nb)) ) ).

% prod.nested_swap'
tff(fact_3718_sum__choose__lower,axiom,
    ! [R3: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_yw(nat,fun(nat,nat),R3)),set_ord_atMost(nat,Nb)) = aa(nat,nat,binomial(aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),R3),Nb))),Nb) ).

% sum_choose_lower
tff(fact_3719_choose__rising__sum_I2_J,axiom,
    ! [Nb: nat,M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_yx(nat,fun(nat,nat),Nb)),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),Nb),M)),one_one(nat))),M) ).

% choose_rising_sum(2)
tff(fact_3720_choose__rising__sum_I1_J,axiom,
    ! [Nb: nat,M: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_yx(nat,fun(nat,nat),Nb)),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),Nb),M)),one_one(nat))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))) ).

% choose_rising_sum(1)
tff(fact_3721_zero__polynom__imp__zero__coeffs,axiom,
    ! [A: $tType] :
      ( ( ab_semigroup_mult(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [C2: fun(nat,A),Nb: nat,K: nat] :
          ( ! [W2: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_yy(fun(nat,A),fun(A,fun(nat,A)),C2),W2)),set_ord_atMost(nat,Nb)) = zero_zero(A)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => ( aa(nat,A,C2,K) = zero_zero(A) ) ) ) ) ).

% zero_polynom_imp_zero_coeffs
tff(fact_3722_polyfun__eq__0,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat] :
          ( ! [X: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_yp(fun(nat,A),fun(A,fun(nat,A)),C2),X)),set_ord_atMost(nat,Nb)) = zero_zero(A)
        <=> ! [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
             => ( aa(nat,A,C2,I2) = zero_zero(A) ) ) ) ) ).

% polyfun_eq_0
tff(fact_3723_sum_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_atMost(nat,Nb)) = 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_up(fun(nat,A),fun(nat,A),G)),set_ord_lessThan(nat,Nb))) ) ).

% sum.atMost_shift
tff(fact_3724_sum__up__index__split,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_ord_atMost(nat,M))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)))) ) ).

% sum_up_index_split
tff(fact_3725_prod_OatMost__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : groups7121269368397514597t_prod(nat,A,G,set_ord_atMost(nat,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,zero_zero(nat))),groups7121269368397514597t_prod(nat,A,aTP_Lamp_xx(fun(nat,A),fun(nat,A),G),set_ord_lessThan(nat,Nb))) ) ).

% prod.atMost_shift
tff(fact_3726_gbinomial__parallel__sum,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_yz(A,fun(nat,A),A2)),set_ord_atMost(nat,Nb)) = 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),A2),aa(nat,A,semiring_1_of_nat(A),Nb))),one_one(A))),Nb) ) ).

% gbinomial_parallel_sum
tff(fact_3727_sum_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,groups7311177749621191930dd_sum(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_za(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_zc(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_atMost(nat,Nb)) ) ).

% sum.triangle_reindex_eq
tff(fact_3728_prod_Otriangle__reindex__eq,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : groups7121269368397514597t_prod(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_za(nat,fun(nat,fun(nat,$o)),Nb)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_ze(fun(nat,fun(nat,A)),fun(nat,A),G),set_ord_atMost(nat,Nb)) ) ).

% prod.triangle_reindex_eq
tff(fact_3729_sum__choose__diagonal,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_zf(nat,fun(nat,fun(nat,nat)),M),Nb)),set_ord_atMost(nat,M)) = aa(nat,nat,binomial(aa(nat,nat,suc,Nb)),M) ) ) ).

% sum_choose_diagonal
tff(fact_3730_vandermonde,axiom,
    ! [M: nat,Nb: nat,R3: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_zg(nat,fun(nat,fun(nat,fun(nat,nat))),M),Nb),R3)),set_ord_atMost(nat,R3)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)),R3) ).

% vandermonde
tff(fact_3731_sum__gp__basic,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: 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,Nb))) = 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,Nb))) ) ).

% sum_gp_basic
tff(fact_3732_polyfun__roots__finite,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),K: nat,Nb: nat] :
          ( ( aa(nat,A,C2,K) != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
           => finite_finite2(A,collect(A,aa(nat,fun(A,$o),aTP_Lamp_zh(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb))) ) ) ) ).

% polyfun_roots_finite
tff(fact_3733_polyfun__finite__roots,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat] :
          ( finite_finite2(A,collect(A,aa(nat,fun(A,$o),aTP_Lamp_zh(fun(nat,A),fun(nat,fun(A,$o)),C2),Nb)))
        <=> ? [I2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I2),Nb)
              & ( aa(nat,A,C2,I2) != zero_zero(A) ) ) ) ) ).

% polyfun_finite_roots
tff(fact_3734_polyfun__linear__factor__root,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),A2: A,Nb: nat] :
          ( ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_zi(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),set_ord_atMost(nat,Nb)) = zero_zero(A) )
         => ~ ! [B3: fun(nat,A)] :
                ~ ! [Z4: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_zi(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),set_ord_atMost(nat,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,Z4),A2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_zi(fun(nat,A),fun(A,fun(nat,A)),B3),Z4)),set_ord_lessThan(nat,Nb))) ) ) ).

% polyfun_linear_factor_root
tff(fact_3735_polyfun__linear__factor,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [C2: fun(nat,A),Nb: nat,A2: A] :
        ? [B3: fun(nat,A)] :
        ! [Z4: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_zi(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),set_ord_atMost(nat,Nb)) = 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),A2)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_zi(fun(nat,A),fun(A,fun(nat,A)),B3),Z4)),set_ord_lessThan(nat,Nb)))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_zi(fun(nat,A),fun(A,fun(nat,A)),C2),A2)),set_ord_atMost(nat,Nb))) ) ).

% polyfun_linear_factor
tff(fact_3736_sum__power__shift,axiom,
    ! [A: $tType] :
      ( ( monoid_mult(A)
        & comm_ring(A) )
     => ! [M: nat,Nb: nat,Xb: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_or1337092689740270186AtMost(nat,M,Nb)) = 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,Nb),M)))) ) ) ) ).

% sum_power_shift
tff(fact_3737_atLeast1__atMost__eq__remove0,axiom,
    ! [Nb: nat] : set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,zero_zero(nat)),Nb) = aa(set(nat),set(nat),minus_minus(set(nat),set_ord_atMost(nat,Nb)),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_atMost_eq_remove0
tff(fact_3738_sum_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : aa(set(product_prod(nat,nat)),A,groups7311177749621191930dd_sum(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G)),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_zj(nat,fun(nat,fun(nat,$o)),Nb)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_zc(fun(nat,fun(nat,A)),fun(nat,A),G)),set_ord_lessThan(nat,Nb)) ) ).

% sum.triangle_reindex
tff(fact_3739_prod_Otriangle__reindex,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,fun(nat,A)),Nb: nat] : groups7121269368397514597t_prod(product_prod(nat,nat),A,product_case_prod(nat,nat,A,G),collect(product_prod(nat,nat),product_case_prod(nat,nat,$o,aTP_Lamp_zj(nat,fun(nat,fun(nat,$o)),Nb)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_ze(fun(nat,fun(nat,A)),fun(nat,A),G),set_ord_lessThan(nat,Nb)) ) ).

% prod.triangle_reindex
tff(fact_3740_choose__row__sum,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,binomial(Nb)),set_ord_atMost(nat,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ).

% choose_row_sum
tff(fact_3741_binomial,axiom,
    ! [A2: nat,B2: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),B2)),Nb) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(nat,fun(nat,fun(nat,nat)),aTP_Lamp_zk(nat,fun(nat,fun(nat,fun(nat,nat))),A2),B2),Nb)),set_ord_atMost(nat,Nb)) ).

% binomial
tff(fact_3742_sum_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: 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))),Nb)))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_uz(fun(nat,A),fun(nat,A),G)),set_ord_atMost(nat,Nb)) ) ).

% sum.in_pairs_0
tff(fact_3743_polynomial__product,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [M: nat,A2: fun(nat,A),Nb: nat,B2: fun(nat,A),Xb: A] :
          ( ! [I3: nat] :
              ( aa(nat,$o,ord_less(nat,M),I3)
             => ( aa(nat,A,A2,I3) = zero_zero(A) ) )
         => ( ! [J2: nat] :
                ( aa(nat,$o,ord_less(nat,Nb),J2)
               => ( aa(nat,A,B2,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_zi(fun(nat,A),fun(A,fun(nat,A)),A2),Xb)),set_ord_atMost(nat,M))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_zi(fun(nat,A),fun(A,fun(nat,A)),B2),Xb)),set_ord_atMost(nat,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(fun(nat,A),fun(A,fun(nat,A)),aTP_Lamp_zm(fun(nat,A),fun(fun(nat,A),fun(A,fun(nat,A))),A2),B2),Xb)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb))) ) ) ) ) ).

% polynomial_product
tff(fact_3744_prod_Oin__pairs__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: 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))),Nb)))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_yi(fun(nat,A),fun(nat,A),G),set_ord_atMost(nat,Nb)) ) ).

% prod.in_pairs_0
tff(fact_3745_polyfun__eq__const,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [C2: fun(nat,A),Nb: nat,K: A] :
          ( ! [X: A] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_yp(fun(nat,A),fun(A,fun(nat,A)),C2),X)),set_ord_atMost(nat,Nb)) = K
        <=> ( ( aa(nat,A,C2,zero_zero(nat)) = K )
            & ! [X: nat] :
                ( member(nat,X,set_or1337092689740270186AtMost(nat,one_one(nat),Nb))
               => ( aa(nat,A,C2,X) = zero_zero(A) ) ) ) ) ) ).

% polyfun_eq_const
tff(fact_3746_binomial__ring,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,B2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),Nb) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_zn(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Nb)),set_ord_atMost(nat,Nb)) ) ).

% binomial_ring
tff(fact_3747_polynomial__product__nat,axiom,
    ! [M: nat,A2: fun(nat,nat),Nb: nat,B2: fun(nat,nat),Xb: nat] :
      ( ! [I3: nat] :
          ( aa(nat,$o,ord_less(nat,M),I3)
         => ( aa(nat,nat,A2,I3) = zero_zero(nat) ) )
     => ( ! [J2: nat] :
            ( aa(nat,$o,ord_less(nat,Nb),J2)
           => ( aa(nat,nat,B2,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_zo(fun(nat,nat),fun(nat,fun(nat,nat)),A2),Xb)),set_ord_atMost(nat,M))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_zo(fun(nat,nat),fun(nat,fun(nat,nat)),B2),Xb)),set_ord_atMost(nat,Nb))) = aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(nat,fun(nat,nat),aa(fun(nat,nat),fun(nat,fun(nat,nat)),aTP_Lamp_zq(fun(nat,nat),fun(fun(nat,nat),fun(nat,fun(nat,nat))),A2),B2),Xb)),set_ord_atMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb))) ) ) ) ).

% polynomial_product_nat
tff(fact_3748_pochhammer__binomial__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,B2: A,Nb: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),Nb) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_zr(A,fun(A,fun(nat,fun(nat,A))),A2),B2),Nb)),set_ord_atMost(nat,Nb)) ) ).

% pochhammer_binomial_sum
tff(fact_3749_choose__square__sum,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_zs(nat,fun(nat,nat),Nb)),set_ord_atMost(nat,Nb)) = aa(nat,nat,binomial(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),Nb) ).

% choose_square_sum
tff(fact_3750_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,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P2)
         => ( aa(nat,$o,aa(nat,fun(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_zt(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_zu(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_3751_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,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),P2)
         => ( aa(nat,$o,aa(nat,fun(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_zv(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_zw(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_3752_sum__gp0,axiom,
    ! [A: $tType] :
      ( ( division_ring(A)
        & comm_ring(A) )
     => ! [Xb: A,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),power_power(A),Xb)),set_ord_atMost(nat,Nb)) = $ite(Xb = one_one(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),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,Nb)))),aa(A,A,minus_minus(A,one_one(A)),Xb))) ) ).

% sum_gp0
tff(fact_3753_gbinomial__sum__nat__pow2,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_zx(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_3754_gbinomial__partial__sum__poly__xpos,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A2: 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_zy(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),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_zz(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),Xb),Y)),set_ord_atMost(nat,M)) ) ).

% gbinomial_partial_sum_poly_xpos
tff(fact_3755_polyfun__diff__alt,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,A2: fun(nat,A),Xb: A,Y: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => ( aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_zi(fun(nat,A),fun(A,fun(nat,A)),A2),Xb)),set_ord_atMost(nat,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_zi(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),set_ord_atMost(nat,Nb))) = 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_aab(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A2),Xb),Y)),set_ord_lessThan(nat,Nb))) ) ) ) ).

% polyfun_diff_alt
tff(fact_3756_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_3757_choose__linear__sum,axiom,
    ! [Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_aac(nat,fun(nat,nat),Nb)),set_ord_atMost(nat,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),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,Nb),one_one(nat)))) ).

% choose_linear_sum
tff(fact_3758_polyfun__extremal__lemma,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [E3: real,C2: fun(nat,A),Nb: nat] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
         => ? [M10: real] :
            ! [Z4: A] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),M10),real_V7770717601297561774m_norm(A,Z4))
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_aad(fun(nat,A),fun(A,fun(nat,A)),C2),Z4)),set_ord_atMost(nat,Nb)))),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,Nb)))) ) ) ) ).

% polyfun_extremal_lemma
tff(fact_3759_polyfun__diff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,A2: fun(nat,A),Xb: A,Y: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => ( aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_zi(fun(nat,A),fun(A,fun(nat,A)),A2),Xb)),set_ord_atMost(nat,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_zi(fun(nat,A),fun(A,fun(nat,A)),A2),Y)),set_ord_atMost(nat,Nb))) = 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_aaf(nat,fun(fun(nat,A),fun(A,fun(A,fun(nat,A)))),Nb),A2),Xb),Y)),set_ord_lessThan(nat,Nb))) ) ) ) ).

% polyfun_diff
tff(fact_3760_gbinomial__code,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] :
          aa(nat,A,gbinomial(A,A2),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_aag(A,fun(nat,fun(A,A)),A2),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_3761_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,aa(int,fun(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_3762_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_3763_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Y: $o] :
      ( ( vEBT_VEBT_minNull(Xb)
      <=> (Y) )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ( (Y)
             => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) ) )
         => ( ! [Uv2: $o] :
                ( ( Xb = vEBT_Leaf($true,(Uv2)) )
               => ( ~ (Y)
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) ) )
           => ( ! [Uu2: $o] :
                  ( ( Xb = vEBT_Leaf((Uu2),$true) )
                 => ( ~ (Y)
                   => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) ) )
             => ( ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                    ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
                   => ( (Y)
                     => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) )
               => ~ ! [Uz2: product_prod(nat,nat),Va: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                      ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va,Vb2,Vc) )
                     => ( ~ (Y)
                       => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va,Vb2,Vc)) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
tff(fact_3764_of__nat__id,axiom,
    ! [Nb: nat] : aa(nat,nat,semiring_1_of_nat(nat),Nb) = Nb ).

% of_nat_id
tff(fact_3765_neg__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% neg_le_iff_le
tff(fact_3766_neg__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,ord_less(A,A2),B2) ) ) ).

% neg_less_iff_less
tff(fact_3767_neg__numeral__eq__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,Nb: 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),Nb)) )
        <=> ( M = Nb ) ) ) ).

% neg_numeral_eq_iff
tff(fact_3768_minus__add__distrib,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_add_distrib
tff(fact_3769_minus__add__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = B2 ) ).

% minus_add_cancel
tff(fact_3770_add__minus__cancel,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),B2)) = B2 ) ).

% add_minus_cancel
tff(fact_3771_mult__minus__left,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% mult_minus_left
tff(fact_3772_minus__mult__minus,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) ) ).

% minus_mult_minus
tff(fact_3773_mult__minus__right,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) ) ).

% mult_minus_right
tff(fact_3774_minus__diff__eq,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,minus_minus(A,A2),B2)) = aa(A,A,minus_minus(A,B2),A2) ) ).

% minus_diff_eq
tff(fact_3775_div__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ).

% div_minus_minus
tff(fact_3776_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_3777_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_3778_mod__minus__minus,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2)) ) ).

% mod_minus_minus
tff(fact_3779_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_3780_neg__0__le__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% neg_0_le_iff_le
tff(fact_3781_neg__le__0__iff__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% neg_le_0_iff_le
tff(fact_3782_less__eq__neg__nonpos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% less_eq_neg_nonpos
tff(fact_3783_neg__less__eq__nonneg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% neg_less_eq_nonneg
tff(fact_3784_neg__less__0__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),A2)),zero_zero(A))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A2) ) ) ).

% neg_less_0_iff_less
tff(fact_3785_neg__0__less__iff__less,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,ord_less(A,A2),zero_zero(A)) ) ) ).

% neg_0_less_iff_less
tff(fact_3786_neg__less__pos,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),A2)),A2)
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A2) ) ) ).

% neg_less_pos
tff(fact_3787_less__neg__neg,axiom,
    ! [A: $tType] :
      ( linord5086331880401160121up_add(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,A2),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,ord_less(A,A2),zero_zero(A)) ) ) ).

% less_neg_neg
tff(fact_3788_add_Oright__inverse,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),A2)) = zero_zero(A) ) ).

% add.right_inverse
tff(fact_3789_ab__left__minus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),A2) = zero_zero(A) ) ).

% ab_left_minus
tff(fact_3790_add__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Nb: 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),Nb))) = 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),Nb))) ) ).

% add_neg_numeral_simps(3)
tff(fact_3791_verit__minus__simplify_I3_J,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [B2: A] : aa(A,A,minus_minus(A,zero_zero(A)),B2) = aa(A,A,uminus_uminus(A),B2) ) ).

% verit_minus_simplify(3)
tff(fact_3792_diff__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(A,A,minus_minus(A,zero_zero(A)),A2) = aa(A,A,uminus_uminus(A),A2) ) ).

% diff_0
tff(fact_3793_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_3794_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_3795_diff__minus__eq__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) ) ).

% diff_minus_eq_add
tff(fact_3796_uminus__add__conv__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,minus_minus(A,B2),A2) ) ).

% uminus_add_conv_diff
tff(fact_3797_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_3798_div__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,uminus_uminus(A),A2) ) ).

% div_minus1_right
tff(fact_3799_minus__mod__self1,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [B2: A,A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,minus_minus(A,B2),A2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% minus_mod_self1
tff(fact_3800_fact__0,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,zero_zero(nat)) = one_one(A) ) ) ).

% fact_0
tff(fact_3801_fact__1,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ( semiring_char_0_fact(A,one_one(nat)) = one_one(A) ) ) ).

% fact_1
tff(fact_3802_negative__zle,axiom,
    ! [Nb: nat,M: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Nb))),aa(nat,int,semiring_1_of_nat(int),M)) ).

% negative_zle
tff(fact_3803_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_3804_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_3805_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_3806_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_3807_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_3808_neg__one__eq__numeral__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] :
          ( ( aa(A,A,uminus_uminus(A),one_one(A)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) )
        <=> ( Nb = one2 ) ) ) ).

% neg_one_eq_numeral_iff
tff(fact_3809_numeral__eq__neg__one__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] :
          ( ( aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( Nb = one2 ) ) ) ).

% numeral_eq_neg_one_iff
tff(fact_3810_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_3811_minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: 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))),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Nb)) = one_one(A) ) ).

% minus_one_mult_self
tff(fact_3812_left__minus__one__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat,A2: 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))),Nb)),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))),Nb)),A2)) = A2 ) ).

% left_minus_one_mult_self
tff(fact_3813_mod__minus1__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,uminus_uminus(A),one_one(A))) = zero_zero(A) ) ).

% mod_minus1_right
tff(fact_3814_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,aa(A,fun(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_3815_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,aa(A,fun(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_3816_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,aa(A,fun(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_3817_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_3818_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_3819_fact__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : semiring_char_0_fact(A,aa(nat,nat,suc,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,Nb))),semiring_char_0_fact(A,Nb)) ) ).

% fact_Suc
tff(fact_3820_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_3821_diff__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Nb: 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),Nb))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Nb)) ) ).

% diff_numeral_simps(2)
tff(fact_3822_diff__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [M: num,Nb: 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),Nb)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),M),Nb))) ) ).

% diff_numeral_simps(3)
tff(fact_3823_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_3824_negative__zless,axiom,
    ! [Nb: 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,Nb)))),aa(nat,int,semiring_1_of_nat(int),M)) ).

% negative_zless
tff(fact_3825_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_3826_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_3827_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_3828_mult__neg__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,Nb: 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),Nb))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),Nb))) ) ).

% mult_neg_numeral_simps(3)
tff(fact_3829_mult__neg__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,Nb: 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),Nb)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),Nb))) ) ).

% mult_neg_numeral_simps(2)
tff(fact_3830_mult__neg__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [M: num,Nb: 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),Nb))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),times_times(num),M),Nb)) ) ).

% mult_neg_numeral_simps(1)
tff(fact_3831_neg__numeral__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Nb: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)))
        <=> aa(num,$o,aa(num,fun(num,$o),ord_less_eq(num),Nb),M) ) ) ).

% neg_numeral_le_iff
tff(fact_3832_neg__numeral__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Nb: num] :
          ( aa(A,$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),Nb)))
        <=> aa(num,$o,ord_less(num,Nb),M) ) ) ).

% neg_numeral_less_iff
tff(fact_3833_not__neg__one__le__neg__numeral__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] :
          ( ~ aa(A,$o,aa(A,fun(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_3834_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_3835_divide__eq__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,W: num,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = A2 )
        <=> $ite(aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) != zero_zero(A),B2 = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),A2 = zero_zero(A)) ) ) ).

% divide_eq_eq_numeral1(2)
tff(fact_3836_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,W: num] :
          ( ( A2 = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))) = B2,A2 = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral1(2)
tff(fact_3837_divide__le__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B2) ) ) ).

% divide_le_eq_numeral1(2)
tff(fact_3838_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ).

% le_divide_eq_numeral1(2)
tff(fact_3839_divide__less__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,W: num,A2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),A2)
        <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))),B2) ) ) ).

% divide_less_eq_numeral1(2)
tff(fact_3840_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,W: num] :
          ( aa(A,$o,ord_less(A,A2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))))
        <=> aa(A,$o,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)))) ) ) ).

% less_divide_eq_numeral1(2)
tff(fact_3841_power2__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_minus
tff(fact_3842_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_3843_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_3844_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_3845_int__div__minus__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,ord_less(int,A2),zero_zero(int))
     => ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) = aa(int,int,uminus_uminus(int),A2) )
      <=> ( B2 = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% int_div_minus_is_minus1
tff(fact_3846_round__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Nb: num] : archimedean_round(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb)) ) ).

% round_neg_numeral
tff(fact_3847_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_3848_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_3849_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_3850_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_3851_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_3852_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_3853_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ) ).

% Power.ring_1_class.power_minus_even
tff(fact_3854_power__minus__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat,A2: A] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb) = aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ) ) ).

% power_minus_odd
tff(fact_3855_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat,A2: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) ) ) ) ).

% Parity.ring_1_class.power_minus_even
tff(fact_3856_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_3857_diff__numeral__special_I3_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : aa(A,A,minus_minus(A,one_one(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(num,A,numeral_numeral(A),aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),Nb)) ) ).

% diff_numeral_special(3)
tff(fact_3858_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,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),one_one(A))) ) ) ).

% ceiling_less_zero
tff(fact_3859_zero__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] :
          ( aa(int,$o,aa(int,fun(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_3860_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_3861_ceiling__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: num] : archimedean_ceiling(real,aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(num,real,numeral_numeral(real),A2)),aa(num,real,numeral_numeral(real),B2))) = 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),A2))),aa(num,int,numeral_numeral(int),B2))) ).

% ceiling_divide_eq_div_numeral
tff(fact_3862_power__minus1__even,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: 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))),Nb)) = one_one(A) ) ).

% power_minus1_even
tff(fact_3863_neg__one__odd__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] :
          ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Nb) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% neg_one_odd_power
tff(fact_3864_neg__one__even__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Nb) = one_one(A) ) ) ) ).

% neg_one_even_power
tff(fact_3865_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Y: int,Xb: num,Nb: 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))),Nb) )
        <=> ( 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))),Nb) ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
tff(fact_3866_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Xb: num,Nb: 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))),Nb) = 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))),Nb) = Y ) ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
tff(fact_3867_neg__numeral__le__floor,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(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,aa(A,fun(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_3868_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_3869_ceiling__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(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,aa(A,fun(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_3870_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_3871_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,$o,aa(A,fun(A,$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_3872_floor__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,V: num] :
          ( aa(int,$o,aa(int,fun(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_3873_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,$o,aa(A,fun(A,$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_3874_neg__numeral__le__ceiling,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [V: num,Xb: A] :
          ( aa(int,$o,aa(int,fun(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_3875_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A2: int] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> aa(int,$o,aa(int,fun(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))),Nb)),A2) ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
tff(fact_3876_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xb: num,Nb: nat] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),Nb)) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
tff(fact_3877_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: num,Nb: nat,A2: int] :
          ( aa(A,$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))),Nb)),aa(int,A,ring_1_of_int(A),A2))
        <=> 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))),Nb)),A2) ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
tff(fact_3878_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: int,Xb: num,Nb: nat] :
          ( aa(A,$o,ord_less(A,aa(int,A,ring_1_of_int(A),A2)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Xb))),Nb))
        <=> aa(int,$o,ord_less(int,A2),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Xb))),Nb)) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
tff(fact_3879_verit__negate__coefficient_I2_J,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% verit_negate_coefficient(2)
tff(fact_3880_less__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),aa(A,A,uminus_uminus(A),B2))
        <=> aa(A,$o,ord_less(A,B2),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% less_minus_iff
tff(fact_3881_minus__less__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),A2)),B2)
        <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B2)),A2) ) ) ).

% minus_less_iff
tff(fact_3882_mod__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A2)),B2)) ) ).

% mod_minus_right
tff(fact_3883_euclidean__ring__cancel__class_Omod__minus__cong,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A,A6: A] :
          ( ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A6),B2) )
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A6)),B2) ) ) ) ).

% euclidean_ring_cancel_class.mod_minus_cong
tff(fact_3884_mod__minus__eq,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: 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),A2),B2))),B2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% mod_minus_eq
tff(fact_3885_numeral__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,Nb: num] : aa(num,A,numeral_numeral(A),M) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) ) ).

% numeral_neq_neg_numeral
tff(fact_3886_neg__numeral__neq__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [M: num,Nb: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)) != aa(num,A,numeral_numeral(A),Nb) ) ).

% neg_numeral_neq_numeral
tff(fact_3887_div__minus__right,axiom,
    ! [A: $tType] :
      ( euclid8851590272496341667cancel(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% div_minus_right
tff(fact_3888_minus__divide__left,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% minus_divide_left
tff(fact_3889_minus__divide__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ).

% minus_divide_divide
tff(fact_3890_minus__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_divide_right
tff(fact_3891_minus__mult__commute,axiom,
    ! [A: $tType] :
      ( ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% minus_mult_commute
tff(fact_3892_square__eq__iff,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),B2) )
        <=> ( ( A2 = B2 )
            | ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% square_eq_iff
tff(fact_3893_group__cancel_Oneg1,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A3: A,K: A,A2: A] :
          ( ( A3 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),A2) )
         => ( aa(A,A,uminus_uminus(A),A3) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,uminus_uminus(A),A2)) ) ) ) ).

% group_cancel.neg1
tff(fact_3894_add_Oinverse__distrib__swap,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ).

% add.inverse_distrib_swap
tff(fact_3895_is__num__normalize_I8_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [A2: A,B2: A] : aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ).

% is_num_normalize(8)
tff(fact_3896_minus__diff__commute,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B2: A,A2: A] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),B2)),A2) = aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A2)),B2) ) ).

% minus_diff_commute
tff(fact_3897_minus__diff__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,minus_minus(A,A2),B2)) ) ).

% minus_diff_minus
tff(fact_3898_le__imp__neg__le,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% le_imp_neg_le
tff(fact_3899_minus__le__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),A2) ) ) ).

% minus_le_iff
tff(fact_3900_le__minus__iff,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),B2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% le_minus_iff
tff(fact_3901_fact__ge__self,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),semiring_char_0_fact(nat,Nb)) ).

% fact_ge_self
tff(fact_3902_fact__mono__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Nb)) ) ).

% fact_mono_nat
tff(fact_3903_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_3904_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_3905_sum__negf,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [F: fun(B,A),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_aah(fun(B,A),fun(B,A),F)),A3) = aa(A,A,uminus_uminus(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F),A3)) ) ).

% sum_negf
tff(fact_3906_sums__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),A2: A] :
          ( aa(A,$o,sums(A,F),A2)
         => aa(A,$o,sums(A,aTP_Lamp_aai(fun(nat,A),fun(nat,A),F)),aa(A,A,uminus_uminus(A),A2)) ) ) ).

% sums_minus
tff(fact_3907_fact__less__mono__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(nat,$o,ord_less(nat,M),Nb)
       => aa(nat,$o,ord_less(nat,semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Nb)) ) ) ).

% fact_less_mono_nat
tff(fact_3908_fact__ge__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_ge_zero
tff(fact_3909_fact__not__neg,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : ~ aa(A,$o,ord_less(A,semiring_char_0_fact(A,Nb)),zero_zero(A)) ) ).

% fact_not_neg
tff(fact_3910_fact__gt__zero,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,ord_less(A,zero_zero(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_gt_zero
tff(fact_3911_fact__ge__1,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),semiring_char_0_fact(A,Nb)) ) ).

% fact_ge_1
tff(fact_3912_zero__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] : zero_zero(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) ) ).

% zero_neq_neg_numeral
tff(fact_3913_neg__numeral__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Nb: num] : aa(A,$o,aa(A,fun(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),Nb)) ) ).

% neg_numeral_le_numeral
tff(fact_3914_not__numeral__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Nb: num] : ~ aa(A,$o,aa(A,fun(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),Nb))) ) ).

% not_numeral_le_neg_numeral
tff(fact_3915_neg__numeral__less__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Nb: num] : aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))),aa(num,A,numeral_numeral(A),Nb)) ) ).

% neg_numeral_less_numeral
tff(fact_3916_not__numeral__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num,Nb: num] : ~ aa(A,$o,ord_less(A,aa(num,A,numeral_numeral(A),M)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_numeral_less_neg_numeral
tff(fact_3917_add__eq__0__iff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) )
        <=> ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% add_eq_0_iff
tff(fact_3918_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),A2)),A2) = zero_zero(A) ) ).

% ab_group_add_class.ab_left_minus
tff(fact_3919_add_Oinverse__unique,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),A2) = B2 ) ) ) ).

% add.inverse_unique
tff(fact_3920_eq__neg__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),B2) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) ) ) ) ).

% eq_neg_iff_add_eq_0
tff(fact_3921_neg__eq__iff__add__eq__0,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,uminus_uminus(A),A2) = B2 )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = zero_zero(A) ) ) ) ).

% neg_eq_iff_add_eq_0
tff(fact_3922_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_3923_le__minus__one__simps_I4_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(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_3924_le__minus__one__simps_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(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_3925_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_3926_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_3927_numeral__neq__neg__one,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] : aa(num,A,numeral_numeral(A),Nb) != aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% numeral_neq_neg_one
tff(fact_3928_one__neq__neg__numeral,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [Nb: num] : one_one(A) != aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)) ) ).

% one_neq_neg_numeral
tff(fact_3929_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_3930_nonzero__minus__divide__right,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% nonzero_minus_divide_right
tff(fact_3931_nonzero__minus__divide__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) ) ) ) ).

% nonzero_minus_divide_divide
tff(fact_3932_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_3933_fact__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,M)),semiring_char_0_fact(A,Nb)) ) ) ).

% fact_mono
tff(fact_3934_group__cancel_Osub2,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [B5: A,K: A,B2: A,A2: A] :
          ( ( B5 = aa(A,A,aa(A,fun(A,A),plus_plus(A),K),B2) )
         => ( aa(A,A,minus_minus(A,A2),B5) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),K)),aa(A,A,minus_minus(A,A2),B2)) ) ) ) ).

% group_cancel.sub2
tff(fact_3935_diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% diff_conv_add_uminus
tff(fact_3936_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,B2: A] : aa(A,A,minus_minus(A,A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,uminus_uminus(A),B2)) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
tff(fact_3937_dvd__neg__div,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,uminus_uminus(A),A2)),B2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% dvd_neg_div
tff(fact_3938_dvd__div__neg,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,dvd_dvd(A,B2),A2)
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,uminus_uminus(A),B2)) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) ) ) ) ).

% dvd_div_neg
tff(fact_3939_fact__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
         => aa(A,$o,dvd_dvd(A,semiring_char_0_fact(A,Nb)),semiring_char_0_fact(A,M)) ) ) ).

% fact_dvd
tff(fact_3940_int__cases,axiom,
    ! [Z: int] :
      ( ! [N2: nat] : Z != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] : Z != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% int_cases
tff(fact_3941_int__of__nat__induct,axiom,
    ! [P: fun(int,$o),Z: int] :
      ( ! [N2: nat] : aa(int,$o,P,aa(nat,int,semiring_1_of_nat(int),N2))
     => ( ! [N2: nat] : aa(int,$o,P,aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))))
       => aa(int,$o,P,Z) ) ) ).

% int_of_nat_induct
tff(fact_3942_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_3943_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,Nb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),Nb) = 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_3944_zmult__eq__1__iff,axiom,
    ! [M: int,Nb: int] :
      ( ( aa(int,int,aa(int,fun(int,int),times_times(int),M),Nb) = one_one(int) )
    <=> ( ( ( M = one_one(int) )
          & ( Nb = one_one(int) ) )
        | ( ( M = aa(int,int,uminus_uminus(int),one_one(int)) )
          & ( Nb = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ) ).

% zmult_eq_1_iff
tff(fact_3945_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_3946_pochhammer__fact,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_semiring_1(A) )
     => ! [Nb: nat] : semiring_char_0_fact(A,Nb) = comm_s3205402744901411588hammer(A,one_one(A),Nb) ) ).

% pochhammer_fact
tff(fact_3947_pochhammer__same,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & comm_ring_1(A)
        & semiri3467727345109120633visors(A) )
     => ! [Nb: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),Nb) = 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))),Nb)),semiring_char_0_fact(A,Nb)) ) ).

% pochhammer_same
tff(fact_3948_fact__ge__Suc__0__nat,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,zero_zero(nat))),semiring_char_0_fact(nat,Nb)) ).

% fact_ge_Suc_0_nat
tff(fact_3949_dvd__fact,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),M)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
       => aa(nat,$o,dvd_dvd(nat,M),semiring_char_0_fact(nat,Nb)) ) ) ).

% dvd_fact
tff(fact_3950_neg__numeral__le__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))),zero_zero(A)) ) ).

% neg_numeral_le_zero
tff(fact_3951_not__zero__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_zero_le_neg_numeral
tff(fact_3952_neg__numeral__less__zero,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))),zero_zero(A)) ) ).

% neg_numeral_less_zero
tff(fact_3953_not__zero__less__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : ~ aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) ) ).

% not_zero_less_neg_numeral
tff(fact_3954_le__minus__one__simps_I3_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ~ aa(A,$o,aa(A,fun(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_3955_le__minus__one__simps_I1_J,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => aa(A,$o,aa(A,fun(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_3956_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_3957_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_3958_neg__numeral__le__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(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_3959_neg__one__le__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(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_3960_neg__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : aa(A,$o,aa(A,fun(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_3961_not__numeral__le__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(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_3962_not__one__le__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: num] : ~ aa(A,$o,aa(A,fun(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_3963_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_3964_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_3965_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_3966_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_3967_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_3968_gbinomial__pochhammer,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),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),A2),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer
tff(fact_3969_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_3970_mult__1s__ring__1_I2_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),B2),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),one2))) = aa(A,A,uminus_uminus(A),B2) ) ).

% mult_1s_ring_1(2)
tff(fact_3971_mult__1s__ring__1_I1_J,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [B2: 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))),B2) = aa(A,A,uminus_uminus(A),B2) ) ).

% mult_1s_ring_1(1)
tff(fact_3972_divide__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( B2 != zero_zero(A) )
            & ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ).

% divide_eq_minus_1_iff
tff(fact_3973_eq__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) )
        <=> $ite(C2 != zero_zero(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2) = aa(A,A,uminus_uminus(A),B2),A2 = zero_zero(A)) ) ) ).

% eq_minus_divide_eq
tff(fact_3974_minus__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)) = A2 )
        <=> $ite(C2 != zero_zero(A),aa(A,A,uminus_uminus(A),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2),A2 = zero_zero(A)) ) ) ).

% minus_divide_eq_eq
tff(fact_3975_nonzero__neg__divide__eq__eq,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,A2: A,C2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = C2 )
          <=> ( aa(A,A,uminus_uminus(A),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq
tff(fact_3976_nonzero__neg__divide__eq__eq2,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( ( C2 = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) )
          <=> ( aa(A,A,aa(A,fun(A,A),times_times(A),C2),B2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
tff(fact_3977_fact__less__mono,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
         => ( aa(nat,$o,ord_less(nat,M),Nb)
           => aa(A,$o,ord_less(A,semiring_char_0_fact(A,M)),semiring_char_0_fact(A,Nb)) ) ) ) ).

% fact_less_mono
tff(fact_3978_power__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb) = 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))),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ).

% power_minus
tff(fact_3979_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_3980_fact__mod,axiom,
    ! [A: $tType] :
      ( ( linordered_semidom(A)
        & semidom_modulo(A) )
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(A,A,aa(A,fun(A,A),modulo_modulo(A),semiring_char_0_fact(A,Nb)),semiring_char_0_fact(A,M)) = zero_zero(A) ) ) ) ).

% fact_mod
tff(fact_3981_fact__fact__dvd__fact,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Nb: nat] : aa(A,$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,Nb))),semiring_char_0_fact(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),Nb))) ) ).

% fact_fact_dvd_fact
tff(fact_3982_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_3983_fact__le__power,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),semiring_char_0_fact(A,Nb)),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),Nb))) ) ).

% fact_le_power
tff(fact_3984_norm__uminus__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Xb: A,Y: A] : real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),Xb)),Y)) = real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) ) ).

% norm_uminus_minus
tff(fact_3985_powr__minus__divide,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: A,A2: A] : powr(A,Xb,aa(A,A,uminus_uminus(A),A2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),powr(A,Xb,A2)) ) ).

% powr_minus_divide
tff(fact_3986_int__cases4,axiom,
    ! [M: int] :
      ( ! [N2: nat] : M != aa(nat,int,semiring_1_of_nat(int),N2)
     => ~ ! [N2: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N2)
           => ( M != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ) ) ).

% int_cases4
tff(fact_3987_int__zle__neg,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,semiring_1_of_nat(int),Nb)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),M)))
    <=> ( ( Nb = zero_zero(nat) )
        & ( M = zero_zero(nat) ) ) ) ).

% int_zle_neg
tff(fact_3988_nonpos__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),zero_zero(int))
     => ~ ! [N2: nat] : K != aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) ) ).

% nonpos_int_cases
tff(fact_3989_negative__zle__0,axiom,
    ! [Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),Nb))),zero_zero(int)) ).

% negative_zle_0
tff(fact_3990_zmod__zminus2__eq__if,axiom,
    ! [A2: int,B2: int] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),aa(int,int,uminus_uminus(int),B2)) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2)),B2)) ).

% zmod_zminus2_eq_if
tff(fact_3991_zmod__zminus1__eq__if,axiom,
    ! [A2: int,B2: int] :
      aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),A2)),B2) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,B2),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2))) ).

% zmod_zminus1_eq_if
tff(fact_3992_fact__prod,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : semiring_char_0_fact(A,Nb) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,aTP_Lamp_vc(nat,nat),set_or1337092689740270186AtMost(nat,one_one(nat),Nb))) ) ).

% fact_prod
tff(fact_3993_fact__diff__Suc,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,suc,M))
     => ( semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,M)),Nb)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,M),Nb))) ) ) ).

% fact_diff_Suc
tff(fact_3994_fact__div__fact__le__pow,axiom,
    ! [R3: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),R3),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Nb)),semiring_char_0_fact(nat,aa(nat,nat,minus_minus(nat,Nb),R3)))),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),Nb),R3)) ) ).

% fact_div_fact_le_pow
tff(fact_3995_binomial__fact__lemma,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( 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,Nb),K)))),aa(nat,nat,binomial(Nb),K)) = semiring_char_0_fact(nat,Nb) ) ) ).

% binomial_fact_lemma
tff(fact_3996_less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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),A2),C2)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,ord_less(A,A2),zero_zero(A))) ) ) ) ).

% less_minus_divide_eq
tff(fact_3997_minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,ord_less(A,zero_zero(A)),A2)) ) ) ) ).

% minus_divide_less_eq
tff(fact_3998_neg__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% neg_less_minus_divide_eq
tff(fact_3999_neg__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divide_less_eq
tff(fact_4000_pos__less__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_less_minus_divide_eq
tff(fact_4001_pos__minus__divide__less__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% pos_minus_divide_less_eq
tff(fact_4002_divide__eq__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [B2: A,C2: A,W: num] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) )
        <=> $ite(C2 != zero_zero(A),B2 = 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_4003_eq__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [W: num,B2: 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),B2),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) = B2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W)) = zero_zero(A)) ) ) ).

% eq_divide_eq_numeral(2)
tff(fact_4004_add__divide__eq__if__simps_I3_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: 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),A2),Z))),B2) = $ite(Z = zero_zero(A),B2,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),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(3)
tff(fact_4005_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_4006_add__divide__eq__if__simps_I6_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z))),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(6)
tff(fact_4007_add__divide__eq__if__simps_I5_J,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Z: A,B2: A] :
          aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),Z)),B2) = $ite(Z = zero_zero(A),aa(A,A,uminus_uminus(A),B2),aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),Z))),Z)) ) ).

% add_divide_eq_if_simps(5)
tff(fact_4008_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_4009_even__minus,axiom,
    ! [A: $tType] :
      ( ring_parity(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,uminus_uminus(A),A2))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% even_minus
tff(fact_4010_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_4011_choose__dvd,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => aa(A,$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,Nb),K)))),semiring_char_0_fact(A,Nb)) ) ) ).

% choose_dvd
tff(fact_4012_int__cases3,axiom,
    ! [K: int] :
      ( ( K != zero_zero(int) )
     => ( ! [N2: nat] :
            ( ( K = aa(nat,int,semiring_1_of_nat(int),N2) )
           => ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),N2) )
       => ~ ! [N2: nat] :
              ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) )
             => ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),N2) ) ) ) ).

% int_cases3
tff(fact_4013_fact__eq__fact__times,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
     => ( semiring_char_0_fact(nat,M) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),semiring_char_0_fact(nat,Nb)),groups7121269368397514597t_prod(nat,nat,aTP_Lamp_vc(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),M))) ) ) ).

% fact_eq_fact_times
tff(fact_4014_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,K: nat] :
          ( aa(nat,$o,ord_less(nat,Nb),K)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K) = zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma
tff(fact_4015_pochhammer__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [Nb: nat,K: nat] :
          ( ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K) = zero_zero(A) )
        <=> aa(nat,$o,ord_less(nat,Nb),K) ) ) ).

% pochhammer_of_nat_eq_0_iff
tff(fact_4016_pochhammer__eq__0__iff,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,Nb: nat] :
          ( ( comm_s3205402744901411588hammer(A,A2,Nb) = zero_zero(A) )
        <=> ? [K4: nat] :
              ( aa(nat,$o,ord_less(nat,K4),Nb)
              & ( A2 = aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),K4)) ) ) ) ) ).

% pochhammer_eq_0_iff
tff(fact_4017_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [A: $tType] :
      ( ( ring_char_0(A)
        & idom(A) )
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),aa(nat,A,semiring_1_of_nat(A),Nb)),K) != zero_zero(A) ) ) ) ).

% pochhammer_of_nat_eq_0_lemma'
tff(fact_4018_not__zle__0__negative,axiom,
    ! [Nb: nat] : ~ aa(int,$o,aa(int,fun(int,$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,Nb)))) ).

% not_zle_0_negative
tff(fact_4019_negative__zless__0,axiom,
    ! [Nb: 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,Nb)))),zero_zero(int)) ).

% negative_zless_0
tff(fact_4020_negD,axiom,
    ! [Xb: int] :
      ( aa(int,$o,ord_less(int,Xb),zero_zero(int))
     => ? [N2: nat] : Xb = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,suc,N2))) ) ).

% negD
tff(fact_4021_verit__less__mono__div__int2,axiom,
    ! [A3: int,B5: int,Nb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A3),B5)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),aa(int,int,uminus_uminus(int),Nb))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),B5),Nb)),aa(int,int,aa(int,fun(int,int),divide_divide(int),A3),Nb)) ) ) ).

% verit_less_mono_div_int2
tff(fact_4022_div__eq__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),one_one(int))),B2) = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ).

% div_eq_minus1
tff(fact_4023_ceiling__divide__eq__div,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: int,B2: int] : archimedean_ceiling(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A2)),aa(int,A,ring_1_of_int(A),B2))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A2)),B2)) ) ).

% ceiling_divide_eq_div
tff(fact_4024_binomial__altdef__nat,axiom,
    ! [K: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
     => ( aa(nat,nat,binomial(Nb),K) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,Nb)),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,Nb),K)))) ) ) ).

% binomial_altdef_nat
tff(fact_4025_le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))) ) ) ) ).

% le_minus_divide_eq
tff(fact_4026_minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)) ) ) ) ).

% minus_divide_le_eq
tff(fact_4027_neg__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% neg_le_minus_divide_eq
tff(fact_4028_neg__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,C2),zero_zero(A))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divide_le_eq
tff(fact_4029_pos__le__minus__divide__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_le_minus_divide_eq
tff(fact_4030_pos__minus__divide__le__eq,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [C2: A,B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),C2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)) ) ) ) ).

% pos_minus_divide_le_eq
tff(fact_4031_divide__less__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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,B2),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,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),B2),aa(A,$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_4032_less__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$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),B2),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)),B2),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,ord_less(A,B2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),W))),C2)),aa(A,$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_4033_power2__eq__1__iff,axiom,
    ! [A: $tType] :
      ( ring_15535105094025558882visors(A)
     => ! [A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) = one_one(A) )
        <=> ( ( A2 = one_one(A) )
            | ( A2 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% power2_eq_1_iff
tff(fact_4034_uminus__power__if,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Nb: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb))) ) ).

% uminus_power_if
tff(fact_4035_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( 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),Nb),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,Nb),K)) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
tff(fact_4036_fact__div__fact,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
     => ( aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),semiring_char_0_fact(nat,M)),semiring_char_0_fact(nat,Nb)) = groups7121269368397514597t_prod(nat,nat,aTP_Lamp_vc(nat,nat),set_or1337092689740270186AtMost(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)),M)) ) ) ).

% fact_div_fact
tff(fact_4037_neg__int__cases,axiom,
    ! [K: int] :
      ( aa(int,$o,ord_less(int,K),zero_zero(int))
     => ~ ! [N2: nat] :
            ( ( K = aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),N2)) )
           => ~ aa(nat,$o,ord_less(nat,zero_zero(nat)),N2) ) ) ).

% neg_int_cases
tff(fact_4038_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z5: int] :
      ( aa(int,$o,aa(int,fun(int,$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_4039_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$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_4040_zmod__minus1,axiom,
    ! [B2: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
     => ( aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),one_one(int))),B2) = aa(int,int,minus_minus(int,B2),one_one(int)) ) ) ).

% zmod_minus1
tff(fact_4041_zdiv__zminus1__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),A2)),B2) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),one_one(int))) ) ) ).

% zdiv_zminus1_eq_if
tff(fact_4042_zdiv__zminus2__eq__if,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),aa(int,int,uminus_uminus(int),B2)) = $ite(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) = zero_zero(int),aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)),aa(int,int,minus_minus(int,aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2))),one_one(int))) ) ) ).

% zdiv_zminus2_eq_if
tff(fact_4043_square__fact__le__2__fact,axiom,
    ! [Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),semiring_char_0_fact(real,Nb)),semiring_char_0_fact(real,Nb))),semiring_char_0_fact(real,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ).

% square_fact_le_2_fact
tff(fact_4044_zminus1__lemma,axiom,
    ! [A2: int,B2: int,Q2: int,R3: int] :
      ( eucl_rel_int(A2,B2,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R3))
     => ( ( B2 != zero_zero(int) )
       => eucl_rel_int(aa(int,int,uminus_uminus(int),A2),B2,
            aa(int,product_prod(int,int),
              aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),
                $ite(R3 = 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(R3 = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,B2),R3)))) ) ) ).

% zminus1_lemma
tff(fact_4045_divide__le__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,C2: A,W: num] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),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,$o,aa(A,fun(A,$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)),B2),aa(A,$o,aa(A,fun(A,$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_4046_le__divide__eq__numeral_I2_J,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [W: num,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(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),B2),C2))
        <=> $ite(
              aa(A,$o,ord_less(A,zero_zero(A)),C2),
              aa(A,$o,aa(A,fun(A,$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)),B2),
              $ite(aa(A,$o,ord_less(A,C2),zero_zero(A)),aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),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,$o,aa(A,fun(A,$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_4047_square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),one_one(A))),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),one_one(A))
           => aa(A,$o,aa(A,fun(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_4048_minus__power__mult__self,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [A2: A,Nb: 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),A2)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ) ).

% minus_power_mult_self
tff(fact_4049_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_4050_fact__code,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : semiring_char_0_fact(A,Nb) = aa(nat,A,semiring_1_of_nat(A),set_fo6178422350223883121st_nat(nat,times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2)),Nb,one_one(nat))) ) ).

% fact_code
tff(fact_4051_minus__one__power__iff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: nat] :
          aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),one_one(A))),Nb) = $ite(aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb),one_one(A),aa(A,A,uminus_uminus(A),one_one(A))) ) ).

% minus_one_power_iff
tff(fact_4052_fact__reduce,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
         => ( semiring_char_0_fact(A,Nb) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))) ) ) ) ).

% fact_reduce
tff(fact_4053_binomial__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,Nb)),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,Nb),K)))) ) ) ) ).

% binomial_fact
tff(fact_4054_fact__binomial,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( 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(Nb),K))) = aa(A,A,aa(A,fun(A,A),divide_divide(A),semiring_char_0_fact(A,Nb)),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Nb),K))) ) ) ) ).

% fact_binomial
tff(fact_4055_pochhammer__absorb__comp,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [R3: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,minus_minus(A,R3),aa(nat,A,semiring_1_of_nat(A),K))),comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),R3),K)) = aa(A,A,aa(A,fun(A,A),times_times(A),R3),comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,uminus_uminus(A),R3)),one_one(A)),K)) ) ).

% pochhammer_absorb_comp
tff(fact_4056_gbinomial__index__swap,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: 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),Nb))),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))),Nb)),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))),Nb)) ) ).

% gbinomial_index_swap
tff(fact_4057_gbinomial__negated__upper,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),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)),A2)),one_one(A))),K)) ) ).

% gbinomial_negated_upper
tff(fact_4058_minus__1__div__exp__eq__int,axiom,
    ! [Nb: 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))),Nb)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% minus_1_div_exp_eq_int
tff(fact_4059_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(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_4060_power__minus1__odd,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: 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))),Nb))) = aa(A,A,uminus_uminus(A),one_one(A)) ) ).

% power_minus1_odd
tff(fact_4061_gbinomial__minus,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,aa(A,A,uminus_uminus(A),A2)),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),A2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A))),K)) ) ).

% gbinomial_minus
tff(fact_4062_pochhammer__minus,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,uminus_uminus(A),B2),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,B2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)) ) ).

% pochhammer_minus
tff(fact_4063_pochhammer__minus_H,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [B2: A,K: nat] : comm_s3205402744901411588hammer(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,minus_minus(A,B2),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),B2),K)) ) ).

% pochhammer_minus'
tff(fact_4064_gbinomial__pochhammer_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),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,A2),aa(nat,A,semiring_1_of_nat(A),K))),one_one(A)),K)),semiring_char_0_fact(A,K)) ) ).

% gbinomial_pochhammer'
tff(fact_4065_int__bit__induct,axiom,
    ! [P: fun(int,$o),K: int] :
      ( aa(int,$o,P,zero_zero(int))
     => ( aa(int,$o,P,aa(int,int,uminus_uminus(int),one_one(int)))
       => ( ! [K3: int] :
              ( aa(int,$o,P,K3)
             => ( ( K3 != zero_zero(int) )
               => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),times_times(int),K3),aa(num,int,numeral_numeral(int),bit0(one2)))) ) )
         => ( ! [K3: int] :
                ( aa(int,$o,P,K3)
               => ( ( K3 != aa(int,int,uminus_uminus(int),one_one(int)) )
                 => aa(int,$o,P,aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),aa(int,int,aa(int,fun(int,int),times_times(int),K3),aa(num,int,numeral_numeral(int),bit0(one2))))) ) )
           => aa(int,$o,P,K) ) ) ) ) ).

% int_bit_induct
tff(fact_4066_m1mod2k,axiom,
    ! [Nb: 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))),Nb)) = 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))),Nb)),one_one(int)) ).

% m1mod2k
tff(fact_4067_gbinomial__sum__lower__neg,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_aaj(A,fun(nat,A),A2)),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,A2),one_one(A))),M)) ) ).

% gbinomial_sum_lower_neg
tff(fact_4068_Maclaurin__zero,axiom,
    ! [A: $tType] :
      ( zero(A)
     => ! [Xb: real,Nb: nat,Diff: fun(nat,fun(A,real))] :
          ( ( Xb = zero_zero(real) )
         => ( ( Nb != zero_zero(nat) )
           => ( aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aa(fun(nat,fun(A,real)),fun(nat,real),aTP_Lamp_aak(real,fun(fun(nat,fun(A,real)),fun(nat,real)),Xb),Diff)),set_ord_lessThan(nat,Nb)) = aa(A,real,aa(nat,fun(A,real),Diff,zero_zero(nat)),zero_zero(A)) ) ) ) ) ).

% Maclaurin_zero
tff(fact_4069_Maclaurin__lemma,axiom,
    ! [H: real,F: fun(real,real),J: fun(nat,real),Nb: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),H)
     => ? [B9: real] : aa(real,real,F,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_aal(real,fun(fun(nat,real),fun(nat,real)),H),J)),set_ord_lessThan(nat,Nb))),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),Nb)),semiring_char_0_fact(real,Nb)))) ) ).

% Maclaurin_lemma
tff(fact_4070_sb__dec__lem_H,axiom,
    ! [K: nat,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$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)),A2)
     => aa(int,$o,aa(int,fun(int,$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),A2),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))),A2)) ) ).

% sb_dec_lem'
tff(fact_4071_m1mod22k,axiom,
    ! [Nb: 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))),Nb))) = 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))),Nb))),one_one(int)) ).

% m1mod22k
tff(fact_4072_gbinomial__Suc,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),aa(nat,nat,suc,K)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_aam(A,fun(nat,A),A2),set_or1337092689740270186AtMost(nat,zero_zero(nat),K))),semiring_char_0_fact(A,aa(nat,nat,suc,K))) ) ).

% gbinomial_Suc
tff(fact_4073_gbinomial__partial__sum__poly,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [M: nat,A2: 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_zy(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),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_aan(nat,fun(A,fun(A,fun(A,fun(nat,A)))),M),A2),Xb),Y)),set_ord_atMost(nat,M)) ) ).

% gbinomial_partial_sum_poly
tff(fact_4074_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( ~ vEBT_VEBT_minNull(Xb)
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),Xb)
       => ( ! [Uv2: $o] :
              ( ( Xb = vEBT_Leaf($true,(Uv2)) )
             => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf($true,(Uv2))) )
         => ( ! [Uu2: $o] :
                ( ( Xb = vEBT_Leaf((Uu2),$true) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf((Uu2),$true)) )
           => ~ ! [Uz2: product_prod(nat,nat),Va: nat,Vb2: list(vEBT_VEBT),Vc: vEBT_VEBT] :
                  ( ( Xb = vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va,Vb2,Vc) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Node(aa(product_prod(nat,nat),option(product_prod(nat,nat)),some(product_prod(nat,nat)),Uz2),Va,Vb2,Vc)) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
tff(fact_4075_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT] :
      ( vEBT_VEBT_minNull(Xb)
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),Xb)
       => ( ( ( Xb = vEBT_Leaf($false,$false) )
           => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Leaf($false,$false)) )
         => ~ ! [Uw2: nat,Ux2: list(vEBT_VEBT),Uy2: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_V6963167321098673237ll_rel),vEBT_Node(none(product_prod(nat,nat)),Uw2,Ux2,Uy2)) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
tff(fact_4076_sb__inc__lem_H,axiom,
    ! [A2: int,K: nat] :
      ( aa(int,$o,ord_less(int,A2),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),K)))
     => aa(int,$o,aa(int,fun(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),A2),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),A2),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_4077_sb__dec__lem,axiom,
    ! [K: nat,A2: int] :
      ( aa(int,$o,aa(int,fun(int,$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))),A2))
     => aa(int,$o,aa(int,fun(int,$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),A2),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))),A2)) ) ).

% sb_dec_lem
tff(fact_4078_root__polyfun,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Nb: nat,Z: A,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),one_one(nat)),Nb)
         => ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Z),Nb) = A2 )
          <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aa(A,fun(A,fun(nat,A)),aTP_Lamp_aao(nat,fun(A,fun(A,fun(nat,A))),Nb),Z),A2)),set_ord_atMost(nat,Nb)) = zero_zero(A) ) ) ) ) ).

% root_polyfun
tff(fact_4079_choose__alternating__linear__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( ( Nb != one_one(nat) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_aap(nat,fun(nat,A),Nb)),set_ord_atMost(nat,Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_linear_sum
tff(fact_4080_fact__double,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Nb: 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))),Nb)) = 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))),Nb))),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))),Nb))),semiring_char_0_fact(A,Nb)) ) ).

% fact_double
tff(fact_4081_binomial__code,axiom,
    ! [Nb: nat,K: nat] :
      aa(nat,nat,binomial(Nb),K) = $ite(
        aa(nat,$o,ord_less(nat,Nb),K),
        zero_zero(nat),
        $ite(aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),K)),aa(nat,nat,binomial(Nb),aa(nat,nat,minus_minus(nat,Nb),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,Nb),K)),one_one(nat)),Nb,one_one(nat))),semiring_char_0_fact(nat,K))) ) ).

% binomial_code
tff(fact_4082_choose__alternating__sum,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_aaq(nat,fun(nat,A),Nb)),set_ord_atMost(nat,Nb)) = zero_zero(A) ) ) ) ).

% choose_alternating_sum
tff(fact_4083_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_4084_one__mod__minus__numeral,axiom,
    ! [Nb: 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),Nb))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),Nb),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,Nb)))) ).

% one_mod_minus_numeral
tff(fact_4085_minus__one__mod__numeral,axiom,
    ! [Nb: 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),Nb)) = adjust_mod(aa(num,int,numeral_numeral(int),Nb),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,one2,Nb))) ).

% minus_one_mod_numeral
tff(fact_4086_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( aa(product_prod(int,int),$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))
     => ( ! [K3: int,L2: int] :
            ( aa(product_prod(int,int),$o,accp(product_prod(int,int),bit_and_int_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),K3),L2))
           => ( ( ~ ( member(int,K3,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int)))))
                    & member(int,L2,aa(set(int),set(int),insert(int,zero_zero(int)),aa(set(int),set(int),insert(int,aa(int,int,uminus_uminus(int),one_one(int))),bot_bot(set(int))))) )
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),divide_divide(int),K3),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),L2),aa(num,int,numeral_numeral(int),bit0(one2)))) )
             => aa(int,$o,aa(int,fun(int,$o),P,K3),L2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% and_int.pinduct
tff(fact_4087_Compl__anti__mono,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),B5)),aa(set(A),set(A),uminus_uminus(set(A)),A3)) ) ).

% Compl_anti_mono
tff(fact_4088_Compl__subset__Compl__iff,axiom,
    ! [A: $tType,A3: set(A),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),uminus_uminus(set(A)),B5))
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3) ) ).

% Compl_subset_Compl_iff
tff(fact_4089_real__add__minus__iff,axiom,
    ! [Xb: real,A2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,uminus_uminus(real),A2)) = zero_zero(real) )
    <=> ( Xb = A2 ) ) ).

% real_add_minus_iff
tff(fact_4090_max__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Nb),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ).

% max_word_max
tff(fact_4091_word__n1__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ).

% word_n1_ge
tff(fact_4092_word__order_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),A2)
        <=> ( A2 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% word_order.extremum_unique
tff(fact_4093_word__order_Oextremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ).

% word_order.extremum
tff(fact_4094_word__minus__one__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Xb)
        <=> ( Xb = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% word_minus_one_le
tff(fact_4095_max__word__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),W)
        <=> ( W = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% max_word_less_eq_iff
tff(fact_4096_subset__Compl__singleton,axiom,
    ! [A: $tType,A3: set(A),B2: A] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))))
    <=> ~ member(A,B2,A3) ) ).

% subset_Compl_singleton
tff(fact_4097_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_4098_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_4099_floor__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: 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),A2)),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),A2))),aa(num,int,numeral_numeral(int),B2)) ).

% floor_minus_divide_eq_div_numeral
tff(fact_4100_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A2: num,B2: 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),A2)),aa(num,real,numeral_numeral(real),B2)))) = aa(int,int,uminus_uminus(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),A2)),aa(num,int,numeral_numeral(int),B2))) ).

% ceiling_minus_divide_eq_div_numeral
tff(fact_4101_numeral__mod__minus__numeral,axiom,
    ! [M: num,Nb: 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),Nb))) = aa(int,int,uminus_uminus(int),adjust_mod(aa(num,int,numeral_numeral(int),Nb),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M,Nb)))) ).

% numeral_mod_minus_numeral
tff(fact_4102_minus__numeral__mod__numeral,axiom,
    ! [M: num,Nb: 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),Nb)) = adjust_mod(aa(num,int,numeral_numeral(int),Nb),aa(product_prod(int,int),int,product_snd(int,int),unique8689654367752047608divmod(int,M,Nb))) ).

% minus_numeral_mod_numeral
tff(fact_4103_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B2: 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),B2)))) = 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),B2)) ).

% floor_minus_one_divide_eq_div_numeral
tff(fact_4104_subset__Compl__self__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),uminus_uminus(set(A)),A3))
    <=> ( A3 = bot_bot(set(A)) ) ) ).

% subset_Compl_self_eq
tff(fact_4105_Compl__eq__Diff__UNIV,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = aa(set(A),set(A),minus_minus(set(A),top_top(set(A))),A3) ).

% Compl_eq_Diff_UNIV
tff(fact_4106_real__minus__mult__self__le,axiom,
    ! [U: real,Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),times_times(real),U),U))),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Xb)) ).

% real_minus_mult_self_le
tff(fact_4107_word__order_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),A2)
         => ( A2 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% word_order.extremum_uniqueI
tff(fact_4108_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_4109_word__lessThan__Suc__atMost,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: word(A)] :
          ( ( K != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( set_ord_lessThan(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),K),one_one(word(A)))) = set_ord_atMost(word(A),K) ) ) ) ).

% word_lessThan_Suc_atMost
tff(fact_4110_word__order_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( ( A2 != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
        <=> aa(word(A),$o,ord_less(word(A),A2),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ) ).

% word_order.not_eq_extremum
tff(fact_4111_word__order_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ~ aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),A2) ) ).

% word_order.extremum_strict
tff(fact_4112_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_4113_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_4114_minus__real__def,axiom,
    ! [Xb: real,Y: real] : aa(real,real,minus_minus(real,Xb),Y) = aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(real,real,uminus_uminus(real),Y)) ).

% minus_real_def
tff(fact_4115_complex__mod__minus__le__complex__mod,axiom,
    ! [Xb: complex] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),real_V7770717601297561774m_norm(complex,Xb))),real_V7770717601297561774m_norm(complex,Xb)) ).

% complex_mod_minus_le_complex_mod
tff(fact_4116_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_4117_Compl__insert,axiom,
    ! [A: $tType,Xb: A,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),insert(A,Xb),A3)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),uminus_uminus(set(A)),A3)),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) ).

% Compl_insert
tff(fact_4118_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_4119_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_4120_real__add__le__0__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,uminus_uminus(real),Xb)) ) ).

% real_add_le_0_iff
tff(fact_4121_real__0__le__add__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),Xb)),Y) ) ).

% real_0_le_add_iff
tff(fact_4122_overflow__plus__one__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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)),one_one(word(A))),P2)),P2)
        <=> ( P2 = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% overflow_plus_one_self
tff(fact_4123_plus__1__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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)))),Xb)
        <=> ( Xb = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% plus_1_less
tff(fact_4124_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_4125_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_4126_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_4127_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),$o,aa(word(A),fun(word(A),$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_4128_divide__powr__uminus,axiom,
    ! [A2: real,B2: real,C2: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),powr(real,B2,C2)) = aa(real,real,aa(real,fun(real,real),times_times(real),A2),powr(real,B2,aa(real,real,uminus_uminus(real),C2))) ).

% divide_powr_uminus
tff(fact_4129_word__neg__numeral__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_neg_numeral_alt
tff(fact_4130_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,aa(word(A),fun(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_4131_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),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),K) ) ) ) ).

% word_Suc_leq
tff(fact_4132_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,aa(word(A),fun(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_4133_realpow__square__minus__le,axiom,
    ! [U: real,Xb: real] : aa(real,$o,aa(real,fun(real,$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_4134_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_4135_Divides_Oadjust__mod__def,axiom,
    ! [L: int,R3: int] :
      adjust_mod(L,R3) = $ite(R3 = zero_zero(int),zero_zero(int),aa(int,int,minus_minus(int,L),R3)) ).

% Divides.adjust_mod_def
tff(fact_4136_Bernoulli__inequality,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => aa(real,$o,aa(real,fun(real,$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),Nb)),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)),Nb)) ) ).

% Bernoulli_inequality
tff(fact_4137_log__minus__eq__powr,axiom,
    ! [B2: real,Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),B2)
     => ( ( B2 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( aa(real,real,minus_minus(real,log(B2,Xb)),Y) = log(B2,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),powr(real,B2,aa(real,real,uminus_uminus(real),Y)))) ) ) ) ) ).

% log_minus_eq_powr
tff(fact_4138_powr__neg__numeral,axiom,
    ! [Xb: real,Nb: 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),Nb))) = 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),Nb))) ) ) ).

% powr_neg_numeral
tff(fact_4139_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_4140_minus__one__div__numeral,axiom,
    ! [Nb: 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),Nb)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,Nb))) ).

% minus_one_div_numeral
tff(fact_4141_one__div__minus__numeral,axiom,
    ! [Nb: 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),Nb))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,one2,Nb))) ).

% one_div_minus_numeral
tff(fact_4142_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_4143_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_4144_compl__le__compl__iff,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xb)),aa(A,A,uminus_uminus(A),Y))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% compl_le_compl_iff
tff(fact_4145_minus__numeral__div__numeral,axiom,
    ! [M: num,Nb: 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),Nb)) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,Nb))) ).

% minus_numeral_div_numeral
tff(fact_4146_numeral__div__minus__numeral,axiom,
    ! [M: num,Nb: 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),Nb))) = aa(int,int,uminus_uminus(int),adjust_div(unique8689654367752047608divmod(int,M,Nb))) ).

% numeral_div_minus_numeral
tff(fact_4147_uminus__set__def,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = collect(A,aa(fun(A,$o),fun(A,$o),uminus_uminus(fun(A,$o)),aTP_Lamp_av(set(A),fun(A,$o),A3))) ).

% uminus_set_def
tff(fact_4148_Collect__neg__eq,axiom,
    ! [A: $tType,P: fun(A,$o)] : collect(A,aTP_Lamp_ss(fun(A,$o),fun(A,$o),P)) = aa(set(A),set(A),uminus_uminus(set(A)),collect(A,P)) ).

% Collect_neg_eq
tff(fact_4149_Compl__eq,axiom,
    ! [A: $tType,A3: set(A)] : aa(set(A),set(A),uminus_uminus(set(A)),A3) = collect(A,aTP_Lamp_aar(set(A),fun(A,$o),A3)) ).

% Compl_eq
tff(fact_4150_compl__le__swap2,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),Xb)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Xb)),Y) ) ) ).

% compl_le_swap2
tff(fact_4151_compl__le__swap1,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),aa(A,A,uminus_uminus(A),Xb))
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,uminus_uminus(A),Y)) ) ) ).

% compl_le_swap1
tff(fact_4152_compl__mono,axiom,
    ! [A: $tType] :
      ( boolea8198339166811842893lgebra(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),Y)),aa(A,A,uminus_uminus(A),Xb)) ) ) ).

% compl_mono
tff(fact_4153_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_4154_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_4155_sin__coeff__Suc,axiom,
    ! [Nb: nat] : sin_coeff(aa(nat,nat,suc,Nb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),cos_coeff(Nb)),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb))) ).

% sin_coeff_Suc
tff(fact_4156_cos__coeff__Suc,axiom,
    ! [Nb: nat] : cos_coeff(aa(nat,nat,suc,Nb)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,uminus_uminus(real),sin_coeff(Nb))),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,Nb))) ).

% cos_coeff_Suc
tff(fact_4157_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,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% diff_shunt_var
tff(fact_4158_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_4159_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_4160_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: fun(int,fun(int,$o))] :
      ( aa(product_prod(int,int),$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))
     => ( ! [I3: int,J2: int] :
            ( aa(product_prod(int,int),$o,accp(product_prod(int,int),upto_rel),aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),I3),J2))
           => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I3),J2)
               => aa(int,$o,aa(int,fun(int,$o),P,aa(int,int,aa(int,fun(int,int),plus_plus(int),I3),one_one(int))),J2) )
             => aa(int,$o,aa(int,fun(int,$o),P,I3),J2) ) )
       => aa(int,$o,aa(int,fun(int,$o),P,A0),A1) ) ) ).

% upto.pinduct
tff(fact_4161_signed__take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2) = $ite(Nb = zero_zero(nat),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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),A2),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,Nb),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))))) ) ).

% signed_take_bit_rec
tff(fact_4162_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_4163_signed__take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,Nb)),one_one(A)) = one_one(A) ) ).

% signed_take_bit_Suc_1
tff(fact_4164_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_4165_signed__take__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),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_4166_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_4167_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_4168_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_4169_signed__take__bit__Suc__bit0,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb)),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,Nb),aa(num,int,numeral_numeral(int),K))),aa(num,int,numeral_numeral(int),bit0(one2))) ).

% signed_take_bit_Suc_bit0
tff(fact_4170_signed__take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb)),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,Nb),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_4171_signed__take__bit__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,zero_zero(nat)),A2) = aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% signed_take_bit_0
tff(fact_4172_signed__take__bit__Suc__bit1,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb)),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,Nb),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_4173_signed__take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,suc,Nb)),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,Nb),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_4174_bot__nat__def,axiom,
    bot_bot(nat) = zero_zero(nat) ).

% bot_nat_def
tff(fact_4175_signed__take__bit__add,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ).

% signed_take_bit_add
tff(fact_4176_signed__take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,minus_minus(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,minus_minus(int,K),L)) ).

% signed_take_bit_diff
tff(fact_4177_signed__take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),L))) = aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% signed_take_bit_mult
tff(fact_4178_bot__option__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ( bot_bot(option(A)) = none(A) ) ) ).

% bot_option_def
tff(fact_4179_one__integer__def,axiom,
    one_one(code_integer) = code_integer_of_int(one_one(int)) ).

% one_integer_def
tff(fact_4180_less__eq__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),code_integer_of_int(Xaa)),code_integer_of_int(Xb))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xaa),Xb) ) ).

% less_eq_integer.abs_eq
tff(fact_4181_plus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),code_integer_of_int(Xaa)),code_integer_of_int(Xb)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xaa),Xb)) ).

% plus_integer.abs_eq
tff(fact_4182_minus__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa(code_integer,code_integer,minus_minus(code_integer,code_integer_of_int(Xaa)),code_integer_of_int(Xb)) = code_integer_of_int(aa(int,int,minus_minus(int,Xaa),Xb)) ).

% minus_integer.abs_eq
tff(fact_4183_times__integer_Oabs__eq,axiom,
    ! [Xaa: int,Xb: int] : aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),times_times(code_integer),code_integer_of_int(Xaa)),code_integer_of_int(Xb)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),times_times(int),Xaa),Xb)) ).

% times_integer.abs_eq
tff(fact_4184_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_4185_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_4186_signed__take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ).

% signed_take_bit_int_less_exp
tff(fact_4187_even__signed__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4674362597316999326ke_bit(A,M),A2))
        <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% even_signed_take_bit_iff
tff(fact_4188_signed__take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(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))),Nb)),K) ) ).

% signed_take_bit_int_less_self_iff
tff(fact_4189_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),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))),Nb)) ) ).

% signed_take_bit_int_greater_eq_self_iff
tff(fact_4190_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$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))),Nb))),K) ) ).

% signed_take_bit_int_less_eq_self_iff
tff(fact_4191_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$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))),Nb))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)) ).

% signed_take_bit_int_greater_eq_minus_exp
tff(fact_4192_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,ord_less(int,K),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K))
    <=> aa(int,$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))),Nb))) ) ).

% signed_take_bit_int_greater_self_iff
tff(fact_4193_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_4194_signed__take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),K)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),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,Nb)))) ) ).

% signed_take_bit_int_less_eq
tff(fact_4195_signed__take__bit__int__eq__self,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$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))),Nb))),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))),Nb))
       => ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = K ) ) ) ).

% signed_take_bit_int_eq_self
tff(fact_4196_signed__take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$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))),Nb))),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))),Nb)) ) ) ).

% signed_take_bit_int_eq_self_iff
tff(fact_4197_signed__take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,ord_less(int,K),aa(int,int,uminus_uminus(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)))
     => aa(int,$o,aa(int,fun(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,Nb)))),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K)) ) ).

% signed_take_bit_int_greater_eq
tff(fact_4198_signed__take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,aa(nat,nat,suc,Nb)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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,Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))))) ) ).

% signed_take_bit_Suc
tff(fact_4199_sin__paired,axiom,
    ! [Xb: real] : aa(real,$o,sums(real,aTP_Lamp_aas(real,fun(nat,real),Xb)),sin(real,Xb)) ).

% sin_paired
tff(fact_4200_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_4201_cos__paired,axiom,
    ! [Xb: real] : aa(real,$o,sums(real,aTP_Lamp_aat(real,fun(nat,real),Xb)),cos(real,Xb)) ).

% cos_paired
tff(fact_4202_ln__one__minus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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_4203_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_4204_ln__one,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ( aa(A,A,ln_ln(A),one_one(A)) = zero_zero(A) ) ) ).

% ln_one
tff(fact_4205_pred__numeral__simps_I1_J,axiom,
    pred_numeral(one2) = zero_zero(nat) ).

% pred_numeral_simps(1)
tff(fact_4206_Suc__eq__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( ( aa(nat,nat,suc,Nb) = aa(num,nat,numeral_numeral(nat),K) )
    <=> ( Nb = pred_numeral(K) ) ) ).

% Suc_eq_numeral
tff(fact_4207_eq__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( ( aa(num,nat,numeral_numeral(nat),K) = aa(nat,nat,suc,Nb) )
    <=> ( pred_numeral(K) = Nb ) ) ).

% eq_numeral_Suc
tff(fact_4208_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_4209_less__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,ord_less(nat,Nb),pred_numeral(K)) ) ).

% less_Suc_numeral
tff(fact_4210_less__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,ord_less(nat,pred_numeral(K)),Nb) ) ).

% less_numeral_Suc
tff(fact_4211_le__Suc__numeral,axiom,
    ! [Nb: nat,K: num] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),pred_numeral(K)) ) ).

% le_Suc_numeral
tff(fact_4212_le__numeral__Suc,axiom,
    ! [K: num,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),pred_numeral(K)),Nb) ) ).

% le_numeral_Suc
tff(fact_4213_diff__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa(nat,nat,minus_minus(nat,aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,minus_minus(nat,Nb),pred_numeral(K)) ).

% diff_Suc_numeral
tff(fact_4214_diff__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa(nat,nat,minus_minus(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb)) = aa(nat,nat,minus_minus(nat,pred_numeral(K)),Nb) ).

% diff_numeral_Suc
tff(fact_4215_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,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ).

% ln_le_cancel_iff
tff(fact_4216_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_4217_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_4218_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_4219_max__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : aa(nat,nat,ord_max(nat,aa(nat,nat,suc,Nb)),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,Nb),pred_numeral(K))) ).

% max_Suc_numeral
tff(fact_4220_max__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : aa(nat,nat,ord_max(nat,aa(num,nat,numeral_numeral(nat),K)),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,pred_numeral(K)),Nb)) ).

% max_numeral_Suc
tff(fact_4221_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_4222_ln__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real)) ) ) ).

% ln_le_zero_iff
tff(fact_4223_ln__ge__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb) ) ) ).

% ln_ge_zero_iff
tff(fact_4224_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_4225_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_4226_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_4227_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_4228_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_4229_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_4230_cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Y))) ) ).

% cos_add
tff(fact_4231_cos__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : cos(A,aa(A,A,minus_minus(A,Xb),Y)) = 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,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),sin(A,Y))) ) ).

% cos_diff
tff(fact_4232_polar__Ex,axiom,
    ! [Xb: real,Y: real] :
    ? [R2: real,A4: real] :
      ( ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),R2),cos(real,A4)) )
      & ( Y = aa(real,real,aa(real,fun(real,real),times_times(real),R2),sin(real,A4)) ) ) ).

% polar_Ex
tff(fact_4233_sin__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sin(A,aa(A,A,minus_minus(A,Xb),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),sin(A,Y))) ) ).

% sin_diff
tff(fact_4234_sin__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sin(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sin(A,Xb)),cos(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cos(A,Xb)),sin(A,Y))) ) ).

% sin_add
tff(fact_4235_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_4236_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_4237_sin__x__le__x,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xb)),Xb) ) ).

% sin_x_le_x
tff(fact_4238_sin__le__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xb)),one_one(real)) ).

% sin_le_one
tff(fact_4239_cos__le__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Xb)),one_one(real)) ).

% cos_le_one
tff(fact_4240_log__def,axiom,
    ! [A2: real,Xb: real] : log(A2,Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Xb)),aa(real,real,ln_ln(real),A2)) ).

% log_def
tff(fact_4241_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_4242_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_4243_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_4244_ln__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,ln_ln(real),Xb)),Xb) ) ).

% ln_bound
tff(fact_4245_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_4246_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_4247_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_4248_ln__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,ln_ln(real),Xb)) ) ).

% ln_ge_zero
tff(fact_4249_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_4250_sin__x__ge__neg__x,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),Xb)),sin(real,Xb)) ) ).

% sin_x_ge_neg_x
tff(fact_4251_sin__ge__minus__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),sin(real,Xb)) ).

% sin_ge_minus_one
tff(fact_4252_cos__ge__minus__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),cos(real,Xb)) ).

% cos_ge_minus_one
tff(fact_4253_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_4254_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_4255_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_4256_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_4257_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_4258_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_4259_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_4260_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_4261_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_4262_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_4263_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_4264_ln__ge__zero__imp__ge__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$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,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb) ) ) ).

% ln_ge_zero_imp_ge_one
tff(fact_4265_ln__add__one__self__le__self,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$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_4266_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_4267_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_4268_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_4269_cos__two__neq__zero,axiom,
    cos(real,aa(num,real,numeral_numeral(real),bit0(one2))) != zero_zero(real) ).

% cos_two_neq_zero
tff(fact_4270_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_4271_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_4272_ln__le__minus__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(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_4273_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,$o,aa(real,fun(real,$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_4274_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,aa(real,fun(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_4275_ln__realpow,axiom,
    ! [Xb: real,Nb: 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),Nb)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(real,real,ln_ln(real),Xb)) ) ) ).

% ln_realpow
tff(fact_4276_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_4277_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_4278_cos__two__le__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,aa(num,real,numeral_numeral(real),bit0(one2)))),zero_zero(real)) ).

% cos_two_le_zero
tff(fact_4279_cos__is__zero,axiom,
    ? [X2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X2)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),aa(num,real,numeral_numeral(real),bit0(one2)))
      & ( cos(real,X2) = zero_zero(real) )
      & ! [Y4: real] :
          ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y4)
            & aa(real,$o,aa(real,fun(real,$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_4280_ln__one__minus__pos__upper__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,ord_less(real,Xb),one_one(real))
       => aa(real,$o,aa(real,fun(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_4281_ln__powr__bound,axiom,
    ! [Xb: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(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,A2)),A2)) ) ) ).

% ln_powr_bound
tff(fact_4282_ln__powr__bound2,axiom,
    ! [Xb: real,A2: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),Xb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),powr(real,aa(real,real,ln_ln(real),Xb),A2)),aa(real,real,aa(real,fun(real,real),times_times(real),powr(real,A2,A2)),Xb)) ) ) ).

% ln_powr_bound2
tff(fact_4283_ln__prod,axiom,
    ! [A: $tType,I5: set(A),F: fun(A,real)] :
      ( finite_finite2(A,I5)
     => ( ! [I3: A] :
            ( member(A,I3,I5)
           => aa(real,$o,ord_less(real,zero_zero(real)),aa(A,real,F,I3)) )
       => ( aa(real,real,ln_ln(real),groups7121269368397514597t_prod(A,real,F,I5)) = aa(set(A),real,groups7311177749621191930dd_sum(A,real,aTP_Lamp_aau(fun(A,real),fun(A,real),F)),I5) ) ) ) ).

% ln_prod
tff(fact_4284_log__eq__div__ln__mult__log,axiom,
    ! [A2: real,B2: real,Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),B2)
         => ( ( B2 != one_one(real) )
           => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
             => ( log(A2,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),B2)),aa(real,real,ln_ln(real),A2))),log(B2,Xb)) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
tff(fact_4285_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_4286_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_4287_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_4288_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_4289_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_4290_ln__one__plus__pos__lower__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => aa(real,$o,aa(real,fun(real,$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_4291_artanh__def,axiom,
    ! [A: $tType] :
      ( ( real_V3459762299906320749_field(A)
        & ln(A) )
     => ! [Xb: 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_4292_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_aaw(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_4293_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_aay(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_4294_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_aba(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_4295_mult__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [A2: real,Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),Y) = aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y)) ) ).

% mult_scaleR_left
tff(fact_4296_mult__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V6157519004096292374lgebra(A)
     => ! [Xb: A,A2: real,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) = aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y)) ) ).

% mult_scaleR_right
tff(fact_4297_scaleR__one,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,one_one(real)),Xb) = Xb ) ).

% scaleR_one
tff(fact_4298_scaleR__scaleR,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),times_times(real),A2),B2)),Xb) ) ).

% scaleR_scaleR
tff(fact_4299_scaleR__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [B2: A,U: real,A2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,U),B2)) )
        <=> ( ( A2 = B2 )
            | ( U = one_one(real) ) ) ) ) ).

% scaleR_eq_iff
tff(fact_4300_scaleR__power,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Xb: real,Y: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,real_V8093663219630862766scaleR(A,Xb),Y)),Nb) = aa(A,A,real_V8093663219630862766scaleR(A,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Y),Nb)) ) ).

% scaleR_power
tff(fact_4301_scaleR__minus1__left,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,uminus_uminus(real),one_one(real))),Xb) = aa(A,A,uminus_uminus(A),Xb) ) ).

% scaleR_minus1_left
tff(fact_4302_scaleR__collapse,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,one_one(real)),U)),A2)),aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) = A2 ) ).

% scaleR_collapse
tff(fact_4303_scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,W: num,A2: 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)),A2)) = 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))),A2) ) ).

% scaleR_times
tff(fact_4304_inverse__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [V: num,W: num,A2: 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)),A2)) = 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))),A2) ) ).

% inverse_scaleR_times
tff(fact_4305_fraction__scaleR__times,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [U: num,V: num,W: num,A2: 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)),A2)) = 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))),A2) ) ).

% fraction_scaleR_times
tff(fact_4306_scaleR__half__double,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: 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),A2),A2)) = A2 ) ).

% scaleR_half_double
tff(fact_4307_scaleR__right__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xb: A,Y: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ).

% scaleR_right_distrib
tff(fact_4308_scaleR__right__diff__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,Xb: A,Y: A] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(A,A,minus_minus(A,Xb),Y)) = aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ).

% scaleR_right_diff_distrib
tff(fact_4309_real__scaleR__def,axiom,
    ! [A2: real,Xb: real] : aa(real,real,real_V8093663219630862766scaleR(real,A2),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),A2),Xb) ).

% real_scaleR_def
tff(fact_4310_scaleR__sum__right,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,F: fun(B,A),A3: set(B)] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_abb(real,fun(fun(B,A),fun(B,A)),A2),F)),A3) ) ).

% scaleR_sum_right
tff(fact_4311_scaleR__right_Osum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,G: fun(B,A),A3: set(B)] : aa(A,A,real_V8093663219630862766scaleR(A,A2),aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(fun(B,A),fun(B,A),aTP_Lamp_abb(real,fun(fun(B,A),fun(B,A)),A2),G)),A3) ) ).

% scaleR_right.sum
tff(fact_4312_sums__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),A2: A,R3: real] :
          ( aa(A,$o,sums(A,X5),A2)
         => aa(A,$o,sums(A,aa(real,fun(nat,A),aTP_Lamp_abc(fun(nat,A),fun(real,fun(nat,A)),X5),R3)),aa(A,A,real_V8093663219630862766scaleR(A,R3),A2)) ) ) ).

% sums_scaleR_right
tff(fact_4313_scaleR__left_Oadd,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: real,Y: real,Xaa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)),Xaa) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,Xb),Xaa)),aa(A,A,real_V8093663219630862766scaleR(A,Y),Xaa)) ) ).

% scaleR_left.add
tff(fact_4314_scaleR__left__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)),Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) ) ).

% scaleR_left_distrib
tff(fact_4315_scaleR__left_Odiff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: real,Y: real,Xaa: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,Xb),Y)),Xaa) = aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,Xb),Xaa)),aa(A,A,real_V8093663219630862766scaleR(A,Y),Xaa)) ) ).

% scaleR_left.diff
tff(fact_4316_scaleR__left__diff__distrib,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [A2: real,B2: real,Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,minus_minus(real,A2),B2)),Xb) = aa(A,A,minus_minus(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) ) ).

% scaleR_left_diff_distrib
tff(fact_4317_scaleR__left_Osum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [G: fun(B,real),A3: set(B),Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(set(B),real,groups7311177749621191930dd_sum(B,real,G),A3)),Xb) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_abd(fun(B,real),fun(A,fun(B,A)),G),Xb)),A3) ) ).

% scaleR_left.sum
tff(fact_4318_scaleR__sum__left,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [F: fun(B,real),A3: set(B),Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,aa(set(B),real,groups7311177749621191930dd_sum(B,real,F),A3)),Xb) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aa(A,fun(B,A),aTP_Lamp_abd(fun(B,real),fun(A,fun(B,A)),F),Xb)),A3) ) ).

% scaleR_sum_left
tff(fact_4319_sums__scaleR__left,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,real),A2: real,Xb: A] :
          ( aa(real,$o,sums(real,X5),A2)
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_abe(fun(nat,real),fun(A,fun(nat,A)),X5),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)) ) ) ).

% sums_scaleR_left
tff(fact_4320_scaleR__right__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Xb)) ) ) ) ).

% scaleR_right_mono
tff(fact_4321_scaleR__right__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: real,A2: real,C2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),C2)) ) ) ) ).

% scaleR_right_mono_neg
tff(fact_4322_scaleR__le__cancel__left,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
        <=> ( ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) )
            & ( aa(real,$o,ord_less(real,C2),zero_zero(real))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% scaleR_le_cancel_left
tff(fact_4323_scaleR__le__cancel__left__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% scaleR_le_cancel_left_neg
tff(fact_4324_scaleR__le__cancel__left__pos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% scaleR_le_cancel_left_pos
tff(fact_4325_scaleR__left__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xb: A,Y: A,A2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Y)) ) ) ) ).

% scaleR_left_mono
tff(fact_4326_scaleR__left__mono__neg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [B2: A,A2: A,C2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),C2),zero_zero(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),B2)) ) ) ) ).

% scaleR_left_mono_neg
tff(fact_4327_vector__fraction__eq__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [U: real,V: real,A2: A,Xb: A] :
          ( ( aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V)),A2) = Xb )
        <=> $ite(V = zero_zero(real),Xb = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,U),A2) = aa(A,A,real_V8093663219630862766scaleR(A,V),Xb)) ) ) ).

% vector_fraction_eq_iff
tff(fact_4328_eq__vector__fraction__iff,axiom,
    ! [A: $tType] :
      ( real_V4867850818363320053vector(A)
     => ! [Xb: A,U: real,V: real,A2: A] :
          ( ( Xb = aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,aa(real,fun(real,real),divide_divide(real),U),V)),A2) )
        <=> $ite(V = zero_zero(real),Xb = zero_zero(A),aa(A,A,real_V8093663219630862766scaleR(A,V),Xb) = aa(A,A,real_V8093663219630862766scaleR(A,U),A2)) ) ) ).

% eq_vector_fraction_iff
tff(fact_4329_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E3: A,C2: A,B2: real,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E3)),D2))
        <=> aa(A,$o,aa(A,fun(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,B2),A2)),E3)),D2)) ) ) ).

% Real_Vector_Spaces.le_add_iff2
tff(fact_4330_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,E3: A,C2: A,B2: real,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),E3)),C2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,real_V8093663219630862766scaleR(A,B2),E3)),D2))
        <=> aa(A,$o,aa(A,fun(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,A2),B2)),E3)),C2)),D2) ) ) ).

% Real_Vector_Spaces.le_add_iff1
tff(fact_4331_zero__le__scaleR__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2))
        <=> ( ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(real,$o,ord_less(real,A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% zero_le_scaleR_iff
tff(fact_4332_scaleR__le__0__iff,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)),zero_zero(A))
        <=> ( ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) )
            | ( aa(real,$o,ord_less(real,A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( A2 = zero_zero(real) ) ) ) ) ).

% scaleR_le_0_iff
tff(fact_4333_scaleR__nonpos__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)) ) ) ) ).

% scaleR_nonpos_nonpos
tff(fact_4334_scaleR__nonpos__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),zero_zero(A)) ) ) ) ).

% scaleR_nonpos_nonneg
tff(fact_4335_scaleR__nonneg__nonpos,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),zero_zero(A)) ) ) ) ).

% scaleR_nonneg_nonpos
tff(fact_4336_scaleR__nonneg__nonneg,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)) ) ) ) ).

% scaleR_nonneg_nonneg
tff(fact_4337_split__scaleR__pos__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,real_V8093663219630862766scaleR(A,A2),B2)) ) ) ).

% split_scaleR_pos_le
tff(fact_4338_split__scaleR__neg__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,Xb: A] :
          ( ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A)) )
            | ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),zero_zero(real))
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb) ) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),zero_zero(A)) ) ) ).

% split_scaleR_neg_le
tff(fact_4339_scaleR__mono_H,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,C2: A,D2: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),D2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),A2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),C2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),C2)),aa(A,A,real_V8093663219630862766scaleR(A,B2),D2)) ) ) ) ) ) ).

% scaleR_mono'
tff(fact_4340_scaleR__mono,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [A2: real,B2: real,Xb: A,Y: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),B2)
             => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),aa(A,A,real_V8093663219630862766scaleR(A,B2),Y)) ) ) ) ) ) ).

% scaleR_mono
tff(fact_4341_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_4342_scaleR__left__le__one__le,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [Xb: A,A2: real] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),one_one(real))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)),Xb) ) ) ) ).

% scaleR_left_le_one_le
tff(fact_4343_sin__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_abf(A,fun(nat,A),Xb)),sin(A,Xb)) ) ).

% sin_converges
tff(fact_4344_cos__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_abg(A,fun(nat,A),Xb)),cos(A,Xb)) ) ).

% cos_converges
tff(fact_4345_sin__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_abh(A,fun(nat,A),Xb)),sin(A,Xb)) ) ).

% sin_minus_converges
tff(fact_4346_cos__minus__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_abi(A,fun(nat,A),Xb)),cos(A,Xb)) ) ).

% cos_minus_converges
tff(fact_4347_Maclaurin__minus__cos__expansion,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,Xb),zero_zero(real))
       => ? [T4: real] :
            ( aa(real,$o,ord_less(real,Xb),T4)
            & aa(real,$o,ord_less(real,T4),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_abj(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),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),T4),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),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
tff(fact_4348_Maclaurin__cos__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => ? [T4: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),T4)
            & aa(real,$o,ord_less(real,T4),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_abj(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),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),T4),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),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_cos_expansion2
tff(fact_4349_Maclaurin__sin__expansion3,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ? [T4: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),T4)
            & aa(real,$o,ord_less(real,T4),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_abk(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),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),T4),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),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_sin_expansion3
tff(fact_4350_Maclaurin__sin__expansion4,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ? [T4: real] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),T4)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),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_abk(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),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),T4),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),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ).

% Maclaurin_sin_expansion4
tff(fact_4351_sin__pi__minus,axiom,
    ! [Xb: real] : sin(real,aa(real,real,minus_minus(real,pi),Xb)) = sin(real,Xb) ).

% sin_pi_minus
tff(fact_4352_cos__pi,axiom,
    cos(real,pi) = aa(real,real,uminus_uminus(real),one_one(real)) ).

% cos_pi
tff(fact_4353_cos__periodic__pi,axiom,
    ! [Xb: real] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),pi)) = aa(real,real,uminus_uminus(real),cos(real,Xb)) ).

% cos_periodic_pi
tff(fact_4354_cos__periodic__pi2,axiom,
    ! [Xb: real] : cos(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),pi),Xb)) = aa(real,real,uminus_uminus(real),cos(real,Xb)) ).

% cos_periodic_pi2
tff(fact_4355_sin__periodic__pi,axiom,
    ! [Xb: real] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),pi)) = aa(real,real,uminus_uminus(real),sin(real,Xb)) ).

% sin_periodic_pi
tff(fact_4356_sin__periodic__pi2,axiom,
    ! [Xb: real] : sin(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),pi),Xb)) = aa(real,real,uminus_uminus(real),sin(real,Xb)) ).

% sin_periodic_pi2
tff(fact_4357_cos__pi__minus,axiom,
    ! [Xb: real] : cos(real,aa(real,real,minus_minus(real,pi),Xb)) = aa(real,real,uminus_uminus(real),cos(real,Xb)) ).

% cos_pi_minus
tff(fact_4358_cos__minus__pi,axiom,
    ! [Xb: real] : cos(real,aa(real,real,minus_minus(real,Xb),pi)) = aa(real,real,uminus_uminus(real),cos(real,Xb)) ).

% cos_minus_pi
tff(fact_4359_sin__minus__pi,axiom,
    ! [Xb: real] : sin(real,aa(real,real,minus_minus(real,Xb),pi)) = aa(real,real,uminus_uminus(real),sin(real,Xb)) ).

% sin_minus_pi
tff(fact_4360_sin__npi2,axiom,
    ! [Nb: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),Nb))) = zero_zero(real) ).

% sin_npi2
tff(fact_4361_sin__npi,axiom,
    ! [Nb: nat] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),pi)) = zero_zero(real) ).

% sin_npi
tff(fact_4362_sin__npi__int,axiom,
    ! [Nb: int] : sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),Nb))) = zero_zero(real) ).

% sin_npi_int
tff(fact_4363_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_4364_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_4365_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_4366_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_4367_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_4368_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_4369_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_4370_cos__npi2,axiom,
    ! [Nb: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(nat,real,semiring_1_of_nat(real),Nb))) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Nb) ).

% cos_npi2
tff(fact_4371_cos__npi,axiom,
    ! [Nb: nat] : cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),pi)) = aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,uminus_uminus(real),one_one(real))),Nb) ).

% cos_npi
tff(fact_4372_sin__2npi,axiom,
    ! [Nb: 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),Nb))),pi)) = zero_zero(real) ).

% sin_2npi
tff(fact_4373_cos__2npi,axiom,
    ! [Nb: 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),Nb))),pi)) = one_one(real) ).

% cos_2npi
tff(fact_4374_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_4375_sin__int__2pin,axiom,
    ! [Nb: 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),Nb))) = zero_zero(real) ).

% sin_int_2pin
tff(fact_4376_cos__int__2pin,axiom,
    ! [Nb: 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),Nb))) = one_one(real) ).

% cos_int_2pin
tff(fact_4377_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_4378_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_4379_cos__npi__int,axiom,
    ! [Nb: int] :
      cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),pi),aa(int,real,ring_1_of_int(real),Nb))) = $ite(aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Nb),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ).

% cos_npi_int
tff(fact_4380_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_4381_sin__cos__npi,axiom,
    ! [Nb: 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))),Nb)))),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))),Nb) ).

% sin_cos_npi
tff(fact_4382_pi__ge__zero,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),pi) ).

% pi_ge_zero
tff(fact_4383_cos__monotone__0__pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Xb)),cos(real,Y)) ) ) ) ).

% cos_monotone_0_pi_le
tff(fact_4384_cos__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Xb)),cos(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xb) ) ) ) ) ) ).

% cos_mono_le_eq
tff(fact_4385_cos__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),pi)
           => ( ( cos(real,Xb) = cos(real,Y) )
             => ( Xb = Y ) ) ) ) ) ) ).

% cos_inj_pi
tff(fact_4386_sin__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sin(real,Xb)) ) ) ).

% sin_ge_zero
tff(fact_4387_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_4388_pi__ge__two,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi) ).

% pi_ge_two
tff(fact_4389_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_4390_cos__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
         => ( aa(real,$o,aa(real,fun(real,$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_4391_cos__monotone__0__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,ord_less(real,Y),Xb)
       => ( aa(real,$o,aa(real,fun(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_4392_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xb)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cos(real,Y)),cos(real,Xb)) ) ) ) ).

% cos_monotone_minus_pi_0'
tff(fact_4393_sin__zero__iff__int2,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
    <=> ? [I2: int] : Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I2)),pi) ) ).

% sin_zero_iff_int2
tff(fact_4394_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_4395_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_4396_pi__half__le__two,axiom,
    aa(real,$o,aa(real,fun(real,$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_4397_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Y)
     => ( aa(real,$o,ord_less(real,Y),Xb)
       => ( aa(real,$o,aa(real,fun(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_4398_cos__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ? [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X2)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),pi)
            & ( cos(real,X2) = Y )
            & ! [Y4: real] :
                ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),pi)
                  & ( cos(real,Y4) = Y ) )
               => ( Y4 = X2 ) ) ) ) ) ).

% cos_total
tff(fact_4399_sincos__principal__value,axiom,
    ! [Xb: real] :
    ? [Y5: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),Y5)
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y5),pi)
      & ( sin(real,Y5) = sin(real,Xb) )
      & ( cos(real,Y5) = cos(real,Xb) ) ) ).

% sincos_principal_value
tff(fact_4400_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_4401_pi__half__ge__zero,axiom,
    aa(real,$o,aa(real,fun(real,$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_4402_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_4403_sin__pi__divide__n__ge__0,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => aa(real,$o,aa(real,fun(real,$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),Nb)))) ) ).

% sin_pi_divide_n_ge_0
tff(fact_4404_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_4405_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_4406_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_4407_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_4408_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_4409_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_4410_sin__monotone__2pi__le,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xb)
       => ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Y)),sin(real,Xb)) ) ) ) ).

% sin_monotone_2pi_le
tff(fact_4411_sin__mono__le__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xb)),sin(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ) ) ).

% sin_mono_le_eq
tff(fact_4412_sin__inj__pi,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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_4413_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_4414_sin__le__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$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,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Xb)),zero_zero(real)) ) ) ).

% sin_le_zero
tff(fact_4415_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_4416_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_4417_cos__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),cos(real,Xb)) ) ) ).

% cos_ge_zero
tff(fact_4418_sin__monotone__2pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$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,aa(real,fun(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_4419_sin__mono__less__eq,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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_4420_sin__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ? [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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_4421_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_4422_sin__pi__divide__n__gt__0,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => 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),Nb)))) ) ).

% sin_pi_divide_n_gt_0
tff(fact_4423_sincos__total__pi,axiom,
    ! [Y: real,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$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) )
       => ? [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
            & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),pi)
            & ( Xb = cos(real,T4) )
            & ( Y = sin(real,T4) ) ) ) ) ).

% sincos_total_pi
tff(fact_4424_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_4425_cos__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = zero_zero(real) )
    <=> ? [I2: int] :
          ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),I2)
          & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I2)),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_4426_sin__zero__iff__int,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
    <=> ? [I2: int] :
          ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),I2)
          & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),I2)),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_4427_cos__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( ( cos(real,Xb) = zero_zero(real) )
       => ? [N2: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)
            & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),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_4428_sin__zero__lemma,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( ( sin(real,Xb) = zero_zero(real) )
       => ? [N2: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N2)
            & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N2)),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_4429_cos__zero__iff,axiom,
    ! [Xb: real] :
      ( ( cos(real,Xb) = zero_zero(real) )
    <=> ( ? [N6: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N6)
            & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N6)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N6: nat] :
            ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N6)
            & ( 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),N6)),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_4430_sin__zero__iff,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
    <=> ( ? [N6: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N6)
            & ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),N6)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2)))) ) )
        | ? [N6: nat] :
            ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),N6)
            & ( 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),N6)),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_4431_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_4432_sincos__total__pi__half,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$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) )
         => ? [T4: real] :
              ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
              & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
              & ( Xb = cos(real,T4) )
              & ( Y = sin(real,T4) ) ) ) ) ) ).

% sincos_total_pi_half
tff(fact_4433_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) )
     => ? [T4: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
          & ( Xb = cos(real,T4) )
          & ( Y = sin(real,T4) ) ) ) ).

% sincos_total_2pi_le
tff(fact_4434_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) )
     => ~ ! [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
           => ( aa(real,$o,ord_less(real,T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( ( Xb = cos(real,T4) )
               => ( Y != sin(real,T4) ) ) ) ) ) ).

% sincos_total_2pi
tff(fact_4435_Maclaurin__sin__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T4: 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_abk(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),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),T4),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),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ).

% Maclaurin_sin_expansion
tff(fact_4436_Maclaurin__cos__expansion,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,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_abj(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),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),T4),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),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ).

% Maclaurin_cos_expansion
tff(fact_4437_Maclaurin__sin__expansion2,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,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_abk(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),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),T4),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),Nb))),pi)))),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ).

% Maclaurin_sin_expansion2
tff(fact_4438_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_abl(real,fun(nat,real),Xb)) ) ) ) ).

% ln_series
tff(fact_4439_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,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_4440_abs__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,abs_abs(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_abs
tff(fact_4441_abs__0,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,abs_abs(A),zero_zero(A)) = zero_zero(A) ) ) ).

% abs_0
tff(fact_4442_abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,A,abs_abs(A),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),Nb) ) ).

% abs_numeral
tff(fact_4443_abs__add__abs,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_add_abs
tff(fact_4444_abs__1,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_1
tff(fact_4445_abs__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),A2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),A2) ) ).

% abs_mult_self_eq
tff(fact_4446_abs__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_divide
tff(fact_4447_abs__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,abs_abs(A),A2) ) ).

% abs_minus
tff(fact_4448_abs__dvd__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( aa(A,$o,dvd_dvd(A,aa(A,A,abs_abs(A),M)),K)
        <=> aa(A,$o,dvd_dvd(A,M),K) ) ) ).

% abs_dvd_iff
tff(fact_4449_dvd__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [M: A,K: A] :
          ( aa(A,$o,dvd_dvd(A,M),aa(A,A,abs_abs(A),K))
        <=> aa(A,$o,dvd_dvd(A,M),K) ) ) ).

% dvd_abs_iff
tff(fact_4450_abs__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : aa(A,A,abs_abs(A),aa(nat,A,semiring_1_of_nat(A),Nb)) = aa(nat,A,semiring_1_of_nat(A),Nb) ) ).

% abs_of_nat
tff(fact_4451_abs__sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F: fun(B,A),A3: set(B)] : aa(A,A,abs_abs(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_abm(fun(B,A),fun(B,A),F)),A3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_abm(fun(B,A),fun(B,A),F)),A3) ) ).

% abs_sum_abs
tff(fact_4452_suminf__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ( suminf(A,aTP_Lamp_abn(nat,A)) = zero_zero(A) ) ) ).

% suminf_zero
tff(fact_4453_abs__le__zero__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),zero_zero(A))
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_le_zero_iff
tff(fact_4454_abs__le__self__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),A2)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% abs_le_self_iff
tff(fact_4455_abs__of__nonneg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_nonneg
tff(fact_4456_zero__less__abs__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,abs_abs(A),A2))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% zero_less_abs_iff
tff(fact_4457_abs__neg__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: num] : aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = aa(num,A,numeral_numeral(A),Nb) ) ).

% abs_neg_numeral
tff(fact_4458_abs__neg__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),aa(A,A,uminus_uminus(A),one_one(A))) = one_one(A) ) ) ).

% abs_neg_one
tff(fact_4459_abs__power__minus,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,uminus_uminus(A),A2)),Nb)) = aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ).

% abs_power_minus
tff(fact_4460_sum__abs,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F: fun(B,A),A3: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(set(B),A,groups7311177749621191930dd_sum(B,A,F),A3))),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_abm(fun(B,A),fun(B,A),F)),A3)) ) ).

% sum_abs
tff(fact_4461_zero__le__divide__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,abs_abs(A),B2)))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            | ( B2 = zero_zero(A) ) ) ) ) ).

% zero_le_divide_abs_iff
tff(fact_4462_divide__le__0__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,abs_abs(A),B2))),zero_zero(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
            | ( B2 = zero_zero(A) ) ) ) ) ).

% divide_le_0_abs_iff
tff(fact_4463_abs__of__nonpos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A))
         => ( aa(A,A,abs_abs(A),A2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% abs_of_nonpos
tff(fact_4464_artanh__minus__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( artanh(real,aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),artanh(real,Xb)) ) ) ).

% artanh_minus_real
tff(fact_4465_norm__scaleR,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: real,Xb: A] : real_V7770717601297561774m_norm(A,aa(A,A,real_V8093663219630862766scaleR(A,A2),Xb)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,abs_abs(real),A2)),real_V7770717601297561774m_norm(A,Xb)) ) ).

% norm_scaleR
tff(fact_4466_sum__abs__ge__zero,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [F: fun(B,A),A3: set(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_abm(fun(B,A),fun(B,A),F)),A3)) ) ).

% sum_abs_ge_zero
tff(fact_4467_zero__less__power__abs__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),Nb))
        <=> ( ( A2 != zero_zero(A) )
            | ( Nb = zero_zero(nat) ) ) ) ) ).

% zero_less_power_abs_iff
tff(fact_4468_power2__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),aa(num,nat,numeral_numeral(nat),bit0(one2))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% power2_abs
tff(fact_4469_abs__power2,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% abs_power2
tff(fact_4470_powser__zero,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [F: fun(nat,A)] : suminf(A,aTP_Lamp_abo(fun(nat,A),fun(nat,A),F)) = aa(nat,A,F,zero_zero(nat)) ) ).

% powser_zero
tff(fact_4471_power__even__abs__numeral,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [W: num,A2: A] :
          ( aa(nat,$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),aa(A,A,abs_abs(A),A2)),aa(num,nat,numeral_numeral(nat),W)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(num,nat,numeral_numeral(nat),W)) ) ) ) ).

% power_even_abs_numeral
tff(fact_4472_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)))) = aa(real,real,abs_abs(real),Xb) ).

% square_powr_half
tff(fact_4473_abs__one,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ( aa(A,A,abs_abs(A),one_one(A)) = one_one(A) ) ) ).

% abs_one
tff(fact_4474_abs__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% abs_eq_0_iff
tff(fact_4475_abs__le__D1,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% abs_le_D1
tff(fact_4476_abs__ge__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_self
tff(fact_4477_abs__minus__commute,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),B2)) = aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B2),A2)) ) ).

% abs_minus_commute
tff(fact_4478_dvd__if__abs__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [L: A,K: A] :
          ( ( aa(A,A,abs_abs(A),L) = aa(A,A,abs_abs(A),K) )
         => aa(A,$o,dvd_dvd(A,L),K) ) ) ).

% dvd_if_abs_eq
tff(fact_4479_abs__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ).

% abs_mult
tff(fact_4480_power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,A,abs_abs(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),Nb) ) ).

% power_abs
tff(fact_4481_abs__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [F: fun(B,A),A3: set(B)] : aa(A,A,abs_abs(A),groups7121269368397514597t_prod(B,A,F,A3)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_abp(fun(B,A),fun(B,A),F),A3) ) ).

% abs_prod
tff(fact_4482_abs__eq__iff,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [Xb: A,Y: A] :
          ( ( aa(A,A,abs_abs(A),Xb) = aa(A,A,abs_abs(A),Y) )
        <=> ( ( Xb = Y )
            | ( Xb = aa(A,A,uminus_uminus(A),Y) ) ) ) ) ).

% abs_eq_iff
tff(fact_4483_abs__ge__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_zero
tff(fact_4484_abs__of__pos,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,A,abs_abs(A),A2) = A2 ) ) ) ).

% abs_of_pos
tff(fact_4485_abs__not__less__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : ~ aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),A2)),zero_zero(A)) ) ).

% abs_not_less_zero
tff(fact_4486_abs__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))) ) ).

% abs_triangle_ineq
tff(fact_4487_abs__mult__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),A2)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),B2)),D2)
           => aa(A,$o,ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D2)) ) ) ) ).

% abs_mult_less
tff(fact_4488_abs__triangle__ineq2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),B2))) ) ).

% abs_triangle_ineq2
tff(fact_4489_abs__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),B2))) ) ).

% abs_triangle_ineq3
tff(fact_4490_abs__triangle__ineq2__sym,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B2),A2))) ) ).

% abs_triangle_ineq2_sym
tff(fact_4491_nonzero__abs__divide,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [B2: A,A2: A] :
          ( ( B2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% nonzero_abs_divide
tff(fact_4492_abs__ge__minus__self,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),aa(A,A,abs_abs(A),A2)) ) ).

% abs_ge_minus_self
tff(fact_4493_abs__le__iff,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ) ) ).

% abs_le_iff
tff(fact_4494_abs__le__D2,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2) ) ) ).

% abs_le_D2
tff(fact_4495_abs__leI,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),A2)),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),B2) ) ) ) ).

% abs_leI
tff(fact_4496_abs__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),A2)),B2)
        <=> ( aa(A,$o,ord_less(A,A2),B2)
            & aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),A2)),B2) ) ) ) ).

% abs_less_iff
tff(fact_4497_abs__sin__x__le__abs__x,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,Xb))),aa(real,real,abs_abs(real),Xb)) ).

% abs_sin_x_le_abs_x
tff(fact_4498_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,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),E2) )
         => ( Xb = zero_zero(A) ) ) ) ).

% dense_eq0_I
tff(fact_4499_abs__eq__mult,axiom,
    ! [A: $tType] :
      ( ordered_ring_abs(A)
     => ! [A2: A,B2: A] :
          ( ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
              | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),zero_zero(A)) ) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% abs_eq_mult
tff(fact_4500_abs__mult__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Xb)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),Y)),Xb) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Xb)) ) ) ) ).

% abs_mult_pos
tff(fact_4501_abs__minus__le__zero,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,abs_abs(A),A2))),zero_zero(A)) ) ).

% abs_minus_le_zero
tff(fact_4502_abs__eq__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,abs_abs(A),A2) = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),B2)
            & ( ( A2 = B2 )
              | ( A2 = aa(A,A,uminus_uminus(A),B2) ) ) ) ) ) ).

% abs_eq_iff'
tff(fact_4503_eq__abs__iff_H,axiom,
    ! [A: $tType] :
      ( linordered_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = aa(A,A,abs_abs(A),B2) )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            & ( ( B2 = A2 )
              | ( B2 = aa(A,A,uminus_uminus(A),A2) ) ) ) ) ) ).

% eq_abs_iff'
tff(fact_4504_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),aa(A,A,abs_abs(A),Xb)),Y) = aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),Y)) ) ) ) ).

% abs_div_pos
tff(fact_4505_zero__le__power__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),Nb)) ) ).

% zero_le_power_abs
tff(fact_4506_abs__if__raw,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [X3: A] :
          aa(A,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_4507_abs__of__neg,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,A2),zero_zero(A))
         => ( aa(A,A,abs_abs(A),A2) = aa(A,A,uminus_uminus(A),A2) ) ) ) ).

% abs_of_neg
tff(fact_4508_abs__if,axiom,
    ! [A: $tType] :
      ( abs_if(A)
     => ! [A2: A] :
          aa(A,A,abs_abs(A),A2) = $ite(aa(A,$o,ord_less(A,A2),zero_zero(A)),aa(A,A,uminus_uminus(A),A2),A2) ) ).

% abs_if
tff(fact_4509_abs__diff__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,A2: A,R3: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Xb),A2))),R3)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,minus_minus(A,A2),R3)),Xb)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R3)) ) ) ) ).

% abs_diff_le_iff
tff(fact_4510_abs__diff__triangle__ineq,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A,C2: A,D2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),D2)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),C2))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,B2),D2)))) ) ).

% abs_diff_triangle_ineq
tff(fact_4511_abs__triangle__ineq4,axiom,
    ! [A: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,A2),B2))),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))) ) ).

% abs_triangle_ineq4
tff(fact_4512_abs__diff__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,A2: A,R3: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Xb),A2))),R3)
        <=> ( aa(A,$o,ord_less(A,aa(A,A,minus_minus(A,A2),R3)),Xb)
            & aa(A,$o,ord_less(A,Xb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),R3)) ) ) ) ).

% abs_diff_less_iff
tff(fact_4513_lemma__interval__lt,axiom,
    ! [A2: real,Xb: real,B2: real] :
      ( aa(real,$o,ord_less(real,A2),Xb)
     => ( aa(real,$o,ord_less(real,Xb),B2)
       => ? [D3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D3)
            & ! [Y4: real] :
                ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Xb),Y4))),D3)
               => ( aa(real,$o,ord_less(real,A2),Y4)
                  & aa(real,$o,ord_less(real,Y4),B2) ) ) ) ) ) ).

% lemma_interval_lt
tff(fact_4514_abs__cos__le__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),cos(real,Xb))),one_one(real)) ).

% abs_cos_le_one
tff(fact_4515_abs__sin__le__one,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),sin(real,Xb))),one_one(real)) ).

% abs_sin_le_one
tff(fact_4516_sin__bound__lemma,axiom,
    ! [Xb: real,Y: real,U: real,V: real] :
      ( ( Xb = Y )
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),U)),V)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),U)),Y))),V) ) ) ).

% sin_bound_lemma
tff(fact_4517_suminf__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [N: set(nat),F: fun(nat,A)] :
          ( finite_finite2(nat,N)
         => ( ! [N2: nat] :
                ( ~ member(nat,N2,N)
               => ( aa(nat,A,F,N2) = zero_zero(A) ) )
           => ( suminf(A,F) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),N) ) ) ) ) ).

% suminf_finite
tff(fact_4518_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)),aa(A,A,abs_abs(A),Xb))) ) ).

% abs_add_one_gt_zero
tff(fact_4519_of__int__leD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: int,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Nb))),Xb)
         => ( ( Nb = zero_zero(int) )
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ) ).

% of_int_leD
tff(fact_4520_of__int__lessD,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: int,Xb: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),aa(int,A,ring_1_of_int(A),Nb))),Xb)
         => ( ( Nb = zero_zero(int) )
            | aa(A,$o,ord_less(A,one_one(A)),Xb) ) ) ) ).

% of_int_lessD
tff(fact_4521_lemma__interval,axiom,
    ! [A2: real,Xb: real,B2: real] :
      ( aa(real,$o,ord_less(real,A2),Xb)
     => ( aa(real,$o,ord_less(real,Xb),B2)
       => ? [D3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D3)
            & ! [Y4: real] :
                ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Xb),Y4))),D3)
               => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),Y4)
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),B2) ) ) ) ) ) ).

% lemma_interval
tff(fact_4522_round__diff__minimal,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Z: A,M: int] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Z),aa(int,A,ring_1_of_int(A),archimedean_round(A,Z))))),aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Z),aa(int,A,ring_1_of_int(A),M)))) ) ).

% round_diff_minimal
tff(fact_4523_norm__triangle__ineq3,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [A2: A,B2: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,real_V7770717601297561774m_norm(A,A2)),real_V7770717601297561774m_norm(A,B2)))),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,A2),B2))) ) ).

% norm_triangle_ineq3
tff(fact_4524_sin__zero__abs__cos__one,axiom,
    ! [Xb: real] :
      ( ( sin(real,Xb) = zero_zero(real) )
     => ( aa(real,real,abs_abs(real),cos(real,Xb)) = one_one(real) ) ) ).

% sin_zero_abs_cos_one
tff(fact_4525_sin__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X3: A] : sin(A,X3) = suminf(A,aTP_Lamp_abf(A,fun(nat,A),X3)) ) ).

% sin_def
tff(fact_4526_cos__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X3: A] : cos(A,X3) = suminf(A,aTP_Lamp_abg(A,fun(nat,A),X3)) ) ).

% cos_def
tff(fact_4527_abs__le__square__iff,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),aa(A,A,abs_abs(A),Y))
        <=> aa(A,$o,aa(A,fun(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_4528_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) )
        <=> ( aa(A,A,abs_abs(A),Xb) = one_one(A) ) ) ) ).

% abs_square_eq_1
tff(fact_4529_power__even__abs,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,abs_abs(A),A2)),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb) ) ) ) ).

% power_even_abs
tff(fact_4530_abs__sqrt__wlog,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: fun(A,fun(A,$o)),Xb: A] :
          ( ! [X2: A] :
              ( aa(A,$o,aa(A,fun(A,$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,aa(A,A,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_4531_power2__le__iff__abs__le,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),Y)
         => ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),Y) ) ) ) ).

% power2_le_iff_abs_le
tff(fact_4532_abs__square__le__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(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,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Xb)),one_one(A)) ) ) ).

% abs_square_le_1
tff(fact_4533_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,aa(A,A,abs_abs(A),Xb)),one_one(A)) ) ) ).

% abs_square_less_1
tff(fact_4534_power__mono__even,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) ) ) ) ).

% power_mono_even
tff(fact_4535_convex__sum__bound__le,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_idom(B)
     => ! [I5: set(A),Xb: fun(A,B),A2: fun(A,B),B2: B,Delta: B] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(A,B,Xb,I3)) )
         => ( ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,Xb),I5) = one_one(B) )
           => ( ! [I3: A] :
                  ( member(A,I3,I5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,B,abs_abs(B),aa(B,B,minus_minus(B,aa(A,B,A2,I3)),B2))),Delta) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(B,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_abq(fun(A,B),fun(fun(A,B),fun(A,B)),Xb),A2)),I5)),B2))),Delta) ) ) ) ) ).

% convex_sum_bound_le
tff(fact_4536_sin__cos__le1,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),sin(real,Xb)),sin(real,Y))),aa(real,real,aa(real,fun(real,real),times_times(real),cos(real,Xb)),cos(real,Y))))),one_one(real)) ).

% sin_cos_le1
tff(fact_4537_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_4538_of__int__round__abs__le,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,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_4539_round__unique_H,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Nb: int] :
          ( aa(A,$o,ord_less(A,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Xb),aa(int,A,ring_1_of_int(A),Nb)))),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) = Nb ) ) ) ).

% round_unique'
tff(fact_4540_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,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_4541_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,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,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,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_4542_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_abr(nat,real)) ).

% pi_series
tff(fact_4543_monoseq__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => topological_monoseq(real,aTP_Lamp_abs(real,fun(nat,real),Xb)) ) ).

% monoseq_arctan_series
tff(fact_4544_arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( arctan(Xb) = suminf(real,aTP_Lamp_abt(real,fun(nat,real),Xb)) ) ) ).

% arctan_series
tff(fact_4545_summable__arctan__series,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => summable(real,aTP_Lamp_abt(real,fun(nat,real),Xb)) ) ).

% summable_arctan_series
tff(fact_4546_tanh__ln__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( 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_4547_zdvd1__eq,axiom,
    ! [Xb: int] :
      ( aa(int,$o,dvd_dvd(int,Xb),one_one(int))
    <=> ( aa(int,int,abs_abs(int),Xb) = one_one(int) ) ) ).

% zdvd1_eq
tff(fact_4548_tanh__real__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),tanh(real,Xb)),tanh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ).

% tanh_real_le_iff
tff(fact_4549_summable__zero,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => summable(A,aTP_Lamp_vh(nat,A)) ) ).

% summable_zero
tff(fact_4550_summable__single,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [I: nat,F: fun(nat,A)] : summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vp(nat,fun(fun(nat,A),fun(nat,A)),I),F)) ) ).

% summable_single
tff(fact_4551_summable__iff__shift,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),K: nat] :
          ( summable(A,aa(nat,fun(nat,A),aTP_Lamp_wr(fun(nat,A),fun(nat,fun(nat,A)),F),K))
        <=> summable(A,F) ) ) ).

% summable_iff_shift
tff(fact_4552_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,abs_abs(int),Z)),one_one(int))
    <=> ( Z = zero_zero(int) ) ) ).

% zabs_less_one_iff
tff(fact_4553_zero__le__arctan__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),arctan(Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb) ) ).

% zero_le_arctan_iff
tff(fact_4554_arctan__le__zero__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arctan(Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ).

% arctan_le_zero_iff
tff(fact_4555_tanh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),tanh(real,Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ).

% tanh_real_nonpos_iff
tff(fact_4556_tanh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),tanh(real,Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb) ) ).

% tanh_real_nonneg_iff
tff(fact_4557_summable__cmult__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wa(A,fun(fun(nat,A),fun(nat,A)),C2),F))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F) ) ) ) ).

% summable_cmult_iff
tff(fact_4558_summable__divide__iff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(nat,A),C2: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_vu(fun(nat,A),fun(A,fun(nat,A)),F),C2))
        <=> ( ( C2 = zero_zero(A) )
            | summable(A,F) ) ) ) ).

% summable_divide_iff
tff(fact_4559_summable__If__finite__set,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [A3: set(nat),F: fun(nat,A)] :
          ( finite_finite2(nat,A3)
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wf(set(nat),fun(fun(nat,A),fun(nat,A)),A3),F)) ) ) ).

% summable_If_finite_set
tff(fact_4560_summable__If__finite,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [P: fun(nat,$o),F: fun(nat,A)] :
          ( finite_finite2(nat,collect(nat,P))
         => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_we(fun(nat,$o),fun(fun(nat,A),fun(nat,A)),P),F)) ) ) ).

% summable_If_finite
tff(fact_4561_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_4562_summable__rabs__cancel,axiom,
    ! [F: fun(nat,real)] :
      ( summable(real,aTP_Lamp_abu(fun(nat,real),fun(nat,real),F))
     => summable(real,F) ) ).

% summable_rabs_cancel
tff(fact_4563_summable__minus__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A)] :
          ( summable(A,aTP_Lamp_aai(fun(nat,A),fun(nat,A),F))
        <=> summable(A,F) ) ) ).

% summable_minus_iff
tff(fact_4564_summable__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A)] :
          ( summable(A,F)
         => summable(A,aTP_Lamp_aai(fun(nat,A),fun(nat,A),F)) ) ) ).

% summable_minus
tff(fact_4565_summable__comparison__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F: fun(nat,A),G: fun(nat,real)] :
          ( ? [N9: nat] :
            ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F,N2))),aa(nat,real,G,N2)) )
         => ( summable(real,G)
           => summable(A,F) ) ) ) ).

% summable_comparison_test
tff(fact_4566_summable__comparison__test_H,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [G: fun(nat,real),N: nat,F: fun(nat,A)] :
          ( summable(real,G)
         => ( ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N2)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F,N2))),aa(nat,real,G,N2)) )
           => summable(A,F) ) ) ) ).

% summable_comparison_test'
tff(fact_4567_summable__const__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [C2: A] :
          ( summable(A,aTP_Lamp_abv(A,fun(nat,A),C2))
        <=> ( C2 = zero_zero(A) ) ) ) ).

% summable_const_iff
tff(fact_4568_summable__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F),G)) ) ) ) ).

% summable_add
tff(fact_4569_summable__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F: fun(nat,A),C2: A] :
          ( summable(A,F)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_vr(fun(nat,A),fun(A,fun(nat,A)),F),C2)) ) ) ).

% summable_mult2
tff(fact_4570_summable__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F: fun(nat,A),C2: A] :
          ( summable(A,F)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_vs(fun(nat,A),fun(A,fun(nat,A)),F),C2)) ) ) ).

% summable_mult
tff(fact_4571_summable__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F)
         => ( summable(A,G)
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vt(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F),G)) ) ) ) ).

% summable_diff
tff(fact_4572_summable__Suc__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A)] :
          ( summable(A,aTP_Lamp_wb(fun(nat,A),fun(nat,A),F))
        <=> summable(A,F) ) ) ).

% summable_Suc_iff
tff(fact_4573_summable__norm__cancel,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F: fun(nat,A)] :
          ( summable(real,aTP_Lamp_abw(fun(nat,A),fun(nat,real),F))
         => summable(A,F) ) ) ).

% summable_norm_cancel
tff(fact_4574_arctan__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arctan(Xb)),arctan(Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ).

% arctan_le_iff
tff(fact_4575_arctan__monotone_H,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arctan(Xb)),arctan(Y)) ) ).

% arctan_monotone'
tff(fact_4576_summable__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),K: nat] :
          ( summable(A,F)
         => summable(A,aa(nat,fun(nat,A),aTP_Lamp_wr(fun(nat,A),fun(nat,fun(nat,A)),F),K)) ) ) ).

% summable_ignore_initial_segment
tff(fact_4577_summable__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(nat,A),C2: A] :
          ( summable(A,F)
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_vu(fun(nat,A),fun(A,fun(nat,A)),F),C2)) ) ) ).

% summable_divide
tff(fact_4578_summable__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F: fun(A,fun(nat,B))] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => summable(B,aa(A,fun(nat,B),F,I3)) )
         => summable(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_vw(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F)) ) ) ).

% summable_sum
tff(fact_4579_summable__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),R3: real] :
          ( summable(A,X5)
         => summable(A,aa(real,fun(nat,A),aTP_Lamp_abc(fun(nat,A),fun(real,fun(nat,A)),X5),R3)) ) ) ).

% summable_scaleR_right
tff(fact_4580_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_abe(fun(nat,real),fun(A,fun(nat,A)),X5),Xb)) ) ) ).

% summable_scaleR_left
tff(fact_4581_powser__insidea,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F: fun(nat,A),Xb: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_aad(fun(nat,A),fun(A,fun(nat,A)),F),Xb))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xb))
           => summable(real,aa(A,fun(nat,real),aTP_Lamp_abx(fun(nat,A),fun(A,fun(nat,real)),F),Z)) ) ) ) ).

% powser_insidea
tff(fact_4582_suminf__le,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A),G: fun(nat,A)] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,F,N2)),aa(nat,A,G,N2))
         => ( summable(A,F)
           => ( summable(A,G)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F)),suminf(A,G)) ) ) ) ) ).

% suminf_le
tff(fact_4583_abs__zmult__eq__1,axiom,
    ! [M: int,Nb: int] :
      ( ( aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),M),Nb)) = one_one(int) )
     => ( aa(int,int,abs_abs(int),M) = one_one(int) ) ) ).

% abs_zmult_eq_1
tff(fact_4584_summable__mult__D,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,F: fun(nat,A)] :
          ( summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_wa(A,fun(fun(nat,A),fun(nat,A)),C2),F))
         => ( ( C2 != zero_zero(A) )
           => summable(A,F) ) ) ) ).

% summable_mult_D
tff(fact_4585_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_4586_abs__div,axiom,
    ! [Y: int,Xb: int] :
      ( aa(int,$o,dvd_dvd(int,Y),Xb)
     => ( aa(int,int,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),aa(int,int,abs_abs(int),Xb)),aa(int,int,abs_abs(int),Y)) ) ) ).

% abs_div
tff(fact_4587_suminf__add,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [F: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F)
         => ( summable(A,G)
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,F)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vq(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F),G)) ) ) ) ) ).

% suminf_add
tff(fact_4588_suminf__mult,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F: fun(nat,A),C2: A] :
          ( summable(A,F)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_vs(fun(nat,A),fun(A,fun(nat,A)),F),C2)) = aa(A,A,aa(A,fun(A,A),times_times(A),C2),suminf(A,F)) ) ) ) ).

% suminf_mult
tff(fact_4589_suminf__mult2,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [F: fun(nat,A),C2: A] :
          ( summable(A,F)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,F)),C2) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_vr(fun(nat,A),fun(A,fun(nat,A)),F),C2)) ) ) ) ).

% suminf_mult2
tff(fact_4590_suminf__diff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),G: fun(nat,A)] :
          ( summable(A,F)
         => ( summable(A,G)
           => ( aa(A,A,minus_minus(A,suminf(A,F)),suminf(A,G)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_vt(fun(nat,A),fun(fun(nat,A),fun(nat,A)),F),G)) ) ) ) ) ).

% suminf_diff
tff(fact_4591_suminf__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(nat,A),C2: A] :
          ( summable(A,F)
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_vu(fun(nat,A),fun(A,fun(nat,A)),F),C2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),suminf(A,F)),C2) ) ) ) ).

% suminf_divide
tff(fact_4592_suminf__minus,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A)] :
          ( summable(A,F)
         => ( suminf(A,aTP_Lamp_aai(fun(nat,A),fun(nat,A),F)) = aa(A,A,uminus_uminus(A),suminf(A,F)) ) ) ) ).

% suminf_minus
tff(fact_4593_suminf__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( topolo5987344860129210374id_add(B)
        & topological_t2_space(B) )
     => ! [I5: set(A),F: fun(A,fun(nat,B))] :
          ( ! [I3: A] :
              ( member(A,I3,I5)
             => summable(B,aa(A,fun(nat,B),F,I3)) )
         => ( suminf(B,aa(fun(A,fun(nat,B)),fun(nat,B),aTP_Lamp_vw(set(A),fun(fun(A,fun(nat,B)),fun(nat,B)),I5),F)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,aTP_Lamp_aby(fun(A,fun(nat,B)),fun(A,B),F)),I5) ) ) ) ).

% suminf_sum
tff(fact_4594_suminf__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A),R3: real] :
          ( summable(A,X5)
         => ( aa(A,A,real_V8093663219630862766scaleR(A,R3),suminf(A,X5)) = suminf(A,aa(real,fun(nat,A),aTP_Lamp_abc(fun(nat,A),fun(real,fun(nat,A)),X5),R3)) ) ) ) ).

% suminf_scaleR_right
tff(fact_4595_summable__Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_abz(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_abz(fun(nat,A),fun(nat,real),B2))
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_acb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ).

% summable_Cauchy_product
tff(fact_4596_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set(int)] :
      ( ~ finite_finite2(int,S2)
    <=> ! [M7: int] :
        ? [N6: int] :
          ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),M7),aa(int,int,abs_abs(int),N6))
          & member(int,N6,S2) ) ) ).

% infinite_int_iff_unbounded_le
tff(fact_4597_tanh__real__lt__1,axiom,
    ! [Xb: real] : aa(real,$o,ord_less(real,tanh(real,Xb)),one_one(real)) ).

% tanh_real_lt_1
tff(fact_4598_suminf__nonneg,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A)] :
          ( summable(A,F)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F,N2))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),suminf(A,F)) ) ) ) ).

% suminf_nonneg
tff(fact_4599_suminf__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A)] :
          ( summable(A,F)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F,N2))
           => ( ( suminf(A,F) = zero_zero(A) )
            <=> ! [N6: nat] : aa(nat,A,F,N6) = zero_zero(A) ) ) ) ) ).

% suminf_eq_zero_iff
tff(fact_4600_suminf__pos,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A)] :
          ( summable(A,F)
         => ( ! [N2: nat] : aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,F,N2))
           => aa(A,$o,ord_less(A,zero_zero(A)),suminf(A,F)) ) ) ) ).

% suminf_pos
tff(fact_4601_summable__0__powser,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F: fun(nat,A)] : summable(A,aTP_Lamp_tv(fun(nat,A),fun(nat,A),F)) ) ).

% summable_0_powser
tff(fact_4602_summable__zero__power_H,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [F: fun(nat,A)] : summable(A,aTP_Lamp_acc(fun(nat,A),fun(nat,A),F)) ) ).

% summable_zero_power'
tff(fact_4603_powser__split__head_I3_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_acd(fun(nat,A),fun(A,fun(nat,A)),F),Z))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_ace(fun(nat,A),fun(A,fun(nat,A)),F),Z)) ) ) ).

% powser_split_head(3)
tff(fact_4604_summable__powser__split__head,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_acf(fun(nat,A),fun(A,fun(nat,A)),F),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_aad(fun(nat,A),fun(A,fun(nat,A)),F),Z)) ) ) ).

% summable_powser_split_head
tff(fact_4605_summable__powser__ignore__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F: fun(nat,A),M: nat,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aa(nat,fun(A,fun(nat,A)),aTP_Lamp_acg(fun(nat,A),fun(nat,fun(A,fun(nat,A))),F),M),Z))
        <=> summable(A,aa(A,fun(nat,A),aTP_Lamp_aad(fun(nat,A),fun(A,fun(nat,A)),F),Z)) ) ) ).

% summable_powser_ignore_initial_segment
tff(fact_4606_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero(int) )
     => aa(int,$o,ord_less(int,aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),K),L))),aa(int,int,abs_abs(int),L)) ) ).

% abs_mod_less
tff(fact_4607_nat__abs__mult__distrib,axiom,
    ! [W: int,Z: int] : nat2(aa(int,int,abs_abs(int),aa(int,int,aa(int,fun(int,int),times_times(int),W),Z))) = aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),nat2(aa(int,int,abs_abs(int),W))),nat2(aa(int,int,abs_abs(int),Z))) ).

% nat_abs_mult_distrib
tff(fact_4608_dvd__imp__le__int,axiom,
    ! [I: int,D2: int] :
      ( ( I != zero_zero(int) )
     => ( aa(int,$o,dvd_dvd(int,D2),I)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),D2)),aa(int,int,abs_abs(int),I)) ) ) ).

% dvd_imp_le_int
tff(fact_4609_summable__norm__comparison__test,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),G: fun(nat,real)] :
          ( ? [N9: nat] :
            ! [N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N2)
             => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F,N2))),aa(nat,real,G,N2)) )
         => ( summable(real,G)
           => summable(real,aTP_Lamp_ach(fun(nat,A),fun(nat,real),F)) ) ) ) ).

% summable_norm_comparison_test
tff(fact_4610_summable__rabs__comparison__test,axiom,
    ! [F: fun(nat,real),G: fun(nat,real)] :
      ( ? [N9: nat] :
        ! [N2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N9),N2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(nat,real,F,N2))),aa(nat,real,G,N2)) )
     => ( summable(real,G)
       => summable(real,aTP_Lamp_abu(fun(nat,real),fun(nat,real),F)) ) ) ).

% summable_rabs_comparison_test
tff(fact_4611_summable__rabs,axiom,
    ! [F: fun(nat,real)] :
      ( summable(real,aTP_Lamp_abu(fun(nat,real),fun(nat,real),F))
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),suminf(real,F))),suminf(real,aTP_Lamp_abu(fun(nat,real),fun(nat,real),F))) ) ).

% summable_rabs
tff(fact_4612_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_abe(fun(nat,real),fun(A,fun(nat,A)),X5),Xb)) ) ) ) ).

% suminf_scaleR_left
tff(fact_4613_suminf__pos__iff,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A)] :
          ( summable(A,F)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F,N2))
           => ( aa(A,$o,ord_less(A,zero_zero(A)),suminf(A,F))
            <=> ? [I2: nat] : aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,F,I2)) ) ) ) ) ).

% suminf_pos_iff
tff(fact_4614_suminf__pos2,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A),I: nat] :
          ( summable(A,F)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F,N2))
           => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,F,I))
             => aa(A,$o,ord_less(A,zero_zero(A)),suminf(A,F)) ) ) ) ) ).

% suminf_pos2
tff(fact_4615_tanh__real__gt__neg1,axiom,
    ! [Xb: real] : aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),one_one(real))),tanh(real,Xb)) ).

% tanh_real_gt_neg1
tff(fact_4616_suminf__le__const,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A),Xb: A] :
          ( summable(A,F)
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_ord_lessThan(nat,N2))),Xb)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),suminf(A,F)),Xb) ) ) ) ).

% suminf_le_const
tff(fact_4617_powser__inside,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F: fun(nat,A),Xb: A,Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_acd(fun(nat,A),fun(A,fun(nat,A)),F),Xb))
         => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),real_V7770717601297561774m_norm(A,Xb))
           => summable(A,aa(A,fun(nat,A),aTP_Lamp_acd(fun(nat,A),fun(A,fun(nat,A)),F),Z)) ) ) ) ).

% powser_inside
tff(fact_4618_suminf__split__head,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A)] :
          ( summable(A,F)
         => ( suminf(A,aTP_Lamp_wb(fun(nat,A),fun(nat,A),F)) = aa(A,A,minus_minus(A,suminf(A,F)),aa(nat,A,F,zero_zero(nat))) ) ) ) ).

% suminf_split_head
tff(fact_4619_summableI__nonneg__bounded,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A),Xb: A] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F,N2))
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_ord_lessThan(nat,N2))),Xb)
           => summable(A,F) ) ) ) ).

% summableI_nonneg_bounded
tff(fact_4620_bounded__imp__summable,axiom,
    ! [A: $tType] :
      ( ( condit6923001295902523014norder(A)
        & linord2810124833399127020strict(A)
        & topolo1944317154257567458pology(A) )
     => ! [A2: fun(nat,A),B5: A] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,A2,N2))
         => ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,A2),set_ord_atMost(nat,N2))),B5)
           => summable(A,A2) ) ) ) ).

% bounded_imp_summable
tff(fact_4621_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_4622_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_4623_nat__abs__triangle__ineq,axiom,
    ! [K: int,L: int] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),nat2(aa(int,int,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(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L)))) ).

% nat_abs_triangle_ineq
tff(fact_4624_zdvd__mult__cancel1,axiom,
    ! [M: int,Nb: int] :
      ( ( M != zero_zero(int) )
     => ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),M),Nb)),M)
      <=> ( aa(int,int,abs_abs(int),Nb) = one_one(int) ) ) ) ).

% zdvd_mult_cancel1
tff(fact_4625_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] : aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) = aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L)))) ).

% div_abs_eq_div_nat
tff(fact_4626_summable__norm,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F: fun(nat,A)] :
          ( summable(real,aTP_Lamp_abw(fun(nat,A),fun(nat,real),F))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,suminf(A,F))),suminf(real,aTP_Lamp_abw(fun(nat,A),fun(nat,real),F))) ) ) ).

% summable_norm
tff(fact_4627_summable__norm__sin,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(real,aTP_Lamp_aci(A,fun(nat,real),Xb)) ) ).

% summable_norm_sin
tff(fact_4628_summable__norm__cos,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(real,aTP_Lamp_acj(A,fun(nat,real),Xb)) ) ).

% summable_norm_cos
tff(fact_4629_sum__le__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere6911136660526730532id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A),I5: set(nat)] :
          ( summable(A,F)
         => ( finite_finite2(nat,I5)
           => ( ! [N2: nat] :
                  ( member(nat,N2,aa(set(nat),set(nat),uminus_uminus(set(nat)),I5))
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F,N2)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),I5)),suminf(A,F)) ) ) ) ) ).

% sum_le_suminf
tff(fact_4630_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,abs_abs(int),K)),L))
    <=> aa(int,$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_4631_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),aa(int,int,abs_abs(int),L)))
    <=> aa(int,$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_4632_suminf__split__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),K: nat] :
          ( summable(A,F)
         => ( suminf(A,F) = aa(A,A,aa(A,fun(A,A),plus_plus(A),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_wr(fun(nat,A),fun(nat,fun(nat,A)),F),K))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_ord_lessThan(nat,K))) ) ) ) ).

% suminf_split_initial_segment
tff(fact_4633_suminf__minus__initial__segment,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [F: fun(nat,A),K: nat] :
          ( summable(A,F)
         => ( suminf(A,aa(nat,fun(nat,A),aTP_Lamp_wr(fun(nat,A),fun(nat,fun(nat,A)),F),K)) = aa(A,A,minus_minus(A,suminf(A,F)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_ord_lessThan(nat,K))) ) ) ) ).

% suminf_minus_initial_segment
tff(fact_4634_nat__abs__int__diff,axiom,
    ! [A2: nat,B2: nat] :
      nat2(aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),A2)),aa(nat,int,semiring_1_of_nat(int),B2)))) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2),aa(nat,nat,minus_minus(nat,B2),A2),aa(nat,nat,minus_minus(nat,A2),B2)) ).

% nat_abs_int_diff
tff(fact_4635_sum__less__suminf,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A),Nb: nat] :
          ( summable(A,F)
         => ( ! [M2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M2)
               => aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,F,M2)) )
           => aa(A,$o,ord_less(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_ord_lessThan(nat,Nb))),suminf(A,F)) ) ) ) ).

% sum_less_suminf
tff(fact_4636_powser__split__head_I1_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_acd(fun(nat,A),fun(A,fun(nat,A)),F),Z))
         => ( suminf(A,aa(A,fun(nat,A),aTP_Lamp_acd(fun(nat,A),fun(A,fun(nat,A)),F),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,F,zero_zero(nat))),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ace(fun(nat,A),fun(A,fun(nat,A)),F),Z))),Z)) ) ) ) ).

% powser_split_head(1)
tff(fact_4637_powser__split__head_I2_J,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V8999393235501362500lgebra(A) )
     => ! [F: fun(nat,A),Z: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_acd(fun(nat,A),fun(A,fun(nat,A)),F),Z))
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,aa(A,fun(nat,A),aTP_Lamp_ace(fun(nat,A),fun(A,fun(nat,A)),F),Z))),Z) = aa(A,A,minus_minus(A,suminf(A,aa(A,fun(nat,A),aTP_Lamp_acd(fun(nat,A),fun(A,fun(nat,A)),F),Z))),aa(nat,A,F,zero_zero(nat))) ) ) ) ).

% powser_split_head(2)
tff(fact_4638_monoseq__realpow,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$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_4639_suminf__exist__split,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R3: real,F: fun(nat,A)] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
         => ( summable(A,F)
           => ? [N10: nat] :
              ! [N8: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N10),N8)
               => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,suminf(A,aa(nat,fun(nat,A),aTP_Lamp_wr(fun(nat,A),fun(nat,fun(nat,A)),F),N8)))),R3) ) ) ) ) ).

% suminf_exist_split
tff(fact_4640_summable__partial__sum__bound,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F: fun(nat,A),E3: real] :
          ( summable(A,F)
         => ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
           => ~ ! [N10: nat] :
                  ~ ! [M3: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N10),M3)
                     => ! [N8: nat] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_or1337092689740270186AtMost(nat,M3,N8)))),E3) ) ) ) ) ).

% summable_partial_sum_bound
tff(fact_4641_summable__power__series,axiom,
    ! [F: fun(nat,real),Z: real] :
      ( ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(nat,real,F,I3)),one_one(real))
     => ( ! [I3: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F,I3))
       => ( aa(real,$o,aa(real,fun(real,$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_ack(fun(nat,real),fun(real,fun(nat,real)),F),Z)) ) ) ) ) ).

% summable_power_series
tff(fact_4642_Abel__lemma,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [R3: real,R0: real,A2: fun(nat,A),M8: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),R3)
         => ( aa(real,$o,ord_less(real,R3),R0)
           => ( ! [N2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),times_times(real),real_V7770717601297561774m_norm(A,aa(nat,A,A2,N2))),aa(nat,real,aa(real,fun(nat,real),power_power(real),R0),N2))),M8)
             => summable(real,aa(fun(nat,A),fun(nat,real),aTP_Lamp_acl(real,fun(fun(nat,A),fun(nat,real)),R3),A2)) ) ) ) ) ).

% Abel_lemma
tff(fact_4643_nat__intermed__int__val,axiom,
    ! [M: nat,Nb: nat,F: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),I3)
            & aa(nat,$o,ord_less(nat,I3),Nb) )
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F,aa(nat,nat,suc,I3))),aa(nat,int,F,I3)))),one_one(int)) )
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F,M)),K)
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F,Nb))
           => ? [I3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),I3)
                & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
                & ( aa(nat,int,F,I3) = K ) ) ) ) ) ) ).

% nat_intermed_int_val
tff(fact_4644_Cauchy__product,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_abz(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_abz(fun(nat,A),fun(nat,real),B2))
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A2)),suminf(A,B2)) = suminf(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_acb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)) ) ) ) ) ).

% Cauchy_product
tff(fact_4645_incr__lemma,axiom,
    ! [D2: int,Z: int,Xb: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D2)
     => aa(int,$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),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,Xb),Z))),one_one(int))),D2))) ) ).

% incr_lemma
tff(fact_4646_decr__lemma,axiom,
    ! [D2: int,Xb: int,Z: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),D2)
     => aa(int,$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),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,Xb),Z))),one_one(int))),D2))),Z) ) ).

% decr_lemma
tff(fact_4647_summable__ratio__test,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [C2: real,N: nat,F: fun(nat,A)] :
          ( aa(real,$o,ord_less(real,C2),one_one(real))
         => ( ! [N2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N),N2)
               => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(nat,A,F,aa(nat,nat,suc,N2)))),aa(real,real,aa(real,fun(real,real),times_times(real),C2),real_V7770717601297561774m_norm(A,aa(nat,A,F,N2)))) )
           => summable(A,F) ) ) ) ).

% summable_ratio_test
tff(fact_4648_sum__less__suminf2,axiom,
    ! [A: $tType] :
      ( ( ordere8940638589300402666id_add(A)
        & topolo1944317154257567458pology(A) )
     => ! [F: fun(nat,A),Nb: nat,I: nat] :
          ( summable(A,F)
         => ( ! [M2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M2)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(nat,A,F,M2)) )
           => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),I)
             => ( aa(A,$o,ord_less(A,zero_zero(A)),aa(nat,A,F,I))
               => aa(A,$o,ord_less(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_ord_lessThan(nat,Nb))),suminf(A,F)) ) ) ) ) ) ).

% sum_less_suminf2
tff(fact_4649_arctan__ubound,axiom,
    ! [Y: real] : aa(real,$o,ord_less(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_4650_arctan__one,axiom,
    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_4651_nat__ivt__aux,axiom,
    ! [Nb: nat,F: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( aa(nat,$o,ord_less(nat,I3),Nb)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F,aa(nat,nat,suc,I3))),aa(nat,int,F,I3)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F,Nb))
         => ? [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
              & ( aa(nat,int,F,I3) = K ) ) ) ) ) ).

% nat_ivt_aux
tff(fact_4652_Cauchy__product__sums,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [A2: fun(nat,A),B2: fun(nat,A)] :
          ( summable(real,aTP_Lamp_abz(fun(nat,A),fun(nat,real),A2))
         => ( summable(real,aTP_Lamp_abz(fun(nat,A),fun(nat,real),B2))
           => aa(A,$o,sums(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_acb(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)),aa(A,A,aa(A,fun(A,A),times_times(A),suminf(A,A2)),suminf(A,B2))) ) ) ) ).

% Cauchy_product_sums
tff(fact_4653_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))))),arctan(Y)) ).

% arctan_lbound
tff(fact_4654_arctan__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),arctan(Y))
      & aa(real,$o,ord_less(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_4655_nat0__intermed__int__val,axiom,
    ! [Nb: nat,F: fun(nat,int),K: int] :
      ( ! [I3: nat] :
          ( aa(nat,$o,ord_less(nat,I3),Nb)
         => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,abs_abs(int),aa(int,int,minus_minus(int,aa(nat,int,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),I3),one_one(nat)))),aa(nat,int,F,I3)))),one_one(int)) )
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,F,zero_zero(nat))),K)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(nat,int,F,Nb))
         => ? [I3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),Nb)
              & ( aa(nat,int,F,I3) = K ) ) ) ) ) ).

% nat0_intermed_int_val
tff(fact_4656_arctan__add,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,real,aa(real,fun(real,real),plus_plus(real),arctan(Xb)),arctan(Y)) = 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_4657_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)))),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))),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_4658_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)))),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))))))),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_4659_sum__pos__lt__pair,axiom,
    ! [F: fun(nat,real),K: nat] :
      ( summable(real,F)
     => ( ! [D3: nat] : aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D3)))),aa(nat,real,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat)))),D3)),one_one(nat))))))
       => aa(real,$o,ord_less(real,aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,F),set_ord_lessThan(nat,K))),suminf(real,F)) ) ) ).

% sum_pos_lt_pair
tff(fact_4660_arctan__double,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,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))),arctan(Xb)) = 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_4661_diffs__equiv,axiom,
    ! [A: $tType] :
      ( ( real_V822414075346904944vector(A)
        & ring_1(A) )
     => ! [C2: fun(nat,A),Xb: A] :
          ( summable(A,aa(A,fun(nat,A),aTP_Lamp_acm(fun(nat,A),fun(A,fun(nat,A)),C2),Xb))
         => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_acn(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)),suminf(A,aa(A,fun(nat,A),aTP_Lamp_acm(fun(nat,A),fun(A,fun(nat,A)),C2),Xb))) ) ) ).

% diffs_equiv
tff(fact_4662_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) )
           => ( 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))),tan(A,Xb))),aa(A,A,minus_minus(A,one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),tan(A,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2))))) ) ) ) ) ).

% tan_double
tff(fact_4663_arcosh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: 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)),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_4664_summable__complex__of__real,axiom,
    ! [F: fun(nat,real)] :
      ( summable(complex,aTP_Lamp_aco(fun(nat,real),fun(nat,complex),F))
    <=> summable(real,F) ) ).

% summable_complex_of_real
tff(fact_4665_tan__periodic__pi,axiom,
    ! [Xb: real] : tan(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),pi)) = tan(real,Xb) ).

% tan_periodic_pi
tff(fact_4666_of__real__eq__1__iff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real] :
          ( ( real_Vector_of_real(A,Xb) = one_one(A) )
        <=> ( Xb = one_one(real) ) ) ) ).

% of_real_eq_1_iff
tff(fact_4667_of__real__1,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ( real_Vector_of_real(A,one_one(real)) = one_one(A) ) ) ).

% of_real_1
tff(fact_4668_of__real__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : real_Vector_of_real(A,aa(num,real,numeral_numeral(real),W)) = aa(num,A,numeral_numeral(A),W) ) ).

% of_real_numeral
tff(fact_4669_of__real__mult,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Y: real] : real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)) = aa(A,A,aa(A,fun(A,A),times_times(A),real_Vector_of_real(A,Xb)),real_Vector_of_real(A,Y)) ) ).

% of_real_mult
tff(fact_4670_of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V5047593784448816457lgebra(A)
     => ! [Xb: real,Y: real] : 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),real_Vector_of_real(A,Xb)),real_Vector_of_real(A,Y)) ) ).

% of_real_divide
tff(fact_4671_of__real__add,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Y: real] : real_Vector_of_real(A,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,Xb)),real_Vector_of_real(A,Y)) ) ).

% of_real_add
tff(fact_4672_of__real__power,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Nb: nat] : real_Vector_of_real(A,aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),real_Vector_of_real(A,Xb)),Nb) ) ).

% of_real_power
tff(fact_4673_of__real__diff,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Xb: real,Y: real] : real_Vector_of_real(A,aa(real,real,minus_minus(real,Xb),Y)) = aa(A,A,minus_minus(A,real_Vector_of_real(A,Xb)),real_Vector_of_real(A,Y)) ) ).

% of_real_diff
tff(fact_4674_of__real__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [F: fun(B,real),S3: set(B)] : real_Vector_of_real(A,aa(set(B),real,groups7311177749621191930dd_sum(B,real,F),S3)) = aa(set(B),A,groups7311177749621191930dd_sum(B,A,aTP_Lamp_acp(fun(B,real),fun(B,A),F)),S3) ) ).

% of_real_sum
tff(fact_4675_of__real__prod,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2191834092415804123ebra_1(A) )
     => ! [F: fun(B,real),S3: set(B)] : real_Vector_of_real(A,groups7121269368397514597t_prod(B,real,F,S3)) = groups7121269368397514597t_prod(B,A,aTP_Lamp_acq(fun(B,real),fun(B,A),F),S3) ) ).

% of_real_prod
tff(fact_4676_tan__npi,axiom,
    ! [Nb: nat] : tan(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),pi)) = zero_zero(real) ).

% tan_npi
tff(fact_4677_tan__periodic__n,axiom,
    ! [Xb: real,Nb: num] : 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),Nb)),pi))) = tan(real,Xb) ).

% tan_periodic_n
tff(fact_4678_tan__periodic__nat,axiom,
    ! [Xb: real,Nb: nat] : 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(nat,real,semiring_1_of_nat(real),Nb)),pi))) = tan(real,Xb) ).

% tan_periodic_nat
tff(fact_4679_tan__periodic__int,axiom,
    ! [Xb: real,I: int] : 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(int,real,ring_1_of_int(real),I)),pi))) = tan(real,Xb) ).

% tan_periodic_int
tff(fact_4680_of__real__neg__numeral,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [W: num] : 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_4681_cos__of__real__pi,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( cos(A,real_Vector_of_real(A,pi)) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ).

% cos_of_real_pi
tff(fact_4682_tan__periodic,axiom,
    ! [Xb: 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))) = tan(real,Xb) ).

% tan_periodic
tff(fact_4683_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),real_Vector_of_real(A,Xb)),one_one(A))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),one_one(real))) ) ).

% norm_of_real_add1
tff(fact_4684_norm__of__real__addn,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: real,B2: num] : real_V7770717601297561774m_norm(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),real_Vector_of_real(A,Xb)),aa(num,A,numeral_numeral(A),B2))) = aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),aa(num,real,numeral_numeral(real),B2))) ) ).

% norm_of_real_addn
tff(fact_4685_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),real_Vector_of_real(A,pi)),aa(num,A,numeral_numeral(A),bit0(one2)))) = zero_zero(A) ) ) ).

% cos_of_real_pi_half
tff(fact_4686_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),real_Vector_of_real(A,pi)),aa(num,A,numeral_numeral(A),bit0(one2)))) = one_one(A) ) ) ).

% sin_of_real_pi_half
tff(fact_4687_diffs__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [F: fun(nat,real),X3: nat] : diffs(A,aTP_Lamp_acr(fun(nat,real),fun(nat,A),F),X3) = real_Vector_of_real(A,diffs(real,F,X3)) ) ).

% diffs_of_real
tff(fact_4688_of__real__def,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R3: real] : real_Vector_of_real(A,R3) = aa(A,A,real_V8093663219630862766scaleR(A,R3),one_one(A)) ) ).

% of_real_def
tff(fact_4689_scaleR__conv__of__real,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [R3: real,Xb: A] : aa(A,A,real_V8093663219630862766scaleR(A,R3),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),real_Vector_of_real(A,R3)),Xb) ) ).

% scaleR_conv_of_real
tff(fact_4690_diffs__minus,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X3: nat] : diffs(A,aTP_Lamp_acs(fun(nat,A),fun(nat,A),C2),X3) = aa(A,A,uminus_uminus(A),diffs(A,C2,X3)) ) ).

% diffs_minus
tff(fact_4691_summable__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real)] :
          ( summable(real,X5)
         => summable(A,aTP_Lamp_act(fun(nat,real),fun(nat,A),X5)) ) ) ).

% summable_of_real
tff(fact_4692_sums__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real),A2: real] :
          ( aa(real,$o,sums(real,X5),A2)
         => aa(A,$o,sums(A,aTP_Lamp_act(fun(nat,real),fun(nat,A),X5)),real_Vector_of_real(A,A2)) ) ) ).

% sums_of_real
tff(fact_4693_sums__of__real__iff,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [F: fun(nat,real),C2: real] :
          ( aa(A,$o,sums(A,aTP_Lamp_acu(fun(nat,real),fun(nat,A),F)),real_Vector_of_real(A,C2))
        <=> aa(real,$o,sums(real,F),C2) ) ) ).

% sums_of_real_iff
tff(fact_4694_nonzero__of__real__divide,axiom,
    ! [A: $tType] :
      ( real_V7773925162809079976_field(A)
     => ! [Y: real,Xb: real] :
          ( ( Y != zero_zero(real) )
         => ( 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),real_Vector_of_real(A,Xb)),real_Vector_of_real(A,Y)) ) ) ) ).

% nonzero_of_real_divide
tff(fact_4695_suminf__of__real,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [X5: fun(nat,real)] :
          ( summable(real,X5)
         => ( real_Vector_of_real(A,suminf(real,X5)) = suminf(A,aTP_Lamp_act(fun(nat,real),fun(nat,A),X5)) ) ) ) ).

% suminf_of_real
tff(fact_4696_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),real_Vector_of_real(A,real_V7770717601297561774m_norm(A,Xb))),one_one(A)))) ) ).

% norm_less_p1
tff(fact_4697_tan__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X3: 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_4698_diffs__def,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [C2: fun(nat,A),X3: nat] : diffs(A,C2,X3) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,suc,X3))),aa(nat,A,C2,aa(nat,nat,suc,X3))) ) ).

% diffs_def
tff(fact_4699_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_acv(fun(nat,A),fun(A,fun(nat,A)),C2),X2))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_acw(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)) ) ) ).

% termdiff_converges_all
tff(fact_4700_norm__of__real__diff,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [B2: real,A2: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,real_Vector_of_real(A,B2)),real_Vector_of_real(A,A2)))),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,B2),A2))) ) ).

% norm_of_real_diff
tff(fact_4701_tan__45,axiom,
    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_4702_cos__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [M: int,Xb: real] : cos(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),M)),real_Vector_of_real(A,Xb))) = real_Vector_of_real(A,cos(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),M)),Xb))) ) ).

% cos_int_times_real
tff(fact_4703_sin__int__times__real,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [M: int,Xb: real] : sin(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(int,A,ring_1_of_int(A),M)),real_Vector_of_real(A,Xb))) = real_Vector_of_real(A,sin(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(int,real,ring_1_of_int(real),M)),Xb))) ) ).

% sin_int_times_real
tff(fact_4704_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),tan(real,X2)) ) ) ).

% lemma_tan_total
tff(fact_4705_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)),tan(real,Xb)) ) ) ).

% tan_gt_zero
tff(fact_4706_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))))
      & ( 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))))
            & ( tan(real,Y4) = Y ) )
         => ( Y4 = X2 ) ) ) ).

% tan_total
tff(fact_4707_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,tan(real,Y)),tan(real,Xb)) ) ) ) ).

% tan_monotone
tff(fact_4708_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,tan(real,Y)),tan(real,Xb)) ) ) ) ) ) ).

% tan_monotone'
tff(fact_4709_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,$o,ord_less(real,tan(real,Xb)),tan(real,Y))
            <=> aa(real,$o,ord_less(real,Xb),Y) ) ) ) ) ) ).

% tan_mono_lt_eq
tff(fact_4710_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))))
      & ( tan(real,X2) = Y ) ) ).

% lemma_tan_total1
tff(fact_4711_tan__minus__45,axiom,
    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_4712_tan__inverse,axiom,
    ! [Y: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),one_one(real)),tan(real,Y)) = 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_4713_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),tan(A,Xb)),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_4714_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),real_Vector_of_real(A,pi)),aa(num,A,numeral_numeral(A),bit0(one2)))),Xb)) ) ).

% sin_cos_eq
tff(fact_4715_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),real_Vector_of_real(A,pi)),aa(num,A,numeral_numeral(A),bit0(one2)))),Xb)) ) ).

% cos_sin_eq
tff(fact_4716_termdiff__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,K6: real,C2: fun(nat,A)] :
          ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Xb)),K6)
         => ( ! [X2: A] :
                ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,X2)),K6)
               => summable(A,aa(A,fun(nat,A),aTP_Lamp_acv(fun(nat,A),fun(A,fun(nat,A)),C2),X2)) )
           => summable(A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_acx(A,fun(fun(nat,A),fun(nat,A)),Xb),C2)) ) ) ) ).

% termdiff_converges
tff(fact_4717_tan__pos__pi2__le,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$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,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),tan(real,Xb)) ) ) ).

% tan_pos_pi2_le
tff(fact_4718_tan__total__pos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ? [X2: real] :
          ( aa(real,$o,aa(real,fun(real,$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))))
          & ( tan(real,X2) = Y ) ) ) ).

% tan_total_pos
tff(fact_4719_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,tan(real,Xb)),zero_zero(real)) ) ) ).

% tan_less_zero
tff(fact_4720_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,aa(real,fun(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,$o,aa(real,fun(real,$o),ord_less_eq(real),tan(real,Xb)),tan(real,Y)) ) ) ) ).

% tan_mono_le
tff(fact_4721_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,$o,aa(real,fun(real,$o),ord_less_eq(real),tan(real,Xb)),tan(real,Y))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ) ) ).

% tan_mono_le_eq
tff(fact_4722_tan__bound__pi2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
     => aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),tan(real,Xb))),one_one(real)) ) ).

% tan_bound_pi2
tff(fact_4723_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))))
       => ( ( tan(real,Xb) = Y )
         => ( arctan(Y) = Xb ) ) ) ) ).

% arctan_unique
tff(fact_4724_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))))
       => ( arctan(tan(real,Xb)) = Xb ) ) ) ).

% arctan_tan
tff(fact_4725_arctan,axiom,
    ! [Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),arctan(Y))
      & aa(real,$o,ord_less(real,arctan(Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))
      & ( tan(real,arctan(Y)) = Y ) ) ).

% arctan
tff(fact_4726_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),tan(A,Xb)),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_4727_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) )
             => ( 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,tan(A,Xb)),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),tan(A,Xb)),tan(A,Y)))) ) ) ) ) ) ).

% tan_diff
tff(fact_4728_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) )
             => ( 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),tan(A,Xb)),tan(A,Y))),aa(A,A,minus_minus(A,one_one(A)),aa(A,A,aa(A,fun(A,A),times_times(A),tan(A,Xb)),tan(A,Y)))) ) ) ) ) ) ).

% tan_add
tff(fact_4729_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),real_Vector_of_real(A,pi)),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% minus_sin_cos_eq
tff(fact_4730_tan__total__pi4,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ? [Z3: real] :
          ( aa(real,$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)))))),Z3)
          & aa(real,$o,ord_less(real,Z3),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(bit0(one2)))))
          & ( tan(real,Z3) = Xb ) ) ) ).

% tan_total_pi4
tff(fact_4731_tan__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: 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_4732_arsinh__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: 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)),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_4733_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( real_V7770717601297561774m_norm(complex,Z) = one_one(real) )
     => ~ ! [T4: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),T4)
           => ( aa(real,$o,ord_less(real,T4),aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),pi))
             => ( Z != complex2(cos(real,T4),sin(real,T4)) ) ) ) ) ).

% complex_unimodular_polar
tff(fact_4734_Maclaurin__exp__lt,axiom,
    ! [Xb: real,Nb: nat] :
      ( ( Xb != zero_zero(real) )
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => ? [T4: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,abs_abs(real),T4))
            & aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xb))
            & ( exp(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_acy(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),exp(real,T4)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ) ) ).

% Maclaurin_exp_lt
tff(fact_4735_exp__le__cancel__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xb)),exp(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ).

% exp_le_cancel_iff
tff(fact_4736_exp__zero,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ( exp(A,zero_zero(A)) = one_one(A) ) ) ).

% exp_zero
tff(fact_4737_exp__eq__one__iff,axiom,
    ! [Xb: real] :
      ( ( exp(real,Xb) = one_one(real) )
    <=> ( Xb = zero_zero(real) ) ) ).

% exp_eq_one_iff
tff(fact_4738_one__less__exp__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,one_one(real)),exp(real,Xb))
    <=> aa(real,$o,ord_less(real,zero_zero(real)),Xb) ) ).

% one_less_exp_iff
tff(fact_4739_exp__less__one__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,exp(real,Xb)),one_one(real))
    <=> aa(real,$o,ord_less(real,Xb),zero_zero(real)) ) ).

% exp_less_one_iff
tff(fact_4740_one__le__exp__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),exp(real,Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb) ) ).

% one_le_exp_iff
tff(fact_4741_exp__le__one__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xb)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ).

% exp_le_one_iff
tff(fact_4742_norm__cos__sin,axiom,
    ! [Ta: real] : real_V7770717601297561774m_norm(complex,complex2(cos(real,Ta),sin(real,Ta))) = one_one(real) ).

% norm_cos_sin
tff(fact_4743_Complex__mult__complex__of__real,axiom,
    ! [Xb: real,Y: real,R3: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(Xb,Y)),real_Vector_of_real(complex,R3)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),Xb),R3),aa(real,real,aa(real,fun(real,real),times_times(real),Y),R3)) ).

% Complex_mult_complex_of_real
tff(fact_4744_complex__of__real__mult__Complex,axiom,
    ! [R3: real,Xb: real,Y: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R3)),complex2(Xb,Y)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R3),Xb),aa(real,real,aa(real,fun(real,real),times_times(real),R3),Y)) ).

% complex_of_real_mult_Complex
tff(fact_4745_complex__diff,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa(complex,complex,minus_minus(complex,complex2(A2,B2)),complex2(C2,D2)) = complex2(aa(real,real,minus_minus(real,A2),C2),aa(real,real,minus_minus(real,B2),D2)) ).

% complex_diff
tff(fact_4746_exp__times__arg__commute,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [A3: A] : aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,A3)),A3) = aa(A,A,aa(A,fun(A,A),times_times(A),A3),exp(A,A3)) ) ).

% exp_times_arg_commute
tff(fact_4747_norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,Xb))),exp(real,real_V7770717601297561774m_norm(A,Xb))) ) ).

% norm_exp
tff(fact_4748_complex__scaleR,axiom,
    ! [R3: real,A2: real,B2: real] : aa(complex,complex,real_V8093663219630862766scaleR(complex,R3),complex2(A2,B2)) = complex2(aa(real,real,aa(real,fun(real,real),times_times(real),R3),A2),aa(real,real,aa(real,fun(real,real),times_times(real),R3),B2)) ).

% complex_scaleR
tff(fact_4749_exp__ge__zero,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),exp(real,Xb)) ).

% exp_ge_zero
tff(fact_4750_not__exp__le__zero,axiom,
    ! [Xb: real] : ~ aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Xb)),zero_zero(real)) ).

% not_exp_le_zero
tff(fact_4751_mult__exp__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,Xb)),exp(A,Y)) = exp(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) ) ).

% mult_exp_exp
tff(fact_4752_exp__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,Y: A] :
          ( ( 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) )
         => ( exp(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),exp(A,Xb)),exp(A,Y)) ) ) ) ).

% exp_add_commuting
tff(fact_4753_exp__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : exp(A,aa(A,A,minus_minus(A,Xb),Y)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),exp(A,Xb)),exp(A,Y)) ) ).

% exp_diff
tff(fact_4754_Complex__eq__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( complex2(A2,B2) = aa(num,complex,numeral_numeral(complex),W) )
    <=> ( ( A2 = aa(num,real,numeral_numeral(real),W) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_numeral
tff(fact_4755_complex__add,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),complex2(A2,B2)),complex2(C2,D2)) = complex2(aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),C2),aa(real,real,aa(real,fun(real,real),plus_plus(real),B2),D2)) ).

% complex_add
tff(fact_4756_Complex__add__complex__of__real,axiom,
    ! [Xb: real,Y: real,R3: real] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),complex2(Xb,Y)),real_Vector_of_real(complex,R3)) = complex2(aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),R3),Y) ).

% Complex_add_complex_of_real
tff(fact_4757_complex__of__real__add__Complex,axiom,
    ! [R3: real,Xb: real,Y: real] : aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,R3)),complex2(Xb,Y)) = complex2(aa(real,real,aa(real,fun(real,real),plus_plus(real),R3),Xb),Y) ).

% complex_of_real_add_Complex
tff(fact_4758_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)),exp(real,Xb)) ) ).

% exp_gt_one
tff(fact_4759_exp__ge__add__one__self,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb)),exp(real,Xb)) ).

% exp_ge_add_one_self
tff(fact_4760_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),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb))) = one_one(A) ) ).

% exp_minus_inverse
tff(fact_4761_exp__of__nat__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Nb: nat,Xb: A] : exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),Xb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,Xb)),Nb) ) ).

% exp_of_nat_mult
tff(fact_4762_exp__of__nat2__mult,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Nb: nat] : exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(nat,A,semiring_1_of_nat(A),Nb))) = aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,Xb)),Nb) ) ).

% exp_of_nat2_mult
tff(fact_4763_Complex__eq__neg__numeral,axiom,
    ! [A2: real,B2: real,W: num] :
      ( ( complex2(A2,B2) = aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W)) )
    <=> ( ( A2 = aa(real,real,uminus_uminus(real),aa(num,real,numeral_numeral(real),W)) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_numeral
tff(fact_4764_log__ln,axiom,
    ! [Xb: real] : aa(real,real,ln_ln(real),Xb) = log(exp(real,one_one(real)),Xb) ).

% log_ln
tff(fact_4765_complex__mult,axiom,
    ! [A2: real,B2: real,C2: real,D2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(A2,B2)),complex2(C2,D2)) = complex2(aa(real,real,minus_minus(real,aa(real,real,aa(real,fun(real,real),times_times(real),A2),C2)),aa(real,real,aa(real,fun(real,real),times_times(real),B2),D2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),A2),D2)),aa(real,real,aa(real,fun(real,real),times_times(real),B2),C2))) ).

% complex_mult
tff(fact_4766_one__complex_Ocode,axiom,
    one_one(complex) = complex2(one_one(real),zero_zero(real)) ).

% one_complex.code
tff(fact_4767_Complex__eq__1,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = one_one(complex) )
    <=> ( ( A2 = one_one(real) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_1
tff(fact_4768_exp__sum,axiom,
    ! [B: $tType,A: $tType] :
      ( ( comm_monoid_mult(B)
        & real_Vector_banach(B)
        & real_V2822296259951069270ebra_1(B) )
     => ! [I5: set(A),F: fun(A,B)] :
          ( finite_finite2(A,I5)
         => ( exp(B,aa(set(A),B,groups7311177749621191930dd_sum(A,B,F),I5)) = groups7121269368397514597t_prod(A,B,aTP_Lamp_acz(fun(A,B),fun(A,B),F),I5) ) ) ) ).

% exp_sum
tff(fact_4769_exp__ge__add__one__self__aux,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),plus_plus(real),one_one(real)),Xb)),exp(real,Xb)) ) ).

% exp_ge_add_one_self_aux
tff(fact_4770_lemma__exp__total,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y)
     => ? [X2: real] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),X2)
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),aa(real,real,minus_minus(real,Y),one_one(real)))
          & ( exp(real,X2) = Y ) ) ) ).

% lemma_exp_total
tff(fact_4771_ln__ge__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),aa(real,real,ln_ln(real),Xb))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,Y)),Xb) ) ) ).

% ln_ge_iff
tff(fact_4772_ln__x__over__x__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Y)),Y)),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),Xb)),Xb)) ) ) ).

% ln_x_over_x_mono
tff(fact_4773_Complex__sum_H,axiom,
    ! [A: $tType,F: fun(A,real),S3: set(A)] : aa(set(A),complex,groups7311177749621191930dd_sum(A,complex,aTP_Lamp_ada(fun(A,real),fun(A,complex),F)),S3) = complex2(aa(set(A),real,groups7311177749621191930dd_sum(A,real,F),S3),zero_zero(real)) ).

% Complex_sum'
tff(fact_4774_powr__def,axiom,
    ! [A: $tType] :
      ( ln(A)
     => ! [Xb: A,A2: A] :
          powr(A,Xb,A2) = $ite(Xb = zero_zero(A),zero_zero(A),exp(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,ln_ln(A),Xb)))) ) ).

% powr_def
tff(fact_4775_Complex__eq__neg__1,axiom,
    ! [A2: real,B2: real] :
      ( ( complex2(A2,B2) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) )
    <=> ( ( A2 = aa(real,real,uminus_uminus(real),one_one(real)) )
        & ( B2 = zero_zero(real) ) ) ) ).

% Complex_eq_neg_1
tff(fact_4776_exp__le,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),exp(real,one_one(real))),aa(num,real,numeral_numeral(real),aa(num,num,bit1,one2))) ).

% exp_le
tff(fact_4777_exp__divide__power__eq,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Nb: nat,Xb: A] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
         => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),exp(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),Xb),aa(nat,A,semiring_1_of_nat(A),Nb)))),Nb) = exp(A,Xb) ) ) ) ).

% exp_divide_power_eq
tff(fact_4778_tanh__altdef,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : tanh(A,Xb) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,minus_minus(A,exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb)))),aa(A,A,aa(A,fun(A,A),plus_plus(A),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb)))) ) ).

% tanh_altdef
tff(fact_4779_exp__half__le2,axiom,
    aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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_4780_exp__double,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: 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),exp(A,Z)),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% exp_double
tff(fact_4781_exp__bound__half,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(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,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,exp(A,Z))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ).

% exp_bound_half
tff(fact_4782_exp__bound,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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_4783_real__exp__bound__lemma,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$o),ord_less_eq(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_4784_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),aa(nat,real,semiring_1_of_nat(real),Nb))),Xb)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => aa(real,$o,aa(real,fun(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),Nb)))),Nb)),exp(real,Xb)) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
tff(fact_4785_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(nat,real,semiring_1_of_nat(real),Nb))
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => aa(real,$o,aa(real,fun(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),Nb)))),Nb)),exp(real,aa(real,real,uminus_uminus(real),Xb))) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
tff(fact_4786_exp__bound__lemma,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Z: A] :
          ( aa(real,$o,aa(real,fun(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,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(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_4787_Maclaurin__exp__le,axiom,
    ! [Xb: real,Nb: nat] :
    ? [T4: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),T4)),aa(real,real,abs_abs(real),Xb))
      & ( exp(real,Xb) = aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_acy(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),exp(real,T4)),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb))) ) ) ).

% Maclaurin_exp_le
tff(fact_4788_exp__lower__Taylor__quadratic,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$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))))),exp(real,Xb)) ) ).

% exp_lower_Taylor_quadratic
tff(fact_4789_log__base__10__eq2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( 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),log(aa(num,real,numeral_numeral(real),bit0(aa(num,num,bit1,bit0(one2)))),exp(real,one_one(real)))),aa(real,real,ln_ln(real),Xb)) ) ) ).

% log_base_10_eq2
tff(fact_4790_tanh__real__altdef,axiom,
    ! [Xb: real] : tanh(real,Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,one_one(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)),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_4791_log__base__10__eq1,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( 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),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_4792_sin__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,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),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),tan(real,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% sin_tan
tff(fact_4793_cos__tan,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,aa(real,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),tan(real,Xb)),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ) ).

% cos_tan
tff(fact_4794_monoI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [M2: nat,N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,M2)),aa(nat,A,X5,N2)) )
         => topological_monoseq(A,X5) ) ) ).

% monoI1
tff(fact_4795_real__sqrt__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa(real,real,sqrt,Xb) = aa(real,real,sqrt,Y) )
    <=> ( Xb = Y ) ) ).

% real_sqrt_eq_iff
tff(fact_4796_real__sqrt__zero,axiom,
    aa(real,real,sqrt,zero_zero(real)) = zero_zero(real) ).

% real_sqrt_zero
tff(fact_4797_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_4798_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_4799_real__sqrt__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ).

% real_sqrt_le_iff
tff(fact_4800_real__sqrt__one,axiom,
    aa(real,real,sqrt,one_one(real)) = one_one(real) ).

% real_sqrt_one
tff(fact_4801_real__sqrt__eq__1__iff,axiom,
    ! [Xb: real] :
      ( ( aa(real,real,sqrt,Xb) = one_one(real) )
    <=> ( Xb = one_one(real) ) ) ).

% real_sqrt_eq_1_iff
tff(fact_4802_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_4803_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_4804_real__sqrt__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ).

% real_sqrt_le_0_iff
tff(fact_4805_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y) ) ).

% real_sqrt_ge_0_iff
tff(fact_4806_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_4807_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_4808_real__sqrt__le__1__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xb)),one_one(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real)) ) ).

% real_sqrt_le_1_iff
tff(fact_4809_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y) ) ).

% real_sqrt_ge_1_iff
tff(fact_4810_real__sqrt__abs2,axiom,
    ! [Xb: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Xb)) = aa(real,real,abs_abs(real),Xb) ).

% real_sqrt_abs2
tff(fact_4811_real__sqrt__mult__self,axiom,
    ! [A2: real] : aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sqrt,A2)),aa(real,real,sqrt,A2)) = aa(real,real,abs_abs(real),A2) ).

% real_sqrt_mult_self
tff(fact_4812_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_4813_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)))) = aa(real,real,abs_abs(real),Xb) ).

% real_sqrt_abs
tff(fact_4814_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,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb) ) ).

% real_sqrt_pow2_iff
tff(fact_4815_real__sqrt__pow2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$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_4816_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_4817_complex__exp__exists,axiom,
    ! [Z: complex] :
    ? [A4: complex,R2: real] : Z = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R2)),exp(complex,A4)) ).

% complex_exp_exists
tff(fact_4818_real__sqrt__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y)) ) ).

% real_sqrt_le_mono
tff(fact_4819_real__sqrt__mult,axiom,
    ! [Xb: real,Y: 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),times_times(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y)) ).

% real_sqrt_mult
tff(fact_4820_real__sqrt__divide,axiom,
    ! [Xb: real,Y: real] : aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,sqrt,Xb)),aa(real,real,sqrt,Y)) ).

% real_sqrt_divide
tff(fact_4821_real__sqrt__power,axiom,
    ! [Xb: real,K: nat] : aa(real,real,sqrt,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,sqrt,Xb)),K) ).

% real_sqrt_power
tff(fact_4822_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_4823_real__sqrt__minus,axiom,
    ! [Xb: real] : aa(real,real,sqrt,aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,sqrt,Xb)) ).

% real_sqrt_minus
tff(fact_4824_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_4825_real__sqrt__eq__zero__cancel,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$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_4826_real__sqrt__ge__zero,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sqrt,Xb)) ) ).

% real_sqrt_ge_zero
tff(fact_4827_real__sqrt__ge__one,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,sqrt,Xb)) ) ).

% real_sqrt_ge_one
tff(fact_4828_real__div__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$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_4829_sqrt__add__le__add__sqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$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_4830_le__real__sqrt__sumsq,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,sqrt,aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Xb)),aa(real,real,aa(real,fun(real,real),times_times(real),Y),Y)))) ).

% le_real_sqrt_sumsq
tff(fact_4831_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_4832_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_4833_real__le__rsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,sqrt,Y)) ) ).

% real_le_rsqrt
tff(fact_4834_sqrt__le__D,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xb)),Y)
     => aa(real,$o,aa(real,fun(real,$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_4835_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,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,real,sqrt,Xb) = Y ) ) ) ).

% real_sqrt_unique
tff(fact_4836_real__le__lsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sqrt,Xb)),Y) ) ) ) ).

% real_le_lsqrt
tff(fact_4837_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_4838_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_4839_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_4840_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A2: real,C2: real,B2: real,D2: real] : aa(real,$o,aa(real,fun(real,$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),A2),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),B2),D2)),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),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),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),D2),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ).

% real_sqrt_sum_squares_triangle_ineq
tff(fact_4841_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,Xb: real] : aa(real,$o,aa(real,fun(real,$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_4842_real__sqrt__sum__squares__ge1,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$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_4843_sqrt__ge__absD,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,sqrt,Y))
     => aa(real,$o,aa(real,fun(real,$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_4844_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_4845_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_4846_tan__60,axiom,
    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_4847_real__less__lsqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$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_4848_sqrt__sum__squares__le__sum,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$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_4849_sqrt__even__pow2,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(real,real,sqrt,aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(num,real,numeral_numeral(real),bit0(one2))),Nb)) = 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),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ).

% sqrt_even_pow2
tff(fact_4850_real__sqrt__ge__abs1,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,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_4851_real__sqrt__ge__abs2,axiom,
    ! [Y: real,Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,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_4852_sqrt__sum__squares__le__sum__abs,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$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),aa(real,real,abs_abs(real),Xb)),aa(real,real,abs_abs(real),Y))) ).

% sqrt_sum_squares_le_sum_abs
tff(fact_4853_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_4854_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_4855_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_4856_arsinh__real__def,axiom,
    ! [Xb: 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_4857_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_4858_real__sqrt__power__even,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(real,$o,aa(real,fun(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)),Nb) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))) ) ) ) ).

% real_sqrt_power_even
tff(fact_4859_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_4860_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [Xb: real,Y: real,Xaa: real,Ya: real] : aa(real,$o,aa(real,fun(real,$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_4861_arith__geo__mean__sqrt,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => aa(real,$o,aa(real,fun(real,$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_4862_powr__half__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$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_4863_tan__30,axiom,
    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_4864_cos__x__y__le__one,axiom,
    ! [Xb: real,Y: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,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_4865_real__sqrt__sum__squares__less,axiom,
    ! [Xb: real,U: real,Y: real] :
      ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),Xb)),aa(real,real,aa(real,fun(real,real),divide_divide(real),U),aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2)))))
     => ( aa(real,$o,ord_less(real,aa(real,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_4866_arcosh__real__def,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
     => ( 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_4867_cos__arctan,axiom,
    ! [Xb: real] : cos(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_4868_sin__arctan,axiom,
    ! [Xb: real] : sin(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_4869_monoseq__minus,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [A2: fun(nat,A)] :
          ( topological_monoseq(A,A2)
         => topological_monoseq(A,aTP_Lamp_adb(fun(nat,A),fun(nat,A),A2)) ) ) ).

% monoseq_minus
tff(fact_4870_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,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$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_4871_sin__cos__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$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_4872_arctan__half,axiom,
    ! [Xb: real] : arctan(Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(num,real,numeral_numeral(real),bit0(one2))),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_4873_monoseq__Suc,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( topological_monoseq(A,X5)
        <=> ( ! [N6: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N6)),aa(nat,A,X5,aa(nat,nat,suc,N6)))
            | ! [N6: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,aa(nat,nat,suc,N6))),aa(nat,A,X5,N6)) ) ) ) ).

% monoseq_Suc
tff(fact_4874_mono__SucI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,aa(nat,nat,suc,N2))),aa(nat,A,X5,N2))
         => topological_monoseq(A,X5) ) ) ).

% mono_SucI2
tff(fact_4875_mono__SucI1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [N2: nat] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N2)),aa(nat,A,X5,aa(nat,nat,suc,N2)))
         => topological_monoseq(A,X5) ) ) ).

% mono_SucI1
tff(fact_4876_monoseq__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( topological_monoseq(A,X5)
        <=> ( ! [M7: nat,N6: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),N6)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,M7)),aa(nat,A,X5,N6)) )
            | ! [M7: nat,N6: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M7),N6)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N6)),aa(nat,A,X5,M7)) ) ) ) ) ).

% monoseq_def
tff(fact_4877_monoI2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [X5: fun(nat,A)] :
          ( ! [M2: nat,N2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M2),N2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,X5,N2)),aa(nat,A,X5,M2)) )
         => topological_monoseq(A,X5) ) ) ).

% monoI2
tff(fact_4878_cos__arcsin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( cos(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_4879_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
     => ( sin(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_4880_sin__arccos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( sin(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_4881_arccos__1,axiom,
    arccos(one_one(real)) = zero_zero(real) ).

% arccos_1
tff(fact_4882_arccos__minus__1,axiom,
    arccos(aa(real,real,uminus_uminus(real),one_one(real))) = pi ).

% arccos_minus_1
tff(fact_4883_cos__arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( cos(real,arccos(Y)) = Y ) ) ) ).

% cos_arccos
tff(fact_4884_sin__arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( sin(real,arcsin(Y)) = Y ) ) ) ).

% sin_arcsin
tff(fact_4885_arccos__0,axiom,
    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_4886_arcsin__1,axiom,
    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_4887_arcsin__minus__1,axiom,
    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_4888_arccos__le__arccos,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arccos(Y)),arccos(Xb)) ) ) ) ).

% arccos_le_arccos
tff(fact_4889_arccos__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real)) )
     => ( ( arccos(Xb) = arccos(Y) )
      <=> ( Xb = Y ) ) ) ).

% arccos_eq_iff
tff(fact_4890_arccos__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arccos(Xb)),arccos(Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xb) ) ) ) ).

% arccos_le_mono
tff(fact_4891_arcsin__le__arcsin,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arcsin(Xb)),arcsin(Y)) ) ) ) ).

% arcsin_le_arcsin
tff(fact_4892_arcsin__minus,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( arcsin(aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),arcsin(Xb)) ) ) ) ).

% arcsin_minus
tff(fact_4893_arcsin__eq__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( ( arcsin(Xb) = arcsin(Y) )
        <=> ( Xb = Y ) ) ) ) ).

% arcsin_eq_iff
tff(fact_4894_arcsin__le__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arcsin(Xb)),arcsin(Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ).

% arcsin_le_mono
tff(fact_4895_arccos__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),arccos(Y)) ) ) ).

% arccos_lbound
tff(fact_4896_arccos__less__arccos,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$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,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,ord_less(real,arccos(Y)),arccos(Xb)) ) ) ) ).

% arccos_less_arccos
tff(fact_4897_arccos__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,ord_less(real,arccos(Xb)),arccos(Y))
        <=> aa(real,$o,ord_less(real,Y),Xb) ) ) ) ).

% arccos_less_mono
tff(fact_4898_arccos__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arccos(Y)),pi) ) ) ).

% arccos_ubound
tff(fact_4899_arccos__cos,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
       => ( arccos(cos(real,Xb)) = Xb ) ) ) ).

% arccos_cos
tff(fact_4900_arcsin__less__arcsin,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$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,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
         => aa(real,$o,ord_less(real,arcsin(Xb)),arcsin(Y)) ) ) ) ).

% arcsin_less_arcsin
tff(fact_4901_arcsin__less__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
       => ( aa(real,$o,ord_less(real,arcsin(Xb)),arcsin(Y))
        <=> aa(real,$o,ord_less(real,Xb),Y) ) ) ) ).

% arcsin_less_mono
tff(fact_4902_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Y)),one_one(real))
     => ( cos(real,arccos(Y)) = Y ) ) ).

% cos_arccos_abs
tff(fact_4903_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Theta)),pi)
     => ( arccos(cos(real,Theta)) = aa(real,real,abs_abs(real),Theta) ) ) ).

% arccos_cos_eq_abs
tff(fact_4904_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,$o,ord_less(real,zero_zero(real)),arccos(Y))
          & aa(real,$o,ord_less(real,arccos(Y)),pi) ) ) ) ).

% arccos_lt_bounded
tff(fact_4905_arccos__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),arccos(Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arccos(Y)),pi) ) ) ) ).

% arccos_bounded
tff(fact_4906_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,arccos(Xb)) != zero_zero(real) ) ) ) ).

% sin_arccos_nonzero
tff(fact_4907_arccos__cos2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),pi)),Xb)
       => ( arccos(cos(real,Xb)) = aa(real,real,uminus_uminus(real),Xb) ) ) ) ).

% arccos_cos2
tff(fact_4908_arccos__minus,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( arccos(aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,minus_minus(real,pi),arccos(Xb)) ) ) ) ).

% arccos_minus
tff(fact_4909_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,arcsin(Xb)) != zero_zero(real) ) ) ) ).

% cos_arcsin_nonzero
tff(fact_4910_arccos,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),arccos(Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arccos(Y)),pi)
          & ( cos(real,arccos(Y)) = Y ) ) ) ) ).

% arccos
tff(fact_4911_arccos__minus__abs,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),one_one(real))
     => ( arccos(aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,minus_minus(real,pi),arccos(Xb)) ) ) ).

% arccos_minus_abs
tff(fact_4912_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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_4913_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,$o,ord_less(real,aa(real,real,uminus_uminus(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),pi),aa(num,real,numeral_numeral(real),bit0(one2))))),arcsin(Y))
          & aa(real,$o,ord_less(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_4914_arcsin__lbound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$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))))),arcsin(Y)) ) ) ).

% arcsin_lbound
tff(fact_4915_arcsin__ubound,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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_4916_arcsin__bounded,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$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))))),arcsin(Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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_4917_arcsin__sin,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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))))
       => ( arcsin(sin(real,Xb)) = Xb ) ) ) ).

% arcsin_sin
tff(fact_4918_le__arcsin__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),arcsin(Xb))
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sin(real,Y)),Xb) ) ) ) ) ) ).

% le_arcsin_iff
tff(fact_4919_arcsin__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$o),ord_less_eq(real),arcsin(Xb)),Y)
            <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),sin(real,Y)) ) ) ) ) ) ).

% arcsin_le_iff
tff(fact_4920_arcsin__pi,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$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))))),arcsin(Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arcsin(Y)),pi)
          & ( sin(real,arcsin(Y)) = Y ) ) ) ) ).

% arcsin_pi
tff(fact_4921_arcsin,axiom,
    ! [Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,uminus_uminus(real),one_one(real))),Y)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),one_one(real))
       => ( aa(real,$o,aa(real,fun(real,$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))))),arcsin(Y))
          & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(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,arcsin(Y)) = Y ) ) ) ) ).

% arcsin
tff(fact_4922_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K3: int] : arccos(cos(real,Theta)) != aa(real,real,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),K3)),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_4923_exp__two__pi__i,axiom,
    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))),real_Vector_of_real(complex,pi))),imaginary_unit)) = one_one(complex) ).

% exp_two_pi_i
tff(fact_4924_exp__two__pi__i_H,axiom,
    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),real_Vector_of_real(complex,pi)),aa(num,complex,numeral_numeral(complex),bit0(one2))))) = one_one(complex) ).

% exp_two_pi_i'
tff(fact_4925_exp__first__two__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: 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_adc(A,fun(nat,A),Xb))) ) ).

% exp_first_two_terms
tff(fact_4926_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_4927_inverse__1,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ( aa(A,A,inverse_inverse(A),one_one(A)) = one_one(A) ) ) ).

% inverse_1
tff(fact_4928_inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) ) ).

% inverse_mult_distrib
tff(fact_4929_inverse__divide,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2) ) ).

% inverse_divide
tff(fact_4930_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),zero_zero(A)) ) ) ).

% inverse_nonpositive_iff_nonpositive
tff(fact_4931_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2) ) ) ).

% inverse_nonnegative_iff_nonnegative
tff(fact_4932_inverse__less__iff__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
           => ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,ord_less(A,B2),A2) ) ) ) ) ).

% inverse_less_iff_less
tff(fact_4933_inverse__less__iff__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,B2),zero_zero(A))
           => ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,ord_less(A,B2),A2) ) ) ) ) ).

% inverse_less_iff_less_neg
tff(fact_4934_inverse__negative__iff__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
        <=> aa(A,$o,ord_less(A,A2),zero_zero(A)) ) ) ).

% inverse_negative_iff_negative
tff(fact_4935_inverse__positive__iff__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A2) ) ) ).

% inverse_positive_iff_positive
tff(fact_4936_norm__ii,axiom,
    real_V7770717601297561774m_norm(complex,imaginary_unit) = one_one(real) ).

% norm_ii
tff(fact_4937_complex__i__mult__minus,axiom,
    ! [Xb: 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),imaginary_unit),Xb)) = aa(complex,complex,uminus_uminus(complex),Xb) ).

% complex_i_mult_minus
tff(fact_4938_divide__i,axiom,
    ! [Xb: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Xb),imaginary_unit) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,uminus_uminus(complex),imaginary_unit)),Xb) ).

% divide_i
tff(fact_4939_inverse__le__iff__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),zero_zero(A))
         => ( aa(A,$o,ord_less(A,B2),zero_zero(A))
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% inverse_le_iff_le_neg
tff(fact_4940_inverse__le__iff__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
            <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ) ).

% inverse_le_iff_le
tff(fact_4941_right__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),A2)) = one_one(A) ) ) ) ).

% right_inverse
tff(fact_4942_left__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),A2) = one_one(A) ) ) ) ).

% left_inverse
tff(fact_4943_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_4944_i__squared,axiom,
    aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),imaginary_unit) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% i_squared
tff(fact_4945_divide__numeral__i,axiom,
    ! [Z: complex,Nb: num] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(num,complex,numeral_numeral(complex),Nb)),imaginary_unit)) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),aa(complex,complex,uminus_uminus(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Z))),aa(num,complex,numeral_numeral(complex),Nb)) ).

% divide_numeral_i
tff(fact_4946_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_4947_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_4948_exp__pi__i,axiom,
    exp(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,pi)),imaginary_unit)) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% exp_pi_i
tff(fact_4949_exp__pi__i_H,axiom,
    exp(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,pi))) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% exp_pi_i'
tff(fact_4950_i__even__power,axiom,
    ! [Nb: 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),Nb),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))),Nb) ).

% i_even_power
tff(fact_4951_complex__i__not__one,axiom,
    imaginary_unit != one_one(complex) ).

% complex_i_not_one
tff(fact_4952_power__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,Nb: nat] : aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,inverse_inverse(A),A2)),Nb) = aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) ) ).

% power_inverse
tff(fact_4953_complex__i__not__numeral,axiom,
    ! [W: num] : imaginary_unit != aa(num,complex,numeral_numeral(complex),W) ).

% complex_i_not_numeral
tff(fact_4954_real__sqrt__inverse,axiom,
    ! [Xb: real] : aa(real,real,sqrt,aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,inverse_inverse(real),aa(real,real,sqrt,Xb)) ).

% real_sqrt_inverse
tff(fact_4955_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_4956_norm__inverse__le__norm,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [R3: real,Xb: A] :
          ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),R3),real_V7770717601297561774m_norm(A,Xb))
         => ( aa(real,$o,ord_less(real,zero_zero(real)),R3)
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),real_V7770717601297561774m_norm(A,aa(A,A,inverse_inverse(A),Xb))),aa(real,real,inverse_inverse(real),R3)) ) ) ) ).

% norm_inverse_le_norm
tff(fact_4957_inverse__less__imp__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
           => aa(A,$o,ord_less(A,B2),A2) ) ) ) ).

% inverse_less_imp_less
tff(fact_4958_less__imp__inverse__less,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
           => aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% less_imp_inverse_less
tff(fact_4959_inverse__less__imp__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,ord_less(A,B2),zero_zero(A))
           => aa(A,$o,ord_less(A,B2),A2) ) ) ) ).

% inverse_less_imp_less_neg
tff(fact_4960_less__imp__inverse__less__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,ord_less(A,B2),zero_zero(A))
           => aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% less_imp_inverse_less_neg
tff(fact_4961_inverse__negative__imp__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A2)),zero_zero(A))
         => ( ( A2 != zero_zero(A) )
           => aa(A,$o,ord_less(A,A2),zero_zero(A)) ) ) ) ).

% inverse_negative_imp_negative
tff(fact_4962_inverse__positive__imp__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,inverse_inverse(A),A2))
         => ( ( A2 != zero_zero(A) )
           => aa(A,$o,ord_less(A,zero_zero(A)),A2) ) ) ) ).

% inverse_positive_imp_positive
tff(fact_4963_negative__imp__inverse__negative,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,A2),zero_zero(A))
         => aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A2)),zero_zero(A)) ) ) ).

% negative_imp_inverse_negative
tff(fact_4964_positive__imp__inverse__positive,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ).

% positive_imp_inverse_positive
tff(fact_4965_nonzero__inverse__mult__distrib,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,inverse_inverse(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ) ).

% nonzero_inverse_mult_distrib
tff(fact_4966_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_4967_inverse__unique,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = one_one(A) )
         => ( aa(A,A,inverse_inverse(A),A2) = B2 ) ) ) ).

% inverse_unique
tff(fact_4968_inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] : aa(A,A,inverse_inverse(A),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ).

% inverse_eq_divide
tff(fact_4969_divide__inverse__commute,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),B2)),A2) ) ).

% divide_inverse_commute
tff(fact_4970_divide__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),B2)) ) ).

% divide_inverse
tff(fact_4971_field__class_Ofield__divide__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,inverse_inverse(A),B2)) ) ).

% field_class.field_divide_inverse
tff(fact_4972_power__mult__power__inverse__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,M: nat,Nb: 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)),Nb)) = 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)),Nb)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),M)) ) ).

% power_mult_power_inverse_commute
tff(fact_4973_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_4974_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_4975_mult__inverse__of__int__commute,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xaa: int,Xb: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xaa))),Xb) = aa(A,A,aa(A,fun(A,A),times_times(A),Xb),aa(A,A,inverse_inverse(A),aa(int,A,ring_1_of_int(A),Xaa))) ) ).

% mult_inverse_of_int_commute
tff(fact_4976_divide__real__def,axiom,
    ! [Xb: real,Y: real] : aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y) = aa(real,real,aa(real,fun(real,real),times_times(real),Xb),aa(real,real,inverse_inverse(real),Y)) ).

% divide_real_def
tff(fact_4977_exp__fdiffs,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X3: nat] : diffs(A,aTP_Lamp_add(nat,A),X3) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,X3)) ) ).

% exp_fdiffs
tff(fact_4978_le__imp__inverse__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,ord_less(A,B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% le_imp_inverse_le_neg
tff(fact_4979_inverse__le__imp__le__neg,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,ord_less(A,B2),zero_zero(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% inverse_le_imp_le_neg
tff(fact_4980_le__imp__inverse__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),B2)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% le_imp_inverse_le
tff(fact_4981_inverse__le__imp__le,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
         => ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% inverse_le_imp_le
tff(fact_4982_i__times__eq__iff,axiom,
    ! [W: complex,Z: complex] :
      ( ( aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),W) = Z )
    <=> ( W = aa(complex,complex,uminus_uminus(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),Z)) ) ) ).

% i_times_eq_iff
tff(fact_4983_inverse__le__1__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),Xb)),one_one(A))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),zero_zero(A))
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),Xb) ) ) ) ).

% inverse_le_1_iff
tff(fact_4984_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_4985_one__less__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,ord_less(A,A2),one_one(A))
           => aa(A,$o,ord_less(A,one_one(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% one_less_inverse
tff(fact_4986_division__ring__inverse__add,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = 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),A2)),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_add
tff(fact_4987_inverse__add,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = 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),A2),B2)),aa(A,A,inverse_inverse(A),A2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% inverse_add
tff(fact_4988_field__class_Ofield__inverse,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,inverse_inverse(A),A2)),A2) = one_one(A) ) ) ) ).

% field_class.field_inverse
tff(fact_4989_division__ring__inverse__diff,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = 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),A2)),aa(A,A,minus_minus(A,B2),A2))),aa(A,A,inverse_inverse(A),B2)) ) ) ) ) ).

% division_ring_inverse_diff
tff(fact_4990_nonzero__inverse__eq__divide,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,inverse_inverse(A),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),A2) ) ) ) ).

% nonzero_inverse_eq_divide
tff(fact_4991_complex__i__not__neg__numeral,axiom,
    ! [W: num] : imaginary_unit != aa(complex,complex,uminus_uminus(complex),aa(num,complex,numeral_numeral(complex),W)) ).

% complex_i_not_neg_numeral
tff(fact_4992_inverse__powr,axiom,
    ! [Y: real,A2: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
     => ( powr(real,aa(real,real,inverse_inverse(real),Y),A2) = aa(real,real,inverse_inverse(real),powr(real,Y,A2)) ) ) ).

% inverse_powr
tff(fact_4993_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_4994_imaginary__unit_Ocode,axiom,
    imaginary_unit = complex2(zero_zero(real),one_one(real)) ).

% imaginary_unit.code
tff(fact_4995_one__le__inverse__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,aa(A,fun(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,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),one_one(A)) ) ) ) ).

% one_le_inverse_iff
tff(fact_4996_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,$o,aa(A,fun(A,$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_4997_one__le__inverse,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),one_one(A))
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,inverse_inverse(A),A2)) ) ) ) ).

% one_le_inverse
tff(fact_4998_inverse__less__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,ord_less(A,B2),A2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
             => aa(A,$o,ord_less(A,A2),B2) ) ) ) ) ).

% inverse_less_iff
tff(fact_4999_inverse__le__iff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2))
        <=> ( ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) )
            & ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),zero_zero(A))
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ) ).

% inverse_le_iff
tff(fact_5000_inverse__diff__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != zero_zero(A) )
         => ( ( B2 != zero_zero(A) )
           => ( aa(A,A,minus_minus(A,aa(A,A,inverse_inverse(A),A2)),aa(A,A,inverse_inverse(A),B2)) = 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),A2)),aa(A,A,minus_minus(A,A2),B2))),aa(A,A,inverse_inverse(A),B2))) ) ) ) ) ).

% inverse_diff_inverse
tff(fact_5001_reals__Archimedean,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xb)
         => ? [N2: 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,N2)))),Xb) ) ) ).

% reals_Archimedean
tff(fact_5002_i__mult__Complex,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),complex2(A2,B2)) = complex2(aa(real,real,uminus_uminus(real),B2),A2) ).

% i_mult_Complex
tff(fact_5003_Complex__mult__i,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),complex2(A2,B2)),imaginary_unit) = complex2(aa(real,real,uminus_uminus(real),B2),A2) ).

% Complex_mult_i
tff(fact_5004_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_5005_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_5006_neg__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_le_divideR_eq
tff(fact_5007_neg__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% neg_divideR_le_eq
tff(fact_5008_pos__le__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% pos_le_divideR_eq
tff(fact_5009_pos__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_divideR_le_eq
tff(fact_5010_pos__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,ord_less(A,B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_divideR_less_eq
tff(fact_5011_pos__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less(A,A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% pos_less_divideR_eq
tff(fact_5012_neg__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)),A2)
          <=> aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),B2) ) ) ) ).

% neg_divideR_less_eq
tff(fact_5013_neg__less__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less(A,A2),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))
          <=> aa(A,$o,ord_less(A,B2),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_less_divideR_eq
tff(fact_5014_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) ) )
     => ( ! [N2: nat] : aa(real,$o,P,aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,N2))))
       => ( aa(real,$o,ord_less(real,zero_zero(real)),E3)
         => aa(real,$o,P,E3) ) ) ) ).

% forall_pos_mono_1
tff(fact_5015_sqrt__divide__self__eq,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$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_5016_summable__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] : summable(A,aTP_Lamp_ade(A,fun(nat,A),Xb)) ) ).

% summable_exp
tff(fact_5017_summable__exp__generic,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(A,aTP_Lamp_adf(A,fun(nat,A),Xb)) ) ).

% summable_exp_generic
tff(fact_5018_ex__inverse__of__nat__less,axiom,
    ! [A: $tType] :
      ( archim462609752435547400_field(A)
     => ! [Xb: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),Xb)
         => ? [N2: nat] :
              ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N2)
              & aa(A,$o,ord_less(A,aa(A,A,inverse_inverse(A),aa(nat,A,semiring_1_of_nat(A),N2))),Xb) ) ) ) ).

% ex_inverse_of_nat_less
tff(fact_5019_power__diff__conv__inverse,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Xb: A,M: nat,Nb: nat] :
          ( ( Xb != zero_zero(A) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
           => ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),aa(nat,nat,minus_minus(nat,Nb),M)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),Xb),Nb)),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_5020_pos__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_le_minus_divideR_eq
tff(fact_5021_pos__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_minus_divideR_le_eq
tff(fact_5022_neg__le__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_le_minus_divideR_eq
tff(fact_5023_neg__minus__divideR__le__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divideR_le_eq
tff(fact_5024_neg__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% neg_minus_divideR_less_eq
tff(fact_5025_neg__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,ord_less(real,C2),zero_zero(real))
         => ( aa(A,$o,ord_less(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% neg_less_minus_divideR_eq
tff(fact_5026_pos__minus__divideR__less__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,B2: A,A2: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2))),A2)
          <=> aa(A,$o,ord_less(A,aa(A,A,uminus_uminus(A),B2)),aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)) ) ) ) ).

% pos_minus_divideR_less_eq
tff(fact_5027_pos__less__minus__divideR__eq,axiom,
    ! [A: $tType] :
      ( real_V5355595471888546746vector(A)
     => ! [C2: real,A2: A,B2: A] :
          ( aa(real,$o,ord_less(real,zero_zero(real)),C2)
         => ( aa(A,$o,ord_less(A,A2),aa(A,A,uminus_uminus(A),aa(A,A,real_V8093663219630862766scaleR(A,aa(real,real,inverse_inverse(real),C2)),B2)))
          <=> aa(A,$o,ord_less(A,aa(A,A,real_V8093663219630862766scaleR(A,C2),A2)),aa(A,A,uminus_uminus(A),B2)) ) ) ) ).

% pos_less_minus_divideR_eq
tff(fact_5028_i__complex__of__real,axiom,
    ! [R3: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,R3)) = complex2(zero_zero(real),R3) ).

% i_complex_of_real
tff(fact_5029_complex__of__real__i,axiom,
    ! [R3: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R3)),imaginary_unit) = complex2(zero_zero(real),R3) ).

% complex_of_real_i
tff(fact_5030_log__inverse,axiom,
    ! [A2: real,Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
     => ( ( A2 != one_one(real) )
       => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
         => ( log(A2,aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,uminus_uminus(real),log(A2,Xb)) ) ) ) ) ).

% log_inverse
tff(fact_5031_Complex__eq,axiom,
    ! [A2: real,B2: real] : complex2(A2,B2) = aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,A2)),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,B2))) ).

% Complex_eq
tff(fact_5032_exp__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_adf(A,fun(nat,A),Xb)),exp(A,Xb)) ) ).

% exp_converges
tff(fact_5033_exp__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [X3: A] : exp(A,X3) = suminf(A,aTP_Lamp_adf(A,fun(nat,A),X3)) ) ).

% exp_def
tff(fact_5034_summable__norm__exp,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : summable(real,aTP_Lamp_adg(A,fun(nat,real),Xb)) ) ).

% summable_norm_exp
tff(fact_5035_exp__plus__inverse__exp,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(num,real,numeral_numeral(real),bit0(one2))),aa(real,real,aa(real,fun(real,real),plus_plus(real),exp(real,Xb)),aa(real,real,inverse_inverse(real),exp(real,Xb)))) ).

% exp_plus_inverse_exp
tff(fact_5036_complex__split__polar,axiom,
    ! [Z: complex] :
    ? [R2: real,A4: real] : Z = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R2)),aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,cos(real,A4))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,sin(real,A4))))) ).

% complex_split_polar
tff(fact_5037_plus__inverse__ge__2,axiom,
    ! [Xb: real] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(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_5038_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_5039_tan__cot,axiom,
    ! [Xb: 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),tan(real,Xb)) ).

% tan_cot
tff(fact_5040_real__le__x__sinh,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,exp(real,Xb)),aa(real,real,inverse_inverse(real),exp(real,Xb)))),aa(num,real,numeral_numeral(real),bit0(one2)))) ) ).

% real_le_x_sinh
tff(fact_5041_real__le__abs__sinh,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),Xb)),aa(real,real,abs_abs(real),aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,exp(real,Xb)),aa(real,real,inverse_inverse(real),exp(real,Xb)))),aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% real_le_abs_sinh
tff(fact_5042_exp__series__add__commuting,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,Y: A,Nb: 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,Nb))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)),Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(A,fun(nat,fun(nat,A)),aTP_Lamp_adh(A,fun(A,fun(nat,fun(nat,A))),Xb),Y),Nb)),set_ord_atMost(nat,Nb)) ) ) ) ).

% exp_series_add_commuting
tff(fact_5043_exp__first__term,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : exp(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),suminf(A,aTP_Lamp_adi(A,fun(nat,A),Xb))) ) ).

% exp_first_term
tff(fact_5044_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),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_5045_cmod__unit__one,axiom,
    ! [A2: real] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,cos(real,A2))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,sin(real,A2))))) = one_one(real) ).

% cmod_unit_one
tff(fact_5046_cmod__complex__polar,axiom,
    ! [R3: real,A2: real] : real_V7770717601297561774m_norm(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,R3)),aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),real_Vector_of_real(complex,cos(real,A2))),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,sin(real,A2)))))) = aa(real,real,abs_abs(real),R3) ).

% cmod_complex_polar
tff(fact_5047_powr__real__of__int,axiom,
    ! [Xb: real,Nb: int] :
      ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
     => ( powr(real,Xb,aa(int,real,ring_1_of_int(real),Nb)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),nat2(Nb)),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),Nb))))) ) ) ).

% powr_real_of_int
tff(fact_5048_exp__first__terms,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A,K: nat] : exp(A,Xb) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_adf(A,fun(nat,A),Xb)),set_ord_lessThan(nat,K))),suminf(A,aa(nat,fun(nat,A),aTP_Lamp_adj(A,fun(nat,fun(nat,A)),Xb),K))) ) ).

% exp_first_terms
tff(fact_5049_Maclaurin__sin__bound,axiom,
    ! [Xb: real,Nb: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,sin(real,Xb)),aa(set(nat),real,groups7311177749621191930dd_sum(nat,real,aTP_Lamp_abk(real,fun(nat,real),Xb)),set_ord_lessThan(nat,Nb))))),aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,inverse_inverse(real),semiring_char_0_fact(real,Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),Xb)),Nb))) ).

% Maclaurin_sin_bound
tff(fact_5050_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_5051_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)),real_Vector_of_real(complex,aa(real,real,sqrt,aa(num,real,numeral_numeral(real),bit0(one2))))) ).

% csqrt_ii
tff(fact_5052_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_5053_csqrt__1,axiom,
    csqrt(one_one(complex)) = one_one(complex) ).

% csqrt_1
tff(fact_5054_csqrt__eq__1,axiom,
    ! [Z: complex] :
      ( ( csqrt(Z) = one_one(complex) )
    <=> ( Z = one_one(complex) ) ) ).

% csqrt_eq_1
tff(fact_5055_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_5056_divide__complex__def,axiom,
    ! [Xb: complex,Y: complex] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Xb),Y) = aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),Xb),aa(complex,complex,inverse_inverse(complex),Y)) ).

% divide_complex_def
tff(fact_5057_of__real__sqrt,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( real_Vector_of_real(complex,aa(real,real,sqrt,Xb)) = csqrt(real_Vector_of_real(complex,Xb)) ) ) ).

% of_real_sqrt
tff(fact_5058_Arg__bounded,axiom,
    ! [Z: complex] :
      ( aa(real,$o,ord_less(real,aa(real,real,uminus_uminus(real),pi)),arg(Z))
      & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),arg(Z)),pi) ) ).

% Arg_bounded
tff(fact_5059_complex__inverse,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,inverse_inverse(complex),complex2(A2,B2)) = complex2(aa(real,real,aa(real,fun(real,real),divide_divide(real),A2),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),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),B2)),aa(real,real,aa(real,fun(real,real),plus_plus(real),aa(nat,real,aa(real,fun(nat,real),power_power(real),A2),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,real,aa(real,fun(nat,real),power_power(real),B2),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% complex_inverse
tff(fact_5060_sinh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_adk(A,fun(nat,A),Xb)),sinh(A,Xb)) ) ).

% sinh_converges
tff(fact_5061_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_5062_cosh__converges,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,$o,sums(A,aTP_Lamp_adl(A,fun(nat,A),Xb)),cosh(A,Xb)) ) ).

% cosh_converges
tff(fact_5063_sinh__real__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,Xb)),sinh(real,Y))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ).

% sinh_real_le_iff
tff(fact_5064_sinh__real__nonneg__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),sinh(real,Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb) ) ).

% sinh_real_nonneg_iff
tff(fact_5065_sinh__real__nonpos__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ).

% sinh_real_nonpos_iff
tff(fact_5066_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_5067_norm__cis,axiom,
    ! [A2: real] : real_V7770717601297561774m_norm(complex,cis(A2)) = one_one(real) ).

% norm_cis
tff(fact_5068_cis__zero,axiom,
    cis(zero_zero(real)) = one_one(complex) ).

% cis_zero
tff(fact_5069_cis__pi,axiom,
    cis(pi) = aa(complex,complex,uminus_uminus(complex),one_one(complex)) ).

% cis_pi
tff(fact_5070_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_5071_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_5072_tanh__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: 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_5073_cosh__minus__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,minus_minus(A,cosh(A,Xb)),sinh(A,Xb)) = exp(A,aa(A,A,uminus_uminus(A),Xb)) ) ).

% cosh_minus_sinh
tff(fact_5074_sinh__minus__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,minus_minus(A,sinh(A,Xb)),cosh(A,Xb)) = aa(A,A,uminus_uminus(A),exp(A,aa(A,A,uminus_uminus(A),Xb))) ) ).

% sinh_minus_cosh
tff(fact_5075_cosh__plus__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),cosh(A,Xb)),sinh(A,Xb)) = exp(A,Xb) ) ).

% cosh_plus_sinh
tff(fact_5076_sinh__plus__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Xb: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),sinh(A,Xb)),cosh(A,Xb)) = exp(A,Xb) ) ).

% sinh_plus_cosh
tff(fact_5077_sinh__le__cosh__real,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),sinh(real,Xb)),cosh(real,Xb)) ).

% sinh_le_cosh_real
tff(fact_5078_cosh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : cosh(A,aa(A,A,minus_minus(A,Xb),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xb)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xb)),sinh(A,Y))) ) ).

% cosh_diff
tff(fact_5079_sinh__diff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sinh(A,aa(A,A,minus_minus(A,Xb),Y)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xb)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xb)),sinh(A,Y))) ) ).

% sinh_diff
tff(fact_5080_sinh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : sinh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xb)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xb)),sinh(A,Y))) ) ).

% sinh_add
tff(fact_5081_cosh__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A,Y: A] : cosh(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),cosh(A,Xb)),cosh(A,Y))),aa(A,A,aa(A,fun(A,A),times_times(A),sinh(A,Xb)),sinh(A,Y))) ) ).

% cosh_add
tff(fact_5082_cosh__real__nonpos__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),zero_zero(real))
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cosh(real,Xb)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y),Xb) ) ) ) ).

% cosh_real_nonpos_le_iff
tff(fact_5083_cosh__real__nonneg__le__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),cosh(real,Xb)),cosh(real,Y))
        <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ) ).

% cosh_real_nonneg_le_iff
tff(fact_5084_cosh__real__nonneg,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),cosh(real,Xb)) ).

% cosh_real_nonneg
tff(fact_5085_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_5086_cosh__real__ge__1,axiom,
    ! [Xb: real] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),cosh(real,Xb)) ).

% cosh_real_ge_1
tff(fact_5087_cis__mult,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),cis(A2)),cis(B2)) = cis(aa(real,real,aa(real,fun(real,real),plus_plus(real),A2),B2)) ).

% cis_mult
tff(fact_5088_cis__divide,axiom,
    ! [A2: real,B2: real] : aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),cis(A2)),cis(B2)) = cis(aa(real,real,minus_minus(real,A2),B2)) ).

% cis_divide
tff(fact_5089_cosh__real__strict__mono,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$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_5090_cosh__real__nonneg__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( aa(real,$o,aa(real,fun(real,$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_5091_cosh__real__nonpos__less__iff,axiom,
    ! [Xb: real,Y: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
     => ( aa(real,$o,aa(real,fun(real,$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_5092_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_5093_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_5094_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_5095_arcosh__cosh__real,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => ( arcosh(real,cosh(real,Xb)) = Xb ) ) ).

% arcosh_cosh_real
tff(fact_5096_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_5097_DeMoivre,axiom,
    ! [A2: real,Nb: nat] : aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),cis(A2)),Nb) = cis(aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),A2)) ).

% DeMoivre
tff(fact_5098_cis__conv__exp,axiom,
    ! [B2: real] : cis(B2) = exp(complex,aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),imaginary_unit),real_Vector_of_real(complex,B2))) ).

% cis_conv_exp
tff(fact_5099_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) )
           => ( 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),tanh(A,Xb)),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),tanh(A,Xb)),tanh(A,Y)))) ) ) ) ) ).

% tanh_add
tff(fact_5100_sinh__zero__iff,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Xb: A] :
          ( ( sinh(A,Xb) = zero_zero(A) )
        <=> member(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_5101_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),exp(A,Z)),exp(A,aa(A,A,uminus_uminus(A),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% cosh_field_def
tff(fact_5102_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,exp(A,Z)),exp(A,aa(A,A,uminus_uminus(A),Z)))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% sinh_field_def
tff(fact_5103_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),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_5104_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),exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb)))) ) ).

% cosh_def
tff(fact_5105_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_5106_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,exp(A,Xb)),exp(A,aa(A,A,uminus_uminus(A),Xb)))) ) ).

% sinh_def
tff(fact_5107_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_5108_bij__betw__roots__unity,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => bij_betw(nat,complex,aTP_Lamp_adm(nat,fun(nat,complex),Nb),set_ord_lessThan(nat,Nb),collect(complex,aTP_Lamp_vo(nat,fun(complex,$o),Nb))) ) ).

% bij_betw_roots_unity
tff(fact_5109_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,cot(real,Xb)),zero_zero(real)) ) ) ).

% cot_less_zero
tff(fact_5110_sint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(int,$o,aa(int,fun(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_5111_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_5112_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_5113_cot__npi,axiom,
    ! [Nb: nat] : cot(real,aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),pi)) = zero_zero(real) ).

% cot_npi
tff(fact_5114_cot__periodic,axiom,
    ! [Xb: 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))) = cot(real,Xb) ).

% cot_periodic
tff(fact_5115_sum_Oreindex__bij__betw,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [H: fun(A,B),S2: set(A),T5: set(B),G: fun(B,C)] :
          ( bij_betw(A,B,H,S2,T5)
         => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_adn(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S2) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),T5) ) ) ) ).

% sum.reindex_bij_betw
tff(fact_5116_prod_Oreindex__bij__betw,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [H: fun(A,B),S2: set(A),T5: set(B),G: fun(B,C)] :
          ( bij_betw(A,B,H,S2,T5)
         => ( groups7121269368397514597t_prod(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ado(fun(A,B),fun(fun(B,C),fun(A,C)),H),G),S2) = groups7121269368397514597t_prod(B,C,G,T5) ) ) ) ).

% prod.reindex_bij_betw
tff(fact_5117_sum_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [S6: set(A),T7: set(B),H: fun(A,B),S2: set(A),T5: set(B),G: fun(B,C)] :
          ( finite_finite2(A,S6)
         => ( finite_finite2(B,T7)
           => ( bij_betw(A,B,H,aa(set(A),set(A),minus_minus(set(A),S2),S6),aa(set(B),set(B),minus_minus(set(B),T5),T7))
             => ( ! [A4: A] :
                    ( member(A,A4,S6)
                   => ( aa(B,C,G,aa(A,B,H,A4)) = zero_zero(C) ) )
               => ( ! [B3: B] :
                      ( member(B,B3,T7)
                     => ( aa(B,C,G,B3) = zero_zero(C) ) )
                 => ( aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_adn(fun(A,B),fun(fun(B,C),fun(A,C)),H),G)),S2) = aa(set(B),C,groups7311177749621191930dd_sum(B,C,G),T5) ) ) ) ) ) ) ) ).

% sum.reindex_bij_betw_not_neutral
tff(fact_5118_prod_Oreindex__bij__betw__not__neutral,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [S6: set(A),T7: set(B),H: fun(A,B),S2: set(A),T5: set(B),G: fun(B,C)] :
          ( finite_finite2(A,S6)
         => ( finite_finite2(B,T7)
           => ( bij_betw(A,B,H,aa(set(A),set(A),minus_minus(set(A),S2),S6),aa(set(B),set(B),minus_minus(set(B),T5),T7))
             => ( ! [A4: A] :
                    ( member(A,A4,S6)
                   => ( aa(B,C,G,aa(A,B,H,A4)) = one_one(C) ) )
               => ( ! [B3: B] :
                      ( member(B,B3,T7)
                     => ( aa(B,C,G,B3) = one_one(C) ) )
                 => ( groups7121269368397514597t_prod(A,C,aa(fun(B,C),fun(A,C),aTP_Lamp_ado(fun(A,B),fun(fun(B,C),fun(A,C)),H),G),S2) = groups7121269368397514597t_prod(B,C,G,T5) ) ) ) ) ) ) ) ).

% prod.reindex_bij_betw_not_neutral
tff(fact_5119_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_5120_cot__def,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [X3: 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_5121_sint__above__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Xb: int] :
          ( aa(int,$o,aa(int,fun(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_5122_sint__below__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int,W: word(A)] :
          ( aa(int,$o,aa(int,fun(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,aa(int,fun(int,$o),ord_less_eq(int),Xb),ring_1_signed(A,int,W)) ) ) ).

% sint_below_size
tff(fact_5123_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)),cot(real,Xb)) ) ) ).

% cot_gt_zero
tff(fact_5124_tan__cot_H,axiom,
    ! [Xb: 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)) = cot(real,Xb) ).

% tan_cot'
tff(fact_5125_infinite__imp__bij__betw,axiom,
    ! [A: $tType,A3: set(A),A2: A] :
      ( ~ finite_finite2(A,A3)
     => ? [H2: fun(A,A)] : bij_betw(A,A,H2,A3,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A))))) ) ).

% infinite_imp_bij_betw
tff(fact_5126_predicate1I,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ! [X2: A] :
          ( aa(A,$o,P,X2)
         => aa(A,$o,Q,X2) )
     => aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q) ) ).

% predicate1I
tff(fact_5127_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_5128_dual__order_Orefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),A2) ) ).

% dual_order.refl
tff(fact_5129_order__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Xb) ) ).

% order_refl
tff(fact_5130_predicate2I,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
      ( ! [X2: A,Y5: B] :
          ( aa(B,$o,aa(A,fun(B,$o),P,X2),Y5)
         => aa(B,$o,aa(A,fun(B,$o),Q,X2),Y5) )
     => aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q) ) ).

% predicate2I
tff(fact_5131_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_5132_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_5133_semiring__norm_I26_J,axiom,
    bitM(one2) = one2 ).

% semiring_norm(26)
tff(fact_5134_scast__n1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( ring_1_signed(B,word(A),aa(word(B),word(B),uminus_uminus(word(B)),one_one(word(B)))) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ).

% scast_n1
tff(fact_5135_semiring__norm_I28_J,axiom,
    ! [Nb: num] : bitM(aa(num,num,bit1,Nb)) = aa(num,num,bit1,bit0(Nb)) ).

% semiring_norm(28)
tff(fact_5136_semiring__norm_I27_J,axiom,
    ! [Nb: num] : bitM(bit0(Nb)) = aa(num,num,bit1,bitM(Nb)) ).

% semiring_norm(27)
tff(fact_5137_order__antisym__conv,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% order_antisym_conv
tff(fact_5138_linorder__le__cases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% linorder_le_cases
tff(fact_5139_ord__le__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( aa(A,B,F,B2) = C2 )
           => ( ! [X2: A,Y5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,F,Y5)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,A2)),C2) ) ) ) ) ).

% ord_le_eq_subst
tff(fact_5140_ord__eq__le__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F,B2) )
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X2: B,Y5: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,F,Y5)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F,C2)) ) ) ) ) ).

% ord_eq_le_subst
tff(fact_5141_linorder__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
          | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% linorder_linear
tff(fact_5142_order__eq__refl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb = Y )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% order_eq_refl
tff(fact_5143_order__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,B2)),C2)
           => ( ! [X2: A,Y5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,F,Y5)) )
             => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,A2)),C2) ) ) ) ) ).

% order_subst2
tff(fact_5144_order__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X2: B,Y5: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,F,Y5)) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F,C2)) ) ) ) ) ).

% order_subst1
tff(fact_5145_Orderings_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% Orderings.order_eq_iff
tff(fact_5146_antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
           => ( A2 = B2 ) ) ) ) ).

% antisym
tff(fact_5147_dual__order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2) ) ) ) ).

% dual_order.trans
tff(fact_5148_dual__order_Oantisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( A2 = B2 ) ) ) ) ).

% dual_order.antisym
tff(fact_5149_dual__order_Oeq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = B2 )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% dual_order.eq_iff
tff(fact_5150_linorder__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A2: A,B2: A] :
          ( ! [A4: A,B3: A] :
              ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B3)
             => aa(A,$o,aa(A,fun(A,$o),P,A4),B3) )
         => ( ! [A4: A,B3: A] :
                ( aa(A,$o,aa(A,fun(A,$o),P,B3),A4)
               => aa(A,$o,aa(A,fun(A,$o),P,A4),B3) )
           => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) ) ) ) ).

% linorder_wlog
tff(fact_5151_order__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z) ) ) ) ).

% order_trans
tff(fact_5152_order_Otrans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% order.trans
tff(fact_5153_order__antisym,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
           => ( Xb = Y ) ) ) ) ).

% order_antisym
tff(fact_5154_ord__le__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( B2 = C2 )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% ord_le_eq_trans
tff(fact_5155_ord__eq__le__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% ord_eq_le_trans
tff(fact_5156_order__class_Oorder__eq__iff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( ( Xb = Y )
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ) ).

% order_class.order_eq_iff
tff(fact_5157_le__cases3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
           => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) )
         => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z) )
           => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z)
               => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y) )
             => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Y)
                 => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) )
               => ( ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
                   => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xb) )
                 => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Z),Xb)
                     => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ) ) ) ) ) ).

% le_cases3
tff(fact_5158_nle__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ( B2 != A2 ) ) ) ) ).

% nle_le
tff(fact_5159_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_5160_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_5161_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_5162_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_5163_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_5164_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_5165_order__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(B,$o,ord_less(B,aa(A,B,F,B2)),C2)
           => ( ! [X2: A,Y5: A] :
                  ( aa(A,$o,ord_less(A,X2),Y5)
                 => aa(B,$o,ord_less(B,aa(A,B,F,X2)),aa(A,B,F,Y5)) )
             => aa(B,$o,ord_less(B,aa(A,B,F,A2)),C2) ) ) ) ) ).

% order_less_subst2
tff(fact_5166_order__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,ord_less(A,A2),aa(B,A,F,B2))
         => ( aa(B,$o,ord_less(B,B2),C2)
           => ( ! [X2: B,Y5: B] :
                  ( aa(B,$o,ord_less(B,X2),Y5)
                 => aa(A,$o,ord_less(A,aa(B,A,F,X2)),aa(B,A,F,Y5)) )
             => aa(A,$o,ord_less(A,A2),aa(B,A,F,C2)) ) ) ) ) ).

% order_less_subst1
tff(fact_5167_order__less__irrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A] : ~ aa(A,$o,ord_less(A,Xb),Xb) ) ).

% order_less_irrefl
tff(fact_5168_ord__less__eq__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( ( aa(A,B,F,B2) = C2 )
           => ( ! [X2: A,Y5: A] :
                  ( aa(A,$o,ord_less(A,X2),Y5)
                 => aa(B,$o,ord_less(B,aa(A,B,F,X2)),aa(A,B,F,Y5)) )
             => aa(B,$o,ord_less(B,aa(A,B,F,A2)),C2) ) ) ) ) ).

% ord_less_eq_subst
tff(fact_5169_ord__eq__less__subst,axiom,
    ! [A: $tType,B: $tType] :
      ( ( ord(B)
        & ord(A) )
     => ! [A2: A,F: fun(B,A),B2: B,C2: B] :
          ( ( A2 = aa(B,A,F,B2) )
         => ( aa(B,$o,ord_less(B,B2),C2)
           => ( ! [X2: B,Y5: B] :
                  ( aa(B,$o,ord_less(B,X2),Y5)
                 => aa(A,$o,ord_less(A,aa(B,A,F,X2)),aa(B,A,F,Y5)) )
             => aa(A,$o,ord_less(A,A2),aa(B,A,F,C2)) ) ) ) ) ).

% ord_eq_less_subst
tff(fact_5170_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_5171_order__less__asym_H,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ~ aa(A,$o,ord_less(A,B2),A2) ) ) ).

% order_less_asym'
tff(fact_5172_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_5173_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_5174_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_5175_dual__order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
         => ( A2 != B2 ) ) ) ).

% dual_order.strict_implies_not_eq
tff(fact_5176_order_Ostrict__implies__not__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( A2 != B2 ) ) ) ).

% order.strict_implies_not_eq
tff(fact_5177_dual__order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
         => ( aa(A,$o,ord_less(A,C2),B2)
           => aa(A,$o,ord_less(A,C2),A2) ) ) ) ).

% dual_order.strict_trans
tff(fact_5178_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_5179_order_Ostrict__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,ord_less(A,B2),C2)
           => aa(A,$o,ord_less(A,A2),C2) ) ) ) ).

% order.strict_trans
tff(fact_5180_linorder__less__wlog,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [P: fun(A,fun(A,$o)),A2: A,B2: A] :
          ( ! [A4: A,B3: A] :
              ( aa(A,$o,ord_less(A,A4),B3)
             => aa(A,$o,aa(A,fun(A,$o),P,A4),B3) )
         => ( ! [A4: A] : aa(A,$o,aa(A,fun(A,$o),P,A4),A4)
           => ( ! [A4: A,B3: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),P,B3),A4)
                 => aa(A,$o,aa(A,fun(A,$o),P,A4),B3) )
             => aa(A,$o,aa(A,fun(A,$o),P,A2),B2) ) ) ) ) ).

% linorder_less_wlog
tff(fact_5181_exists__least__iff,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o)] :
          ( ? [X_1: A] : aa(A,$o,P,X_1)
        <=> ? [N6: A] :
              ( aa(A,$o,P,N6)
              & ! [M7: A] :
                  ( aa(A,$o,ord_less(A,M7),N6)
                 => ~ aa(A,$o,P,M7) ) ) ) ) ).

% exists_least_iff
tff(fact_5182_dual__order_Oirrefl,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A] : ~ aa(A,$o,ord_less(A,A2),A2) ) ).

% dual_order.irrefl
tff(fact_5183_dual__order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
         => ~ aa(A,$o,ord_less(A,A2),B2) ) ) ).

% dual_order.asym
tff(fact_5184_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_5185_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_5186_less__induct,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A2: 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,A2) ) ) ).

% less_induct
tff(fact_5187_ord__less__eq__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( ( B2 = C2 )
           => aa(A,$o,ord_less(A,A2),C2) ) ) ) ).

% ord_less_eq_trans
tff(fact_5188_ord__eq__less__trans,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A,C2: A] :
          ( ( A2 = B2 )
         => ( aa(A,$o,ord_less(A,B2),C2)
           => aa(A,$o,ord_less(A,A2),C2) ) ) ) ).

% ord_eq_less_trans
tff(fact_5189_order_Oasym,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ~ aa(A,$o,ord_less(A,B2),A2) ) ) ).

% order.asym
tff(fact_5190_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_5191_dense,axiom,
    ! [A: $tType] :
      ( dense_order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
         => ? [Z3: A] :
              ( aa(A,$o,ord_less(A,Xb),Z3)
              & aa(A,$o,ord_less(A,Z3),Y) ) ) ) ).

% dense
tff(fact_5192_gt__ex,axiom,
    ! [A: $tType] :
      ( no_top(A)
     => ! [Xb: A] :
        ? [X_13: A] : aa(A,$o,ord_less(A,Xb),X_13) ) ).

% gt_ex
tff(fact_5193_lt__ex,axiom,
    ! [A: $tType] :
      ( no_bot(A)
     => ! [Xb: A] :
        ? [Y5: A] : aa(A,$o,ord_less(A,Y5),Xb) ) ).

% lt_ex
tff(fact_5194_eval__nat__numeral_I2_J,axiom,
    ! [Nb: num] : aa(num,nat,numeral_numeral(nat),bit0(Nb)) = aa(nat,nat,suc,aa(num,nat,numeral_numeral(nat),bitM(Nb))) ).

% eval_nat_numeral(2)
tff(fact_5195_one__plus__BitM,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),one2),bitM(Nb)) = bit0(Nb) ).

% one_plus_BitM
tff(fact_5196_BitM__plus__one,axiom,
    ! [Nb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),bitM(Nb)),one2) = bit0(Nb) ).

% BitM_plus_one
tff(fact_5197_less__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,ord_less(fun(A,B),F),G)
        <=> ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G)
            & ~ aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),G),F) ) ) ) ).

% less_fun_def
tff(fact_5198_predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o)),Xb: A,Y: B] :
      ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
     => ( aa(B,$o,aa(A,fun(B,$o),P,Xb),Y)
       => aa(B,$o,aa(A,fun(B,$o),Q,Xb),Y) ) ) ).

% predicate2D
tff(fact_5199_rev__predicate2D,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Xb: A,Y: B,Q: fun(A,fun(B,$o))] :
      ( aa(B,$o,aa(A,fun(B,$o),P,Xb),Y)
     => ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
       => aa(B,$o,aa(A,fun(B,$o),Q,Xb),Y) ) ) ).

% rev_predicate2D
tff(fact_5200_predicate1D,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o),Xb: A] :
      ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q)
     => ( aa(A,$o,P,Xb)
       => aa(A,$o,Q,Xb) ) ) ).

% predicate1D
tff(fact_5201_rev__predicate1D,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,Xb)
     => ( aa(fun(A,$o),$o,aa(fun(A,$o),fun(fun(A,$o),$o),ord_less_eq(fun(A,$o)),P),Q)
       => aa(A,$o,Q,Xb) ) ) ).

% rev_predicate1D
tff(fact_5202_numeral__BitM,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : aa(num,A,numeral_numeral(A),bitM(Nb)) = aa(A,A,minus_minus(A,aa(num,A,numeral_numeral(A),bit0(Nb))),one_one(A)) ) ).

% numeral_BitM
tff(fact_5203_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_5204_order__le__imp__less__or__eq,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(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_5205_linorder__le__less__linear,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
          | aa(A,$o,ord_less(A,Y),Xb) ) ) ).

% linorder_le_less_linear
tff(fact_5206_order__less__le__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,B2)),C2)
           => ( ! [X2: A,Y5: A] :
                  ( aa(A,$o,ord_less(A,X2),Y5)
                 => aa(B,$o,ord_less(B,aa(A,B,F,X2)),aa(A,B,F,Y5)) )
             => aa(B,$o,ord_less(B,aa(A,B,F,A2)),C2) ) ) ) ) ).

% order_less_le_subst2
tff(fact_5207_order__less__le__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,ord_less(A,A2),aa(B,A,F,B2))
         => ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),B2),C2)
           => ( ! [X2: B,Y5: B] :
                  ( aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),X2),Y5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(B,A,F,X2)),aa(B,A,F,Y5)) )
             => aa(A,$o,ord_less(A,A2),aa(B,A,F,C2)) ) ) ) ) ).

% order_less_le_subst1
tff(fact_5208_order__le__less__subst2,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,B2: A,F: fun(A,B),C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(B,$o,ord_less(B,aa(A,B,F,B2)),C2)
           => ( ! [X2: A,Y5: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y5)
                 => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,F,Y5)) )
             => aa(B,$o,ord_less(B,aa(A,B,F,A2)),C2) ) ) ) ) ).

% order_le_less_subst2
tff(fact_5209_order__le__less__subst1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( order(B)
        & order(A) )
     => ! [A2: A,F: fun(B,A),B2: B,C2: B] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),aa(B,A,F,B2))
         => ( aa(B,$o,ord_less(B,B2),C2)
           => ( ! [X2: B,Y5: B] :
                  ( aa(B,$o,ord_less(B,X2),Y5)
                 => aa(A,$o,ord_less(A,aa(B,A,F,X2)),aa(B,A,F,Y5)) )
             => aa(A,$o,ord_less(A,A2),aa(B,A,F,C2)) ) ) ) ) ).

% order_le_less_subst1
tff(fact_5210_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,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z)
           => aa(A,$o,ord_less(A,Xb),Z) ) ) ) ).

% order_less_le_trans
tff(fact_5211_order__le__less__trans,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(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_5212_order__neq__le__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ( A2 != B2 )
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => aa(A,$o,ord_less(A,A2),B2) ) ) ) ).

% order_neq_le_trans
tff(fact_5213_order__le__neq__trans,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ( A2 != B2 )
           => aa(A,$o,ord_less(A,A2),B2) ) ) ) ).

% order_le_neq_trans
tff(fact_5214_order__less__imp__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y) ) ) ).

% order_less_imp_le
tff(fact_5215_linorder__not__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,Xb),Y)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% linorder_not_less
tff(fact_5216_linorder__not__le,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
        <=> aa(A,$o,ord_less(A,Y),Xb) ) ) ).

% linorder_not_le
tff(fact_5217_order__less__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & ( Xb != Y ) ) ) ) ).

% order_less_le
tff(fact_5218_order__le__less,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
        <=> ( aa(A,$o,ord_less(A,Xb),Y)
            | ( Xb = Y ) ) ) ) ).

% order_le_less
tff(fact_5219_dual__order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ).

% dual_order.strict_implies_order
tff(fact_5220_order_Ostrict__implies__order,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% order.strict_implies_order
tff(fact_5221_dual__order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ) ).

% dual_order.strict_iff_not
tff(fact_5222_dual__order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),B2)
           => aa(A,$o,ord_less(A,C2),A2) ) ) ) ).

% dual_order.strict_trans2
tff(fact_5223_dual__order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [B2: A,A2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( aa(A,$o,ord_less(A,C2),B2)
           => aa(A,$o,ord_less(A,C2),A2) ) ) ) ).

% dual_order.strict_trans1
tff(fact_5224_dual__order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            & ( A2 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
tff(fact_5225_dual__order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( aa(A,$o,ord_less(A,B2),A2)
            | ( A2 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
tff(fact_5226_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,$o,aa(A,fun(A,$o),ord_less_eq(A),W2),Z) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% dense_le_bounded
tff(fact_5227_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,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),W2) ) )
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% dense_ge_bounded
tff(fact_5228_order_Ostrict__iff__not,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2) ) ) ) ).

% order.strict_iff_not
tff(fact_5229_order_Ostrict__trans2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
           => aa(A,$o,ord_less(A,A2),C2) ) ) ) ).

% order.strict_trans2
tff(fact_5230_order_Ostrict__trans1,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,ord_less(A,B2),C2)
           => aa(A,$o,ord_less(A,A2),C2) ) ) ) ).

% order.strict_trans1
tff(fact_5231_order_Ostrict__iff__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & ( A2 != B2 ) ) ) ) ).

% order.strict_iff_order
tff(fact_5232_order_Oorder__iff__strict,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( aa(A,$o,ord_less(A,A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% order.order_iff_strict
tff(fact_5233_not__le__imp__less,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Y: A,Xb: A] :
          ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => aa(A,$o,ord_less(A,Xb),Y) ) ) ).

% not_le_imp_less
tff(fact_5234_less__le__not__le,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,ord_less(A,Xb),Y)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
            & ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ) ).

% less_le_not_le
tff(fact_5235_dense__le,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Y: A,Z: A] :
          ( ! [X2: A] :
              ( aa(A,$o,ord_less(A,X2),Y)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Z) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ).

% dense_le
tff(fact_5236_dense__ge,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [Z: A,Y: A] :
          ( ! [X2: A] :
              ( aa(A,$o,ord_less(A,Z),X2)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),X2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ).

% dense_ge
tff(fact_5237_antisym__conv2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( ~ aa(A,$o,ord_less(A,Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% antisym_conv2
tff(fact_5238_antisym__conv1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,Xb),Y)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
          <=> ( Xb = Y ) ) ) ) ).

% antisym_conv1
tff(fact_5239_nless__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( ~ aa(A,$o,ord_less(A,A2),B2)
        <=> ( ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            | ( A2 = B2 ) ) ) ) ).

% nless_le
tff(fact_5240_leI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A] :
          ( ~ aa(A,$o,ord_less(A,Xb),Y)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb) ) ) ).

% leI
tff(fact_5241_leD,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ~ aa(A,$o,ord_less(A,Xb),Y) ) ) ).

% leD
tff(fact_5242_bot_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),bot_bot(A))
         => ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_uniqueI
tff(fact_5243_bot_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),bot_bot(A))
        <=> ( A2 = bot_bot(A) ) ) ) ).

% bot.extremum_unique
tff(fact_5244_bot_Oextremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),bot_bot(A)),A2) ) ).

% bot.extremum
tff(fact_5245_top__greatest,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),top_top(A)) ) ).

% top_greatest
tff(fact_5246_top_Oextremum__unique,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A2)
        <=> ( A2 = top_top(A) ) ) ) ).

% top.extremum_unique
tff(fact_5247_top_Oextremum__uniqueI,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),top_top(A)),A2)
         => ( A2 = top_top(A) ) ) ) ).

% top.extremum_uniqueI
tff(fact_5248_bot_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] : ~ aa(A,$o,ord_less(A,A2),bot_bot(A)) ) ).

% bot.extremum_strict
tff(fact_5249_bot_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_bot(A)
     => ! [A2: A] :
          ( ( A2 != bot_bot(A) )
        <=> aa(A,$o,ord_less(A,bot_bot(A)),A2) ) ) ).

% bot.not_eq_extremum
tff(fact_5250_top_Oextremum__strict,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] : ~ aa(A,$o,ord_less(A,top_top(A)),A2) ) ).

% top.extremum_strict
tff(fact_5251_top_Onot__eq__extremum,axiom,
    ! [A: $tType] :
      ( order_top(A)
     => ! [A2: A] :
          ( ( A2 != top_top(A) )
        <=> aa(A,$o,ord_less(A,A2),top_top(A)) ) ) ).

% top.not_eq_extremum
tff(fact_5252_dependent__nat__choice,axiom,
    ! [A: $tType,P: fun(nat,fun(A,$o)),Q: fun(nat,fun(A,fun(A,$o)))] :
      ( ? [X_12: A] : aa(A,$o,aa(nat,fun(A,$o),P,zero_zero(nat)),X_12)
     => ( ! [X2: A,N2: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N2),X2)
           => ? [Y4: A] :
                ( aa(A,$o,aa(nat,fun(A,$o),P,aa(nat,nat,suc,N2)),Y4)
                & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N2),X2),Y4) ) )
       => ? [F2: fun(nat,A)] :
          ! [N8: nat] :
            ( aa(A,$o,aa(nat,fun(A,$o),P,N8),aa(nat,A,F2,N8))
            & aa(A,$o,aa(A,fun(A,$o),aa(nat,fun(A,fun(A,$o)),Q,N8),aa(nat,A,F2,N8)),aa(nat,A,F2,aa(nat,nat,suc,N8))) ) ) ) ).

% dependent_nat_choice
tff(fact_5253_max__absorb2,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( aa(A,A,ord_max(A,Xb),Y) = Y ) ) ) ).

% max_absorb2
tff(fact_5254_max__absorb1,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( aa(A,A,ord_max(A,Xb),Y) = Xb ) ) ) ).

% max_absorb1
tff(fact_5255_max__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          aa(A,A,ord_max(A,A2),B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2),B2,A2) ) ).

% max_def
tff(fact_5256_le__fun__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),G: fun(A,B)] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G)
        <=> ! [X: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X)),aa(A,B,G,X)) ) ) ).

% le_fun_def
tff(fact_5257_le__funI,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),G: fun(A,B)] :
          ( ! [X2: A] : aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,X2)),aa(A,B,G,X2))
         => aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G) ) ) ).

% le_funI
tff(fact_5258_le__funE,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),G: fun(A,B),Xb: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Xb)),aa(A,B,G,Xb)) ) ) ).

% le_funE
tff(fact_5259_le__funD,axiom,
    ! [B: $tType,A: $tType] :
      ( ord(B)
     => ! [F: fun(A,B),G: fun(A,B),Xb: A] :
          ( aa(fun(A,B),$o,aa(fun(A,B),fun(fun(A,B),$o),ord_less_eq(fun(A,B)),F),G)
         => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,F,Xb)),aa(A,B,G,Xb)) ) ) ).

% le_funD
tff(fact_5260_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,Nb: nat] :
      ( ( C2 != zero_zero(complex) )
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => bij_betw(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),real_Vector_of_real(complex,aa(real,real,root(Nb),real_V7770717601297561774m_norm(complex,C2)))),cis(aa(real,real,aa(real,fun(real,real),divide_divide(real),arg(C2)),aa(nat,real,semiring_1_of_nat(real),Nb))))),collect(complex,aTP_Lamp_vo(nat,fun(complex,$o),Nb)),collect(complex,aa(nat,fun(complex,$o),aTP_Lamp_adp(complex,fun(nat,fun(complex,$o)),C2),Nb))) ) ) ).

% bij_betw_nth_root_unity
tff(fact_5261_assnle,axiom,
    ! [TreeLista: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),X13a: array(vEBT_VEBTi),Summarya: vEBT_VEBT,X14a: 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,TreeLista),Tree_is)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13a),Tree_is)),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summarya),X14a)))),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,Summarya),X14a)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13a),Tree_is))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeLista),Tree_is))) ).

% assnle
tff(fact_5262_int__ge__less__than2__def,axiom,
    ! [D2: int] : int_ge_less_than2(D2) = collect(product_prod(int,int),product_case_prod(int,int,$o,aTP_Lamp_adq(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than2_def
tff(fact_5263_real__root__zero,axiom,
    ! [Nb: nat] : aa(real,real,root(Nb),zero_zero(real)) = zero_zero(real) ).

% real_root_zero
tff(fact_5264_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_5265_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_5266_real__root__eq__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( ( aa(real,real,root(Nb),Xb) = aa(real,real,root(Nb),Y) )
      <=> ( Xb = Y ) ) ) ).

% real_root_eq_iff
tff(fact_5267_root__0,axiom,
    ! [Xb: real] : aa(real,real,root(zero_zero(nat)),Xb) = zero_zero(real) ).

% root_0
tff(fact_5268_ent__pure__pre__iff,axiom,
    ! [P: assn,B2: $o,Q: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),pure_assn((B2)))),Q)
    <=> ( (B2)
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,P),Q) ) ) ).

% ent_pure_pre_iff
tff(fact_5269_ent__pure__pre__iff__sng,axiom,
    ! [B2: $o,Q: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,pure_assn((B2))),Q)
    <=> ( (B2)
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,one_one(assn)),Q) ) ) ).

% ent_pure_pre_iff_sng
tff(fact_5270_real__root__eq__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( ( aa(real,real,root(Nb),Xb) = zero_zero(real) )
      <=> ( Xb = zero_zero(real) ) ) ) ).

% real_root_eq_0_iff
tff(fact_5271_real__root__less__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,ord_less(real,Xb),Y) ) ) ).

% real_root_less_iff
tff(fact_5272_real__root__le__iff,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y) ) ) ).

% real_root_le_iff
tff(fact_5273_real__root__eq__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( ( aa(real,real,root(Nb),Xb) = one_one(real) )
      <=> ( Xb = one_one(real) ) ) ) ).

% real_root_eq_1_iff
tff(fact_5274_real__root__one,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),one_one(real)) = one_one(real) ) ) ).

% real_root_one
tff(fact_5275_ent__pure__post__iff,axiom,
    ! [P: assn,Q: assn,B2: $o] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),pure_assn((B2))))
    <=> ( ! [H4: product_prod(heap_ext(product_unit),set(nat))] :
            ( rep_assn(P,H4)
           => (B2) )
        & aa(assn,$o,aa(assn,fun(assn,$o),entails,P),Q) ) ) ).

% ent_pure_post_iff
tff(fact_5276_ent__pure__post__iff__sng,axiom,
    ! [P: assn,B2: $o] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),pure_assn((B2)))
    <=> ( ! [H4: product_prod(heap_ext(product_unit),set(nat))] :
            ( rep_assn(P,H4)
           => (B2) )
        & aa(assn,$o,aa(assn,fun(assn,$o),entails,P),one_one(assn)) ) ) ).

% ent_pure_post_iff_sng
tff(fact_5277_real__root__lt__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,aa(real,real,root(Nb),Xb)),zero_zero(real))
      <=> aa(real,$o,ord_less(real,Xb),zero_zero(real)) ) ) ).

% real_root_lt_0_iff
tff(fact_5278_real__root__gt__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,ord_less(real,zero_zero(real)),Y) ) ) ).

% real_root_gt_0_iff
tff(fact_5279_real__root__ge__0__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y) ) ) ).

% real_root_ge_0_iff
tff(fact_5280_real__root__le__0__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),zero_zero(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ) ).

% real_root_le_0_iff
tff(fact_5281_real__root__lt__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,aa(real,real,root(Nb),Xb)),one_one(real))
      <=> aa(real,$o,ord_less(real,Xb),one_one(real)) ) ) ).

% real_root_lt_1_iff
tff(fact_5282_real__root__gt__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,one_one(real)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,ord_less(real,one_one(real)),Y) ) ) ).

% real_root_gt_1_iff
tff(fact_5283_real__root__le__1__iff,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),one_one(real))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real)) ) ) ).

% real_root_le_1_iff
tff(fact_5284_real__root__ge__1__iff,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),aa(real,real,root(Nb),Y))
      <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Y) ) ) ).

% real_root_ge_1_iff
tff(fact_5285_real__root__pow__pos2,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(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(Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos2
tff(fact_5286_real__root__inverse,axiom,
    ! [Nb: nat,Xb: real] : aa(real,real,root(Nb),aa(real,real,inverse_inverse(real),Xb)) = aa(real,real,inverse_inverse(real),aa(real,real,root(Nb),Xb)) ).

% real_root_inverse
tff(fact_5287_fr__rot,axiom,
    ! [A3: assn,B5: assn,C5: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),B5)),C5)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B5),A3)),C5) ) ).

% fr_rot
tff(fact_5288_fr__refl,axiom,
    ! [A3: assn,B5: assn,C5: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,A3),B5)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),C5)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B5),C5)) ) ).

% fr_refl
tff(fact_5289_fr__rot__rhs,axiom,
    ! [A3: assn,B5: assn,C5: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,A3),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B5),C5))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,A3),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),C5),B5)) ) ).

% fr_rot_rhs
tff(fact_5290_ent__frame__fwd,axiom,
    ! [P: assn,R: assn,Ps: assn,F5: assn,Q: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),R)
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Ps),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),F5))
       => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R),F5)),Q)
         => aa(assn,$o,aa(assn,fun(assn,$o),entails,Ps),Q) ) ) ) ).

% ent_frame_fwd
tff(fact_5291_ent__star__mono,axiom,
    ! [P: assn,P3: assn,Q: assn,Q6: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),P3)
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Q),Q6)
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),Q)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P3),Q6)) ) ) ).

% ent_star_mono
tff(fact_5292_real__root__mult,axiom,
    ! [Nb: nat,Xb: real,Y: real] : aa(real,real,root(Nb),aa(real,real,aa(real,fun(real,real),times_times(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y)) ).

% real_root_mult
tff(fact_5293_real__root__mult__exp,axiom,
    ! [M: nat,Nb: nat,Xb: real] : aa(real,real,root(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb)),Xb) = aa(real,real,root(M),aa(real,real,root(Nb),Xb)) ).

% real_root_mult_exp
tff(fact_5294_real__root__minus,axiom,
    ! [Nb: nat,Xb: real] : aa(real,real,root(Nb),aa(real,real,uminus_uminus(real),Xb)) = aa(real,real,uminus_uminus(real),aa(real,real,root(Nb),Xb)) ).

% real_root_minus
tff(fact_5295_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_5296_cons__rule,axiom,
    ! [A: $tType,P: assn,P3: assn,Q: fun(A,assn),Q6: fun(A,assn),C2: heap_Time_Heap(A)] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),P3)
     => ( ! [X2: A] : aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(A,assn,Q,X2)),aa(A,assn,Q6,X2))
       => ( hoare_hoare_triple(A,P3,C2,Q)
         => hoare_hoare_triple(A,P,C2,Q6) ) ) ) ).

% cons_rule
tff(fact_5297_cons__post__rule,axiom,
    ! [A: $tType,P: assn,C2: heap_Time_Heap(A),Q: fun(A,assn),Q6: fun(A,assn)] :
      ( hoare_hoare_triple(A,P,C2,Q)
     => ( ! [X2: A] : aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(A,assn,Q,X2)),aa(A,assn,Q6,X2))
       => hoare_hoare_triple(A,P,C2,Q6) ) ) ).

% cons_post_rule
tff(fact_5298_is__entails,axiom,
    ! [P: assn,Q: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),Q)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,P),Q) ) ).

% is_entails
tff(fact_5299_real__root__commute,axiom,
    ! [M: nat,Nb: nat,Xb: real] : aa(real,real,root(M),aa(real,real,root(Nb),Xb)) = aa(real,real,root(Nb),aa(real,real,root(M),Xb)) ).

% real_root_commute
tff(fact_5300_list__assn__mono,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,assn)),P3: fun(A,fun(B,assn)),L: list(A),L3: list(B)] :
      ( ! [X2: A,X6: B] : aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(B,assn,aa(A,fun(B,assn),P,X2),X6)),aa(B,assn,aa(A,fun(B,assn),P3,X2),X6))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(list(B),assn,vEBT_List_list_assn(A,B,P,L),L3)),aa(list(B),assn,vEBT_List_list_assn(A,B,P3,L),L3)) ) ).

% list_assn_mono
tff(fact_5301_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_5302_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_5303_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,$o),entails,P),aa(A,assn,Q,X2)) )
     => ( ? [X_12: A] : aa(A,$o,Z6,X_12)
       => aa(assn,$o,aa(assn,fun(assn,$o),entails,P),ex_assn(A,Q)) ) ) ).

% enorm_exI'
tff(fact_5304_real__root__divide,axiom,
    ! [Nb: nat,Xb: real,Y: real] : aa(real,real,root(Nb),aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),Y)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y)) ).

% real_root_divide
tff(fact_5305_real__root__pos__pos__le,axiom,
    ! [Xb: real,Nb: nat] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
     => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Nb),Xb)) ) ).

% real_root_pos_pos_le
tff(fact_5306_mod__frame__fwd,axiom,
    ! [Ps: assn,H: product_prod(heap_ext(product_unit),set(nat)),P: assn,R: assn,F5: assn] :
      ( rep_assn(Ps,H)
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),R)
       => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,Ps),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),F5))
         => rep_assn(aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),R),F5),H) ) ) ) ).

% mod_frame_fwd
tff(fact_5307_ent__star__mono__true,axiom,
    ! [A3: assn,A9: assn,B5: assn,B10: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,A3),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A9),top_top(assn)))
     => ( aa(assn,$o,aa(assn,fun(assn,$o),entails,B5),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B10),top_top(assn)))
       => 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),A3),B5)),top_top(assn))),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A9),B10)),top_top(assn))) ) ) ).

% ent_star_mono_true
tff(fact_5308_ent__refl__true,axiom,
    ! [A3: assn] : aa(assn,$o,aa(assn,fun(assn,$o),entails,A3),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A3),top_top(assn))) ).

% ent_refl_true
tff(fact_5309_ent__true__drop_I1_J,axiom,
    ! [P: assn,Q: assn,R: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),top_top(assn)))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P),R)),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),top_top(assn))) ) ).

% ent_true_drop(1)
tff(fact_5310_ent__true__drop_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),Q)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),top_top(assn))) ) ).

% ent_true_drop(2)
tff(fact_5311_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_5312_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_bf(fun(A,assn),fun(assn,fun(A,assn)),Q),F5)) ) ) ).

% fi_rule
tff(fact_5313_htt__cons__rule,axiom,
    ! [A: $tType,P3: assn,C2: heap_Time_Heap(A),Q6: fun(A,assn),T3: nat,P: assn,Q: fun(A,assn),Ta: nat] :
      ( time_htt(A,P3,C2,Q6,T3)
     => ( aa(assn,$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,aa(nat,fun(nat,$o),ord_less_eq(nat),T3),Ta)
           => time_htt(A,P,C2,Q,Ta) ) ) ) ) ).

% htt_cons_rule
tff(fact_5314_real__root__less__mono,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,Xb),Y)
       => aa(real,$o,ord_less(real,aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y)) ) ) ).

% real_root_less_mono
tff(fact_5315_real__root__le__mono,axiom,
    ! [Nb: nat,Xb: real,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),Y)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(Nb),Y)) ) ) ).

% real_root_le_mono
tff(fact_5316_real__root__power,axiom,
    ! [Nb: nat,Xb: real,K: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),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(Nb),Xb)),K) ) ) ).

% real_root_power
tff(fact_5317_real__root__abs,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),aa(real,real,abs_abs(real),Xb)) = aa(real,real,abs_abs(real),aa(real,real,root(Nb),Xb)) ) ) ).

% real_root_abs
tff(fact_5318_real__root__gt__zero,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => aa(real,$o,ord_less(real,zero_zero(real)),aa(real,real,root(Nb),Xb)) ) ) ).

% real_root_gt_zero
tff(fact_5319_real__root__strict__decreasing,axiom,
    ! [Nb: nat,N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,ord_less(nat,Nb),N)
       => ( aa(real,$o,ord_less(real,one_one(real)),Xb)
         => aa(real,$o,ord_less(real,aa(real,real,root(N),Xb)),aa(real,real,root(Nb),Xb)) ) ) ) ).

% real_root_strict_decreasing
tff(fact_5320_sqrt__def,axiom,
    sqrt = root(aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% sqrt_def
tff(fact_5321_root__abs__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,real,abs_abs(real),aa(real,real,root(Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),Nb))) = aa(real,real,abs_abs(real),Y) ) ) ).

% root_abs_power
tff(fact_5322_real__root__pos__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,root(Nb),Xb)) ) ) ).

% real_root_pos_pos
tff(fact_5323_real__root__strict__increasing,axiom,
    ! [Nb: nat,N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,ord_less(nat,Nb),N)
       => ( 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(Nb),Xb)),aa(real,real,root(N),Xb)) ) ) ) ) ).

% real_root_strict_increasing
tff(fact_5324_real__root__decreasing,axiom,
    ! [Nb: nat,N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),one_one(real)),Xb)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(N),Xb)),aa(real,real,root(Nb),Xb)) ) ) ) ).

% real_root_decreasing
tff(fact_5325_odd__real__root__pow,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),Nb) = Xb ) ) ).

% odd_real_root_pow
tff(fact_5326_odd__real__root__unique,axiom,
    ! [Nb: nat,Y: real,Xb: real] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),Nb) = Xb )
       => ( aa(real,real,root(Nb),Xb) = Y ) ) ) ).

% odd_real_root_unique
tff(fact_5327_odd__real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( ~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
     => ( aa(real,real,root(Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb)) = Xb ) ) ).

% odd_real_root_power_cancel
tff(fact_5328_real__root__pow__pos,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),Nb) = Xb ) ) ) ).

% real_root_pow_pos
tff(fact_5329_real__root__pos__unique,axiom,
    ! [Nb: nat,Y: real,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y)
       => ( ( aa(nat,real,aa(real,fun(nat,real),power_power(real),Y),Nb) = Xb )
         => ( aa(real,real,root(Nb),Xb) = Y ) ) ) ) ).

% real_root_pos_unique
tff(fact_5330_real__root__power__cancel,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
       => ( aa(real,real,root(Nb),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),Nb)) = Xb ) ) ) ).

% real_root_power_cancel
tff(fact_5331_real__root__increasing,axiom,
    ! [Nb: nat,N: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N)
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb)
         => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),one_one(real))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,root(Nb),Xb)),aa(real,real,root(N),Xb)) ) ) ) ) ).

% real_root_increasing
tff(fact_5332_log__root,axiom,
    ! [Nb: nat,A2: real,B2: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),A2)
       => ( log(B2,aa(real,real,root(Nb),A2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),log(B2,A2)),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% log_root
tff(fact_5333_log__base__root,axiom,
    ! [Nb: nat,B2: real,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),B2)
       => ( log(aa(real,real,root(Nb),B2),Xb) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),log(B2,Xb)) ) ) ) ).

% log_base_root
tff(fact_5334_ln__root,axiom,
    ! [Nb: nat,B2: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),B2)
       => ( aa(real,real,ln_ln(real),aa(real,real,root(Nb),B2)) = aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,ln_ln(real),B2)),aa(nat,real,semiring_1_of_nat(real),Nb)) ) ) ) ).

% ln_root
tff(fact_5335_root__powr__inverse,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,$o,ord_less(real,zero_zero(real)),Xb)
       => ( aa(real,real,root(Nb),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),Nb))) ) ) ) ).

% root_powr_inverse
tff(fact_5336_listI__assn__wrap__insert,axiom,
    ! [A: $tType,P: assn,Uu: vEBT_VEBT,Uua: nat,Xi2: vEBT_VEBTi,I5: set(nat),I: nat,Xs: list(vEBT_VEBT),Xsi: list(vEBT_VEBTi),F5: assn,C2: heap_Time_Heap(A),Q: fun(A,assn)] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,vEBT_vebt_insert(Uu,Uua)),Xi2)),vEBT_List_listI_assn(vEBT_VEBT,vEBT_VEBTi,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,Xs,Xsi))),F5))
     => ( aa(nat,$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,Xi2))),F5),C2,Q)
           => hoare_hoare_triple(A,P,C2,Q) ) ) ) ) ).

% listI_assn_wrap_insert
tff(fact_5337_int__ge__less__than__def,axiom,
    ! [D2: int] : int_ge_less_than(D2) = collect(product_prod(int,int),product_case_prod(int,int,$o,aTP_Lamp_adr(int,fun(int,fun(int,$o)),D2))) ).

% int_ge_less_than_def
tff(fact_5338_big__assn__simp,axiom,
    ! [H: nat,TreeLista: list(vEBT_VEBT),L: nat,Xb: vEBT_VEBTi,Xaa: option(nat),X13a: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,H),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => aa(assn,$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,TreeLista),H),L)),Xb)),pure_assn(Xaa = vEBT_vebt_mint(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),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,X13a),list_update(vEBT_VEBTi,Tree_is,H,Xb))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summarya),X14a))),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)),TreeLista))),aa(set(nat),set(nat),insert(nat,H),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeLista,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,X13a),list_update(vEBT_VEBTi,Tree_is,H,Xb))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summarya),X14a))),pure_assn(Xaa = vEBT_vebt_mint(vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),H),L))))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeLista,H,vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),H),L))),list_update(vEBT_VEBTi,Tree_is,H,Xb)))) ) ).

% big_assn_simp
tff(fact_5339_atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [I: A,L: A,U: A] :
          ( member(A,I,set_or7035219750837199246ssThan(A,L,U))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),L),I)
            & aa(A,$o,ord_less(A,I),U) ) ) ) ).

% atLeastLessThan_iff
tff(fact_5340_atLeastLessThan__empty,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) ) ) ) ).

% atLeastLessThan_empty
tff(fact_5341_atLeastLessThan__empty__iff2,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( bot_bot(set(A)) = set_or7035219750837199246ssThan(A,A2,B2) )
        <=> ~ aa(A,$o,ord_less(A,A2),B2) ) ) ).

% atLeastLessThan_empty_iff2
tff(fact_5342_atLeastLessThan__empty__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = bot_bot(set(A)) )
        <=> ~ aa(A,$o,ord_less(A,A2),B2) ) ) ).

% atLeastLessThan_empty_iff
tff(fact_5343_infinite__Ico__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( ~ finite_finite2(A,set_or7035219750837199246ssThan(A,A2,B2))
        <=> aa(A,$o,ord_less(A,A2),B2) ) ) ).

% infinite_Ico_iff
tff(fact_5344_ivl__subset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,J: A,M: A,Nb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,I,J)),set_or7035219750837199246ssThan(A,M,Nb))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),I)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),I)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),J),Nb) ) ) ) ) ).

% ivl_subset
tff(fact_5345_ivl__diff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [I: A,Nb: A,M: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),I),Nb)
         => ( aa(set(A),set(A),minus_minus(set(A),set_or7035219750837199246ssThan(A,I,M)),set_or7035219750837199246ssThan(A,I,Nb)) = set_or7035219750837199246ssThan(A,Nb,M) ) ) ) ).

% ivl_diff
tff(fact_5346_lessThan__minus__lessThan,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Nb: A,M: A] : aa(set(A),set(A),minus_minus(set(A),set_ord_lessThan(A,Nb)),set_ord_lessThan(A,M)) = set_or7035219750837199246ssThan(A,M,Nb) ) ).

% lessThan_minus_lessThan
tff(fact_5347_local_Oext,axiom,
    ! [Y: nat,TreeLista: list(vEBT_VEBT),X13a: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,Y),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => aa(assn,$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,X13a),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,Summarya),X14a)),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,TreeLista),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)),TreeLista))),aa(set(nat),set(nat),insert(nat,Y),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeLista,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,X13a),Tree_is)),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summarya),X14a))),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)),TreeLista))),aa(set(nat),set(nat),insert(nat,Y),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeLista,Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),Y)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),Y)))) ) ).

% local.ext
tff(fact_5348_atLeastLessThan__singleton,axiom,
    ! [M: nat] : set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,M)) = aa(set(nat),set(nat),insert(nat,M),bot_bot(set(nat))) ).

% atLeastLessThan_singleton
tff(fact_5349_txe,axiom,
    ! [Y: nat,TreeLista: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),X13a: array(vEBT_VEBTi),Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,Y),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => aa(assn,$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,TreeLista),Y)),aa(nat,vEBT_VEBTi,nth(vEBT_VEBTi,Tree_is),Y))),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13a),Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summarya),X14a))),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)),TreeLista))),aa(set(nat),set(nat),insert(nat,Y),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeLista,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,Summarya),X14a)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13a),Tree_is))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeLista),Tree_is))) ) ).

% txe
tff(fact_5350_repack,axiom,
    ! [I: nat,TreeLista: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),Rest: assn,X13a: array(vEBT_VEBTi),Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => aa(assn,$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,TreeLista),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,X13a),Tree_is)),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summarya),X14a))),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)),TreeLista))),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeLista,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,Summarya),X14a))),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13a),Tree_is))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeLista),Tree_is))) ) ).

% repack
tff(fact_5351_recomp,axiom,
    ! [I: nat,TreeLista: list(vEBT_VEBT),Tree_is: list(vEBT_VEBTi),X13a: array(vEBT_VEBTi),Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => aa(assn,$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,TreeLista),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)),TreeLista))),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeLista,Tree_is))),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13a),Tree_is))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summarya),X14a))),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,Summarya),X14a)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,X13a),Tree_is))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,TreeLista),Tree_is))) ) ).

% recomp
tff(fact_5352_tcd,axiom,
    ! [A: $tType,I: nat,TreeLista: list(vEBT_VEBT),TreeList3: list(A),Y: vEBT_VEBT,Xb: vEBT_VEBTi,X13a: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => ( ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(list(A),nat,size_size(list(A)),TreeList3) )
       => 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,X13a),list_update(vEBT_VEBTi,Tree_is,I,Xb)))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summarya),X14a))),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)),TreeLista))),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeLista,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,X13a),list_update(vEBT_VEBTi,Tree_is,I,Xb))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summarya),X14a))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,list_update(vEBT_VEBT,TreeLista,I,Y)),list_update(vEBT_VEBTi,Tree_is,I,Xb)))) ) ) ).

% tcd
tff(fact_5353_big__assn__simp_H,axiom,
    ! [H: nat,TreeLista: list(vEBT_VEBT),Xaa: vEBT_VEBT,L: nat,Xb: vEBT_VEBTi,Xba: option(nat),X13a: array(vEBT_VEBTi),Tree_is: list(vEBT_VEBTi),Summarya: vEBT_VEBT,X14a: vEBT_VEBTi] :
      ( aa(nat,$o,ord_less(nat,H),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista))
     => ( ( Xaa = vEBT_vebt_delete(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,TreeLista),H),L) )
       => aa(assn,$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,X13a),list_update(vEBT_VEBTi,Tree_is,H,Xb))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summarya),X14a))),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)),TreeLista))),aa(set(nat),set(nat),insert(nat,H),bot_bot(set(nat)))),vEBT_vebt_assn_raw,TreeLista,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,X13a),list_update(vEBT_VEBTi,Tree_is,H,Xb))),aa(vEBT_VEBTi,assn,aa(vEBT_VEBT,fun(vEBT_VEBTi,assn),vEBT_vebt_assn_raw,Summarya),X14a))),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,TreeLista,H,Xaa)),list_update(vEBT_VEBTi,Tree_is,H,Xb)))) ) ) ).

% big_assn_simp'
tff(fact_5354_sum_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,ord_less(nat,Nb),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,Nb))),aa(nat,A,G,Nb))) ) ).

% sum.op_ivl_Suc
tff(fact_5355_prod_Oop__ivl__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] :
          groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Nb))) = $ite(aa(nat,$o,ord_less(nat,Nb),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Nb))),aa(nat,A,G,Nb))) ) ).

% prod.op_ivl_Suc
tff(fact_5356_atLeastLessThan__inj_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
         => ( aa(A,$o,ord_less(A,A2),B2)
           => ( aa(A,$o,ord_less(A,C2),D2)
             => ( B2 = D2 ) ) ) ) ) ).

% atLeastLessThan_inj(2)
tff(fact_5357_atLeastLessThan__inj_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
         => ( aa(A,$o,ord_less(A,A2),B2)
           => ( aa(A,$o,ord_less(A,C2),D2)
             => ( A2 = C2 ) ) ) ) ) ).

% atLeastLessThan_inj(1)
tff(fact_5358_atLeastLessThan__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( aa(A,$o,ord_less(A,C2),D2)
           => ( ( set_or7035219750837199246ssThan(A,A2,B2) = set_or7035219750837199246ssThan(A,C2,D2) )
            <=> ( ( A2 = C2 )
                & ( B2 = D2 ) ) ) ) ) ) ).

% atLeastLessThan_eq_iff
tff(fact_5359_infinite__Ico,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ~ finite_finite2(A,set_or7035219750837199246ssThan(A,A2,B2)) ) ) ).

% infinite_Ico
tff(fact_5360_atLeastLessThan__subset__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2))
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
            | ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% atLeastLessThan_subset_iff
tff(fact_5361_list__assn__conv__idx,axiom,
    ! [A: $tType,B: $tType,A3: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B)] : aa(list(B),assn,vEBT_List_list_assn(A,B,A3,Xs),Xsi) = vEBT_List_listI_assn(A,B,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(A),nat,size_size(list(A)),Xs)),A3,Xs,Xsi) ).

% list_assn_conv_idx
tff(fact_5362_listI__assn__conv,axiom,
    ! [A: $tType,B: $tType,Nb: nat,Xs: list(A),A3: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( ( Nb = aa(list(A),nat,size_size(list(A)),Xs) )
     => ( vEBT_List_listI_assn(A,B,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb),A3,Xs,Xsi) = aa(list(B),assn,vEBT_List_list_assn(A,B,A3,Xs),Xsi) ) ) ).

% listI_assn_conv
tff(fact_5363_ex__nat__less__eq,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ? [M7: nat] :
          ( aa(nat,$o,ord_less(nat,M7),Nb)
          & aa(nat,$o,P,M7) )
    <=> ? [X: nat] :
          ( member(nat,X,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
          & aa(nat,$o,P,X) ) ) ).

% ex_nat_less_eq
tff(fact_5364_all__nat__less__eq,axiom,
    ! [Nb: nat,P: fun(nat,$o)] :
      ( ! [M7: nat] :
          ( aa(nat,$o,ord_less(nat,M7),Nb)
         => aa(nat,$o,P,M7) )
    <=> ! [X: nat] :
          ( member(nat,X,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
         => aa(nat,$o,P,X) ) ) ).

% all_nat_less_eq
tff(fact_5365_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : set_or7035219750837199246ssThan(nat,L,aa(nat,nat,suc,U)) = set_or1337092689740270186AtMost(nat,L,U) ).

% atLeastLessThanSuc_atLeastAtMost
tff(fact_5366_listI__assn__conv_H,axiom,
    ! [B: $tType,A: $tType,Nb: nat,Xs: list(A),A3: fun(A,fun(B,assn)),Xsi: list(B),F5: assn] :
      ( ( Nb = 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),Nb),A3,Xs,Xsi)),F5) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,A3,Xs),Xsi)),F5) ) ) ).

% listI_assn_conv'
tff(fact_5367_sum_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_up(fun(nat,A),fun(nat,A),G)),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% sum.shift_bounds_Suc_ivl
tff(fact_5368_sum_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Nb: 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),Nb),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_uq(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% sum.shift_bounds_nat_ivl
tff(fact_5369_prod_Oshift__bounds__Suc__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_xx(fun(nat,A),fun(nat,A),G),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% prod.shift_bounds_Suc_ivl
tff(fact_5370_prod_Oshift__bounds__nat__ivl,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Nb: 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),Nb),K))) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_xz(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% prod.shift_bounds_nat_ivl
tff(fact_5371_sum_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_add(B) )
     => ! [A2: A,C2: A,B2: A,D2: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X2)
                 => ( aa(A,$o,ord_less(A,X2),D2)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) ) )
             => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),set_or7035219750837199246ssThan(A,A2,B2)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),set_or7035219750837199246ssThan(A,C2,D2)) ) ) ) ) ) ).

% sum.ivl_cong
tff(fact_5372_prod_Oivl__cong,axiom,
    ! [B: $tType,A: $tType] :
      ( ( ord(A)
        & comm_monoid_mult(B) )
     => ! [A2: A,C2: A,B2: A,D2: A,G: fun(A,B),H: fun(A,B)] :
          ( ( A2 = C2 )
         => ( ( B2 = D2 )
           => ( ! [X2: A] :
                  ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),X2)
                 => ( aa(A,$o,ord_less(A,X2),D2)
                   => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) ) )
             => ( groups7121269368397514597t_prod(A,B,G,set_or7035219750837199246ssThan(A,A2,B2)) = groups7121269368397514597t_prod(A,B,H,set_or7035219750837199246ssThan(A,C2,D2)) ) ) ) ) ) ).

% prod.ivl_cong
tff(fact_5373_sum_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,P2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),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,Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Nb,P2))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,P2)) ) ) ) ) ).

% sum.atLeastLessThan_concat
tff(fact_5374_sum__diff__nat__ivl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Nb: nat,P2: nat,F: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),P2)
           => ( aa(A,A,minus_minus(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_or7035219750837199246ssThan(nat,M,P2))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_or7035219750837199246ssThan(nat,M,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_or7035219750837199246ssThan(nat,Nb,P2)) ) ) ) ) ).

% sum_diff_nat_ivl
tff(fact_5375_prod_OatLeastLessThan__concat,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,P2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),P2)
           => ( aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Nb))),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,Nb,P2))) = groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,P2)) ) ) ) ) ).

% prod.atLeastLessThan_concat
tff(fact_5376_atLeast0__lessThan__Suc,axiom,
    ! [Nb: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,Nb),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ).

% atLeast0_lessThan_Suc
tff(fact_5377_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N: set(nat),Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),N),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))
     => finite_finite2(nat,N) ) ).

% subset_eq_atLeast0_lessThan_finite
tff(fact_5378_listI__assn__weak__cong,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),I6: set(nat),A3: fun(A,fun(B,assn)),A9: fun(A,fun(B,assn)),Xs: list(A),Xs4: list(A),Xsi: list(B),Xsi2: list(B)] :
      ( ( I5 = I6 )
     => ( ( A3 = A9 )
       => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Xs4) )
         => ( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(B),nat,size_size(list(B)),Xsi2) )
           => ( ! [I3: nat] :
                  ( member(nat,I3,I5)
                 => ( aa(nat,$o,ord_less(nat,I3),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),I3) = aa(nat,A,nth(A,Xs4),I3) )
                        & ( aa(nat,B,nth(B,Xsi),I3) = aa(nat,B,nth(B,Xsi2),I3) ) ) ) ) )
             => ( vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) = vEBT_List_listI_assn(A,B,I6,A9,Xs4,Xsi2) ) ) ) ) ) ) ).

% listI_assn_weak_cong
tff(fact_5379_listI__assn__cong,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),I6: set(nat),Xs: list(A),Xs4: list(A),Xsi: list(B),Xsi2: list(B),A3: fun(A,fun(B,assn)),A9: fun(A,fun(B,assn))] :
      ( ( I5 = I6 )
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(list(A),nat,size_size(list(A)),Xs4) )
       => ( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(B),nat,size_size(list(B)),Xsi2) )
         => ( ! [I3: nat] :
                ( member(nat,I3,I5)
               => ( aa(nat,$o,ord_less(nat,I3),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),I3) = aa(nat,A,nth(A,Xs4),I3) )
                      & ( aa(nat,B,nth(B,Xsi),I3) = aa(nat,B,nth(B,Xsi2),I3) )
                      & ( aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I3)),aa(nat,B,nth(B,Xsi),I3)) = aa(B,assn,aa(A,fun(B,assn),A9,aa(nat,A,nth(A,Xs4),I3)),aa(nat,B,nth(B,Xsi2),I3)) ) ) ) ) )
           => ( vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) = vEBT_List_listI_assn(A,B,I6,A9,Xs4,Xsi2) ) ) ) ) ) ).

% listI_assn_cong
tff(fact_5380_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
    ! [A: $tType] :
      ( dense_linorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or7035219750837199246ssThan(A,A2,B2)),set_or1337092689740270186AtMost(A,C2,D2))
        <=> ( aa(A,$o,ord_less(A,A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2) ) ) ) ) ).

% atLeastLessThan_subseteq_atLeastAtMost_iff
tff(fact_5381_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),set_or1337092689740270186AtMost(A,A2,B2)),set_or7035219750837199246ssThan(A,C2,D2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),C2),A2)
              & aa(A,$o,ord_less(A,B2),D2) ) ) ) ) ).

% atLeastAtMost_subseteq_atLeastLessThan_iff
tff(fact_5382_subst__not__in,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A3: fun(A,fun(B,assn)),X1: A,Xsi: list(B),X23: B] :
      ( ~ member(nat,I,I5)
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,I5,A3,list_update(A,Xs,I,X1),list_update(B,Xsi,I,X23)) = vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) ) ) ) ).

% subst_not_in
tff(fact_5383_sum__shift__lb__Suc0__0__upt,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [F: fun(nat,A),K: nat] :
          ( ( aa(nat,A,F,zero_zero(nat)) = zero_zero(A) )
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),K)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% sum_shift_lb_Suc0_0_upt
tff(fact_5384_sum_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = 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),Nb))),aa(nat,A,G,Nb)) ) ).

% sum.atLeast0_lessThan_Suc
tff(fact_5385_sum_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less(nat,M),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,Nb)) = 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),Nb))) ) ) ) ).

% sum.atLeast_Suc_lessThan
tff(fact_5386_sum_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% sum.atLeastLessThan_Suc
tff(fact_5387_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] : set_or7035219750837199246ssThan(A,A2,B2) = aa(set(A),set(A),minus_minus(set(A),set_or1337092689740270186AtMost(A,A2,B2)),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))) ) ).

% atLeastLessThan_eq_atLeastAtMost_diff
tff(fact_5388_prod_OatLeast0__lessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(nat,A,G,Nb)) ) ).

% prod.atLeast0_lessThan_Suc
tff(fact_5389_prod_OatLeast__Suc__lessThan,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,ord_less(nat,M),Nb)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Nb)) = 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),Nb))) ) ) ) ).

% prod.atLeast_Suc_lessThan
tff(fact_5390_prod_OatLeastLessThan__Suc,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: nat,B2: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,A2,aa(nat,nat,suc,B2))) = aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,A2,B2))),aa(nat,A,G,B2)) ) ) ) ).

% prod.atLeastLessThan_Suc
tff(fact_5391_sum_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Nb)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(nat,A,G,Nb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,Nb))) ) ) ) ).

% sum.last_plus
tff(fact_5392_prod_Olast__plus,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,G,Nb)),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Nb))) ) ) ) ).

% prod.last_plus
tff(fact_5393_atLeastLessThanSuc,axiom,
    ! [M: nat,Nb: nat] :
      set_or7035219750837199246ssThan(nat,M,aa(nat,nat,suc,Nb)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb),aa(set(nat),set(nat),insert(nat,Nb),set_or7035219750837199246ssThan(nat,M,Nb)),bot_bot(set(nat))) ).

% atLeastLessThanSuc
tff(fact_5394_sum__Suc__diff_H,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [M: nat,Nb: nat,F: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_uu(fun(nat,A),fun(nat,A),F)),set_or7035219750837199246ssThan(nat,M,Nb)) = aa(A,A,minus_minus(A,aa(nat,A,F,Nb)),aa(nat,A,F,M)) ) ) ) ).

% sum_Suc_diff'
tff(fact_5395_sum_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Nb,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ads(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),M)),set_or7035219750837199246ssThan(nat,Nb,M)) ) ).

% sum.atLeastLessThan_rev
tff(fact_5396_sum_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_adt(fun(nat,fun(nat,A)),fun(nat,A),A2)),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_ys(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% sum.nested_swap
tff(fact_5397_prod_OatLeastLessThan__rev,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,Nb,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_adu(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),M),set_or7035219750837199246ssThan(nat,Nb,M)) ) ).

% prod.atLeastLessThan_rev
tff(fact_5398_prod_Onested__swap,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [A2: fun(nat,fun(nat,A)),Nb: nat] : groups7121269368397514597t_prod(nat,A,aTP_Lamp_adv(fun(nat,fun(nat,A)),fun(nat,A),A2),set_or1337092689740270186AtMost(nat,zero_zero(nat),Nb)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_yv(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),A2),Nb),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% prod.nested_swap
tff(fact_5399_sum_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),K: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aTP_Lamp_adw(fun(nat,A),fun(nat,fun(nat,A)),G),K)),set_ord_lessThan(nat,Nb)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))) ) ).

% sum.nat_group
tff(fact_5400_prod_Onat__group,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),K: nat,Nb: nat] : groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_adx(fun(nat,A),fun(nat,fun(nat,A)),G),K),set_ord_lessThan(nat,Nb)) = groups7121269368397514597t_prod(nat,A,G,set_ord_lessThan(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),K))) ) ).

% prod.nat_group
tff(fact_5401_prod__Suc__Suc__fact,axiom,
    ! [Nb: nat] : groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),Nb)) = semiring_char_0_fact(nat,Nb) ).

% prod_Suc_Suc_fact
tff(fact_5402_prod__Suc__fact,axiom,
    ! [Nb: nat] : groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) = semiring_char_0_fact(nat,Nb) ).

% prod_Suc_fact
tff(fact_5403_sum_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] :
          aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Nb)) = $ite(aa(nat,$o,ord_less(nat,Nb),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,Nb))),aa(nat,A,G,Nb))) ) ).

% sum.head_if
tff(fact_5404_prod_Ohead__if,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] :
          groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Nb)) = $ite(aa(nat,$o,ord_less(nat,Nb),M),one_one(A),aa(A,A,aa(A,fun(A,A),times_times(A),groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Nb))),aa(nat,A,G,Nb))) ) ).

% prod.head_if
tff(fact_5405_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      set_or7035219750837199246ssThan(nat,M,aa(num,nat,numeral_numeral(nat),K)) = $ite(aa(nat,$o,aa(nat,fun(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_5406_fact__prod__Suc,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : semiring_char_0_fact(A,Nb) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,suc,set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% fact_prod_Suc
tff(fact_5407_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat,M: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,Nb,M)) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_ut(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),M)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),M)) ) ).

% sum.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_5408_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat,M: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,Nb,M)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aa(nat,fun(nat,fun(nat,A)),aTP_Lamp_yc(fun(nat,A),fun(nat,fun(nat,fun(nat,A))),G),Nb),M),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Nb),M)) ) ).

% prod.atLeastLessThan_rev_at_least_Suc_atMost
tff(fact_5409_pochhammer__prod,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [A2: A,Nb: nat] : comm_s3205402744901411588hammer(A,A2,Nb) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_yg(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% pochhammer_prod
tff(fact_5410_fact__prod__rev,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [Nb: nat] : semiring_char_0_fact(A,Nb) = aa(nat,A,semiring_1_of_nat(A),groups7121269368397514597t_prod(nat,nat,minus_minus(nat,Nb),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))) ) ).

% fact_prod_rev
tff(fact_5411_summable__Cauchy,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [F: fun(nat,A)] :
          ( summable(A,F)
        <=> ! [E4: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E4)
             => ? [N7: nat] :
                ! [M7: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N7),M7)
                 => ! [N6: nat] : aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,F),set_or7035219750837199246ssThan(nat,M7,N6)))),E4) ) ) ) ) ).

% summable_Cauchy
tff(fact_5412_sums__group,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F: fun(nat,A),S3: A,K: nat] :
          ( aa(A,$o,sums(A,F),S3)
         => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),K)
           => aa(A,$o,sums(A,aa(nat,fun(nat,A),aTP_Lamp_ady(fun(nat,A),fun(nat,fun(nat,A)),F),K)),S3) ) ) ) ).

% sums_group
tff(fact_5413_listI__assn__insert,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A3: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( ~ member(nat,I,I5)
     => ( aa(nat,$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),A3,Xs,Xsi) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi)) ) ) ) ).

% listI_assn_insert
tff(fact_5414_atLeast1__lessThan__eq__remove0,axiom,
    ! [Nb: nat] : set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,zero_zero(nat)),Nb) = aa(set(nat),set(nat),minus_minus(set(nat),set_ord_lessThan(nat,Nb)),aa(set(nat),set(nat),insert(nat,zero_zero(nat)),bot_bot(set(nat)))) ).

% atLeast1_lessThan_eq_remove0
tff(fact_5415_listI__assn__subst,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A3: fun(A,fun(B,assn)),X1: A,Xsi: list(B),X23: B] :
      ( ~ member(nat,I,I5)
     => ( aa(nat,$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),A3,list_update(A,Xs,I,X1),list_update(B,Xsi,I,X23)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,X1),X23)),vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi)) ) ) ) ).

% listI_assn_subst
tff(fact_5416_rule__at__index,axiom,
    ! [B: $tType,A: $tType,C: $tType,P: assn,A3: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B),F5: assn,I: nat,C2: heap_Time_Heap(C),Q6: fun(C,assn),F7: fun(C,assn)] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,A3,Xs),Xsi)),F5))
     => ( aa(nat,$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),A3,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)))),A3,Xs,Xsi))),F5),C2,Q6)
         => ( ! [R2: C] : aa(assn,$o,aa(assn,fun(assn,$o),entails,aa(C,assn,Q6,R2)),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),A3,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)))),A3,Xs,Xsi))),aa(C,assn,F7,R2)))
           => 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_adz(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(fun(C,assn),fun(C,assn)))),A3),Xs),Xsi),F7)) ) ) ) ) ).

% rule_at_index
tff(fact_5417_fact__split,axiom,
    ! [A: $tType] :
      ( semiring_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( semiring_char_0_fact(A,Nb) = 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,Nb),K),Nb)))),semiring_char_0_fact(A,aa(nat,nat,minus_minus(nat,Nb),K))) ) ) ) ).

% fact_split
tff(fact_5418_binomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),Nb)
         => ( aa(nat,A,semiring_1_of_nat(A),aa(nat,nat,binomial(Nb),K)) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aea(nat,fun(nat,fun(nat,A)),K),Nb),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ) ) ).

% binomial_altdef_of_nat
tff(fact_5419_gbinomial__altdef__of__nat,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aeb(A,fun(nat,fun(nat,A)),A2),K),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_altdef_of_nat
tff(fact_5420_gbinomial__mult__fact_H,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [A2: A,K: nat] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,gbinomial(A,A2),K)),semiring_char_0_fact(A,K)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_aec(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact'
tff(fact_5421_gbinomial__mult__fact,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [K: nat,A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),semiring_char_0_fact(A,K)),aa(nat,A,gbinomial(A,A2),K)) = groups7121269368397514597t_prod(nat,A,aTP_Lamp_aec(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K)) ) ).

% gbinomial_mult_fact
tff(fact_5422_gbinomial__prod__rev,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [A2: A,K: nat] : aa(nat,A,gbinomial(A,A2),K) = aa(A,A,aa(A,fun(A,A),divide_divide(A),groups7121269368397514597t_prod(nat,A,aTP_Lamp_aam(A,fun(nat,A),A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),K))),semiring_char_0_fact(A,K)) ) ).

% gbinomial_prod_rev
tff(fact_5423_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_5424_Sum__Ico__nat,axiom,
    ! [M: nat,Nb: nat] : aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aTP_Lamp_vc(nat,nat)),set_or7035219750837199246ssThan(nat,M,Nb)) = 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),Nb),aa(nat,nat,minus_minus(nat,Nb),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_5425_listI__assn__extract,axiom,
    ! [A: $tType,B: $tType,I: nat,I5: set(nat),Xs: list(A),A3: fun(A,fun(B,assn)),Xsi: list(B)] :
      ( member(nat,I,I5)
     => ( aa(nat,$o,ord_less(nat,I),aa(list(A),nat,size_size(list(A)),Xs))
       => ( vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,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)))),A3,Xs,Xsi)) ) ) ) ).

% listI_assn_extract
tff(fact_5426_listI__assn__reinsert,axiom,
    ! [B: $tType,A: $tType,P: assn,A3: fun(A,fun(B,assn)),Xs: list(A),I: nat,Xsi: list(B),I5: set(nat),F5: assn,Q: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A3,Xs,Xsi))),F5))
     => ( aa(nat,$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,A3,Xs,Xsi)),F5)),Q)
           => aa(assn,$o,aa(assn,fun(assn,$o),entails,P),Q) ) ) ) ) ).

% listI_assn_reinsert
tff(fact_5427_listI__assn__reinsert__upd,axiom,
    ! [B: $tType,A: $tType,P: assn,A3: fun(A,fun(B,assn)),Xb: A,Xi2: B,I5: set(nat),I: nat,Xs: list(A),Xsi: list(B),F5: assn,Q: assn] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,Xb),Xi2)),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A3,Xs,Xsi))),F5))
     => ( aa(nat,$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,A3,list_update(A,Xs,I,Xb),list_update(B,Xsi,I,Xi2))),F5)),Q)
           => aa(assn,$o,aa(assn,fun(assn,$o),entails,P),Q) ) ) ) ) ).

% listI_assn_reinsert_upd
tff(fact_5428_listI__assn__reinsert_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,A3: fun(A,fun(B,assn)),Xs: list(A),I: nat,Xsi: list(B),I5: set(nat),F5: assn,C2: heap_Time_Heap(C),Q: fun(C,assn)] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,aa(nat,A,nth(A,Xs),I)),aa(nat,B,nth(B,Xsi),I))),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A3,Xs,Xsi))),F5))
     => ( aa(nat,$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,A3,Xs,Xsi)),F5),C2,Q)
           => hoare_hoare_triple(C,P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert'
tff(fact_5429_listI__assn__reinsert__upd_H,axiom,
    ! [A: $tType,B: $tType,C: $tType,P: assn,A3: fun(A,fun(B,assn)),Xb: A,Xi2: B,I5: set(nat),I: nat,Xs: list(A),Xsi: list(B),F5: assn,C2: heap_Time_Heap(C),Q: fun(C,assn)] :
      ( aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),A3,Xb),Xi2)),vEBT_List_listI_assn(A,B,aa(set(nat),set(nat),minus_minus(set(nat),I5),aa(set(nat),set(nat),insert(nat,I),bot_bot(set(nat)))),A3,Xs,Xsi))),F5))
     => ( aa(nat,$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,A3,list_update(A,Xs,I,Xb),list_update(B,Xsi,I,Xi2))),F5),C2,Q)
           => hoare_hoare_triple(C,P,C2,Q) ) ) ) ) ).

% listI_assn_reinsert_upd'
tff(fact_5430_Chebyshev__sum__upper,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat,A2: fun(nat,A),B2: fun(nat,A)] :
          ( ! [I3: nat,J2: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J2)
             => ( aa(nat,$o,ord_less(nat,J2),Nb)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,A2,I3)),aa(nat,A,A2,J2)) ) )
         => ( ! [I3: nat,J2: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J2)
               => ( aa(nat,$o,ord_less(nat,J2),Nb)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(nat,A,B2,J2)),aa(nat,A,B2,I3)) ) )
           => aa(A,$o,aa(A,fun(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),Nb)),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(fun(nat,A),fun(nat,A),aTP_Lamp_aed(fun(nat,A),fun(fun(nat,A),fun(nat,A)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))) ) ) ) ).

% Chebyshev_sum_upper
tff(fact_5431_Chebyshev__sum__upper__nat,axiom,
    ! [Nb: nat,A2: fun(nat,nat),B2: fun(nat,nat)] :
      ( ! [I3: nat,J2: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J2)
         => ( aa(nat,$o,ord_less(nat,J2),Nb)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,A2,I3)),aa(nat,nat,A2,J2)) ) )
     => ( ! [I3: nat,J2: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I3),J2)
           => ( aa(nat,$o,ord_less(nat,J2),Nb)
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,B2,J2)),aa(nat,nat,B2,I3)) ) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_aee(fun(nat,nat),fun(fun(nat,nat),fun(nat,nat)),A2),B2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,A2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb))),aa(set(nat),nat,groups7311177749621191930dd_sum(nat,nat,B2),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)))) ) ) ).

% Chebyshev_sum_upper_nat
tff(fact_5432_option_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,nat),X23: A] : size_option(A,Xb,aa(A,option(A),some(A),X23)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(A,nat,Xb,X23)),aa(nat,nat,suc,zero_zero(nat))) ).

% option.size_gen(2)
tff(fact_5433_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_5434_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_5435_atLeastLessThanPlusOne__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : set_or7035219750837199246ssThan(code_integer,L,aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),plus_plus(code_integer),U),one_one(code_integer))) = set_or1337092689740270186AtMost(code_integer,L,U) ).

% atLeastLessThanPlusOne_atLeastAtMost_integer
tff(fact_5436_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_5437_word__atLeastLessThan__Suc__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),L: word(A)] :
          ( ( U != aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) )
         => ( set_or7035219750837199246ssThan(word(A),L,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),U),one_one(word(A)))) = set_or1337092689740270186AtMost(word(A),L,U) ) ) ) ).

% word_atLeastLessThan_Suc_atLeastAtMost
tff(fact_5438_word__range__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( ( B2 != zero_zero(word(A)) )
         => ( set_or1337092689740270186AtMost(word(A),A2,aa(word(A),word(A),minus_minus(word(A),B2),one_one(word(A)))) = set_or7035219750837199246ssThan(word(A),A2,B2) ) ) ) ).

% word_range_minus_1
tff(fact_5439_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_5440_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_5441_flip__bit__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : bit_se8732182000553998342ip_bit(A,zero_zero(nat),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))),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),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% flip_bit_0
tff(fact_5442_forall__finite_I3_J,axiom,
    ! [Xb: nat,P: fun(nat,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,suc,aa(nat,nat,suc,Xb)))
         => aa(nat,$o,P,I2) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I2: nat] :
            ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,suc,Xb))
           => aa(nat,$o,P,aa(nat,nat,suc,I2)) ) ) ) ).

% forall_finite(3)
tff(fact_5443_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_5444_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_5445_of__bool__less__eq__iff,axiom,
    ! [A: $tType] :
      ( linord181362715937106298miring(A)
     => ! [P: $o,Q: $o] :
          ( aa(A,$o,aa(A,fun(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_5446_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_5447_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_5448_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_5449_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_5450_abs__bool__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [P: $o] : aa(A,A,abs_abs(A),aa($o,A,zero_neq_one_of_bool(A),(P))) = aa($o,A,zero_neq_one_of_bool(A),(P)) ) ).

% abs_bool_eq
tff(fact_5451_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_5452_div__word__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),one_one(word(A))),W) = aa($o,word(A),zero_neq_one_of_bool(word(A)),W = one_one(word(A))) ) ).

% div_word_one
tff(fact_5453_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_5454_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_5455_Suc__0__mod__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) = aa($o,nat,zero_neq_one_of_bool(nat),Nb != aa(nat,nat,suc,zero_zero(nat))) ).

% Suc_0_mod_eq
tff(fact_5456_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_5457_finite__atLeastLessThan__integer,axiom,
    ! [L: code_integer,U: code_integer] : finite_finite2(code_integer,set_or7035219750837199246ssThan(code_integer,L,U)) ).

% finite_atLeastLessThan_integer
tff(fact_5458_div__word__by__minus__1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),W),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa($o,word(A),zero_neq_one_of_bool(word(A)),W = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ).

% div_word_by_minus_1_eq
tff(fact_5459_finite__atLeastAtMost__integer,axiom,
    ! [L: code_integer,U: code_integer] : finite_finite2(code_integer,set_or1337092689740270186AtMost(code_integer,L,U)) ).

% finite_atLeastAtMost_integer
tff(fact_5460_mod__word__one,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),one_one(word(A))),W) = aa(word(A),word(A),minus_minus(word(A),one_one(word(A))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),W),aa($o,word(A),zero_neq_one_of_bool(word(A)),W = one_one(word(A))))) ) ).

% mod_word_one
tff(fact_5461_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R3: int] : adjust_div(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R3)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Q2),aa($o,int,zero_neq_one_of_bool(int),R3 != zero_zero(int))) ).

% Divides.adjust_div_eq
tff(fact_5462_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_5463_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_5464_of__bool__half__eq__0,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [B2: $o] : aa(A,A,aa(A,fun(A,A),divide_divide(A),aa($o,A,zero_neq_one_of_bool(A),(B2))),aa(num,A,numeral_numeral(A),bit0(one2))) = zero_zero(A) ) ).

% of_bool_half_eq_0
tff(fact_5465_set__decode__Suc,axiom,
    ! [Nb: nat,Xb: nat] :
      ( member(nat,aa(nat,nat,suc,Nb),nat_set_decode(Xb))
    <=> member(nat,Nb,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_5466_bits__1__div__exp,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% bits_1_div_exp
tff(fact_5467_one__div__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% one_div_2_pow_eq
tff(fact_5468_one__mod__2__pow__eq,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),one_one(A)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)) ) ).

% one_mod_2_pow_eq
tff(fact_5469_zero__less__eq__of__bool,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,aa(A,fun(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_5470_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_5471_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_5472_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_5473_of__bool__less__eq__one,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [P: $o] : aa(A,$o,aa(A,fun(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_5474_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_5475_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_5476_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_aef(int,fun(int,int))),Qr) ).

% Divides.adjust_div_def
tff(fact_5477_subset__decode__imp__le,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(set(nat),$o,aa(set(nat),fun(set(nat),$o),ord_less_eq(set(nat)),nat_set_decode(M)),nat_set_decode(Nb))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% subset_decode_imp_le
tff(fact_5478_of__bool__odd__eq__mod__2,axiom,
    ! [A: $tType] :
      ( semiring_parity(A)
     => ! [A2: A] : aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% of_bool_odd_eq_mod_2
tff(fact_5479_bits__induct,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [P: fun(A,$o),A2: A] :
          ( ! [A4: A] :
              ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A4),aa(num,A,numeral_numeral(A),bit0(one2))) = A4 )
             => aa(A,$o,P,A4) )
         => ( ! [A4: A,B3: $o] :
                ( aa(A,$o,P,A4)
               => ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4))),aa(num,A,numeral_numeral(A),bit0(one2))) = A4 )
                 => aa(A,$o,P,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa($o,A,zero_neq_one_of_bool(A),(B3))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A4))) ) )
           => aa(A,$o,P,A2) ) ) ) ).

% bits_induct
tff(fact_5480_exp__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [M: nat,Nb: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,ord_less(nat,M),Nb))),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_5481_forall__finite_I1_J,axiom,
    ! [P: fun(nat,$o),I4: nat] :
      ( aa(nat,$o,ord_less(nat,I4),zero_zero(nat))
     => aa(nat,$o,P,I4) ) ).

% forall_finite(1)
tff(fact_5482_exp__div__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Nb: nat] :
          aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),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),Nb))) ) ).

% exp_div_exp_eq
tff(fact_5483_set__decode__plus__power__2,axiom,
    ! [Nb: nat,Z: nat] :
      ( ~ member(nat,Nb,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))),Nb)),Z)) = aa(set(nat),set(nat),insert(nat,Nb),nat_set_decode(Z)) ) ) ).

% set_decode_plus_power_2
tff(fact_5484_set__decode__def,axiom,
    ! [Xb: nat] : nat_set_decode(Xb) = collect(nat,aTP_Lamp_aeg(nat,fun(nat,$o),Xb)) ).

% set_decode_def
tff(fact_5485_Comparator__Generator_OAll__less__Suc,axiom,
    ! [Xb: nat,P: fun(nat,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,suc,Xb))
         => aa(nat,$o,P,I2) )
    <=> ( aa(nat,$o,P,zero_zero(nat))
        & ! [I2: nat] :
            ( aa(nat,$o,ord_less(nat,I2),Xb)
           => aa(nat,$o,P,aa(nat,nat,suc,I2)) ) ) ) ).

% Comparator_Generator.All_less_Suc
tff(fact_5486_forall__finite_I2_J,axiom,
    ! [P: fun(nat,$o)] :
      ( ! [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,suc,zero_zero(nat)))
         => aa(nat,$o,P,I2) )
    <=> aa(nat,$o,P,zero_zero(nat)) ) ).

% forall_finite(2)
tff(fact_5487_and__int_Opsimps,axiom,
    ! [K: int,L: int] :
      ( aa(product_prod(int,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),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_5488_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 )
     => ( aa(product_prod(int,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),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,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Xaa) ))),
                    aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xb),aa(num,int,numeral_numeral(int),bit0(one2)))),aa(int,int,aa(int,fun(int,int),divide_divide(int),Xaa),aa(num,int,numeral_numeral(int),bit0(one2)))))) ) )
           => ~ aa(product_prod(int,int),$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),Xb),Xaa)) ) ) ) ).

% and_int.pelims
tff(fact_5489_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_5490_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_5491_and_Oright__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,uminus_uminus(A),one_one(A))) = A2 ) ).

% and.right_neutral
tff(fact_5492_and_Oleft__neutral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,uminus_uminus(A),one_one(A))),A2) = A2 ) ).

% and.left_neutral
tff(fact_5493_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        | aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% and_nonnegative_int_iff
tff(fact_5494_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_5495_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_5496_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_5497_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_5498_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_5499_and__minus__numerals_I6_J,axiom,
    ! [Nb: 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,Nb)))),one_one(int)) = one_one(int) ).

% and_minus_numerals(6)
tff(fact_5500_and__minus__numerals_I2_J,axiom,
    ! [Nb: 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,Nb)))) = one_one(int) ).

% and_minus_numerals(2)
tff(fact_5501_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_5502_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_5503_and__minus__numerals_I1_J,axiom,
    ! [Nb: 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(Nb)))) = zero_zero(int) ).

% and_minus_numerals(1)
tff(fact_5504_and__minus__numerals_I5_J,axiom,
    ! [Nb: 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(Nb)))),one_one(int)) = zero_zero(int) ).

% and_minus_numerals(5)
tff(fact_5505_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_5506_and__eq__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> ( ( A2 = aa(A,A,uminus_uminus(A),one_one(A)) )
            & ( B2 = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ) ).

% and_eq_minus_1_iff
tff(fact_5507_AND__upper2_H,axiom,
    ! [Y: int,Z: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Y)),Z) ) ) ).

% AND_upper2'
tff(fact_5508_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Y),Z)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Y),Ya)),Z) ) ) ).

% AND_upper1'
tff(fact_5509_AND__upper2,axiom,
    ! [Y: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Y)),Y) ) ).

% AND_upper2
tff(fact_5510_AND__upper1,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),bit_se5824344872417868541ns_and(int),Xb),Y)),Xb) ) ).

% AND_upper1
tff(fact_5511_AND__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => aa(int,$o,aa(int,fun(int,$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_5512_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,Xb: int] :
      ( aa(int,$o,aa(int,fun(int,$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_5513_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( aa(int,$o,aa(int,fun(int,$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_5514_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,ord_less(int,L),zero_zero(int))
     => aa(int,$o,aa(int,fun(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_5515_even__and__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),B2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ) ).

% even_and_iff
tff(fact_5516_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_5517_and__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),one_one(A)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% and_one_eq
tff(fact_5518_one__and__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),one_one(A)),A2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% one_and_eq
tff(fact_5519_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_5520_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_5521_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_5522_take__bit__word__Bit1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num,M: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),Nb)),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(Nb)),aa(num,word(A),numeral_numeral(word(A)),M)))) ) ).

% take_bit_word_Bit1_eq
tff(fact_5523_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_5524_add__scale__eq__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [R3: A,A2: A,B2: A,C2: A,D2: A] :
          ( ( R3 != zero_zero(A) )
         => ( ( ( A2 = B2 )
              & ( C2 != D2 ) )
           => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),R3),C2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),aa(A,A,aa(A,fun(A,A),times_times(A),R3),D2)) ) ) ) ) ).

% add_scale_eq_noteq
tff(fact_5525_word__and__max,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),uminus_uminus(word(A)),one_one(word(A)))) = Xb ) ).

% word_and_max
tff(fact_5526_word__bitwise__m1__simps_I2_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)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Xb) = Xb ) ).

% word_bitwise_m1_simps(2)
tff(fact_5527_take__bit__Suc__1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),one_one(A)) = one_one(A) ) ).

% take_bit_Suc_1
tff(fact_5528_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_5529_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_5530_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_5531_take__bit__of__1__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),one_one(A)) = zero_zero(A) )
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% take_bit_of_1_eq_0_iff
tff(fact_5532_word__no__log__defs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2)),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(3)
tff(fact_5533_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_5534_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_5535_take__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)) ) ).

% take_bit_of_1
tff(fact_5536_even__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))
        <=> ( ( Nb = zero_zero(nat) )
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% even_take_bit_eq
tff(fact_5537_and__Suc__0__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),Nb),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% and_Suc_0_eq
tff(fact_5538_Suc__0__and__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))) ).

% Suc_0_and_eq
tff(fact_5539_word__bitwise__1__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2)) = 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),B2))) ) ).

% word_bitwise_1_simps(2)
tff(fact_5540_word__bitwise__1__simps_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: 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)),A2)),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),A2)),one_one(int))) ) ).

% word_bitwise_1_simps(4)
tff(fact_5541_word__no__log__defs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(6)
tff(fact_5542_word__no__log__defs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2))),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(5)
tff(fact_5543_word__no__log__defs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(4)
tff(fact_5544_take__bit__Suc__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,zero_zero(nat))),A2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_0
tff(fact_5545_take__bit__word__Bit0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num,M: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),Nb)),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(Nb)),aa(num,word(A),numeral_numeral(word(A)),M))) ) ).

% take_bit_word_Bit0_eq
tff(fact_5546_take__bit__of__exp,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,ord_less(nat,Nb),M))),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) ) ).

% take_bit_of_exp
tff(fact_5547_take__bit__of__2,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(num,A,numeral_numeral(A),bit0(one2))) = aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_of_2
tff(fact_5548_word__bitwise__1__simps_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2))) = 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),B2)))) ) ).

% word_bitwise_1_simps(3)
tff(fact_5549_word__bitwise__1__simps_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: 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)),A2))),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),A2))),one_one(int))) ) ).

% word_bitwise_1_simps(5)
tff(fact_5550_take__bit__word__minus__Bit0__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num,M: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),Nb)),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(Nb)),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_5551_word__and__le1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),A2: word(A)] : aa(word(A),$o,aa(word(A),fun(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)),Y),A2)),A2) ) ).

% word_and_le1
tff(fact_5552_word__and__le2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Y: word(A)] : aa(word(A),$o,aa(word(A),fun(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)),A2),Y)),A2) ) ).

% word_and_le2
tff(fact_5553_take__bit__unset__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,M: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),bit_se2638667681897837118et_bit(A,M,A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2),bit_se2638667681897837118et_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))) ) ).

% take_bit_unset_bit_eq
tff(fact_5554_take__bit__set__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,M: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),bit_se5668285175392031749et_bit(A,M,A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2),bit_se5668285175392031749et_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))) ) ).

% take_bit_set_bit_eq
tff(fact_5555_take__bit__flip__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,M: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),bit_se8732182000553998342ip_bit(A,M,A2)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2),bit_se8732182000553998342ip_bit(A,M,aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2))) ) ).

% take_bit_flip_bit_eq
tff(fact_5556_take__bit__tightened,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A,M: nat] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2) )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
           => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),B2) ) ) ) ) ).

% take_bit_tightened
tff(fact_5557_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A,B2: A] :
          ( ( aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2) = aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),B2) )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),A2) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),B2) ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
tff(fact_5558_take__bit__add,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),B2))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) ) ).

% take_bit_add
tff(fact_5559_signed__take__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Nb: nat,A2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,M),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) = aa(A,A,
            $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M),bit_se2584673776208193580ke_bit(A,Nb),bit_ri4674362597316999326ke_bit(A,M)),
            A2) ) ).

% signed_take_bit_take_bit
tff(fact_5560_word__and__max__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),A2: word(A)] :
          ( ( Xb = 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)),bit_se5824344872417868541ns_and(word(A)),A2),Xb) = A2 ) ) ) ).

% word_and_max_word
tff(fact_5561_take__bit__signed__take__bit,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Nb: nat,A2: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),aa(nat,nat,suc,Nb))
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2) ) ) ) ).

% take_bit_signed_take_bit
tff(fact_5562_even__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,even_word(A),A2)
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),one_one(word(A))) = zero_zero(word(A)) ) ) ) ).

% even_word_iff
tff(fact_5563_take__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),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,Nb),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% take_bit_Suc_bit0
tff(fact_5564_take__bit__eq__mod,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) ) ).

% take_bit_eq_mod
tff(fact_5565_take__bit__eq__0__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = zero_zero(A) )
        <=> aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),A2) ) ) ).

% take_bit_eq_0_iff
tff(fact_5566_bin__last__bintrunc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [L: nat,Nb: A] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_se2584673776208193580ke_bit(A,L),Nb))
        <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),L)
            & ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb) ) ) ) ).

% bin_last_bintrunc
tff(fact_5567_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_5568_add__0__iff,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [B2: A,A2: A] :
          ( ( B2 = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% add_0_iff
tff(fact_5569_crossproduct__noteq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [A2: A,B2: A,C2: A,D2: A] :
          ( ( ( A2 != B2 )
            & ( C2 != D2 ) )
        <=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),D2)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),D2)),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ) ) ).

% crossproduct_noteq
tff(fact_5570_crossproduct__eq,axiom,
    ! [A: $tType] :
      ( semiri1453513574482234551roduct(A)
     => ! [W: A,Y: A,Xb: A,Z: A] :
          ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W),Y)),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),W),Z)),aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y)) )
        <=> ( ( W = Xb )
            | ( Y = Z ) ) ) ) ).

% crossproduct_eq
tff(fact_5571_take__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),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,Nb),aa(num,A,numeral_numeral(A),K))),aa(num,A,numeral_numeral(A),bit0(one2)))),one_one(A)) ) ).

% take_bit_Suc_bit1
tff(fact_5572_take__bit__Suc__minus__1__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),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,Nb))),one_one(A)) ) ).

% take_bit_Suc_minus_1_eq
tff(fact_5573_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_5574_take__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),A2) = 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,Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),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),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% take_bit_Suc
tff(fact_5575_and__nat__unfold,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),Nb) = $ite(
        ( ( M = zero_zero(nat) )
        | ( Nb = 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),Nb),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),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ) ).

% and_nat_unfold
tff(fact_5576_and__nat__rec,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344872417868541ns_and(nat),M),Nb) = 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))),Nb) ))),
        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),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% and_nat_rec
tff(fact_5577_stable__imp__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = $ite(aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2),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))),Nb)),one_one(A))) ) ) ) ).

% stable_imp_take_bit_eq
tff(fact_5578_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_5579_take__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = $ite(Nb = 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,Nb),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),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),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% take_bit_rec
tff(fact_5580_take__bit__word__minus__Bit1__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num,M: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),Nb)),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(Nb)),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_5581_take__bit__numeral__minus__numeral__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Nb: 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)),Nb))) = case_option(word(A),num,zero_zero(word(A)),aTP_Lamp_aeh(num,fun(num,word(A)),M),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),Nb)) ) ).

% take_bit_numeral_minus_numeral_word
tff(fact_5582_slice__nth,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A),I: nat] :
      ( aa(nat,$o,ord_less(nat,From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$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_5583_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_5584_take__bit__num__simps_I2_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(aa(nat,nat,suc,Nb),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(2)
tff(fact_5585_take__bit__num__simps_I5_J,axiom,
    ! [R3: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R3),one2) = aa(num,option(num),some(num),one2) ).

% take_bit_num_simps(5)
tff(fact_5586_pred__numeral__inc,axiom,
    ! [K: num] : pred_numeral(inc(K)) = aa(num,nat,numeral_numeral(nat),K) ).

% pred_numeral_inc
tff(fact_5587_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_5588_take__bit__num__simps_I3_J,axiom,
    ! [Nb: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,Nb),bit0(M)) = case_option(option(num),num,none(num),aTP_Lamp_aei(num,option(num)),bit_take_bit_num(Nb,M)) ).

% take_bit_num_simps(3)
tff(fact_5589_take__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),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)),Nb)) ).

% take_bit_of_Suc_0
tff(fact_5590_slice__len,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),From),To)
     => ( aa(nat,$o,aa(nat,fun(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_5591_take__bit__num__simps_I4_J,axiom,
    ! [Nb: nat,M: num] : bit_take_bit_num(aa(nat,nat,suc,Nb),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(Nb,M))) ).

% take_bit_num_simps(4)
tff(fact_5592_take__bit__num__simps_I6_J,axiom,
    ! [R3: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R3),bit0(M)) = case_option(option(num),num,none(num),aTP_Lamp_aei(num,option(num)),bit_take_bit_num(pred_numeral(R3),M)) ).

% take_bit_num_simps(6)
tff(fact_5593_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_5594_add__neg__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: 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),Nb))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Nb))) ) ).

% add_neg_numeral_special(5)
tff(fact_5595_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_5596_diff__numeral__special_I5_J,axiom,
    ! [A: $tType] :
      ( neg_numeral(A)
     => ! [Nb: num] : aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),one_one(A))),aa(num,A,numeral_numeral(A),Nb)) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Nb))) ) ).

% diff_numeral_special(5)
tff(fact_5597_take__bit__num__simps_I7_J,axiom,
    ! [R3: num,M: num] : bit_take_bit_num(aa(num,nat,numeral_numeral(nat),R3),aa(num,num,bit1,M)) = aa(num,option(num),some(num),case_option(num,num,one2,bit1,bit_take_bit_num(pred_numeral(R3),M))) ).

% take_bit_num_simps(7)
tff(fact_5598_take__bit__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,Nb: num] : aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),M)),aa(num,A,numeral_numeral(A),Nb)) = case_option(A,num,zero_zero(A),numeral_numeral(A),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),Nb)) ) ).

% take_bit_numeral_numeral
tff(fact_5599_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,Nb: 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),Nb))) = case_option(int,num,zero_zero(int),aTP_Lamp_aej(num,fun(num,int),M),bit_take_bit_num(aa(num,nat,numeral_numeral(nat),M),Nb)) ).

% take_bit_numeral_minus_numeral_int
tff(fact_5600_take__bit__nat__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),nat2(K)) = nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ) ).

% take_bit_nat_eq
tff(fact_5601_nat__take__bit__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
     => ( nat2(aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) = aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),nat2(K)) ) ) ).

% nat_take_bit_eq
tff(fact_5602_take__bit__nat__less__eq__self,axiom,
    ! [Nb: nat,M: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),M)),M) ).

% take_bit_nat_less_eq_self
tff(fact_5603_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,Nb: nat,Q2: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,M),Q2)),aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),Q2)) ) ).

% take_bit_tightened_less_eq_nat
tff(fact_5604_take__bit__mult,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,aa(int,fun(int,int),times_times(int),K),L)) ).

% take_bit_mult
tff(fact_5605_take__bit__diff,axiom,
    ! [Nb: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),L))) = aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,minus_minus(int,K),L)) ).

% take_bit_diff
tff(fact_5606_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_5607_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_5608_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,Nb: nat,K: int] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% take_bit_tightened_less_eq_int
tff(fact_5609_take__bit__nonnegative,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ).

% take_bit_nonnegative
tff(fact_5610_take__bit__int__less__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% take_bit_int_less_eq_self_iff
tff(fact_5611_inc_Osimps_I1_J,axiom,
    inc(one2) = bit0(one2) ).

% inc.simps(1)
tff(fact_5612_inc_Osimps_I2_J,axiom,
    ! [Xb: num] : inc(bit0(Xb)) = aa(num,num,bit1,Xb) ).

% inc.simps(2)
tff(fact_5613_inc_Osimps_I3_J,axiom,
    ! [Xb: num] : inc(aa(num,num,bit1,Xb)) = bit0(inc(Xb)) ).

% inc.simps(3)
tff(fact_5614_add__One,axiom,
    ! [Xb: num] : aa(num,num,aa(num,fun(num,num),plus_plus(num),Xb),one2) = inc(Xb) ).

% add_One
tff(fact_5615_inc__BitM__eq,axiom,
    ! [Nb: num] : inc(bitM(Nb)) = bit0(Nb) ).

% inc_BitM_eq
tff(fact_5616_BitM__inc__eq,axiom,
    ! [Nb: num] : bitM(inc(Nb)) = aa(num,num,bit1,Nb) ).

% BitM_inc_eq
tff(fact_5617_take__bit__num__eq__Some__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: num,Q2: num] :
          ( ( bit_take_bit_num(M,Nb) = aa(num,option(num),some(num),Q2) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(num,A,numeral_numeral(A),Nb)) = aa(num,A,numeral_numeral(A),Q2) ) ) ) ).

% take_bit_num_eq_Some_imp
tff(fact_5618_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_5619_take__bit__decr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) != zero_zero(int) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,minus_minus(int,K),one_one(int))) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),one_one(int)) ) ) ).

% take_bit_decr_eq
tff(fact_5620_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_5621_take__bit__num__eq__None__imp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: num] :
          ( ( bit_take_bit_num(M,Nb) = none(num) )
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(num,A,numeral_numeral(A),Nb)) = zero_zero(A) ) ) ) ).

% take_bit_num_eq_None_imp
tff(fact_5622_take__bit__nat__eq__self,axiom,
    ! [M: nat,Nb: 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))),Nb))
     => ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),M) = M ) ) ).

% take_bit_nat_eq_self
tff(fact_5623_take__bit__nat__less__exp,axiom,
    ! [Nb: nat,M: nat] : aa(nat,$o,ord_less(nat,aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),M)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% take_bit_nat_less_exp
tff(fact_5624_take__bit__nat__eq__self__iff,axiom,
    ! [Nb: nat,M: nat] :
      ( ( aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),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))),Nb)) ) ).

% take_bit_nat_eq_self_iff
tff(fact_5625_take__bit__nat__def,axiom,
    ! [Nb: nat,M: nat] : aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),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))),Nb)) ).

% take_bit_nat_def
tff(fact_5626_take__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),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,Nb),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_5627_take__bit__int__less__exp,axiom,
    ! [Nb: nat,K: int] : aa(int,$o,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ).

% take_bit_int_less_exp
tff(fact_5628_take__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),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))),Nb)) ).

% take_bit_int_def
tff(fact_5629_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_5630_take__bit__nat__less__self__iff,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),M)),M)
    <=> aa(nat,$o,aa(nat,fun(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))),Nb)),M) ) ).

% take_bit_nat_less_self_iff
tff(fact_5631_take__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),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,Nb),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_5632_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),K),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),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))),Nb)) ) ).

% take_bit_int_greater_eq_self_iff
tff(fact_5633_take__bit__int__less__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,ord_less(int,aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),K)
    <=> aa(int,$o,aa(int,fun(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))),Nb)),K) ) ).

% take_bit_int_less_self_iff
tff(fact_5634_take__bit__int__eq__self__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = K )
    <=> ( aa(int,$o,aa(int,fun(int,$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))),Nb)) ) ) ).

% take_bit_int_eq_self_iff
tff(fact_5635_take__bit__int__eq__self,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,aa(int,fun(int,$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))),Nb))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = K ) ) ) ).

% take_bit_int_eq_self
tff(fact_5636_take__bit__incr__eq,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),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))),Nb)),one_one(int)) )
     => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),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,Nb),K)) ) ) ).

% take_bit_incr_eq
tff(fact_5637_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_5638_take__bit__int__less__eq,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),K)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),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))),Nb))) ) ) ).

% take_bit_int_less_eq
tff(fact_5639_take__bit__int__greater__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,ord_less(int,K),zero_zero(int))
     => aa(int,$o,aa(int,fun(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))),Nb))),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ) ).

% take_bit_int_greater_eq
tff(fact_5640_signed__take__bit__eq__take__bit__shift,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),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))),Nb)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ).

% signed_take_bit_eq_take_bit_shift
tff(fact_5641_take__bit__minus__small__eq,axiom,
    ! [K: int,Nb: nat] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),K)
     => ( aa(int,$o,aa(int,fun(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))),Nb))
       => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),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))),Nb)),K) ) ) ) ).

% take_bit_minus_small_eq
tff(fact_5642_and__mask__arith_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb)) = 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)),Nb)))),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)),Nb))) ) ) ) ).

% and_mask_arith'
tff(fact_5643_listI__assn__def,axiom,
    ! [A: $tType,B: $tType,I5: set(nat),A3: fun(A,fun(B,assn)),Xs: list(A),Xsi: list(B)] :
      vEBT_List_listI_assn(A,B,I5,A3,Xs,Xsi) = aa(assn,assn,
        aa(assn,fun(assn,assn),times_times(assn),
          pure_assn(( ( aa(list(B),nat,size_size(list(B)),Xsi) = aa(list(A),nat,size_size(list(A)),Xs) )
            & aa(set(nat),$o,aa(set(nat),fun(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_aek(fun(A,fun(B,assn)),fun(list(A),fun(list(B),fun(nat,fun(assn,assn)))),A3),Xs),Xsi),one_one(assn),I5)) ).

% listI_assn_def
tff(fact_5644_word__2p__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),aa(word(A),nat,size_size(word(A)),W))
         => ( aa(word(A),$o,ord_less(word(A),W),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb))
          <=> aa(int,$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))),Nb)) ) ) ) ).

% word_2p_lem
tff(fact_5645_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_5646_uint__ge__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),Xb)) ) ).

% uint_ge_0
tff(fact_5647_uint__le__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(int,$o,aa(int,fun(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_5648_uint__nonnegative,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(word(A),int,semiring_1_unsigned(A,int),W)) ) ).

% uint_nonnegative
tff(fact_5649_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_5650_take__bit__minus__one__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,uminus_uminus(A),one_one(A))) = bit_se2239418461657761734s_mask(A,Nb) ) ).

% take_bit_minus_one_eq_mask
tff(fact_5651_word__le__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B2))) ) ) ).

% word_le_no
tff(fact_5652_word__less__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,ord_less(int,aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B2))) ) ) ).

% word_less_no
tff(fact_5653_word__div__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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)),A2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B2)))) ) ).

% word_div_no
tff(fact_5654_word__mod__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B2)))) ) ).

% word_mod_no
tff(fact_5655_mask__eqs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat,B2: 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)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),bit_se2239418461657761734s_mask(word(A),Nb))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),Nb)))),bit_se2239418461657761734s_mask(word(A),Nb)) = 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)),plus_plus(word(A)),A2),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(7)
tff(fact_5656_mask__eqs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),Nb: 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)),plus_plus(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),Nb)))),bit_se2239418461657761734s_mask(word(A),Nb)) = 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)),plus_plus(word(A)),A2),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(2)
tff(fact_5657_mask__eqs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat,B2: 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)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),bit_se2239418461657761734s_mask(word(A),Nb))),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) = 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)),plus_plus(word(A)),A2),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(1)
tff(fact_5658_and__mask__eq__iff__le__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb)) = W )
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb)) ) ) ).

% and_mask_eq_iff_le_mask
tff(fact_5659_le__mask__imp__and__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),bit_se2239418461657761734s_mask(word(A),Nb))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),bit_se2239418461657761734s_mask(word(A),Nb)) = Xb ) ) ) ).

% le_mask_imp_and_mask
tff(fact_5660_mask__twice2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat,Xb: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),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),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),bit_se2239418461657761734s_mask(word(A),Nb)) ) ) ) ).

% mask_twice2
tff(fact_5661_ex__mask__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ? [X2: nat] : bit_se2239418461657761734s_mask(word(A),X2) = one_one(word(A)) ) ).

% ex_mask_1
tff(fact_5662_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_5663_word__le__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)) ) ) ).

% word_le_def
tff(fact_5664_mask__eqs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat,B2: 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),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),bit_se2239418461657761734s_mask(word(A),Nb))),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),minus_minus(word(A),A2),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(3)
tff(fact_5665_mask__eqs_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),Nb: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),minus_minus(word(A),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),Nb)))),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),minus_minus(word(A),A2),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(4)
tff(fact_5666_mask__eqs_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat,B2: 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),minus_minus(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),bit_se2239418461657761734s_mask(word(A),Nb))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),Nb)))),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),minus_minus(word(A),A2),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(8)
tff(fact_5667_mask__power__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: nat,K: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(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),Nb))),K)),bit_se2239418461657761734s_mask(word(A),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),Xb),K)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_power_eq
tff(fact_5668_mask__eqs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A),Nb: 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)),times_times(word(A)),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),Nb)))),bit_se2239418461657761734s_mask(word(A),Nb)) = 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)),times_times(word(A)),A2),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(5)
tff(fact_5669_mask__eqs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),Nb: nat,A2: 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)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),Nb))),A2)),bit_se2239418461657761734s_mask(word(A),Nb)) = 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)),times_times(word(A)),B2),A2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(6)
tff(fact_5670_mask__eqs_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat,B2: 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)),times_times(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),A2),bit_se2239418461657761734s_mask(word(A),Nb))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),B2),bit_se2239418461657761734s_mask(word(A),Nb)))),bit_se2239418461657761734s_mask(word(A),Nb)) = 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)),times_times(word(A)),A2),B2)),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% mask_eqs(9)
tff(fact_5671_unsigned__greater__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & unique1627219031080169319umeral(A) )
     => ! [W: word(B)] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),aa(word(B),A,semiring_1_unsigned(B,A),W)) ) ).

% unsigned_greater_eq
tff(fact_5672_less__eq__mask__iff__take__bit__eq__self,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb))
        <=> ( aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),Nb),W) = W ) ) ) ).

% less_eq_mask_iff_take_bit_eq_self
tff(fact_5673_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_5674_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_5675_word__less__eq__iff__unsigned,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & linordered_semidom(B) )
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
        <=> aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(word(A),B,semiring_1_unsigned(A,B),A2)),aa(word(A),B,semiring_1_unsigned(A,B),B2)) ) ) ).

% word_less_eq_iff_unsigned
tff(fact_5676_word__less__iff__unsigned,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & linordered_semidom(B) )
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A2),B2)
        <=> aa(B,$o,ord_less(B,aa(word(A),B,semiring_1_unsigned(A,B),A2)),aa(word(A),B,semiring_1_unsigned(A,B),B2)) ) ) ).

% word_less_iff_unsigned
tff(fact_5677_mask__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ( bit_se2239418461657761734s_mask(A,one_one(nat)) = one_one(A) ) ) ).

% mask_1
tff(fact_5678_mask__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb)) = W )
        <=> aa(int,$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))),Nb)) ) ) ).

% mask_eq_iff
tff(fact_5679_and__mask__lt__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] : aa(int,$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),Nb)))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)) ) ).

% and_mask_lt_2p
tff(fact_5680_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_5681_mask__bin,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : bit_se2239418461657761734s_mask(word(A),Nb) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(int,int,uminus_uminus(int),one_one(int)))) ) ).

% mask_bin
tff(fact_5682_uint__add__ge0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xaa: word(A),Xb: word(B)] : aa(int,$o,aa(int,fun(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_5683_uint__mult__ge0,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xaa: word(A),Xb: word(B)] : aa(int,$o,aa(int,fun(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_5684_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_5685_uint__add__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] : aa(int,$o,aa(int,fun(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_5686_uint__plus__simple__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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),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_plus_simple_iff
tff(fact_5687_uint__plus__simple,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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),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_plus_simple
tff(fact_5688_no__ulen__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Xb),Y)),Xb)
        <=> aa(int,$o,aa(int,fun(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_5689_word__add__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2) = 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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_add_def
tff(fact_5690_uint__sub__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A)] :
          ( aa(int,$o,aa(int,fun(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_5691_uint__sub__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] : aa(int,$o,aa(int,fun(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_5692_uint__minus__simple__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),minus_minus(word(A),Xb),Y)),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_minus_simple_iff
tff(fact_5693_uint__minus__simple__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),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_minus_simple_alt
tff(fact_5694_word__sub__wi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),minus_minus(word(A),A2),B2) = 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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_sub_wi
tff(fact_5695_word__arith__power__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat] : aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),A2),Nb) = 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),A2)),Nb)) ) ).

% word_arith_power_alt
tff(fact_5696_word__mult__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_mult_def
tff(fact_5697_word__div__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),B2) = 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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% word_div_def
tff(fact_5698_and__mask__mod__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: 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),Nb)) = 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))),Nb))) ) ).

% and_mask_mod_2p
tff(fact_5699_and__mask__dvd,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(int,$o,dvd_dvd(int,aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb)),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),Nb)) = zero_zero(word(A)) ) ) ) ).

% and_mask_dvd
tff(fact_5700_and__mask__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: num,Nb: 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),Nb)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(num,int,numeral_numeral(int),I))) ) ).

% and_mask_no
tff(fact_5701_udvd__incr__lem0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Up: int,Uq: int,Nb: int,K6: word(A),N3: int] :
          ( aa(int,$o,ord_less(int,Up),Uq)
         => ( ( Up = aa(int,int,aa(int,fun(int,int),times_times(int),Nb),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
           => ( ( Uq = aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
             => aa(int,$o,aa(int,fun(int,$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),K6))),Uq) ) ) ) ) ).

% udvd_incr_lem0
tff(fact_5702_udvd__incr__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Up: int,Uq: int,Ua: int,Nb: int,K6: word(A),N3: int] :
          ( aa(int,$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),Nb),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
           => ( ( Uq = aa(int,int,aa(int,fun(int,int),plus_plus(int),Ua),aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
             => aa(int,$o,aa(int,fun(int,$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),K6))),Uq) ) ) ) ) ).

% udvd_incr_lem
tff(fact_5703_udvd__incr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q2: word(A),Nb: int,K6: word(A),N3: int] :
          ( aa(word(A),$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),Nb),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),K6)),Q2) ) ) ) ) ).

% udvd_incr0
tff(fact_5704_udvd__decr0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q2: word(A),Nb: int,K6: word(A),N3: int] :
          ( aa(word(A),$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),Nb),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
           => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
             => ( ( aa(word(A),int,semiring_1_unsigned(A,int),Q2) = aa(int,int,aa(int,fun(int,int),times_times(int),N3),aa(word(A),int,semiring_1_unsigned(A,int),K6)) )
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),minus_minus(word(A),Q2),K6)) ) ) ) ) ) ).

% udvd_decr0
tff(fact_5705_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_5706_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_5707_udvd__incr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q2: word(A),Ua: int,Nb: int,K6: word(A),N3: int] :
          ( aa(word(A),$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),Nb),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
           => ( ( 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),N3),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
             => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),P2),K6)),Q2) ) ) ) ) ).

% udvd_incr'
tff(fact_5708_udvd__decr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Q2: word(A),Ua: int,Nb: int,K6: word(A),N3: int] :
          ( aa(word(A),$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),Nb),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
           => ( ( 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),N3),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
             => ( ( 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),N3),aa(word(A),int,semiring_1_unsigned(A,int),K6))) )
               => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),minus_minus(word(A),Q2),K6)) ) ) ) ) ) ).

% udvd_decr'
tff(fact_5709_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),bit_se2239418461657761734s_mask(A,Nb))
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% semiring_bit_operations_class.even_mask_iff
tff(fact_5710_mask__plus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),bit_se2239418461657761734s_mask(word(A),Nb)),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))),Nb) ) ).

% mask_plus_1
tff(fact_5711_less__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: nat] :
          ( aa(word(A),$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))),Nb))
         => ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),bit_se2239418461657761734s_mask(word(A),Nb)) = Xb ) ) ) ).

% less_mask_eq
tff(fact_5712_mask__eq__decr__exp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : bit_se2239418461657761734s_mask(word(A),Nb) = 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))),Nb)),one_one(word(A))) ) ).

% mask_eq_decr_exp
tff(fact_5713_mask__eq__exp__minus__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_se2239418461657761734s_mask(A,Nb) = 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))),Nb)),one_one(A)) ) ).

% mask_eq_exp_minus_1
tff(fact_5714_mask__Suc__rec,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : bit_se2239418461657761734s_mask(word(A),aa(nat,nat,suc,Nb)) = 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),Nb))),one_one(word(A))) ) ).

% mask_Suc_rec
tff(fact_5715_is__aligned__AND__less__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),Nb: nat,V: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),U),bit_se2239418461657761734s_mask(word(A),Nb)) = zero_zero(word(A)) )
         => ( aa(word(A),$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))),Nb))
           => ( 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_5716_add__mask__fold,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: 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))),Nb))),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),Nb)) ) ).

% add_mask_fold
tff(fact_5717_mask__eq__iff__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),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),Nb)) = 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))),Nb)) ) ) ) ).

% mask_eq_iff_w2p
tff(fact_5718_and__mask__less__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),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),Nb))),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))),Nb)) ) ) ).

% and_mask_less_size
tff(fact_5719_word__and__mask__le__2pm1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] : aa(word(A),$o,aa(word(A),fun(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),Nb))),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))),Nb)),one_one(word(A)))) ) ).

% word_and_mask_le_2pm1
tff(fact_5720_uint__range__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          ( aa(int,$o,aa(int,fun(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_5721_word__mod__2p__is__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xb: word(A)] :
          ( aa(word(A),$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))),Nb))
         => ( 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))),Nb)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),bit_se2239418461657761734s_mask(word(A),Nb)) ) ) ) ).

% word_mod_2p_is_mask
tff(fact_5722_uint__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(word(A),$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))),Nb))
         => ( 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))),Nb)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb) ) ) ) ).

% uint_2p
tff(fact_5723_no__plus__overflow__uint__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_5724_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_5725_and__mask__arith,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: 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),Nb)) = 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)),Nb)))),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)),Nb))) ) ).

% and_mask_arith
tff(fact_5726_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,aa(int,fun(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_5727_mask__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: num] : bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),Nb)) = 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(Nb)))) ) ).

% mask_numeral
tff(fact_5728_and__minus__numerals_I7_J,axiom,
    ! [Nb: 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(Nb)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(Nb))) ).

% and_minus_numerals(7)
tff(fact_5729_and__minus__numerals_I3_J,axiom,
    ! [M: num,Nb: 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(Nb)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bitM(Nb))) ).

% and_minus_numerals(3)
tff(fact_5730_and__minus__numerals_I8_J,axiom,
    ! [Nb: 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,Nb)))),aa(num,int,numeral_numeral(int),M)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bit0(Nb))) ).

% and_minus_numerals(8)
tff(fact_5731_mask__nat__positive__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),bit_se2239418461657761734s_mask(nat,Nb))
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ).

% mask_nat_positive_iff
tff(fact_5732_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_5733_and__minus__numerals_I4_J,axiom,
    ! [M: num,Nb: 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,Nb)))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,bit0(Nb))) ).

% and_minus_numerals(4)
tff(fact_5734_word__unat__and__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: nat,Y: word(A)] :
          ( ( aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),Nb)
            | aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Y)),Nb) )
         => aa(nat,$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))),Nb) ) ) ).

% word_unat_and_lt
tff(fact_5735_le__ucast__ucast__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xb: word(A),Y: word(B)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Y))
         => aa(word(B),$o,aa(word(B),fun(word(B),$o),ord_less_eq(word(B)),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),Xb)),Y) ) ) ).

% le_ucast_ucast_le
tff(fact_5736_less__eq__mask,axiom,
    ! [Nb: nat] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),bit_se2239418461657761734s_mask(nat,Nb)) ).

% less_eq_mask
tff(fact_5737_ucast__1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ( aa(word(B),word(A),semiring_1_unsigned(B,word(A)),one_one(word(B))) = one_one(word(A)) ) ) ).

% ucast_1
tff(fact_5738_word__unat__Rep__inject1,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A)] :
          ( ( aa(word(A),nat,semiring_1_unsigned(A,nat),Xb) = aa(word(B),nat,semiring_1_unsigned(B,nat),one_one(word(B))) )
        <=> ( Xb = one_one(word(A)) ) ) ) ).

% word_unat_Rep_inject1
tff(fact_5739_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_5740_word__le__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),B2)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) ) ).

% word_le_nat_alt
tff(fact_5741_unat__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A2),B2)
         => aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) ) ).

% unat_mono
tff(fact_5742_word__less__nat__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( aa(word(A),$o,ord_less(word(A),A2),B2)
        <=> aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) ) ).

% word_less_nat_alt
tff(fact_5743_le__unat__uoi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: nat,Z: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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_5744_uno__simps_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Z: word(A),Nb: 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)),Nb))) = aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),Z)),Nb) ) ).

% uno_simps(2)
tff(fact_5745_mask__nonnegative__int,axiom,
    ! [Nb: nat] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_se2239418461657761734s_mask(int,Nb)) ).

% mask_nonnegative_int
tff(fact_5746_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_5747_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_5748_max__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: 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),A2),B2)),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),A2),B2))),aa(word(A),nat,semiring_1_unsigned(A,nat),C2)) ) ).

% max_lt
tff(fact_5749_and__not__num_Osimps_I1_J,axiom,
    bit_and_not_num(one2,one2) = none(num) ).

% and_not_num.simps(1)
tff(fact_5750_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_5751_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_5752_un__ui__le,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [A2: word(A),B2: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(B),nat,semiring_1_unsigned(B,nat),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(B),int,semiring_1_unsigned(B,int),B2)) ) ) ).

% un_ui_le
tff(fact_5753_unat__plus__simple,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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),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_plus_simple
tff(fact_5754_unat__sub,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: word(A),A2: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),B2),A2)
         => ( aa(word(A),nat,semiring_1_unsigned(A,nat),aa(word(A),word(A),minus_minus(word(A),A2),B2)) = aa(nat,nat,minus_minus(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) ) ) ).

% unat_sub
tff(fact_5755_word__of__nat__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),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)),Nb)),Xb) ) ) ).

% word_of_nat_less
tff(fact_5756_unat__less__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: nat] :
          ( aa(word(A),$o,ord_less(word(A),Xb),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb))
         => aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)),Nb) ) ) ).

% unat_less_helper
tff(fact_5757_word__unat__less__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),aa(nat,word(A),semiring_1_of_nat(word(A)),B2))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),B2) ) ) ).

% word_unat_less_le
tff(fact_5758_word__of__nat__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xb: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(word(A),nat,semiring_1_unsigned(A,nat),Xb))
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb)),Xb) ) ) ).

% word_of_nat_le
tff(fact_5759_word__arith__nat__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))) ) ).

% word_arith_nat_add
tff(fact_5760_less__mask,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,aa(nat,nat,suc,zero_zero(nat))),Nb)
     => aa(nat,$o,ord_less(nat,Nb),bit_se2239418461657761734s_mask(nat,Nb)) ) ).

% less_mask
tff(fact_5761_word__arith__nat__mult,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))) ) ).

% word_arith_nat_mult
tff(fact_5762_word__arith__nat__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),A2),B2) = 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),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))) ) ).

% word_arith_nat_div
tff(fact_5763_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_5764_and__not__num_Osimps_I2_J,axiom,
    ! [Nb: num] : bit_and_not_num(one2,bit0(Nb)) = aa(num,option(num),some(num),one2) ).

% and_not_num.simps(2)
tff(fact_5765_and__not__num_Osimps_I3_J,axiom,
    ! [Nb: num] : bit_and_not_num(one2,aa(num,num,bit1,Nb)) = none(num) ).

% and_not_num.simps(3)
tff(fact_5766_unat__1__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_5767_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_5768_unatSuc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),Nb) != 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))),Nb)) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Nb)) ) ) ) ).

% unatSuc
tff(fact_5769_unatSuc2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Nb),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)),Nb),one_one(word(A)))) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Nb)) ) ) ) ).

% unatSuc2
tff(fact_5770_Suc__unat__diff__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),one_one(word(A))),Xb)
         => ( 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_diff_1
tff(fact_5771_unat__Suc2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A)] :
          ( ( Nb != aa(word(A),word(A),uminus_uminus(word(A)),one_one(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)),Nb),one_one(word(A)))) = aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Nb)) ) ) ) ).

% unat_Suc2
tff(fact_5772_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_5773_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_5774_take__bit__eq__mask__iff,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = bit_se2239418461657761734s_mask(int,Nb) )
    <=> ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),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_5775_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_5776_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_5777_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_5778_Suc__mask__eq__exp,axiom,
    ! [Nb: nat] : aa(nat,nat,suc,bit_se2239418461657761734s_mask(nat,Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ).

% Suc_mask_eq_exp
tff(fact_5779_mask__nat__less__exp,axiom,
    ! [Nb: nat] : aa(nat,$o,ord_less(nat,bit_se2239418461657761734s_mask(nat,Nb)),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)) ).

% mask_nat_less_exp
tff(fact_5780_lt__plus__1__le__word,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,MaxBound: word(A),Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),aa(word(A),nat,semiring_1_unsigned(A,nat),MaxBound))
         => ( aa(word(A),$o,ord_less(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),one_one(word(A))),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb)))
          <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb)) ) ) ) ).

% lt_plus_1_le_word
tff(fact_5781_mask__half__int,axiom,
    ! [Nb: nat] : aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_se2239418461657761734s_mask(int,Nb)),aa(num,int,numeral_numeral(int),bit0(one2))) = bit_se2239418461657761734s_mask(int,aa(nat,nat,minus_minus(nat,Nb),one_one(nat))) ).

% mask_half_int
tff(fact_5782_mask__int__def,axiom,
    ! [Nb: nat] : bit_se2239418461657761734s_mask(int,Nb) = 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))),Nb)),one_one(int)) ).

% mask_int_def
tff(fact_5783_mask__nat__def,axiom,
    ! [Nb: nat] : bit_se2239418461657761734s_mask(nat,Nb) = 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))),Nb)),one_one(nat)) ).

% mask_nat_def
tff(fact_5784_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_5785_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_5786_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,Nb: num] : bit_and_not_num(aa(num,num,bit1,M),bit0(Nb)) = case_option(option(num),num,aa(num,option(num),some(num),one2),aTP_Lamp_ael(num,option(num)),bit_and_not_num(M,Nb)) ).

% and_not_num.simps(8)
tff(fact_5787_word__div__eq__1__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),M: word(A)] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),Nb),M) = one_one(word(A)) )
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),M),Nb)
            & aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),Nb)),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_5788_and__mask__dvd__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,dvd_dvd(nat,aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)),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),Nb)) = zero_zero(word(A)) ) ) ) ).

% and_mask_dvd_nat
tff(fact_5789_of__nat__eq__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Nb) = W )
        <=> ? [Q4: nat] : Nb = 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_5790_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [Nb: nat,K: int] :
      ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K) = bit_se2239418461657761734s_mask(int,Nb) )
    <=> 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))),Nb)),aa(int,int,aa(int,fun(int,int),plus_plus(int),K),one_one(int))) ) ).

% take_bit_eq_mask_iff_exp_dvd
tff(fact_5791_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_5792_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,aa(nat,fun(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_5793_no__plus__overflow__unat__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_5794_word__unat__mask__lt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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_5795_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [Nb: nat,M: num] : bit_take_bit_num(Nb,aa(num,num,bit1,M)) = case_nat(option(num),none(num),aTP_Lamp_aem(num,fun(nat,option(num)),M),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(3)
tff(fact_5796_arctan__def,axiom,
    ! [Y: real] : arctan(Y) = the(real,aTP_Lamp_aen(real,fun(real,$o),Y)) ).

% arctan_def
tff(fact_5797_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,
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L))))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,minus_minus(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),L)),aa($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(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L))))))) ) ).

% modulo_int_def
tff(fact_5798_sgn__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,sgn_sgn(A),aa(A,A,sgn_sgn(A),A2)) = aa(A,A,sgn_sgn(A),A2) ) ).

% sgn_sgn
tff(fact_5799_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_5800_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_5801_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_5802_sgn__divide,axiom,
    ! [A: $tType] :
      ( field_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,sgn_sgn(A),B2)) ) ).

% sgn_divide
tff(fact_5803_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,sgn_sgn(A),aa(A,A,uminus_uminus(A),A2)) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),A2)) ) ).

% idom_abs_sgn_class.sgn_minus
tff(fact_5804_power__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A,Nb: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,sgn_sgn(A),A2)),Nb) ) ).

% power_sgn
tff(fact_5805_sgn__less,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,aa(A,A,sgn_sgn(A),A2)),zero_zero(A))
        <=> aa(A,$o,ord_less(A,A2),zero_zero(A)) ) ) ).

% sgn_less
tff(fact_5806_sgn__greater,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),aa(A,A,sgn_sgn(A),A2))
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A2) ) ) ).

% sgn_greater
tff(fact_5807_divide__sgn,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [A2: A,B2: A] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,sgn_sgn(A),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,sgn_sgn(A),B2)) ) ).

% divide_sgn
tff(fact_5808_case__nat__numeral,axiom,
    ! [A: $tType,A2: A,F: fun(nat,A),V: num] : case_nat(A,A2,F,aa(num,nat,numeral_numeral(nat),V)) = aa(nat,A,F,pred_numeral(V)) ).

% case_nat_numeral
tff(fact_5809_sgn__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),A2)
         => ( aa(A,A,sgn_sgn(A),A2) = one_one(A) ) ) ) ).

% sgn_pos
tff(fact_5810_abs__sgn__eq__1,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( A2 != zero_zero(A) )
         => ( aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = one_one(A) ) ) ) ).

% abs_sgn_eq_1
tff(fact_5811_sgn__mult__self__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,sgn_sgn(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% sgn_mult_self_eq
tff(fact_5812_sgn__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% sgn_abs
tff(fact_5813_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,sgn_sgn(A),aa(A,A,abs_abs(A),A2)) = aa($o,A,zero_neq_one_of_bool(A),A2 != zero_zero(A)) ) ).

% idom_abs_sgn_class.abs_sgn
tff(fact_5814_sgn__mult__dvd__iff,axiom,
    ! [R3: int,L: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R3)),L)),K)
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        & ( ( R3 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% sgn_mult_dvd_iff
tff(fact_5815_mult__sgn__dvd__iff,axiom,
    ! [L: int,R3: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,aa(int,int,aa(int,fun(int,int),times_times(int),L),aa(int,int,sgn_sgn(int),R3))),K)
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        & ( ( R3 = zero_zero(int) )
         => ( K = zero_zero(int) ) ) ) ) ).

% mult_sgn_dvd_iff
tff(fact_5816_dvd__sgn__mult__iff,axiom,
    ! [L: int,R3: int,K: int] :
      ( aa(int,$o,dvd_dvd(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),R3)),K))
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        | ( R3 = zero_zero(int) ) ) ) ).

% dvd_sgn_mult_iff
tff(fact_5817_dvd__mult__sgn__iff,axiom,
    ! [L: int,K: int,R3: int] :
      ( aa(int,$o,dvd_dvd(int,L),aa(int,int,aa(int,fun(int,int),times_times(int),K),aa(int,int,sgn_sgn(int),R3)))
    <=> ( aa(int,$o,dvd_dvd(int,L),K)
        | ( R3 = zero_zero(int) ) ) ) ).

% dvd_mult_sgn_iff
tff(fact_5818_case__nat__add__eq__if,axiom,
    ! [A: $tType,A2: A,F: fun(nat,A),V: num,Nb: nat] : case_nat(A,A2,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(num,nat,numeral_numeral(nat),V)),Nb)) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),pred_numeral(V)),Nb)) ).

% case_nat_add_eq_if
tff(fact_5819_sgn__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( aa(A,$o,ord_less(A,A2),zero_zero(A))
         => ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),one_one(A)) ) ) ) ).

% sgn_neg
tff(fact_5820_sgn__of__nat,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Nb: nat] : aa(A,A,sgn_sgn(A),aa(nat,A,semiring_1_of_nat(A),Nb)) = aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)) ) ).

% sgn_of_nat
tff(fact_5821_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),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) ) ) ).

% div_eq_sgn_abs
tff(fact_5822_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N2: nat,L2: int] : K != aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L2)),aa(nat,int,semiring_1_of_nat(int),N2)) ).

% int_sgnE
tff(fact_5823_same__sgn__abs__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A2) )
         => ( aa(A,A,abs_abs(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,abs_abs(A),A2)),aa(A,A,abs_abs(A),B2)) ) ) ) ).

% same_sgn_abs_add
tff(fact_5824_linordered__idom__class_Oabs__sgn,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [K: A] : aa(A,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_5825_abs__mult__sgn,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,abs_abs(A),A2)),aa(A,A,sgn_sgn(A),A2)) = A2 ) ).

% abs_mult_sgn
tff(fact_5826_sgn__mult__abs,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,abs_abs(A),A2)) = A2 ) ).

% sgn_mult_abs
tff(fact_5827_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)),aa(A,A,abs_abs(A),Xb)) = Xb ) ).

% mult_sgn_abs
tff(fact_5828_old_Onat_Osimps_I5_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A),X23: nat] : case_nat(A,F1,F22,aa(nat,nat,suc,X23)) = aa(nat,A,F22,X23) ).

% old.nat.simps(5)
tff(fact_5829_old_Onat_Osimps_I4_J,axiom,
    ! [A: $tType,F1: A,F22: fun(nat,A)] : case_nat(A,F1,F22,zero_zero(nat)) = F1 ).

% old.nat.simps(4)
tff(fact_5830_sgn__0__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_0_0
tff(fact_5831_sgn__eq__0__iff,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = zero_zero(A) )
        <=> ( A2 = zero_zero(A) ) ) ) ).

% sgn_eq_0_iff
tff(fact_5832_same__sgn__sgn__add,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) = aa(A,A,sgn_sgn(A),A2) )
         => ( aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,sgn_sgn(A),A2) ) ) ) ).

% same_sgn_sgn_add
tff(fact_5833_nat_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: B,F22: fun(nat,B),Nat: nat] : aa(B,A,H,case_nat(B,F1,F22,Nat)) = case_nat(A,aa(B,A,H,F1),aa(fun(nat,B),fun(nat,A),aTP_Lamp_aeo(fun(B,A),fun(fun(nat,B),fun(nat,A)),H),F22),Nat) ).

% nat.case_distrib
tff(fact_5834_sgn__mult,axiom,
    ! [A: $tType] :
      ( idom_abs_sgn(A)
     => ! [A2: A,B2: A] : aa(A,A,sgn_sgn(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,sgn_sgn(A),A2)),aa(A,A,sgn_sgn(A),B2)) ) ).

% sgn_mult
tff(fact_5835_Real__Vector__Spaces_Osgn__mult,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Xb: A,Y: A] : aa(A,A,sgn_sgn(A),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,sgn_sgn(A),Xb)),aa(A,A,sgn_sgn(A),Y)) ) ).

% Real_Vector_Spaces.sgn_mult
tff(fact_5836_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_5837_sgn__not__eq__imp,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [B2: A,A2: A] :
          ( ( aa(A,A,sgn_sgn(A),B2) != aa(A,A,sgn_sgn(A),A2) )
         => ( ( aa(A,A,sgn_sgn(A),A2) != zero_zero(A) )
           => ( ( aa(A,A,sgn_sgn(A),B2) != zero_zero(A) )
             => ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),aa(A,A,sgn_sgn(A),B2)) ) ) ) ) ) ).

% sgn_not_eq_imp
tff(fact_5838_ln__real__def,axiom,
    ! [Xb: real] : aa(real,real,ln_ln(real),Xb) = the(real,aTP_Lamp_aep(real,fun(real,$o),Xb)) ).

% ln_real_def
tff(fact_5839_suminf__def,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [F: fun(nat,A)] : suminf(A,F) = the(A,sums(A,F)) ) ).

% suminf_def
tff(fact_5840_sgn__1__pos,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = one_one(A) )
        <=> aa(A,$o,ord_less(A,zero_zero(A)),A2) ) ) ).

% sgn_1_pos
tff(fact_5841_abs__sgn__eq,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          aa(A,A,abs_abs(A),aa(A,A,sgn_sgn(A),A2)) = $ite(A2 = zero_zero(A),zero_zero(A),one_one(A)) ) ).

% abs_sgn_eq
tff(fact_5842_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_5843_ln__neg__is__const,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real))
     => ( aa(real,real,ln_ln(real),Xb) = the(real,aTP_Lamp_aeq(real,$o)) ) ) ).

% ln_neg_is_const
tff(fact_5844_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_5845_sgn__1__neg,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( ( aa(A,A,sgn_sgn(A),A2) = aa(A,A,uminus_uminus(A),one_one(A)) )
        <=> aa(A,$o,ord_less(A,A2),zero_zero(A)) ) ) ).

% sgn_1_neg
tff(fact_5846_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_5847_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_5848_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)),aa(int,int,abs_abs(int),K))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),V)),aa(int,int,abs_abs(int),L))) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L)) ) ) ).

% div_sgn_abs_cancel
tff(fact_5849_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),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L))) ) ) ).

% div_dvd_sgn_abs
tff(fact_5850_arccos__def,axiom,
    ! [Y: real] : arccos(Y) = the(real,aTP_Lamp_aer(real,fun(real,$o),Y)) ).

% arccos_def
tff(fact_5851_eucl__rel__int__remainderI,axiom,
    ! [R3: int,L: int,K: int,Q2: int] :
      ( ( aa(int,int,sgn_sgn(int),R3) = aa(int,int,sgn_sgn(int),L) )
     => ( aa(int,$o,ord_less(int,aa(int,int,abs_abs(int),R3)),aa(int,int,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)),R3) )
         => eucl_rel_int(K,L,aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q2),R3)) ) ) ) ).

% eucl_rel_int_remainderI
tff(fact_5852_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),aa(int,int,abs_abs(int),K)),aa(int,int,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_5853_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) ) ) )
         => ~ ! [R2: int,Q3: int] :
                ( ( A32 = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Q3),R2) )
               => ( ( aa(int,int,sgn_sgn(int),R2) = aa(int,int,sgn_sgn(int),A22) )
                 => ( aa(int,$o,ord_less(int,aa(int,int,abs_abs(int),R2)),aa(int,int,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)),R2) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
tff(fact_5854_eucl__rel__int_Osimps,axiom,
    ! [A1: int,A22: int,A32: product_prod(int,int)] :
      ( eucl_rel_int(A1,A22,A32)
    <=> ( ? [K4: int] :
            ( ( A1 = K4 )
            & ( 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)),K4) ) )
        | ? [L4: int,K4: int,Q4: int] :
            ( ( A1 = K4 )
            & ( 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) )
            & ( K4 = aa(int,int,aa(int,fun(int,int),times_times(int),Q4),L4) ) )
        | ? [R6: int,L4: int,K4: int,Q4: int] :
            ( ( A1 = K4 )
            & ( 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,aa(int,int,abs_abs(int),R6)),aa(int,int,abs_abs(int),L4))
            & ( K4 = 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_5855_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [Nb: nat] : bit_take_bit_num(Nb,one2) = case_nat(option(num),none(num),aTP_Lamp_aes(nat,option(num)),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(1)
tff(fact_5856_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_aet(real,$o)) ).

% pi_half
tff(fact_5857_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_aet(real,$o))) ).

% pi_def
tff(fact_5858_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [Nb: nat,M: num] : bit_take_bit_num(Nb,bit0(M)) = case_nat(option(num),none(num),aTP_Lamp_aeu(num,fun(nat,option(num)),M),Nb) ).

% Code_Abstract_Nat.take_bit_num_code(2)
tff(fact_5859_divide__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Nb: 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),Nb))) = $ite(
        ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
        | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
        | ( Nb = zero_zero(nat) ) ),
        zero_zero(int),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb)),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,dvd_dvd(nat,Nb),M)))))) ) ).

% divide_int_unfold
tff(fact_5860_modulo__int__unfold,axiom,
    ! [K: int,M: nat,L: int,Nb: 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),Nb))) = $ite(
        ( ( aa(int,int,sgn_sgn(int),L) = zero_zero(int) )
        | ( aa(int,int,sgn_sgn(int),K) = zero_zero(int) )
        | ( Nb = 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)),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),L)),aa(int,int,minus_minus(int,aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,dvd_dvd(nat,Nb),M))))),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),M),Nb))))) ) ).

% modulo_int_unfold
tff(fact_5861_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),
        $ite(aa(int,int,sgn_sgn(int),K) = aa(int,int,sgn_sgn(int),L),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L)))),aa(int,int,uminus_uminus(int),aa(nat,int,semiring_1_of_nat(int),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),nat2(aa(int,int,abs_abs(int),K))),nat2(aa(int,int,abs_abs(int),L)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(int,$o,dvd_dvd(int,L),K)))))) ) ).

% divide_int_def
tff(fact_5862_arcsin__def,axiom,
    ! [Y: real] : arcsin(Y) = the(real,aTP_Lamp_aev(real,fun(real,$o),Y)) ).

% arcsin_def
tff(fact_5863_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_5864_Bit__Operations_Otake__bit__num__code,axiom,
    ! [Nb: nat,M: num] : bit_take_bit_num(Nb,M) = aa(product_prod(nat,num),option(num),product_case_prod(nat,num,option(num),aTP_Lamp_aez(nat,fun(num,option(num)))),aa(num,product_prod(nat,num),aa(nat,fun(num,product_prod(nat,num)),product_Pair(nat,num),Nb),M)) ).

% Bit_Operations.take_bit_num_code
tff(fact_5865_sgn__div__eq__sgn__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2) != zero_zero(int) )
     => ( aa(int,int,sgn_sgn(int),aa(int,int,aa(int,fun(int,int),divide_divide(int),A2),B2)) = aa(int,int,sgn_sgn(int),aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2)) ) ) ).

% sgn_div_eq_sgn_mult
tff(fact_5866_zero__le__sgn__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,sgn_sgn(real),Xb))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Xb) ) ).

% zero_le_sgn_iff
tff(fact_5867_sgn__le__0__iff,axiom,
    ! [Xb: real] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,sgn_sgn(real),Xb)),zero_zero(real))
    <=> aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Xb),zero_zero(real)) ) ).

% sgn_le_0_iff
tff(fact_5868_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,$o)),aTP_Lamp_afa(A,fun(B,fun(A,fun(B,$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_5869_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero(nat) )
    <=> case_nat($o,$false,aTP_Lamp_afb(nat,$o),Nat) ) ).

% nat.disc_eq_case(2)
tff(fact_5870_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero(nat) )
    <=> case_nat($o,$true,aTP_Lamp_afc(nat,$o),Nat) ) ).

% nat.disc_eq_case(1)
tff(fact_5871_real__sgn__eq,axiom,
    ! [Xb: real] : aa(real,real,sgn_sgn(real),Xb) = aa(real,real,aa(real,fun(real,real),divide_divide(real),Xb),aa(real,real,abs_abs(real),Xb)) ).

% real_sgn_eq
tff(fact_5872_num_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,H: fun(B,A),F1: B,F22: fun(num,B),F32: fun(num,B),Num: num] : aa(B,A,H,case_num(B,F1,F22,F32,Num)) = case_num(A,aa(B,A,H,F1),aa(fun(num,B),fun(num,A),aTP_Lamp_afd(fun(B,A),fun(fun(num,B),fun(num,A)),H),F22),aa(fun(num,B),fun(num,A),aTP_Lamp_afd(fun(B,A),fun(fun(num,B),fun(num,A)),H),F32),Num) ).

% num.case_distrib
tff(fact_5873_verit__eq__simplify_I17_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A),X23: num] : case_num(A,F1,F22,F32,bit0(X23)) = aa(num,A,F22,X23) ).

% verit_eq_simplify(17)
tff(fact_5874_verit__eq__simplify_I16_J,axiom,
    ! [A: $tType,F1: A,F22: fun(num,A),F32: fun(num,A)] : case_num(A,F1,F22,F32,one2) = F1 ).

% verit_eq_simplify(16)
tff(fact_5875_sgn__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,real,sgn_sgn(real),aa(real,real,root(Nb),Xb)) = aa(real,real,sgn_sgn(real),Xb) ) ) ).

% sgn_root
tff(fact_5876_sgn__eq,axiom,
    ! [Z: complex] : aa(complex,complex,sgn_sgn(complex),Z) = aa(complex,complex,aa(complex,fun(complex,complex),divide_divide(complex),Z),real_Vector_of_real(complex,real_V7770717601297561774m_norm(complex,Z))) ).

% sgn_eq
tff(fact_5877_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,M)),Nb)
    <=> case_nat($o,$false,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ).

% less_eq_nat.simps(2)
tff(fact_5878_max__Suc2,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,ord_max(nat,M),aa(nat,nat,suc,Nb)) = case_nat(nat,aa(nat,nat,suc,Nb),aTP_Lamp_afe(nat,fun(nat,nat),Nb),M) ).

% max_Suc2
tff(fact_5879_max__Suc1,axiom,
    ! [Nb: nat,M: nat] : aa(nat,nat,ord_max(nat,aa(nat,nat,suc,Nb)),M) = case_nat(nat,aa(nat,nat,suc,Nb),aTP_Lamp_aff(nat,fun(nat,nat),Nb),M) ).

% max_Suc1
tff(fact_5880_The__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : the(product_prod(A,B),product_case_prod(A,B,$o,P)) = the(product_prod(A,B),aTP_Lamp_afg(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P)) ).

% The_case_prod
tff(fact_5881_sgn__real__def,axiom,
    ! [A2: real] :
      aa(real,real,sgn_sgn(real),A2) = $ite(
        A2 = zero_zero(real),
        zero_zero(real),
        $ite(aa(real,$o,ord_less(real,zero_zero(real)),A2),one_one(real),aa(real,real,uminus_uminus(real),one_one(real))) ) ).

% sgn_real_def
tff(fact_5882_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_5883_diff__Suc,axiom,
    ! [M: nat,Nb: nat] : aa(nat,nat,minus_minus(nat,M),aa(nat,nat,suc,Nb)) = case_nat(nat,zero_zero(nat),aTP_Lamp_vc(nat,nat),aa(nat,nat,minus_minus(nat,M),Nb)) ).

% diff_Suc
tff(fact_5884_sgn__power__injE,axiom,
    ! [A2: real,Nb: nat,Xb: real,B2: real] :
      ( ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),A2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),A2)),Nb)) = Xb )
     => ( ( Xb = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),B2)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),B2)),Nb)) )
       => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
         => ( A2 = B2 ) ) ) ) ).

% sgn_power_injE
tff(fact_5885_sgn__power__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,sgn_sgn(real),aa(real,real,root(Nb),Xb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,abs_abs(real),aa(real,real,root(Nb),Xb))),Nb)) = Xb ) ) ).

% sgn_power_root
tff(fact_5886_root__sgn__power,axiom,
    ! [Nb: nat,Y: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(real,real,root(Nb),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),aa(real,real,abs_abs(real),Y)),Nb))) = Y ) ) ).

% root_sgn_power
tff(fact_5887_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,aa(real,fun(real,$o),ord_less_eq(real),Xb),pi)
         => ( arg(Z) = Xb ) ) ) ) ).

% cis_Arg_unique
tff(fact_5888_split__root,axiom,
    ! [P: fun(real,$o),Nb: nat,Xb: real] :
      ( aa(real,$o,P,aa(real,real,root(Nb),Xb))
    <=> ( ( ( Nb = zero_zero(nat) )
         => aa(real,$o,P,zero_zero(real)) )
        & ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
         => ! [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),aa(real,real,abs_abs(real),Y3)),Nb)) = Xb )
             => aa(real,$o,P,Y3) ) ) ) ) ).

% split_root
tff(fact_5889_floor__real__def,axiom,
    ! [Xb: real] : archim6421214686448440834_floor(real,Xb) = the(int,aTP_Lamp_afh(real,fun(int,$o),Xb)) ).

% floor_real_def
tff(fact_5890_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,aa(real,fun(real,$o),ord_less_eq(real),arg(Z)),pi) ) ) ).

% Arg_correct
tff(fact_5891_arctan__inverse,axiom,
    ! [Xb: real] :
      ( ( Xb != zero_zero(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)))),arctan(Xb)) ) ) ).

% arctan_inverse
tff(fact_5892_Nitpick_Ocase__nat__unfold,axiom,
    ! [A: $tType,Xb: A,F: fun(nat,A),Nb: nat] :
      case_nat(A,Xb,F,Nb) = $ite(Nb = zero_zero(nat),Xb,aa(nat,A,F,aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))) ).

% Nitpick.case_nat_unfold
tff(fact_5893_the__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( aa(A,$o,P,A2)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = A2 ) )
       => ( the(A,P) = A2 ) ) ) ).

% the_equality
tff(fact_5894_the__eq__trivial,axiom,
    ! [A: $tType,A2: A] : the(A,aTP_Lamp_dc(A,fun(A,$o),A2)) = A2 ).

% the_eq_trivial
tff(fact_5895_the__sym__eq__trivial,axiom,
    ! [A: $tType,Xb: A] : the(A,aa(A,fun(A,$o),fequal(A),Xb)) = Xb ).

% the_sym_eq_trivial
tff(fact_5896_the1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y5: A] :
              ( aa(A,$o,P,Y5)
             => ( Y5 = X3 ) ) )
     => ( aa(A,$o,P,A2)
       => ( the(A,P) = A2 ) ) ) ).

% the1_equality
tff(fact_5897_the1I2,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y5: A] :
              ( aa(A,$o,P,Y5)
             => ( Y5 = X3 ) ) )
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,the(A,P)) ) ) ).

% the1I2
tff(fact_5898_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,$o)),aTP_Lamp_afi($o,fun(A,fun(A,fun(A,$o))),(P)),Xb),Y)) ).

% If_def
tff(fact_5899_theI2,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,A2)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = A2 ) )
       => ( ! [X2: A] :
              ( aa(A,$o,P,X2)
             => aa(A,$o,Q,X2) )
         => aa(A,$o,Q,the(A,P)) ) ) ) ).

% theI2
tff(fact_5900_theI_H,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y5: A] :
              ( aa(A,$o,P,Y5)
             => ( Y5 = X3 ) ) )
     => aa(A,$o,P,the(A,P)) ) ).

% theI'
tff(fact_5901_theI,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( aa(A,$o,P,A2)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = A2 ) )
       => aa(A,$o,P,the(A,P)) ) ) ).

% theI
tff(fact_5902_floor__rat__def,axiom,
    ! [Xb: rat] : archim6421214686448440834_floor(rat,Xb) = the(int,aTP_Lamp_afj(rat,fun(int,$o),Xb)) ).

% floor_rat_def
tff(fact_5903_root__def,axiom,
    ! [Nb: nat,Xb: real] :
      aa(real,real,root(Nb),Xb) = $ite(Nb = zero_zero(nat),zero_zero(real),the_inv_into(real,real,top_top(set(real)),aTP_Lamp_afk(nat,fun(real,real),Nb),Xb)) ).

% root_def
tff(fact_5904_Greatest__def,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] : order_Greatest(A,P) = the(A,aTP_Lamp_afl(fun(A,$o),fun(A,$o),P)) ) ).

% Greatest_def
tff(fact_5905_sgn__rat__def,axiom,
    ! [A2: rat] :
      aa(rat,rat,sgn_sgn(rat),A2) = $ite(
        A2 = zero_zero(rat),
        zero_zero(rat),
        $ite(aa(rat,$o,ord_less(rat,zero_zero(rat)),A2),one_one(rat),aa(rat,rat,uminus_uminus(rat),one_one(rat))) ) ).

% sgn_rat_def
tff(fact_5906_obtain__pos__sum,axiom,
    ! [R3: rat] :
      ( aa(rat,$o,ord_less(rat,zero_zero(rat)),R3)
     => ~ ! [S: rat] :
            ( aa(rat,$o,ord_less(rat,zero_zero(rat)),S)
           => ! [T4: rat] :
                ( aa(rat,$o,ord_less(rat,zero_zero(rat)),T4)
               => ( R3 != aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),S),T4) ) ) ) ) ).

% obtain_pos_sum
tff(fact_5907_less__eq__rat__def,axiom,
    ! [Xb: rat,Y: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),Xb),Y)
    <=> ( aa(rat,$o,ord_less(rat,Xb),Y)
        | ( Xb = Y ) ) ) ).

% less_eq_rat_def
tff(fact_5908_GreatestI__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y5: nat] :
            ( aa(nat,$o,P,Y5)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y5),B2) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_nat
tff(fact_5909_Greatest__le__nat,axiom,
    ! [P: fun(nat,$o),K: nat,B2: nat] :
      ( aa(nat,$o,P,K)
     => ( ! [Y5: nat] :
            ( aa(nat,$o,P,Y5)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y5),B2) )
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),order_Greatest(nat,P)) ) ) ).

% Greatest_le_nat
tff(fact_5910_GreatestI__ex__nat,axiom,
    ! [P: fun(nat,$o),B2: nat] :
      ( ? [X_12: nat] : aa(nat,$o,P,X_12)
     => ( ! [Y5: nat] :
            ( aa(nat,$o,P,Y5)
           => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y5),B2) )
       => aa(nat,$o,P,order_Greatest(nat,P)) ) ) ).

% GreatestI_ex_nat
tff(fact_5911_Greatest__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y5: A] :
                ( aa(A,$o,P,Y5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y5),Xb) )
           => ( order_Greatest(A,P) = Xb ) ) ) ) ).

% Greatest_equality
tff(fact_5912_GreatestI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y5: A] :
                ( aa(A,$o,P,Y5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y5),Xb) )
           => ( ! [X2: A] :
                  ( aa(A,$o,P,X2)
                 => ( ! [Y4: A] :
                        ( aa(A,$o,P,Y4)
                       => aa(A,$o,aa(A,fun(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_5913_Arg__def,axiom,
    ! [Z: complex] :
      arg(Z) = $ite(Z = zero_zero(complex),zero_zero(real),fChoice(real,aTP_Lamp_afm(complex,fun(real,$o),Z))) ).

% Arg_def
tff(fact_5914_the__inv__into__def,axiom,
    ! [B: $tType,A: $tType,A3: set(B),F: fun(B,A),X3: A] : the_inv_into(B,A,A3,F,X3) = the(B,aa(A,fun(B,$o),aa(fun(B,A),fun(A,fun(B,$o)),aTP_Lamp_afn(set(B),fun(fun(B,A),fun(A,fun(B,$o))),A3),F),X3)) ).

% the_inv_into_def
tff(fact_5915_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))
       => ? [H5: heap_ext(product_unit),R2: A,T4: nat] :
            ( heap_Time_effect(A,C2,H,H5,R2,T4)
            & rep_assn(aa(A,assn,Q,R2),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),hoare_new_addrs(H,As3,H5))) ) ) ) ).

% hoare_triple_effect
tff(fact_5916_some__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( aa(A,$o,P,A2)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => ( X2 = A2 ) )
       => ( fChoice(A,P) = A2 ) ) ) ).

% some_equality
tff(fact_5917_some__eq__trivial,axiom,
    ! [A: $tType,Xb: A] : fChoice(A,aTP_Lamp_dc(A,fun(A,$o),Xb)) = Xb ).

% some_eq_trivial
tff(fact_5918_some__sym__eq__trivial,axiom,
    ! [A: $tType,Xb: A] : fChoice(A,aa(A,fun(A,$o),fequal(A),Xb)) = Xb ).

% some_sym_eq_trivial
tff(fact_5919_new__addr__refl,axiom,
    ! [H: heap_ext(product_unit),As3: set(nat)] : hoare_new_addrs(H,As3,H) = As3 ).

% new_addr_refl
tff(fact_5920_some__insert__self,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( S2 != bot_bot(set(A)) )
     => ( aa(set(A),set(A),insert(A,fChoice(A,aTP_Lamp_av(set(A),fun(A,$o),S2))),S2) = S2 ) ) ).

% some_insert_self
tff(fact_5921_diff__rat__def,axiom,
    ! [Q2: rat,R3: rat] : aa(rat,rat,minus_minus(rat,Q2),R3) = aa(rat,rat,aa(rat,fun(rat,rat),plus_plus(rat),Q2),aa(rat,rat,uminus_uminus(rat),R3)) ).

% diff_rat_def
tff(fact_5922_verit__sko__ex_H,axiom,
    ! [A: $tType,P: fun(A,$o),A3: $o] :
      ( ( aa(A,$o,P,fChoice(A,P))
      <=> (A3) )
     => ( ? [X_1: A] : aa(A,$o,P,X_1)
      <=> (A3) ) ) ).

% verit_sko_ex'
tff(fact_5923_verit__sko__forall,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ! [X_1: A] : aa(A,$o,P,X_1)
    <=> aa(A,$o,P,fChoice(A,aTP_Lamp_ss(fun(A,$o),fun(A,$o),P))) ) ).

% verit_sko_forall
tff(fact_5924_verit__sko__forall_H,axiom,
    ! [A: $tType,P: fun(A,$o),A3: $o] :
      ( ( aa(A,$o,P,fChoice(A,aTP_Lamp_ss(fun(A,$o),fun(A,$o),P)))
      <=> (A3) )
     => ( ! [X_1: A] : aa(A,$o,P,X_1)
      <=> (A3) ) ) ).

% verit_sko_forall'
tff(fact_5925_verit__sko__forall_H_H,axiom,
    ! [A: $tType,B5: A,A3: A,P: fun(A,$o)] :
      ( ( B5 = A3 )
     => ( ( fChoice(A,P) = A3 )
      <=> ( fChoice(A,P) = B5 ) ) ) ).

% verit_sko_forall''
tff(fact_5926_verit__sko__ex__indirect,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o)] :
      ( ( Xb = fChoice(A,P) )
     => ( ? [X_1: A] : aa(A,$o,P,X_1)
      <=> aa(A,$o,P,Xb) ) ) ).

% verit_sko_ex_indirect
tff(fact_5927_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,$o,P3,X2) )
       => ( ? [X_1: A] : aa(A,$o,P3,X_1)
        <=> aa(A,$o,P,Xb) ) ) ) ).

% verit_sko_ex_indirect2
tff(fact_5928_verit__sko__forall__indirect,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o)] :
      ( ( Xb = fChoice(A,aTP_Lamp_ss(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X_1: A] : aa(A,$o,P,X_1)
      <=> aa(A,$o,P,Xb) ) ) ).

% verit_sko_forall_indirect
tff(fact_5929_verit__sko__forall__indirect2,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o),P3: fun(A,$o)] :
      ( ( Xb = fChoice(A,aTP_Lamp_ss(fun(A,$o),fun(A,$o),P)) )
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
          <=> aa(A,$o,P3,X2) )
       => ( ! [X_1: A] : aa(A,$o,P3,X_1)
        <=> aa(A,$o,P,Xb) ) ) ) ).

% verit_sko_forall_indirect2
tff(fact_5930_someI2,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A,Q: fun(A,$o)] :
      ( aa(A,$o,P,A2)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2
tff(fact_5931_someI__ex,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ? [X_12: A] : aa(A,$o,P,X_12)
     => aa(A,$o,P,fChoice(A,P)) ) ).

% someI_ex
tff(fact_5932_someI2__ex,axiom,
    ! [A: $tType,P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X_12: A] : aa(A,$o,P,X_12)
     => ( ! [X2: A] :
            ( aa(A,$o,P,X2)
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,fChoice(A,P)) ) ) ).

% someI2_ex
tff(fact_5933_someI2__bex,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,A3)
          & aa(A,$o,P,X3) )
     => ( ! [X2: A] :
            ( ( member(A,X2,A3)
              & aa(A,$o,P,X2) )
           => aa(A,$o,Q,X2) )
       => aa(A,$o,Q,fChoice(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cv(set(A),fun(fun(A,$o),fun(A,$o)),A3),P))) ) ) ).

% someI2_bex
tff(fact_5934_some__eq__ex,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
    <=> ? [X_1: A] : aa(A,$o,P,X_1) ) ).

% some_eq_ex
tff(fact_5935_some1__equality,axiom,
    ! [A: $tType,P: fun(A,$o),A2: A] :
      ( ? [X3: A] :
          ( aa(A,$o,P,X3)
          & ! [Y5: A] :
              ( aa(A,$o,P,Y5)
             => ( Y5 = X3 ) ) )
     => ( aa(A,$o,P,A2)
       => ( fChoice(A,P) = A2 ) ) ) ).

% some1_equality
tff(fact_5936_some__in__eq,axiom,
    ! [A: $tType,A3: set(A)] :
      ( member(A,fChoice(A,aTP_Lamp_av(set(A),fun(A,$o),A3)),A3)
    <=> ( A3 != bot_bot(set(A)) ) ) ).

% some_in_eq
tff(fact_5937_some__elem,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ( S2 != bot_bot(set(A)) )
     => member(A,fChoice(A,aTP_Lamp_av(set(A),fun(A,$o),S2)),S2) ) ).

% some_elem
tff(fact_5938_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_afo(set(product_prod(B,A)),fun(B,fun(A,$o)),R),Xb)) ).

% fun_of_rel_def
tff(fact_5939_TBOUND__assert_H__bind,axiom,
    ! [A: $tType,P: $o,P3: $o,M: heap_Time_Heap(A),Ta: nat] :
      ( time_EQ($o,(P),(P3))
     => ( ( (P)
         => time_TBOUND(A,M,Ta) )
       => time_TBOUND(A,heap_Time_bind(product_unit,A,refine_Imp_assert((P)),aTP_Lamp_cy(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),M)),
            $ite((P3),Ta,zero_zero(nat))) ) ) ).

% TBOUND_assert'_bind
tff(fact_5940_neg__mask__is__div_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),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),Nb))) = 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))),Nb))),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))),Nb)) ) ) ) ).

% neg_mask_is_div'
tff(fact_5941_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,$o)),aTP_Lamp_afa(A,fun(B,fun(A,fun(B,$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_5942_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_5943_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_5944_minus__not__numeral__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : aa(A,A,uminus_uminus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),Nb))) = aa(num,A,numeral_numeral(A),inc(Nb)) ) ).

% minus_not_numeral_eq
tff(fact_5945_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_5946_word__add__not,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),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xb)) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% word_add_not
tff(fact_5947_even__not__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,bit_ri4277139882892585799ns_not(A),A2))
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% even_not_iff
tff(fact_5948_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_5949_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_5950_word__plus__and__or__coroll2,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)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),W)),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 ) ).

% word_plus_and_or_coroll2
tff(fact_5951_not__add__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,minus_minus(A,aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),B2) ) ).

% not_add_distrib
tff(fact_5952_not__diff__distrib,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,B2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,minus_minus(A,A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),B2) ) ).

% not_diff_distrib
tff(fact_5953_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_5954_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,$o,aTP_Lamp_afp(fun(product_prod(A,B),$o),fun(A,fun(B,$o)),P))) ).

% split_paired_Eps
tff(fact_5955_minus__eq__not__plus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),one_one(A)) ) ).

% minus_eq_not_plus_1
tff(fact_5956_minus__eq__not__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,uminus_uminus(A),A2) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,minus_minus(A,A2),one_one(A))) ) ).

% minus_eq_not_minus_1
tff(fact_5957_not__eq__complement,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A] : aa(A,A,bit_ri4277139882892585799ns_not(A),A2) = aa(A,A,minus_minus(A,aa(A,A,uminus_uminus(A),A2)),one_one(A)) ) ).

% not_eq_complement
tff(fact_5958_take__bit__not__eq__mask__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)) = aa(A,A,minus_minus(A,bit_se2239418461657761734s_mask(A,Nb)),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) ) ).

% take_bit_not_eq_mask_diff
tff(fact_5959_mask__lower__twice,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat,Xb: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),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),Nb)))),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_5960_mask__out__first__mask__some,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: nat,Y: word(A),M: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Nb))) = Y )
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),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_5961_NOT__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xb) = aa(word(A),word(A),minus_minus(word(A),aa(word(A),word(A),uminus_uminus(word(A)),Xb)),one_one(word(A))) ) ).

% NOT_eq
tff(fact_5962_AND__NOT__mask__plus__AND__mask__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: 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)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Nb)))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb))) = W ) ).

% AND_NOT_mask_plus_AND_mask_eq
tff(fact_5963_minus__numeral__inc__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),inc(Nb))) = aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),Nb)) ) ).

% minus_numeral_inc_eq
tff(fact_5964_subtract__mask_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Nb: nat] : aa(word(A),word(A),minus_minus(word(A),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),P2),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Nb)))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),P2),bit_se2239418461657761734s_mask(word(A),Nb)) ) ).

% subtract_mask(2)
tff(fact_5965_subtract__mask_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Nb: nat] : aa(word(A),word(A),minus_minus(word(A),P2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),P2),bit_se2239418461657761734s_mask(word(A),Nb))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),P2),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Nb))) ) ).

% subtract_mask(1)
tff(fact_5966_mask__out__sub__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: 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),Nb))) = aa(word(A),word(A),minus_minus(word(A),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),bit_se2239418461657761734s_mask(word(A),Nb))) ) ).

% mask_out_sub_mask
tff(fact_5967_Eps__case__prod,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,$o))] : fChoice(product_prod(A,B),product_case_prod(A,B,$o,P)) = fChoice(product_prod(A,B),aTP_Lamp_afg(fun(A,fun(B,$o)),fun(product_prod(A,B),$o),P)) ).

% Eps_case_prod
tff(fact_5968_not__numeral__Bit0__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bit0(Nb))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,Nb))) ) ).

% not_numeral_Bit0_eq
tff(fact_5969_take__bit__not__mask__eq__0,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb))) = zero_zero(A) ) ) ) ).

% take_bit_not_mask_eq_0
tff(fact_5970_not__numeral__BitM__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : aa(A,A,bit_ri4277139882892585799ns_not(A),aa(num,A,numeral_numeral(A),bitM(Nb))) = aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(Nb))) ) ).

% not_numeral_BitM_eq
tff(fact_5971_multiple__mask__trivia,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Nb: nat,Xb: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( 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),Nb)))),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),Nb)),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_5972_and__mask__0__iff__le__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Nb))) = zero_zero(word(A)) )
        <=> aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb)) ) ) ).

% and_mask_0_iff_le_mask
tff(fact_5973_TBOUND__Let__strong,axiom,
    ! [B: $tType,A: $tType,V: A,V4: A,F: fun(A,heap_Time_Heap(B)),Bnd: fun(A,nat)] :
      ( time_EQ(A,V,V4)
     => ( time_TBOUND(B,aa(A,heap_Time_Heap(B),F,V),aa(A,nat,Bnd,V))
       => time_TBOUND(B,aa(A,heap_Time_Heap(B),F,V),aa(A,nat,Bnd,V4)) ) ) ).

% TBOUND_Let_strong
tff(fact_5974_minus__exp__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: 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))),Nb)) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb)) ) ).

% minus_exp_eq_not_mask
tff(fact_5975_NOT__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Nb)) = 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))),Nb)) ) ).

% NOT_mask
tff(fact_5976_neg__mask__is__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: 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),Nb))) = 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))),Nb))),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))),Nb)) ) ).

% neg_mask_is_div
tff(fact_5977_signed__take__bit__eq__take__bit__minus,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K) = aa(int,int,minus_minus(int,aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),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,Nb))),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Nb)))) ).

% signed_take_bit_eq_take_bit_minus
tff(fact_5978_neg__mask__add__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: 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),Nb)))),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))),Nb)),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),Nb)) ) ).

% neg_mask_add_mask
tff(fact_5979_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_5980_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_5981_word__plus__and__or,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)),plus_plus(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),Y)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xb),Y)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Y) ) ).

% word_plus_and_or
tff(fact_5982_word__or__max,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),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% word_or_max
tff(fact_5983_word__bitwise__m1__simps_I4_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)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Xb) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% word_bitwise_m1_simps(4)
tff(fact_5984_bit__numeral__Bit0__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),bit0(M))),aa(nat,nat,suc,Nb))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),M)),Nb) ) ) ).

% bit_numeral_Bit0_Suc_iff
tff(fact_5985_bit__numeral__Bit1__Suc__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,M))),aa(nat,nat,suc,Nb))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),M)),Nb) ) ) ).

% bit_numeral_Bit1_Suc_iff
tff(fact_5986_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_5987_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_5988_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_5989_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_5990_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,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% not_negative_int_iff
tff(fact_5991_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( aa(int,$o,aa(int,fun(int,$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_5992_signed__take__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,bit_ri4674362597316999326ke_bit(int,Nb),K))
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Nb) ) ).

% signed_take_bit_nonnegative_iff
tff(fact_5993_word__or__not,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),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xb)) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% word_or_not
tff(fact_5994_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_5995_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_5996_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_5997_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,Nb: 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,Nb))
    <=> 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))),Nb) ) ).

% bit_minus_numeral_Bit0_Suc_iff
tff(fact_5998_bit__numeral__simps_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Nb: num] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),bit0(W))),aa(num,nat,numeral_numeral(nat),Nb))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),W)),pred_numeral(Nb)) ) ) ).

% bit_numeral_simps(2)
tff(fact_5999_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,Nb: 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,Nb))
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W)),Nb) ) ).

% bit_minus_numeral_Bit1_Suc_iff
tff(fact_6000_bit__numeral__simps_I3_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Nb: num] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,W))),aa(num,nat,numeral_numeral(nat),Nb))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),W)),pred_numeral(Nb)) ) ) ).

% bit_numeral_simps(3)
tff(fact_6001_word__no__log__defs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num] : aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),A2))) ) ).

% word_no_log_defs(1)
tff(fact_6002_bit__0,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),zero_zero(nat))
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ).

% bit_0
tff(fact_6003_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,Nb: 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),Nb))
    <=> 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(Nb)) ) ).

% bit_minus_numeral_int(1)
tff(fact_6004_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,Nb: 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),Nb))
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W)),pred_numeral(Nb)) ) ).

% bit_minus_numeral_int(2)
tff(fact_6005_bit__mod__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),Nb)
        <=> ( ( Nb = zero_zero(nat) )
            & ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% bit_mod_2_iff
tff(fact_6006_bin__nth__minus__Bit0,axiom,
    ! [Nb: nat,W: num] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),bit0(W))),Nb)
      <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W)),aa(nat,nat,minus_minus(nat,Nb),one_one(nat))) ) ) ).

% bin_nth_minus_Bit0
tff(fact_6007_bin__nth__minus__Bit1,axiom,
    ! [Nb: nat,W: num] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,W))),Nb)
      <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(num,int,numeral_numeral(int),W)),aa(nat,nat,minus_minus(nat,Nb),one_one(nat))) ) ) ).

% bin_nth_minus_Bit1
tff(fact_6008_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_6009_word__no__log__defs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: 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)),A2))) = 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),A2)))) ) ).

% word_no_log_defs(2)
tff(fact_6010_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_6011_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_6012_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_6013_bit__minus__int__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(int,int,uminus_uminus(int),K)),Nb)
    <=> 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)))),Nb) ) ).

% bit_minus_int_iff
tff(fact_6014_bit__take__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,bit_se2584673776208193580ke_bit(A,M),A2)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),M)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Nb) ) ) ) ).

% bit_take_bit_iff
tff(fact_6015_le__word__or1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [C2: word(A),Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),C2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Y),C2)) ) ).

% le_word_or1
tff(fact_6016_le__word__or2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xb),Y)) ) ).

% le_word_or2
tff(fact_6017_bit__numeral__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: num,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),M)),Nb)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(num,nat,numeral_numeral(nat),M)),Nb) ) ) ).

% bit_numeral_iff
tff(fact_6018_disjunctive__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( ! [N2: nat] :
              ( ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),N2)
              | ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B2),N2) )
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2) ) ) ) ).

% disjunctive_add
tff(fact_6019_bit__disjunctive__add__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( ! [N2: nat] :
              ( ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),N2)
              | ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B2),N2) )
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)),Nb)
          <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Nb)
              | aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B2),Nb) ) ) ) ) ).

% bit_disjunctive_add_iff
tff(fact_6020_bit__1__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),one_one(A)),Nb)
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% bit_1_iff
tff(fact_6021_bit__numeral__simps_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: num] : ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),one_one(A)),aa(num,nat,numeral_numeral(nat),Nb)) ) ).

% bit_numeral_simps(1)
tff(fact_6022_not__bit__1__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),one_one(A)),aa(nat,nat,suc,Nb)) ) ).

% not_bit_1_Suc
tff(fact_6023_signed__take__bit__def,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)),aa(A,A,aa(A,fun(A,A),times_times(A),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Nb))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb)))) ) ).

% signed_take_bit_def
tff(fact_6024_disjunctive__diff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [B2: A,A2: A] :
          ( ! [N2: nat] :
              ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),B2),N2)
             => aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),N2) )
         => ( aa(A,A,minus_minus(A,A2),B2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,bit_ri4277139882892585799ns_not(A),B2)) ) ) ) ).

% disjunctive_diff
tff(fact_6025_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_6026_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_6027_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_6028_bit__not__int__iff_H,axiom,
    ! [K: int,Nb: 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))),Nb)
    <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Nb) ) ).

% bit_not_int_iff'
tff(fact_6029_even__or__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),B2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
            & aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ) ).

% even_or_iff
tff(fact_6030_bit_Ocomplement__unique,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Xb: A,Y: A] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),Xb) = zero_zero(A) )
         => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Xb) = aa(A,A,uminus_uminus(A),one_one(A)) )
           => ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),Y) = zero_zero(A) )
             => ( ( aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),Y) = aa(A,A,uminus_uminus(A),one_one(A)) )
               => ( Xb = Y ) ) ) ) ) ) ).

% bit.complement_unique
tff(fact_6031_bit__numeral__rec_I1_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),bit0(W))),Nb)
        <=> case_nat($o,$false,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),W)),Nb) ) ) ).

% bit_numeral_rec(1)
tff(fact_6032_bit__numeral__rec_I2_J,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [W: num,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),aa(num,num,bit1,W))),Nb)
        <=> case_nat($o,$true,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),W)),Nb) ) ) ).

% bit_numeral_rec(2)
tff(fact_6033_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_6034_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_6035_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_6036_and__not__numerals_I2_J,axiom,
    ! [Nb: 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(Nb)))) = one_one(int) ).

% and_not_numerals(2)
tff(fact_6037_bit__imp__take__bit__positive,axiom,
    ! [Nb: nat,M: nat,K: int] :
      ( aa(nat,$o,ord_less(nat,Nb),M)
     => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Nb)
       => aa(int,$o,ord_less(int,zero_zero(int)),aa(int,int,bit_se2584673776208193580ke_bit(int,M),K)) ) ) ).

% bit_imp_take_bit_positive
tff(fact_6038_mask__subsume,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat,Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),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),Nb)))),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_6039_and__not__num__eq__Some__iff,axiom,
    ! [M: num,Nb: num,Q2: num] :
      ( ( bit_and_not_num(M,Nb) = 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),Nb))) = aa(num,int,numeral_numeral(int),Q2) ) ) ).

% and_not_num_eq_Some_iff
tff(fact_6040_exp__eq__0__imp__not__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat,A2: A] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) = zero_zero(A) )
         => ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Nb) ) ) ).

% exp_eq_0_imp_not_bit
tff(fact_6041_bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,nat,suc,Nb))
        <=> aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),Nb) ) ) ).

% bit_Suc
tff(fact_6042_stable__imp__bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Nb)
          <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% stable_imp_bit_iff_odd
tff(fact_6043_bit__iff__idd__imp__stable,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A] :
          ( ! [N2: nat] :
              ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),N2)
            <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) )
         => ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 ) ) ) ).

% bit_iff_idd_imp_stable
tff(fact_6044_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_6045_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N2: nat] :
          ( ! [M3: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),M3)
             => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),M3)
              <=> aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N2) ) )
         => ~ ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N2)
             => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),aa(nat,nat,minus_minus(nat,N2),one_one(nat)))
              <=> ~ aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),N2) ) ) ) ).

% int_bit_bound
tff(fact_6046_and__not__numerals_I5_J,axiom,
    ! [M: num,Nb: 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(Nb)))) = 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),Nb)))) ).

% and_not_numerals(5)
tff(fact_6047_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_6048_and__not__numerals_I3_J,axiom,
    ! [Nb: 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,Nb)))) = zero_zero(int) ).

% and_not_numerals(3)
tff(fact_6049_and__not__num__eq__None__iff,axiom,
    ! [M: num,Nb: num] :
      ( ( bit_and_not_num(M,Nb) = 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),Nb))) = zero_zero(int) ) ) ).

% and_not_num_eq_None_iff
tff(fact_6050_bit__iff__odd,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Nb)
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb))) ) ) ).

% bit_iff_odd
tff(fact_6051_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = zero_zero(A) )
        <=> ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Nb) ) ) ).

% and_exp_eq_0_iff_not_bit
tff(fact_6052_bit__not__iff__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,bit_ri4277139882892585799ns_not(A),A2)),Nb)
        <=> ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) != zero_zero(A) )
            & ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Nb) ) ) ) ).

% bit_not_iff_eq
tff(fact_6053_mask__Suc__exp,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,Nb)) = 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))),Nb)),bit_se2239418461657761734s_mask(A,Nb)) ) ).

% mask_Suc_exp
tff(fact_6054_and__not__numerals_I9_J,axiom,
    ! [M: num,Nb: 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,Nb)))) = 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),Nb)))) ).

% and_not_numerals(9)
tff(fact_6055_and__not__numerals_I6_J,axiom,
    ! [M: num,Nb: 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,Nb)))) = 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),Nb)))) ).

% and_not_numerals(6)
tff(fact_6056_int__numeral__not__and__num,axiom,
    ! [M: num,Nb: 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),Nb)) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(Nb,M)) ).

% int_numeral_not_and_num
tff(fact_6057_int__numeral__and__not__num,axiom,
    ! [M: num,Nb: 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),Nb))) = case_option(int,num,zero_zero(int),numeral_numeral(int),bit_and_not_num(M,Nb)) ).

% int_numeral_and_not_num
tff(fact_6058_bit__int__def,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Nb)
    <=> ~ 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))),Nb))) ) ).

% bit_int_def
tff(fact_6059_even__bit__succ__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( 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)),A2)),Nb)
          <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Nb)
              | ( Nb = zero_zero(nat) ) ) ) ) ) ).

% even_bit_succ_iff
tff(fact_6060_odd__bit__iff__bit__pred,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Nb)
          <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,minus_minus(A,A2),one_one(A))),Nb)
              | ( Nb = zero_zero(nat) ) ) ) ) ) ).

% odd_bit_iff_bit_pred
tff(fact_6061_mask__Suc__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_se2239418461657761734s_mask(A,aa(nat,nat,suc,Nb)) = 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,Nb))) ) ).

% mask_Suc_double
tff(fact_6062_or__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),one_one(A)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ).

% or_one_eq
tff(fact_6063_one__or__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),one_one(A)),A2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ).

% one_or_eq
tff(fact_6064_bit__sum__mult__2__cases,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A,Nb: nat] :
          ( ! [J2: nat] : ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,nat,suc,J2))
         => ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),B2))),Nb)
          <=> $ite(Nb = zero_zero(nat),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2),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))),B2)),Nb)) ) ) ) ).

% bit_sum_mult_2_cases
tff(fact_6065_bit__rec,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Nb)
        <=> $ite(Nb = zero_zero(nat),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))),aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))) ) ) ).

% bit_rec
tff(fact_6066_and__not__numerals_I8_J,axiom,
    ! [M: num,Nb: 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(Nb)))) = 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),Nb))))) ).

% and_not_numerals(8)
tff(fact_6067_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_6068_Bit__Operations_Oset__bit__eq,axiom,
    ! [Nb: nat,K: int] : bit_se5668285175392031749et_bit(int,Nb,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),Nb))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))) ).

% Bit_Operations.set_bit_eq
tff(fact_6069_unset__bit__eq,axiom,
    ! [Nb: nat,K: int] : bit_se2638667681897837118et_bit(int,Nb,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),Nb))),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))) ).

% unset_bit_eq
tff(fact_6070_take__bit__Suc__from__most,axiom,
    ! [Nb: nat,K: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,suc,Nb)),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))),Nb)),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Nb)))),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)) ).

% take_bit_Suc_from_most
tff(fact_6071_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_6072_bitNOT__integer__code,axiom,
    ! [I: code_integer] : aa(code_integer,code_integer,bit_ri4277139882892585799ns_not(code_integer),I) = aa(code_integer,code_integer,minus_minus(code_integer,aa(code_integer,code_integer,uminus_uminus(code_integer),I)),one_one(code_integer)) ).

% bitNOT_integer_code
tff(fact_6073_test__bit__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),one_one(word(A))),Nb)
        <=> ( Nb = zero_zero(nat) ) ) ) ).

% test_bit_1
tff(fact_6074_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se1065995026697491101ons_or(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% or_nonnegative_int_iff
tff(fact_6075_map__nth__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xs: list(nat)] : aa(list(nat),list($o),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_6076_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_6077_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_6078_or__minus__numerals_I6_J,axiom,
    ! [Nb: 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,Nb)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Nb))) ).

% or_minus_numerals(6)
tff(fact_6079_or__minus__numerals_I2_J,axiom,
    ! [Nb: 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,Nb)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),aa(num,num,bit1,Nb))) ).

% or_minus_numerals(2)
tff(fact_6080_word__no__log__defs_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2)),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(7)
tff(fact_6081_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_6082_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_6083_word__bitwise__1__simps_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: 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)),A2)),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),A2)),one_one(int))) ) ).

% word_bitwise_1_simps(8)
tff(fact_6084_word__bitwise__1__simps_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2)) = 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),B2))) ) ).

% word_bitwise_1_simps(6)
tff(fact_6085_word__no__log__defs_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(8)
tff(fact_6086_word__no__log__defs_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2))),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(9)
tff(fact_6087_word__no__log__defs_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(10)
tff(fact_6088_and__minus__minus__numerals,axiom,
    ! [M: num,Nb: 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),Nb))) = 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),Nb)),one_one(int)))) ).

% and_minus_minus_numerals
tff(fact_6089_or__minus__minus__numerals,axiom,
    ! [M: num,Nb: 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),Nb))) = 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),Nb)),one_one(int)))) ).

% or_minus_minus_numerals
tff(fact_6090_word__bitwise__1__simps_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: 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)),A2))),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),A2))),one_one(int))) ) ).

% word_bitwise_1_simps(9)
tff(fact_6091_word__bitwise__1__simps_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2))) = 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),B2)))) ) ).

% word_bitwise_1_simps(7)
tff(fact_6092_test__bit__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Nb)
         => aa(nat,$o,ord_less(nat,Nb),aa(word(A),nat,size_size(word(A)),W)) ) ) ).

% test_bit_size
tff(fact_6093_word__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),V: word(A)] :
          ( ! [N2: nat] :
              ( aa(nat,$o,ord_less(nat,N2),aa(word(A),nat,size_size(word(A)),U))
             => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),U),N2)
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),V),N2) ) )
         => ( U = V ) ) ) ).

% word_eqI
tff(fact_6094_test__bit__over,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),Xb)),Nb)
         => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),Nb) ) ) ).

% test_bit_over
tff(fact_6095_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_6096_not__bit__Suc__0__Suc,axiom,
    ! [Nb: nat] : ~ aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,suc,zero_zero(nat))),aa(nat,nat,suc,Nb)) ).

% not_bit_Suc_0_Suc
tff(fact_6097_bit__Suc__0__iff,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),aa(nat,nat,suc,zero_zero(nat))),Nb)
    <=> ( Nb = zero_zero(nat) ) ) ).

% bit_Suc_0_iff
tff(fact_6098_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_6099_OR__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => aa(int,$o,aa(int,fun(int,$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_6100_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L)
     => aa(int,$o,aa(int,fun(int,$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_6101_not__bit__Suc__0__numeral,axiom,
    ! [Nb: 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),Nb)) ).

% not_bit_Suc_0_numeral
tff(fact_6102_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_6103_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_6104_word__leI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [U: word(A),V: word(A)] :
          ( ! [N2: nat] :
              ( aa(nat,$o,ord_less(nat,N2),aa(word(A),nat,size_size(word(A)),U))
             => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),U),N2)
               => aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),V),N2) ) )
         => aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),U),V) ) ) ).

% word_leI
tff(fact_6105_nth__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,I: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_se2239418461657761734s_mask(word(A),Nb)),I)
        <=> ( aa(nat,$o,ord_less(nat,I),Nb)
            & aa(nat,$o,ord_less(nat,I),aa(word(A),nat,size_size(word(A)),bit_se2239418461657761734s_mask(word(A),Nb))) ) ) ) ).

% nth_mask
tff(fact_6106_or__not__numerals_I2_J,axiom,
    ! [Nb: 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(Nb)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Nb))) ).

% or_not_numerals(2)
tff(fact_6107_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_6108_bit__nat__iff,axiom,
    ! [K: int,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),nat2(K)),Nb)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
        & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Nb) ) ) ).

% bit_nat_iff
tff(fact_6109_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_6110_word__and__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A)] :
          aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Nb),one_one(word(A))) = $ite(aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Nb),zero_zero(nat)),one_one(word(A)),zero_zero(word(A))) ) ).

% word_and_1
tff(fact_6111_test__bit__bin_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),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)),Nb) ) ) ) ).

% test_bit_bin'
tff(fact_6112_le__mask__high__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb))
        <=> ! [X: nat] :
              ( member(nat,X,set_or7035219750837199246ssThan(nat,Nb,aa(word(A),nat,size_size(word(A)),W)))
             => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),X) ) ) ) ).

% le_mask_high_bits
tff(fact_6113_or__not__numerals_I3_J,axiom,
    ! [Nb: 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,Nb)))) = aa(int,int,bit_ri4277139882892585799ns_not(int),aa(num,int,numeral_numeral(int),bit0(Nb))) ).

% or_not_numerals(3)
tff(fact_6114_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_6115_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,aa(word(A),fun(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_6116_bit__nat__def,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),M),Nb)
    <=> ~ 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))),Nb))) ) ).

% bit_nat_def
tff(fact_6117_or__not__numerals_I6_J,axiom,
    ! [M: num,Nb: 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,Nb)))) = 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),Nb)))) ).

% or_not_numerals(6)
tff(fact_6118_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_6119_and__neq__0__is__nth,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Nb: nat,Xb: word(A)] :
          ( ( Y = aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb) )
         => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),Y) != zero_zero(word(A)) )
          <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),Nb) ) ) ) ).

% and_neq_0_is_nth
tff(fact_6120_nth__is__and__neq__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),Nb)
        <=> ( 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))),Nb)) != zero_zero(word(A)) ) ) ) ).

% nth_is_and_neq_0
tff(fact_6121_OR__upper,axiom,
    ! [Xb: int,Nb: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,ord_less(int,Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
       => ( aa(int,$o,ord_less(int,Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
         => 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))),Nb)) ) ) ) ).

% OR_upper
tff(fact_6122_or__not__numerals_I5_J,axiom,
    ! [M: num,Nb: 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(Nb)))) = 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),Nb))))) ).

% or_not_numerals(5)
tff(fact_6123_Suc__0__or__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ).

% Suc_0_or_eq
tff(fact_6124_or__Suc__0__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Nb),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ).

% or_Suc_0_eq
tff(fact_6125_or__nat__rec,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),Nb) = 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))),Nb) ))),
        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),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% or_nat_rec
tff(fact_6126_or__not__numerals_I9_J,axiom,
    ! [M: num,Nb: 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,Nb)))) = 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),Nb))))) ).

% or_not_numerals(9)
tff(fact_6127_or__not__numerals_I8_J,axiom,
    ! [M: num,Nb: 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(Nb)))) = 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),Nb))))) ).

% or_not_numerals(8)
tff(fact_6128_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_6129_or__nat__unfold,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),M),Nb) = $ite(
        M = zero_zero(nat),
        Nb,
        $ite(Nb = 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),Nb),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),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% or_nat_unfold
tff(fact_6130_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_6131_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_6132_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),aa(int,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_6133_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(int,fun(int,int),bit_se5824344971392196577ns_xor(int),K),L))
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K)
      <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ) ).

% xor_nonnegative_int_iff
tff(fact_6134_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_6135_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_6136_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_6137_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_6138_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_6139_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_6140_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_6141_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_6142_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_6143_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_6144_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_6145_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_6146_XOR__lower,axiom,
    ! [Xb: int,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
       => aa(int,$o,aa(int,fun(int,$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_6147_even__xor__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),B2))
        <=> ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)
          <=> aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),B2) ) ) ) ).

% even_xor_iff
tff(fact_6148_XOR__upper,axiom,
    ! [Xb: int,Nb: nat,Y: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
     => ( aa(int,$o,ord_less(int,Xb),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
       => ( aa(int,$o,ord_less(int,Y),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))
         => 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))),Nb)) ) ) ) ).

% XOR_upper
tff(fact_6149_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_6150_xor__one__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),one_one(A)) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)))),aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ).

% xor_one_eq
tff(fact_6151_one__xor__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A] : aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),one_one(A)),A2) = aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa($o,A,zero_neq_one_of_bool(A),aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2)))),aa($o,A,zero_neq_one_of_bool(A),~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2))) ) ).

% one_xor_eq
tff(fact_6152_or__minus__numerals_I1_J,axiom,
    ! [Nb: 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(Nb)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(Nb)))) ).

% or_minus_numerals(1)
tff(fact_6153_or__minus__numerals_I5_J,axiom,
    ! [Nb: 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(Nb)))),one_one(int)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(one2,bitM(Nb)))) ).

% or_minus_numerals(5)
tff(fact_6154_word__bitwise__m1__simps_I6_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)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Xb) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xb) ) ).

% word_bitwise_m1_simps(6)
tff(fact_6155_word__bitwise__m1__simps_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_se5824344971392196577ns_xor(word(A)),Xb),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xb) ) ).

% word_bitwise_m1_simps(7)
tff(fact_6156_word__no__log__defs_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2)),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(11)
tff(fact_6157_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_6158_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_6159_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_6160_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_6161_or__minus__numerals_I8_J,axiom,
    ! [Nb: 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,Nb)))),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(Nb)))) ).

% or_minus_numerals(8)
tff(fact_6162_or__minus__numerals_I4_J,axiom,
    ! [M: num,Nb: 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,Nb)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bit0(Nb)))) ).

% or_minus_numerals(4)
tff(fact_6163_word__bitwise__1__simps_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: 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)),A2)),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),A2)),one_one(int))) ) ).

% word_bitwise_1_simps(12)
tff(fact_6164_word__bitwise__1__simps_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2)) = 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),B2))) ) ).

% word_bitwise_1_simps(10)
tff(fact_6165_word__no__log__defs_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2)),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(12)
tff(fact_6166_word__no__log__defs_I13_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2))),aa(num,int,numeral_numeral(int),B2))) ) ).

% word_no_log_defs(13)
tff(fact_6167_word__no__log__defs_I14_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2)))) ) ).

% word_no_log_defs(14)
tff(fact_6168_or__minus__numerals_I7_J,axiom,
    ! [Nb: 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(Nb)))),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(Nb)))) ).

% or_minus_numerals(7)
tff(fact_6169_or__minus__numerals_I3_J,axiom,
    ! [M: num,Nb: 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(Nb)))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,bitM(Nb)))) ).

% or_minus_numerals(3)
tff(fact_6170_word__bitwise__1__simps_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2))) = 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),B2)))) ) ).

% word_bitwise_1_simps(11)
tff(fact_6171_word__bitwise__1__simps_I13_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: 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)),A2))),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),A2))),one_one(int))) ) ).

% word_bitwise_1_simps(13)
tff(fact_6172_or__not__num__neg_Osimps_I1_J,axiom,
    bit_or_not_num_neg(one2,one2) = one2 ).

% or_not_num_neg.simps(1)
tff(fact_6173_or__not__num__neg_Osimps_I4_J,axiom,
    ! [Nb: num] : bit_or_not_num_neg(bit0(Nb),one2) = bit0(one2) ).

% or_not_num_neg.simps(4)
tff(fact_6174_or__not__num__neg_Osimps_I6_J,axiom,
    ! [Nb: num,M: num] : bit_or_not_num_neg(bit0(Nb),aa(num,num,bit1,M)) = bit0(bit_or_not_num_neg(Nb,M)) ).

% or_not_num_neg.simps(6)
tff(fact_6175_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_6176_or__not__num__neg_Osimps_I7_J,axiom,
    ! [Nb: num] : bit_or_not_num_neg(aa(num,num,bit1,Nb),one2) = one2 ).

% or_not_num_neg.simps(7)
tff(fact_6177_or__not__num__neg_Osimps_I5_J,axiom,
    ! [Nb: num,M: num] : bit_or_not_num_neg(bit0(Nb),bit0(M)) = bitM(bit_or_not_num_neg(Nb,M)) ).

% or_not_num_neg.simps(5)
tff(fact_6178_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_6179_or__not__num__neg_Osimps_I8_J,axiom,
    ! [Nb: num,M: num] : bit_or_not_num_neg(aa(num,num,bit1,Nb),bit0(M)) = bitM(bit_or_not_num_neg(Nb,M)) ).

% or_not_num_neg.simps(8)
tff(fact_6180_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 )
           => ! [M2: num] :
                ( ( Xaa = bit0(M2) )
               => ( Y != aa(num,num,bit1,M2) ) ) )
         => ( ( ( Xb = one2 )
             => ! [M2: num] :
                  ( ( Xaa = aa(num,num,bit1,M2) )
                 => ( Y != aa(num,num,bit1,M2) ) ) )
           => ( ( ? [N2: num] : Xb = bit0(N2)
               => ( ( Xaa = one2 )
                 => ( Y != bit0(one2) ) ) )
             => ( ! [N2: num] :
                    ( ( Xb = bit0(N2) )
                   => ! [M2: num] :
                        ( ( Xaa = bit0(M2) )
                       => ( Y != bitM(bit_or_not_num_neg(N2,M2)) ) ) )
               => ( ! [N2: num] :
                      ( ( Xb = bit0(N2) )
                     => ! [M2: num] :
                          ( ( Xaa = aa(num,num,bit1,M2) )
                         => ( Y != bit0(bit_or_not_num_neg(N2,M2)) ) ) )
                 => ( ( ? [N2: num] : Xb = aa(num,num,bit1,N2)
                     => ( ( Xaa = one2 )
                       => ( Y != one2 ) ) )
                   => ( ! [N2: num] :
                          ( ( Xb = aa(num,num,bit1,N2) )
                         => ! [M2: num] :
                              ( ( Xaa = bit0(M2) )
                             => ( Y != bitM(bit_or_not_num_neg(N2,M2)) ) ) )
                     => ~ ! [N2: num] :
                            ( ( Xb = aa(num,num,bit1,N2) )
                           => ! [M2: num] :
                                ( ( Xaa = aa(num,num,bit1,M2) )
                               => ( Y != bitM(bit_or_not_num_neg(N2,M2)) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
tff(fact_6181_int__numeral__or__not__num__neg,axiom,
    ! [M: num,Nb: 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),Nb))) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,Nb))) ).

% int_numeral_or_not_num_neg
tff(fact_6182_int__numeral__not__or__num__neg,axiom,
    ! [M: num,Nb: 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),Nb)) = aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),bit_or_not_num_neg(Nb,M))) ).

% int_numeral_not_or_num_neg
tff(fact_6183_numeral__or__not__num__eq,axiom,
    ! [M: num,Nb: num] : aa(num,int,numeral_numeral(int),bit_or_not_num_neg(M,Nb)) = 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),Nb)))) ).

% numeral_or_not_num_eq
tff(fact_6184_word__ops__nth__size,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),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)),Nb)
            <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),Nb)
                | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Nb) ) )
            & ( 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)),Nb)
            <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),Nb)
                & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Nb) ) )
            & ( 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)),Nb)
            <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),Nb)
                <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),Nb) ) )
            & ( 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)),Nb)
            <=> ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),Nb) ) ) ) ) ).

% word_ops_nth_size
tff(fact_6185_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_6186_xor__nat__unfold,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),Nb) = $ite(
        M = zero_zero(nat),
        Nb,
        $ite(Nb = 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),Nb),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),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))))))) ) ).

% xor_nat_unfold
tff(fact_6187_xor__nat__rec,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),M),Nb) = 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))),Nb))),
        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),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2)))))) ).

% xor_nat_rec
tff(fact_6188_Suc__0__xor__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),aa(nat,nat,suc,zero_zero(nat))),Nb) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ).

% Suc_0_xor_eq
tff(fact_6189_xor__Suc__0__eq,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Nb),aa(nat,nat,suc,zero_zero(nat))) = aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa($o,nat,zero_neq_one_of_bool(nat),aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)))),aa($o,nat,zero_neq_one_of_bool(nat),~ aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb))) ).

% xor_Suc_0_eq
tff(fact_6190_or__not__num__neg_Opelims,axiom,
    ! [Xb: num,Xaa: num,Y: num] :
      ( ( bit_or_not_num_neg(Xb,Xaa) = Y )
     => ( aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),Xb),Xaa))
       => ( ( ( Xb = one2 )
           => ( ( Xaa = one2 )
             => ( ( Y = one2 )
               => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),one2)) ) ) )
         => ( ( ( Xb = one2 )
             => ! [M2: num] :
                  ( ( Xaa = bit0(M2) )
                 => ( ( Y = aa(num,num,bit1,M2) )
                   => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),bit0(M2))) ) ) )
           => ( ( ( Xb = one2 )
               => ! [M2: num] :
                    ( ( Xaa = aa(num,num,bit1,M2) )
                   => ( ( Y = aa(num,num,bit1,M2) )
                     => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),one2),aa(num,num,bit1,M2))) ) ) )
             => ( ! [N2: num] :
                    ( ( Xb = bit0(N2) )
                   => ( ( Xaa = one2 )
                     => ( ( Y = bit0(one2) )
                       => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(N2)),one2)) ) ) )
               => ( ! [N2: num] :
                      ( ( Xb = bit0(N2) )
                     => ! [M2: num] :
                          ( ( Xaa = bit0(M2) )
                         => ( ( Y = bitM(bit_or_not_num_neg(N2,M2)) )
                           => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(N2)),bit0(M2))) ) ) )
                 => ( ! [N2: num] :
                        ( ( Xb = bit0(N2) )
                       => ! [M2: num] :
                            ( ( Xaa = aa(num,num,bit1,M2) )
                           => ( ( Y = bit0(bit_or_not_num_neg(N2,M2)) )
                             => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),bit0(N2)),aa(num,num,bit1,M2))) ) ) )
                   => ( ! [N2: num] :
                          ( ( Xb = aa(num,num,bit1,N2) )
                         => ( ( Xaa = one2 )
                           => ( ( Y = one2 )
                             => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N2)),one2)) ) ) )
                     => ( ! [N2: num] :
                            ( ( Xb = aa(num,num,bit1,N2) )
                           => ! [M2: num] :
                                ( ( Xaa = bit0(M2) )
                               => ( ( Y = bitM(bit_or_not_num_neg(N2,M2)) )
                                 => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N2)),bit0(M2))) ) ) )
                       => ~ ! [N2: num] :
                              ( ( Xb = aa(num,num,bit1,N2) )
                             => ! [M2: num] :
                                  ( ( Xaa = aa(num,num,bit1,M2) )
                                 => ( ( Y = bitM(bit_or_not_num_neg(N2,M2)) )
                                   => ~ aa(product_prod(num,num),$o,accp(product_prod(num,num),bit_or3848514188828904588eg_rel),aa(num,product_prod(num,num),aa(num,fun(num,product_prod(num,num)),product_Pair(num,num),aa(num,num,bit1,N2)),aa(num,num,bit1,M2))) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
tff(fact_6191_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_6192_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,aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),Xb)) = aa(nat,nat,suc,zero_zero(nat)) ).

% Heap.size_gen
tff(fact_6193_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_6194_num_Osize__gen_I1_J,axiom,
    size_num(one2) = zero_zero(nat) ).

% num.size_gen(1)
tff(fact_6195_num_Osize__gen_I2_J,axiom,
    ! [X23: num] : size_num(bit0(X23)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_num(X23)),aa(nat,nat,suc,zero_zero(nat))) ).

% num.size_gen(2)
tff(fact_6196_new__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,Xb: A] : hoare_hoare_triple(array(A),one_one(assn),array_new(A,Nb,Xb),aa(A,fun(array(A),assn),aTP_Lamp_afq(nat,fun(A,fun(array(A),assn)),Nb),Xb)) ) ).

% new_rule
tff(fact_6197_aux,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn)),A2: 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_afr(fun(A,fun(B,assn)),fun(A,fun(list(A),fun(B,fun(list(B),fun(nat,fun(assn,assn)))))),P),A2),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,A2),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_aek(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_6198_length__nth__simps_I4_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(nat,nat,suc,Nb)) = aa(nat,A,nth(A,Xs),Nb) ).

% length_nth_simps(4)
tff(fact_6199_nth__Cons__Suc,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(nat,nat,suc,Nb)) = aa(nat,A,nth(A,Xs),Nb) ).

% nth_Cons_Suc
tff(fact_6200_length__nth__simps_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),zero_zero(nat)) = Xb ).

% length_nth_simps(3)
tff(fact_6201_nth__Cons__0,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),zero_zero(nat)) = Xb ).

% nth_Cons_0
tff(fact_6202_list_Osimps_I15_J,axiom,
    ! [A: $tType,X21: A,X22: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)) = aa(set(A),set(A),insert(A,X21),aa(list(A),set(A),set2(A),X22)) ).

% list.simps(15)
tff(fact_6203_enumerate__simps_I2_J,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] : enumerate(A,Nb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),aa(product_prod(nat,A),fun(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),Nb),Xb)),enumerate(A,aa(nat,nat,suc,Nb),Xs)) ).

% enumerate_simps(2)
tff(fact_6204_nth__Cons__numeral,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),V: num] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(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_6205_nth__Cons__pos,axiom,
    ! [A: $tType,Nb: nat,Xb: A,Xs: list(A)] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Nb) = aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,Nb),one_one(nat))) ) ) ).

% nth_Cons_pos
tff(fact_6206_set__subset__Cons,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs))) ).

% set_subset_Cons
tff(fact_6207_impossible__Cons,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Xb: A] :
      ( aa(nat,$o,aa(nat,fun(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),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys) ) ) ).

% impossible_Cons
tff(fact_6208_list__update__code_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),I: nat,Y: A] : list_update(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),aa(nat,nat,suc,I),Y) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),list_update(A,Xs,I,Y)) ).

% list_update_code(3)
tff(fact_6209_set__ConsD,axiom,
    ! [A: $tType,Y: A,Xb: A,Xs: list(A)] :
      ( member(A,Y,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)))
     => ( ( Y = Xb )
        | member(A,Y,aa(list(A),set(A),set2(A),Xs)) ) ) ).

% set_ConsD
tff(fact_6210_list_Oset__cases,axiom,
    ! [A: $tType,E3: A,A2: list(A)] :
      ( member(A,E3,aa(list(A),set(A),set2(A),A2))
     => ( ! [Z23: list(A)] : A2 != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),Z23)
       => ~ ! [Z12: A,Z23: list(A)] :
              ( ( A2 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z12),Z23) )
             => ~ member(A,E3,aa(list(A),set(A),set2(A),Z23)) ) ) ) ).

% list.set_cases
tff(fact_6211_list_Oset__intros_I1_J,axiom,
    ! [A: $tType,X21: A,X22: list(A)] : member(A,X21,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22))) ).

% list.set_intros(1)
tff(fact_6212_list_Oset__intros_I2_J,axiom,
    ! [A: $tType,Y: A,X22: list(A),X21: A] :
      ( member(A,Y,aa(list(A),set(A),set2(A),X22))
     => member(A,Y,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22))) ) ).

% list.set_intros(2)
tff(fact_6213_list__tail__coinc,axiom,
    ! [A: $tType,N12: A,R12: list(A),N23: A,R23: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N12),R12) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),N23),R23) )
     => ( ( N12 = N23 )
        & ( R12 = R23 ) ) ) ).

% list_tail_coinc
tff(fact_6214_length__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(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_6215_Suc__length__conv,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( ( aa(nat,nat,suc,Nb) = aa(list(A),nat,size_size(list(A)),Xs) )
    <=> ? [Y3: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = Nb ) ) ) ).

% Suc_length_conv
tff(fact_6216_length__Suc__conv,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,Nb) )
    <=> ? [Y3: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys3) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = Nb ) ) ) ).

% length_Suc_conv
tff(fact_6217_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),aa(A,fun(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_6218_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),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),I,V) = case_nat(list(A),aa(list(A),list(A),aa(A,fun(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_afs(A,fun(list(A),fun(A,fun(nat,list(A)))),Xb),Xs),V),I) ).

% list_update.simps(2)
tff(fact_6219_list__assn_Osimps_I2_J,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,assn)),A2: A,As3: list(A),C2: B,Cs: list(B)] : aa(list(B),assn,vEBT_List_list_assn(A,B,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),As3)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C2),Cs)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),P,A2),C2)),aa(list(B),assn,vEBT_List_list_assn(A,B,P,As3),Cs)) ).

% list_assn.simps(2)
tff(fact_6220_list__assn__simps_I2_J,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,assn)),A2: A,As3: list(A),C2: B,Cs: list(B)] : aa(list(B),assn,vEBT_List_list_assn(A,B,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),As3)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C2),Cs)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),P,A2),C2)),aa(list(B),assn,vEBT_List_list_assn(A,B,P,As3),Cs)) ).

% list_assn_simps(2)
tff(fact_6221_map__eq__consE,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),Ls: list(B),Fa: A,Fl: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F),Ls) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Fa),Fl) )
     => ~ ! [A4: B,L2: list(B)] :
            ( ( Ls = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A4),L2) )
           => ( ( aa(B,A,F,A4) = Fa )
             => ( aa(list(B),list(A),map(B,A,F),L2) != Fl ) ) ) ) ).

% map_eq_consE
tff(fact_6222_map__consI_I1_J,axiom,
    ! [A: $tType,B: $tType,W: list(A),F: fun(B,A),Ww: list(B),A2: B] :
      ( ( W = aa(list(B),list(A),map(B,A,F),Ww) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F,A2)),W) = aa(list(B),list(A),map(B,A,F),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),Ww)) ) ) ).

% map_consI(1)
tff(fact_6223_replicate__Suc,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : replicate(A,aa(nat,nat,suc,Nb),Xb) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),replicate(A,Nb,Xb)) ).

% replicate_Suc
tff(fact_6224_VEBT__internal_Oreplicatei_Osimps_I2_J,axiom,
    ! [A: $tType,Nb: nat,Xb: heap_Time_Heap(A)] : vEBT_VEBT_replicatei(A,aa(nat,nat,suc,Nb),Xb) = heap_Time_bind(A,list(A),Xb,aa(heap_Time_Heap(A),fun(A,heap_Time_Heap(list(A))),aTP_Lamp_afu(nat,fun(heap_Time_Heap(A),fun(A,heap_Time_Heap(list(A)))),Nb),Xb)) ).

% VEBT_internal.replicatei.simps(2)
tff(fact_6225_nth__Cons,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Nb) = case_nat(A,Xb,nth(A,Xs),Nb) ).

% nth_Cons
tff(fact_6226_Suc__le__length__iff,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,suc,Nb)),aa(list(A),nat,size_size(list(A)),Xs))
    <=> ? [X: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Ys3) )
          & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Ys3)) ) ) ).

% Suc_le_length_iff
tff(fact_6227_list_Osize_I4_J,axiom,
    ! [A: $tType,X21: A,X22: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),X22)),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size(4)
tff(fact_6228_nth__Cons_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Nb) = $ite(Nb = zero_zero(nat),Xb,aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))) ).

% nth_Cons'
tff(fact_6229_nth__non__equal__first__eq,axiom,
    ! [A: $tType,Xb: A,Y: A,Xs: list(A),Nb: nat] :
      ( ( Xb != Y )
     => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Nb) = Y )
      <=> ( ( aa(nat,A,nth(A,Xs),aa(nat,nat,minus_minus(nat,Nb),one_one(nat))) = Y )
          & aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ) ) ).

% nth_non_equal_first_eq
tff(fact_6230_nth__equal__first__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))
       => ( ( aa(nat,A,nth(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Nb) = Xb )
        <=> ( Nb = zero_zero(nat) ) ) ) ) ).

% nth_equal_first_eq
tff(fact_6231_Cons__replicate__eq,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Nb: nat,Y: A] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) = replicate(A,Nb,Y) )
    <=> ( ( Xb = Y )
        & aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
        & ( Xs = replicate(A,aa(nat,nat,minus_minus(nat,Nb),one_one(nat)),Xb) ) ) ) ).

% Cons_replicate_eq
tff(fact_6232_slice__Cons,axiom,
    ! [A: $tType,Begin: nat,End: nat,Xb: A,Xs: list(A)] :
      slice(A,Begin,End,aa(list(A),list(A),aa(A,fun(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),aa(A,fun(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_6233_TBOUND__new,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,Xb: A] : time_TBOUND(array(A),array_new(A,Nb,Xb),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))) ) ).

% TBOUND_new
tff(fact_6234_time__array__new,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,Xb: A,H: heap_ext(product_unit)] : time_time(array(A),array_new(A,Nb,Xb),H) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)) ) ).

% time_array_new
tff(fact_6235_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_6236_sum__count__set,axiom,
    ! [A: $tType,Xs: list(A),X5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),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_6237_count__notin,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(A,nat,count_list(A,Xs),Xb) = zero_zero(nat) ) ) ).

% count_notin
tff(fact_6238_count__le__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] : aa(nat,$o,aa(nat,fun(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_6239_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),aa(A,fun(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_6240_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_6241_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),aa(A,fun(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_6242_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_6243_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)) ) )
       => ~ ! [N2: nat] :
              ( ( Xb = aa(nat,nat,suc,N2) )
             => ( Y != heap_Time_bind(A,list(A),Xaa,aa(nat,fun(A,heap_Time_Heap(list(A))),aTP_Lamp_afv(heap_Time_Heap(A),fun(nat,fun(A,heap_Time_Heap(list(A)))),Xaa),N2)) ) ) ) ) ).

% VEBT_internal.replicatei.elims
tff(fact_6244_call__mono,axiom,
    ! [B: $tType,A: $tType,Ord: fun(B,fun(B,$o)),Ta: A] : comple7038119648293358887notone(fun(A,B),B,partial_fun_ord(B,B,A,Ord),Ord,aTP_Lamp_afw(A,fun(fun(A,B),B),Ta)) ).

% call_mono
tff(fact_6245_slice__eq__bounds__empty,axiom,
    ! [A: $tType,I: nat,Xs: list(A)] : slice(A,I,I,Xs) = nil(A) ).

% slice_eq_bounds_empty
tff(fact_6246_slice__Nil,axiom,
    ! [A: $tType,Begin: nat,End: nat] : slice(A,Begin,End,nil(A)) = nil(A) ).

% slice_Nil
tff(fact_6247_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_6248_set__empty2,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( bot_bot(set(A)) = aa(list(A),set(A),set2(A),Xs) )
    <=> ( Xs = nil(A) ) ) ).

% set_empty2
tff(fact_6249_set__empty,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( aa(list(A),set(A),set2(A),Xs) = bot_bot(set(A)) )
    <=> ( Xs = nil(A) ) ) ).

% set_empty
tff(fact_6250_list__assn__aux__simps_I2_J,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn)),L: list(A)] : aa(list(B),assn,vEBT_List_list_assn(A,B,P,L),nil(B)) = pure_assn(L = nil(A)) ).

% list_assn_aux_simps(2)
tff(fact_6251_list__assn__aux__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,assn)),L3: list(B)] : aa(list(B),assn,vEBT_List_list_assn(A,B,P,nil(A)),L3) = pure_assn(L3 = nil(B)) ).

% list_assn_aux_simps(1)
tff(fact_6252_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_6253_length__ge__1__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( aa(nat,$o,aa(nat,fun(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_6254_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_6255_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_6256_empty__set,axiom,
    ! [A: $tType] : bot_bot(set(A)) = aa(list(A),set(A),set2(A),nil(A)) ).

% empty_set
tff(fact_6257_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_6258_list__induct4,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list(A),Ys: list(B),Zs: list(C),Ws: list(D),P: fun(list(A),fun(list(B),fun(list(C),fun(list(D),$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(D),nat,size_size(list(D)),Ws) )
         => ( aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),$o))),P,nil(A)),nil(B)),nil(C)),nil(D))
           => ( ! [X2: A,Xs2: list(A),Y5: B,Ys4: list(B),Z3: C,Zs2: list(C),W2: D,Ws2: list(D)] :
                  ( ( 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(D),nat,size_size(list(D)),Ws2) )
                     => ( aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),$o))),P,Xs2),Ys4),Zs2),Ws2)
                       => aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),$o))),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z3),Zs2)),aa(list(D),list(D),aa(D,fun(list(D),list(D)),cons(D),W2),Ws2)) ) ) ) )
             => aa(list(D),$o,aa(list(C),fun(list(D),$o),aa(list(B),fun(list(C),fun(list(D),$o)),aa(list(A),fun(list(B),fun(list(C),fun(list(D),$o))),P,Xs),Ys),Zs),Ws) ) ) ) ) ) ).

% list_induct4
tff(fact_6259_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),$o)),P,nil(A)),nil(B)),nil(C))
         => ( ! [X2: A,Xs2: list(A),Y5: B,Ys4: list(B),Z3: C,Zs2: list(C)] :
                ( ( aa(list(A),nat,size_size(list(A)),Xs2) = aa(list(B),nat,size_size(list(B)),Ys4) )
               => ( ( aa(list(B),nat,size_size(list(B)),Ys4) = aa(list(C),nat,size_size(list(C)),Zs2) )
                 => ( aa(list(C),$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),$o)),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4)),aa(list(C),list(C),aa(C,fun(list(C),list(C)),cons(C),Z3),Zs2)) ) ) )
           => aa(list(C),$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_6260_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),Y5: 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),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4)) ) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ).

% list_induct2
tff(fact_6261_Misc_Ofoldr__Cons,axiom,
    ! [A: $tType,Xs: list(A)] : foldr(A,list(A),cons(A),Xs,nil(A)) = Xs ).

% Misc.foldr_Cons
tff(fact_6262_neq__NilE,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => ~ ! [X2: A,Xs2: list(A)] : L != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) ) ).

% neq_NilE
tff(fact_6263_list__2pre__induct,axiom,
    ! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),$o)),W1: list(A),W22: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),nil(B))
     => ( ! [E2: A,W12: list(A),W23: list(B)] :
            ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,W12),W23)
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),W12)),W23) )
       => ( ! [E2: B,W13: list(A),W24: list(B)] :
              ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,W13),W24)
             => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,W13),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),E2),W24)) )
         => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,W1),W22) ) ) ) ).

% list_2pre_induct
tff(fact_6264_list__induct__first2,axiom,
    ! [A: $tType,P: fun(list(A),$o),Xs: list(A)] :
      ( aa(list(A),$o,P,nil(A))
     => ( ! [X2: A] : aa(list(A),$o,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
       => ( ! [X15: A,X24: A,Xs2: list(A)] :
              ( aa(list(A),$o,P,Xs2)
             => aa(list(A),$o,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X24),Xs2))) )
         => aa(list(A),$o,P,Xs) ) ) ) ).

% list_induct_first2
tff(fact_6265_mergesort__by__rel__merge__induct,axiom,
    ! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),$o)),R: fun(A,fun(B,$o)),Xs: list(A),Ys: list(B)] :
      ( ! [Xs2: list(A)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs2),nil(B))
     => ( ! [Ys4: list(B)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),Ys4)
       => ( ! [X2: A,Xs2: list(A),Y5: B,Ys4: list(B)] :
              ( aa(B,$o,aa(A,fun(B,$o),R,X2),Y5)
             => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs2),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4))
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4)) ) )
         => ( ! [X2: A,Xs2: list(A),Y5: B,Ys4: list(B)] :
                ( ~ aa(B,$o,aa(A,fun(B,$o),R,X2),Y5)
               => ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Ys4)
                 => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),Ys4)) ) )
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ) ).

% mergesort_by_rel_merge_induct
tff(fact_6266_subset__eq__mset__impl_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(list(A),list(A))] :
      ( ! [Ys4: list(A)] : Xb != 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)),nil(A)),Ys4)
     => ~ ! [X2: A,Xs2: list(A),Ys4: list(A)] : Xb != 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),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Ys4) ) ).

% subset_eq_mset_impl.cases
tff(fact_6267_zipf_Ocases,axiom,
    ! [C: $tType,A: $tType,B: $tType,Xb: product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))] :
      ( ! [F2: fun(A,fun(B,C))] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
     => ( ! [F2: fun(A,fun(B,C)),A4: A,As: list(A),B3: B,Bs: list(B)] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),F2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs)))
       => ( ! [A4: fun(A,fun(B,C)),V3: A,Va2: list(A)] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va2)),nil(B)))
         => ~ ! [A4: fun(A,fun(B,C)),V3: B,Va2: list(B)] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B))),aa(fun(A,fun(B,C)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,C)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,C)),product_prod(list(A),list(B))),A4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va2))) ) ) ) ).

% zipf.cases
tff(fact_6268_merge_Ocases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: product_prod(list(A),list(A))] :
          ( ! [L22: list(A)] : Xb != 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)),nil(A)),L22)
         => ( ! [V3: A,Va2: list(A)] : Xb != 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),aa(A,fun(list(A),list(A)),cons(A),V3),Va2)),nil(A))
           => ~ ! [X15: A,L1: list(A),X24: A,L22: list(A)] : Xb != 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),aa(A,fun(list(A),list(A)),cons(A),X15),L1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X24),L22)) ) ) ) ).

% merge.cases
tff(fact_6269_list__all__zip_Ocases,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))] :
      ( ! [P4: fun(A,fun(B,$o))] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
     => ( ! [P4: fun(A,fun(B,$o)),A4: A,As: list(A),B3: B,Bs: list(B)] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),B3),Bs)))
       => ( ! [P4: fun(A,fun(B,$o)),V3: A,Va2: list(A)] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va2)),nil(B)))
         => ~ ! [P4: fun(A,fun(B,$o)),V3: B,Va2: list(B)] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B))),aa(fun(A,fun(B,$o)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,$o)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,$o)),product_prod(list(A),list(B))),P4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va2))) ) ) ) ).

% list_all_zip.cases
tff(fact_6270_partition__rev_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))] :
      ( ! [P4: fun(A,$o),Yes: list(A),No: list(A)] : Xb != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),P4),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Yes),No)),nil(A)))
     => ~ ! [P4: fun(A,$o),Yes: list(A),No: list(A),X2: A,Xs2: list(A)] : Xb != aa(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),aa(fun(A,$o),fun(product_prod(product_prod(list(A),list(A)),list(A)),product_prod(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A)))),product_Pair(fun(A,$o),product_prod(product_prod(list(A),list(A)),list(A))),P4),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),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)),Yes),No)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2))) ) ).

% partition_rev.cases
tff(fact_6271_quicksort__by__rel_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))] :
      ( ! [R7: fun(A,fun(A,$o)),Sl: list(A)] : Xb != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Sl),nil(A)))
     => ~ ! [R7: fun(A,fun(A,$o)),Sl: list(A),X2: A,Xs2: list(A)] : Xb != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R7),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)),Sl),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2))) ) ).

% quicksort_by_rel.cases
tff(fact_6272_mergesort__by__rel__merge_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))] :
      ( ! [R7: fun(A,fun(A,$o)),X2: A,Xs2: list(A),Y5: A,Ys4: list(A)] : Xb != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R7),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4)))
     => ( ! [R7: fun(A,fun(A,$o)),Xs2: list(A)] : Xb != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R7),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)),Xs2),nil(A)))
       => ~ ! [R7: fun(A,fun(A,$o)),V3: A,Va2: list(A)] : Xb != aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),R7),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)),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va2))) ) ) ).

% mergesort_by_rel_merge.cases
tff(fact_6273_mergesort__by__rel__split_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(product_prod(list(A),list(A)),list(A))] :
      ( ! [Xs1: list(A),Xs22: list(A)] : Xb != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),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)),Xs1),Xs22)),nil(A))
     => ( ! [Xs1: list(A),Xs22: list(A),X2: A] : Xb != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),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)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
       => ~ ! [Xs1: list(A),Xs22: list(A),X15: A,X24: A,Xs2: list(A)] : Xb != aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),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)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X24),Xs2))) ) ) ).

% mergesort_by_rel_split.cases
tff(fact_6274_shuffles_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(list(A),list(A))] :
      ( ! [Ys4: list(A)] : Xb != 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)),nil(A)),Ys4)
     => ( ! [Xs2: list(A)] : Xb != 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)),Xs2),nil(A))
       => ~ ! [X2: A,Xs2: list(A),Y5: A,Ys4: list(A)] : Xb != 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),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4)) ) ) ).

% shuffles.cases
tff(fact_6275_sorted__wrt_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,$o)),list(A))] :
      ( ! [P4: fun(A,fun(A,$o))] : Xb != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P4),nil(A))
     => ~ ! [P4: fun(A,fun(A,$o)),X2: A,Ys4: list(A)] : Xb != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Ys4)) ) ).

% sorted_wrt.cases
tff(fact_6276_arg__min__list_Ocases,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Xb: product_prod(fun(A,B),list(A))] :
          ( ! [F2: fun(A,B),X2: A] : Xb != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
         => ( ! [F2: fun(A,B),X2: A,Y5: A,Zs2: list(A)] : Xb != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),F2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Zs2)))
           => ~ ! [A4: fun(A,B)] : Xb != aa(list(A),product_prod(fun(A,B),list(A)),aa(fun(A,B),fun(list(A),product_prod(fun(A,B),list(A))),product_Pair(fun(A,B),list(A)),A4),nil(A)) ) ) ) ).

% arg_min_list.cases
tff(fact_6277_successively_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,$o)),list(A))] :
      ( ! [P4: fun(A,fun(A,$o))] : Xb != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P4),nil(A))
     => ( ! [P4: fun(A,fun(A,$o)),X2: A] : Xb != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))
       => ~ ! [P4: fun(A,fun(A,$o)),X2: A,Y5: A,Xs2: list(A)] : Xb != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),P4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Xs2))) ) ) ).

% successively.cases
tff(fact_6278_map__tailrec__rev_Ocases,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(fun(A,B),product_prod(list(A),list(B)))] :
      ( ! [F2: fun(A,B),Bs: list(B)] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),Bs))
     => ~ ! [F2: fun(A,B),A4: A,As: list(A),Bs: list(B)] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B))),aa(fun(A,B),fun(product_prod(list(A),list(B)),product_prod(fun(A,B),product_prod(list(A),list(B)))),product_Pair(fun(A,B),product_prod(list(A),list(B))),F2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As)),Bs)) ) ).

% map_tailrec_rev.cases
tff(fact_6279_list__assn_Ocases,axiom,
    ! [A: $tType,B: $tType,Xb: product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))] :
      ( ! [P4: fun(A,fun(B,assn))] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),P4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))
     => ( ! [P4: fun(A,fun(B,assn)),A4: A,As: list(A),C3: B,Cs2: list(B)] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),P4),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C3),Cs2)))
       => ( ! [Uu2: fun(A,fun(B,assn)),V3: A,Va2: list(A)] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Uu2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va2)),nil(B)))
         => ~ ! [Uu2: fun(A,fun(B,assn)),V3: B,Va2: list(B)] : Xb != aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Uu2),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va2))) ) ) ) ).

% list_assn.cases
tff(fact_6280_list__assn__simps_I1_J,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn))] : aa(list(B),assn,vEBT_List_list_assn(A,B,P,nil(A)),nil(B)) = one_one(assn) ).

% list_assn_simps(1)
tff(fact_6281_list__assn_Osimps_I1_J,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn))] : aa(list(B),assn,vEBT_List_list_assn(A,B,P,nil(A)),nil(B)) = one_one(assn) ).

% list_assn.simps(1)
tff(fact_6282_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,aa(nat,fun(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),$o,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),L2)) )
       => aa(list(A),$o,P,L) ) ) ).

% length_compl_induct
tff(fact_6283_list__decomp__1,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = one_one(nat) )
     => ? [A4: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),nil(A)) ) ).

% list_decomp_1
tff(fact_6284_list__assn_Osimps_I4_J,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),V: B,Vaa: list(B)] : aa(list(B),assn,vEBT_List_list_assn(A,B,Uu,nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V),Vaa)) = bot_bot(assn) ).

% list_assn.simps(4)
tff(fact_6285_list__assn_Osimps_I3_J,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),V: A,Vaa: list(A)] : aa(list(B),assn,vEBT_List_list_assn(A,B,Uu,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Vaa)),nil(B)) = bot_bot(assn) ).

% list_assn.simps(3)
tff(fact_6286_list__assn__simps_I4_J,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn)),C2: B,Cs: list(B)] : aa(list(B),assn,vEBT_List_list_assn(A,B,P,nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C2),Cs)) = bot_bot(assn) ).

% list_assn_simps(4)
tff(fact_6287_list__assn__simps_I3_J,axiom,
    ! [B: $tType,A: $tType,P: fun(A,fun(B,assn)),A2: A,As3: list(A)] : aa(list(B),assn,vEBT_List_list_assn(A,B,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),As3)),nil(B)) = bot_bot(assn) ).

% list_assn_simps(3)
tff(fact_6288_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_6289_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)) )
     => ? [A4: A,B3: A] : L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),B3),nil(A))) ) ).

% list_decomp_2
tff(fact_6290_let__mono,axiom,
    ! [B: $tType,C: $tType,A: $tType,Orda: fun(B,fun(B,$o)),Ordb: fun(C,fun(C,$o)),B2: fun(B,fun(A,C)),Ta: A] :
      ( ! [X2: A] : comple7038119648293358887notone(B,C,Orda,Ordb,aa(A,fun(B,C),aTP_Lamp_afx(fun(B,fun(A,C)),fun(A,fun(B,C)),B2),X2))
     => comple7038119648293358887notone(B,C,Orda,Ordb,aa(A,fun(B,C),aTP_Lamp_afx(fun(B,fun(A,C)),fun(A,fun(B,C)),B2),Ta)) ) ).

% let_mono
tff(fact_6291_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($o,fun(A,B)),aTP_Lamp_afy(fun(A,B),fun(fun(A,B),fun($o,fun(A,B))),F5),G4),(C2))) ) ) ).

% if_mono
tff(fact_6292_list__assn_Oelims,axiom,
    ! [B: $tType,A: $tType,Xb: fun(A,fun(B,assn)),Xaa: list(A),Xba: list(B),Y: assn] :
      ( ( aa(list(B),assn,vEBT_List_list_assn(A,B,Xb,Xaa),Xba) = Y )
     => ( ( ( Xaa = nil(A) )
         => ( ( Xba = nil(B) )
           => ( Y != one_one(assn) ) ) )
       => ( ! [A4: A,As: list(A)] :
              ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As) )
             => ! [C3: B,Cs2: list(B)] :
                  ( ( Xba = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C3),Cs2) )
                 => ( Y != aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Xb,A4),C3)),aa(list(B),assn,vEBT_List_list_assn(A,B,Xb,As),Cs2)) ) ) )
         => ( ( ? [V3: A,Va2: list(A)] : Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va2)
             => ( ( Xba = nil(B) )
               => ( Y != bot_bot(assn) ) ) )
           => ~ ( ( Xaa = nil(A) )
               => ( ? [V3: B,Va2: list(B)] : Xba = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va2)
                 => ( Y != bot_bot(assn) ) ) ) ) ) ) ) ).

% list_assn.elims
tff(fact_6293_VEBT__internal_Oreplicatei_Opelims,axiom,
    ! [A: $tType,Xb: nat,Xaa: heap_Time_Heap(A),Y: heap_Time_Heap(list(A))] :
      ( ( vEBT_VEBT_replicatei(A,Xb,Xaa) = Y )
     => ( aa(product_prod(nat,heap_Time_Heap(A)),$o,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)) )
             => ~ aa(product_prod(nat,heap_Time_Heap(A)),$o,accp(product_prod(nat,heap_Time_Heap(A)),vEBT_V8535202610096940344ei_rel(A)),aa(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A)),aa(nat,fun(heap_Time_Heap(A),product_prod(nat,heap_Time_Heap(A))),product_Pair(nat,heap_Time_Heap(A)),zero_zero(nat)),Xaa)) ) )
         => ~ ! [N2: nat] :
                ( ( Xb = aa(nat,nat,suc,N2) )
               => ( ( Y = heap_Time_bind(A,list(A),Xaa,aa(nat,fun(A,heap_Time_Heap(list(A))),aTP_Lamp_afv(heap_Time_Heap(A),fun(nat,fun(A,heap_Time_Heap(list(A)))),Xaa),N2)) )
                 => ~ aa(product_prod(nat,heap_Time_Heap(A)),$o,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,N2)),Xaa)) ) ) ) ) ) ).

% VEBT_internal.replicatei.pelims
tff(fact_6294_list__assn_Opelims,axiom,
    ! [A: $tType,B: $tType,Xb: fun(A,fun(B,assn)),Xaa: list(A),Xba: list(B),Y: assn] :
      ( ( aa(list(B),assn,vEBT_List_list_assn(A,B,Xb,Xaa),Xba) = Y )
     => ( aa(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),vEBT_L4249061453398456502sn_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Xb),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),Xaa),Xba)))
       => ( ( ( Xaa = nil(A) )
           => ( ( Xba = nil(B) )
             => ( ( Y = one_one(assn) )
               => ~ aa(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),vEBT_L4249061453398456502sn_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Xb),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),nil(B)))) ) ) )
         => ( ! [A4: A,As: list(A)] :
                ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As) )
               => ! [C3: B,Cs2: list(B)] :
                    ( ( Xba = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C3),Cs2) )
                   => ( ( Y = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(B,assn,aa(A,fun(B,assn),Xb,A4),C3)),aa(list(B),assn,vEBT_List_list_assn(A,B,Xb,As),Cs2)) )
                     => ~ aa(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),vEBT_L4249061453398456502sn_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Xb),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A4),As)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),C3),Cs2)))) ) ) )
           => ( ! [V3: A,Va2: list(A)] :
                  ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va2) )
                 => ( ( Xba = nil(B) )
                   => ( ( Y = bot_bot(assn) )
                     => ~ aa(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),vEBT_L4249061453398456502sn_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Xb),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va2)),nil(B)))) ) ) )
             => ~ ( ( Xaa = nil(A) )
                 => ! [V3: B,Va2: list(B)] :
                      ( ( Xba = aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va2) )
                     => ( ( Y = bot_bot(assn) )
                       => ~ aa(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),$o,accp(product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),vEBT_L4249061453398456502sn_rel(A,B)),aa(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B))),aa(fun(A,fun(B,assn)),fun(product_prod(list(A),list(B)),product_prod(fun(A,fun(B,assn)),product_prod(list(A),list(B)))),product_Pair(fun(A,fun(B,assn)),product_prod(list(A),list(B))),Xb),aa(list(B),product_prod(list(A),list(B)),aa(list(A),fun(list(B),product_prod(list(A),list(B))),product_Pair(list(A),list(B)),nil(A)),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),V3),Va2)))) ) ) ) ) ) ) ) ) ).

% list_assn.pelims
tff(fact_6295_mergesort__by__rel_Ocases,axiom,
    ! [A: $tType,Xb: product_prod(fun(A,fun(A,$o)),list(A))] :
      ~ ! [R7: fun(A,fun(A,$o)),Xs2: list(A)] : Xb != aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),R7),Xs2) ).

% mergesort_by_rel.cases
tff(fact_6296_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),F: C,P: fun(fun(B,A),$o)] :
      ( partia7178651479351089652itions(A,Leq,Lub)
     => ( ! [X2: B] : comple7038119648293358887notone(fun(B,A),A,partial_fun_ord(A,A,B,Leq),Leq,aa(B,fun(fun(B,A),A),aa(fun(fun(B,A),C),fun(B,fun(fun(B,A),A)),aa(fun(C,C),fun(fun(fun(B,A),C),fun(B,fun(fun(B,A),A))),aTP_Lamp_afz(fun(C,fun(B,A)),fun(fun(C,C),fun(fun(fun(B,A),C),fun(B,fun(fun(B,A),A)))),U2),F5),C5),X2))
       => ( ( F = aa(fun(B,A),C,C5,comple187402453842119260l_fixp(fun(B,A),partial_fun_lub(A,A,B,Lub),partial_fun_ord(A,A,B,Leq),aa(fun(fun(B,A),C),fun(fun(B,A),fun(B,A)),aa(fun(C,C),fun(fun(fun(B,A),C),fun(fun(B,A),fun(B,A))),aTP_Lamp_aga(fun(C,fun(B,A)),fun(fun(C,C),fun(fun(fun(B,A),C),fun(fun(B,A),fun(B,A)))),U2),F5),C5))) )
         => ( ! [F2: fun(B,A)] : aa(C,fun(B,A),U2,aa(fun(B,A),C,C5,F2)) = F2
           => ( comple1908693960933563346ssible(fun(B,A),partial_fun_lub(A,A,B,Lub),partial_fun_ord(A,A,B,Leq),P)
             => ( aa(fun(B,A),$o,P,aTP_Lamp_agb(fun(set(A),A),fun(B,A),Lub))
               => ( ! [F2: C] :
                      ( aa(fun(B,A),$o,P,aa(C,fun(B,A),U2,F2))
                     => aa(fun(B,A),$o,P,aa(C,fun(B,A),U2,aa(C,C,F5,F2))) )
                 => aa(fun(B,A),$o,P,aa(C,fun(B,A),U2,F)) ) ) ) ) ) ) ) ).

% partial_function_definitions.fixp_induct_uc
tff(fact_6297_in__set__simps_I2_J,axiom,
    ! [A: $tType,Xb: A,Y: A] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),nil(A))))
    <=> ( Xb = Y ) ) ).

% in_set_simps(2)
tff(fact_6298_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_eq(A,fun(B,A)),C2)) ) ).

% partial_function_definitions.const_mono
tff(fact_6299_in__set__simps_I1_J,axiom,
    ! [A: $tType,Xb: A,Y: A,Z: A,Ys: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z),Ys))))
    <=> ( ( Xb = Y )
        | member(A,Xb,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Z),Ys))) ) ) ).

% in_set_simps(1)
tff(fact_6300_in__set__simps_I3_J,axiom,
    ! [A: $tType,Xb: A] : ~ member(A,Xb,aa(list(A),set(A),set2(A),nil(A))) ).

% in_set_simps(3)
tff(fact_6301_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),F: 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_afz(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))
       => ( ( F = 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_aga(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))) )
         => ( ! [F2: C] : aa(fun(B,A),C,C5,aa(C,fun(B,A),U2,F2)) = F2
           => ( F = aa(C,C,F5,F) ) ) ) ) ) ).

% partial_function_definitions.fixp_rule_uc
tff(fact_6302_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_agc(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_6303_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_agd(fun(B,fun(A,$o)),fun(fun(B,A),$o),Q)) ) ) ).

% admissible_fun
tff(fact_6304_mergesort__by__rel__split__length,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: 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)),Xs12),Xs23),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)),Xs12)),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)),Xs12),Xs23),Xs))) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(list(A),nat,size_size(list(A)),Xs23)),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_6305_concat__inth,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),append(A,Xs),aa(list(A),list(A),append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xb ).

% concat_inth
tff(fact_6306_empty__append__eq__id,axiom,
    ! [A: $tType,X3: list(A)] : aa(list(A),list(A),append(A,nil(A)),X3) = X3 ).

% empty_append_eq_id
tff(fact_6307_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) ) )
     => ( ( aa(list(A),list(A),append(A,Xs),Us) = aa(list(A),list(A),append(A,Ys),Vs) )
      <=> ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
tff(fact_6308_list__ee__eq__leel_I1_J,axiom,
    ! [A: $tType,E1: A,E22: A,L12: list(A),E12: A,E23: A,L23: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E22),nil(A))) = aa(list(A),list(A),append(A,L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),L23))) )
    <=> ( ( L12 = nil(A) )
        & ( E1 = E12 )
        & ( E22 = E23 )
        & ( L23 = nil(A) ) ) ) ).

% list_ee_eq_leel(1)
tff(fact_6309_list__ee__eq__leel_I2_J,axiom,
    ! [A: $tType,L12: list(A),E12: A,E23: A,L23: list(A),E1: A,E22: A] :
      ( ( aa(list(A),list(A),append(A,L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E23),L23))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E22),nil(A))) )
    <=> ( ( L12 = nil(A) )
        & ( E1 = E12 )
        & ( E22 = E23 )
        & ( L23 = nil(A) ) ) ) ).

% list_ee_eq_leel(2)
tff(fact_6310_list__se__match_I1_J,axiom,
    ! [A: $tType,L12: list(A),L23: list(A),A2: A] :
      ( ( L12 != nil(A) )
     => ( ( aa(list(A),list(A),append(A,L12),L23) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A)) )
      <=> ( ( L12 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A)) )
          & ( L23 = nil(A) ) ) ) ) ).

% list_se_match(1)
tff(fact_6311_list__se__match_I2_J,axiom,
    ! [A: $tType,L23: list(A),L12: list(A),A2: A] :
      ( ( L23 != nil(A) )
     => ( ( aa(list(A),list(A),append(A,L12),L23) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A)) )
      <=> ( ( L12 = nil(A) )
          & ( L23 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A)) ) ) ) ) ).

% list_se_match(2)
tff(fact_6312_list__se__match_I3_J,axiom,
    ! [A: $tType,L12: list(A),A2: A,L23: list(A)] :
      ( ( L12 != nil(A) )
     => ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A)) = aa(list(A),list(A),append(A,L12),L23) )
      <=> ( ( L12 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A)) )
          & ( L23 = nil(A) ) ) ) ) ).

% list_se_match(3)
tff(fact_6313_list__se__match_I4_J,axiom,
    ! [A: $tType,L23: list(A),A2: A,L12: list(A)] :
      ( ( L23 != nil(A) )
     => ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A)) = aa(list(A),list(A),append(A,L12),L23) )
      <=> ( ( L12 = nil(A) )
          & ( L23 = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),A2),nil(A)) ) ) ) ) ).

% list_se_match(4)
tff(fact_6314_list__e__eq__lel_I1_J,axiom,
    ! [A: $tType,E3: A,L12: list(A),E5: A,L23: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),nil(A)) = aa(list(A),list(A),append(A,L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E5),L23)) )
    <=> ( ( L12 = nil(A) )
        & ( E5 = E3 )
        & ( L23 = nil(A) ) ) ) ).

% list_e_eq_lel(1)
tff(fact_6315_list__e__eq__lel_I2_J,axiom,
    ! [A: $tType,L12: list(A),E5: A,L23: list(A),E3: A] :
      ( ( aa(list(A),list(A),append(A,L12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E5),L23)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),nil(A)) )
    <=> ( ( L12 = nil(A) )
        & ( E5 = E3 )
        & ( L23 = nil(A) ) ) ) ).

% list_e_eq_lel(2)
tff(fact_6316_length__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),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_6317_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,aa(list(A),list(A),append(A,L),L3)),I) = aa(nat,A,nth(A,L),I) ) ) ).

% nth_append_first
tff(fact_6318_nth__append__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A)] : aa(nat,A,nth(A,aa(list(A),list(A),append(A,Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys))),aa(list(A),nat,size_size(list(A)),Xs)) = Xb ).

% nth_append_length
tff(fact_6319_nth__append__length__plus,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat] : aa(nat,A,nth(A,aa(list(A),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)),Nb)) = aa(nat,A,nth(A,Ys),Nb) ).

% nth_append_length_plus
tff(fact_6320_list__update__length,axiom,
    ! [A: $tType,Xs: list(A),Xb: A,Ys: list(A),Y: A] : list_update(A,aa(list(A),list(A),append(A,Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys)),aa(list(A),nat,size_size(list(A)),Xs),Y) = aa(list(A),list(A),append(A,Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) ).

% list_update_length
tff(fact_6321_list__assn__aux__append2,axiom,
    ! [A: $tType,B: $tType,L23: list(A),L24: list(B),P: fun(A,fun(B,assn)),L12: list(A),L13: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L23) = aa(list(B),nat,size_size(list(B)),L24) )
     => ( aa(list(B),assn,vEBT_List_list_assn(A,B,P,aa(list(A),list(A),append(A,L12),L23)),aa(list(B),list(B),append(B,L13),L24)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,P,L12),L13)),aa(list(B),assn,vEBT_List_list_assn(A,B,P,L23),L24)) ) ) ).

% list_assn_aux_append2
tff(fact_6322_list__assn__aux__append,axiom,
    ! [A: $tType,B: $tType,L12: list(A),L13: list(B),P: fun(A,fun(B,assn)),L23: list(A),L24: list(B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L12) = aa(list(B),nat,size_size(list(B)),L13) )
     => ( aa(list(B),assn,vEBT_List_list_assn(A,B,P,aa(list(A),list(A),append(A,L12),L23)),aa(list(B),list(B),append(B,L13),L24)) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),aa(list(B),assn,vEBT_List_list_assn(A,B,P,L12),L13)),aa(list(B),assn,vEBT_List_list_assn(A,B,P,L23),L24)) ) ) ).

% list_assn_aux_append
tff(fact_6323_neq__Nil__revE,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
     => ~ ! [Ll2: list(A),E2: A] : L != aa(list(A),list(A),append(A,Ll2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A))) ) ).

% neq_Nil_revE
tff(fact_6324_rev__induct2_H,axiom,
    ! [A: $tType,B: $tType,P: fun(list(A),fun(list(B),$o)),Xs: list(A),Ys: list(B)] :
      ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),nil(B))
     => ( ! [X2: A,Xs2: list(A)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),append(A,Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))),nil(B))
       => ( ! [Y5: B,Ys4: list(B)] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,nil(A)),aa(list(B),list(B),append(B,Ys4),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),nil(B))))
         => ( ! [X2: A,Xs2: list(A),Y5: B,Ys4: list(B)] :
                ( 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),append(A,Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))),aa(list(B),list(B),append(B,Ys4),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),nil(B)))) )
           => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ) ).

% rev_induct2'
tff(fact_6325_neq__Nil__rev__conv,axiom,
    ! [A: $tType,L: list(A)] :
      ( ( L != nil(A) )
    <=> ? [Xs3: list(A),X: A] : L = aa(list(A),list(A),append(A,Xs3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),nil(A))) ) ).

% neq_Nil_rev_conv
tff(fact_6326_rev__nonempty__induct2_H,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Ys: list(B),P: fun(list(A),fun(list(B),$o))] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(B) )
       => ( ! [X2: A,Y5: B] : aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),nil(B)))
         => ( ! [X2: A,Xs2: list(A),Y5: B] :
                ( ( Xs2 != nil(A) )
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),append(A,Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),nil(B))) )
           => ( ! [X2: A,Y5: B,Ys4: list(B)] :
                  ( ( Ys4 != nil(B) )
                 => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))),aa(list(B),list(B),append(B,Ys4),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),nil(B)))) )
             => ( ! [X2: A,Xs2: list(A),Y5: B,Ys4: list(B)] :
                    ( aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs2),Ys4)
                   => ( ( Xs2 != nil(A) )
                     => ( ( Ys4 != nil(B) )
                       => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,aa(list(A),list(A),append(A,Xs2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))),aa(list(B),list(B),append(B,Ys4),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Y5),nil(B)))) ) ) )
               => aa(list(B),$o,aa(list(A),fun(list(B),$o),P,Xs),Ys) ) ) ) ) ) ) ).

% rev_nonempty_induct2'
tff(fact_6327_list__Cons__eq__append__cases,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A),Zs: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) = aa(list(A),list(A),append(A,Ys),Zs) )
     => ( ( ( Ys = nil(A) )
         => ( Zs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) ) )
       => ~ ! [Ys5: list(A)] :
              ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys5) )
             => ( aa(list(A),list(A),append(A,Ys5),Zs) != Xs ) ) ) ) ).

% list_Cons_eq_append_cases
tff(fact_6328_list__append__eq__Cons__cases,axiom,
    ! [A: $tType,Ys: list(A),Zs: list(A),Xb: A,Xs: list(A)] :
      ( ( aa(list(A),list(A),append(A,Ys),Zs) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) )
     => ( ( ( Ys = nil(A) )
         => ( Zs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) ) )
       => ~ ! [Ys5: list(A)] :
              ( ( Ys = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys5) )
             => ( aa(list(A),list(A),append(A,Ys5),Zs) != Xs ) ) ) ) ).

% list_append_eq_Cons_cases
tff(fact_6329_list__match__lel__lel,axiom,
    ! [A: $tType,C1: list(A),Qs: A,C22: list(A),C12: list(A),Qs2: A,C23: list(A)] :
      ( ( aa(list(A),list(A),append(A,C1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C22)) = aa(list(A),list(A),append(A,C12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C23)) )
     => ( ! [C21: list(A)] :
            ( ( C1 = aa(list(A),list(A),append(A,C12),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C21)) )
           => ( C23 != aa(list(A),list(A),append(A,C21),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C22)) ) )
       => ( ( ( C12 = C1 )
           => ( ( Qs2 = Qs )
             => ( C23 != C22 ) ) )
         => ~ ! [C212: list(A)] :
                ( ( C12 = aa(list(A),list(A),append(A,C1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs),C212)) )
               => ( C22 != aa(list(A),list(A),append(A,C212),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Qs2),C23)) ) ) ) ) ) ).

% list_match_lel_lel
tff(fact_6330_split__list__first__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X) )
    <=> ? [Ys3: list(A),X: A] :
          ( ? [Zs3: list(A)] : Xs = aa(list(A),list(A),append(A,Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs3))
          & aa(A,$o,P,X)
          & ! [Xa3: A] :
              ( member(A,Xa3,aa(list(A),set(A),set2(A),Ys3))
             => ~ aa(A,$o,P,Xa3) ) ) ) ).

% split_list_first_prop_iff
tff(fact_6331_split__list__last__prop__iff,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X) )
    <=> ? [Ys3: list(A),X: A,Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),append(A,Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Zs3)) )
          & aa(A,$o,P,X)
          & ! [Xa3: A] :
              ( member(A,Xa3,aa(list(A),set(A),set2(A),Zs3))
             => ~ aa(A,$o,P,Xa3) ) ) ) ).

% split_list_last_prop_iff
tff(fact_6332_in__set__conv__decomp__first,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
    <=> ? [Ys3: list(A),Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),append(A,Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Zs3)) )
          & ~ member(A,Xb,aa(list(A),set(A),set2(A),Ys3)) ) ) ).

% in_set_conv_decomp_first
tff(fact_6333_in__set__conv__decomp__last,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
    <=> ? [Ys3: list(A),Zs3: list(A)] :
          ( ( Xs = aa(list(A),list(A),append(A,Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Zs3)) )
          & ~ member(A,Xb,aa(list(A),set(A),set2(A),Zs3)) ) ) ).

% in_set_conv_decomp_last
tff(fact_6334_split__list__first__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X3) )
     => ~ ! [Ys4: list(A),X2: A] :
            ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),append(A,Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Zs2))
           => ( aa(A,$o,P,X2)
             => ~ ! [Xa: A] :
                    ( member(A,Xa,aa(list(A),set(A),set2(A),Ys4))
                   => ~ aa(A,$o,P,Xa) ) ) ) ) ).

% split_list_first_propE
tff(fact_6335_split__list__last__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X3) )
     => ~ ! [Ys4: list(A),X2: A,Zs2: list(A)] :
            ( ( Xs = aa(list(A),list(A),append(A,Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Zs2)) )
           => ( aa(A,$o,P,X2)
             => ~ ! [Xa: A] :
                    ( member(A,Xa,aa(list(A),set(A),set2(A),Zs2))
                   => ~ aa(A,$o,P,Xa) ) ) ) ) ).

% split_list_last_propE
tff(fact_6336_split__list__first__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X3) )
     => ? [Ys4: list(A),X2: A] :
          ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),append(A,Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Zs2))
          & aa(A,$o,P,X2)
          & ! [Xa: A] :
              ( member(A,Xa,aa(list(A),set(A),set2(A),Ys4))
             => ~ aa(A,$o,P,Xa) ) ) ) ).

% split_list_first_prop
tff(fact_6337_split__list__last__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X3) )
     => ? [Ys4: list(A),X2: A,Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),append(A,Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Zs2)) )
          & aa(A,$o,P,X2)
          & ! [Xa: A] :
              ( member(A,Xa,aa(list(A),set(A),set2(A),Zs2))
             => ~ aa(A,$o,P,Xa) ) ) ) ).

% split_list_last_prop
tff(fact_6338_in__set__conv__decomp,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
    <=> ? [Ys3: list(A),Zs3: list(A)] : Xs = aa(list(A),list(A),append(A,Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Zs3)) ) ).

% in_set_conv_decomp
tff(fact_6339_append__Cons__eq__iff,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A),Xs4: list(A),Ys6: list(A)] :
      ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( ~ member(A,Xb,aa(list(A),set(A),set2(A),Ys))
       => ( ( aa(list(A),list(A),append(A,Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys)) = aa(list(A),list(A),append(A,Xs4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Ys6)) )
        <=> ( ( Xs = Xs4 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
tff(fact_6340_split__list__propE,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X3) )
     => ~ ! [Ys4: list(A),X2: A] :
            ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),append(A,Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Zs2))
           => ~ aa(A,$o,P,X2) ) ) ).

% split_list_propE
tff(fact_6341_split__list__first,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ? [Ys4: list(A),Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),append(A,Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Zs2)) )
          & ~ member(A,Xb,aa(list(A),set(A),set2(A),Ys4)) ) ) ).

% split_list_first
tff(fact_6342_split__list__prop,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ? [X3: A] :
          ( member(A,X3,aa(list(A),set(A),set2(A),Xs))
          & aa(A,$o,P,X3) )
     => ? [Ys4: list(A),X2: A] :
          ( ? [Zs2: list(A)] : Xs = aa(list(A),list(A),append(A,Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Zs2))
          & aa(A,$o,P,X2) ) ) ).

% split_list_prop
tff(fact_6343_split__list__last,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ? [Ys4: list(A),Zs2: list(A)] :
          ( ( Xs = aa(list(A),list(A),append(A,Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Zs2)) )
          & ~ member(A,Xb,aa(list(A),set(A),set2(A),Zs2)) ) ) ).

% split_list_last
tff(fact_6344_split__list,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ? [Ys4: list(A),Zs2: list(A)] : Xs = aa(list(A),list(A),append(A,Ys4),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Zs2)) ) ).

% split_list
tff(fact_6345_xy__in__set__cases,axiom,
    ! [A: $tType,Xb: A,L: list(A),Y: A] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),L))
     => ( member(A,Y,aa(list(A),set(A),set2(A),L))
       => ( ( ( Xb = Y )
           => ! [L1: list(A),L22: list(A)] : L != aa(list(A),list(A),append(A,L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),L22)) )
         => ( ( ( Xb != Y )
             => ! [L1: list(A),L22: list(A),L32: list(A)] : L != aa(list(A),list(A),append(A,L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),append(A,L22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),L32)))) )
           => ~ ( ( Xb != Y )
               => ! [L1: list(A),L22: list(A),L32: list(A)] : L != aa(list(A),list(A),append(A,L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),aa(list(A),list(A),append(A,L22),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),L32)))) ) ) ) ) ) ).

% xy_in_set_cases
tff(fact_6346_in__set__list__format,axiom,
    ! [A: $tType,E3: A,L: list(A)] :
      ( member(A,E3,aa(list(A),set(A),set2(A),L))
     => ~ ! [L1: list(A),L22: list(A)] : L != aa(list(A),list(A),append(A,L1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E3),L22)) ) ).

% in_set_list_format
tff(fact_6347_map__consI_I2_J,axiom,
    ! [B: $tType,A: $tType,W: list(A),L: list(A),F: fun(B,A),Ww: list(B),A2: B] :
      ( ( aa(list(A),list(A),append(A,W),L) = aa(list(A),list(A),append(A,aa(list(B),list(A),map(B,A,F),Ww)),L) )
     => ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,F,A2)),aa(list(A),list(A),append(A,W),L)) = aa(list(A),list(A),append(A,aa(list(B),list(A),map(B,A,F),aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),A2),Ww))),L) ) ) ).

% map_consI(2)
tff(fact_6348_replicate__add,axiom,
    ! [A: $tType,Nb: nat,M: nat,Xb: A] : replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M),Xb) = aa(list(A),list(A),append(A,replicate(A,Nb,Xb)),replicate(A,M,Xb)) ).

% replicate_add
tff(fact_6349_enumerate__append__eq,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A),Ys: list(A)] : enumerate(A,Nb,aa(list(A),list(A),append(A,Xs),Ys)) = aa(list(product_prod(nat,A)),list(product_prod(nat,A)),append(product_prod(nat,A),enumerate(A,Nb,Xs)),enumerate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs)),Ys)) ).

% enumerate_append_eq
tff(fact_6350_list__rest__coinc,axiom,
    ! [A: $tType,S22: list(A),S1: list(A),R12: list(A),R23: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),S22)),aa(list(A),nat,size_size(list(A)),S1))
     => ( ( aa(list(A),list(A),append(A,S1),R12) = aa(list(A),list(A),append(A,S22),R23) )
       => ? [R1p: list(A)] : R23 = aa(list(A),list(A),append(A,R1p),R12) ) ) ).

% list_rest_coinc
tff(fact_6351_append__eq__mapE,axiom,
    ! [B: $tType,A: $tType,Fl: list(A),Fl2: list(A),F: fun(B,A),Ls: list(B)] :
      ( ( aa(list(A),list(A),append(A,Fl),Fl2) = aa(list(B),list(A),map(B,A,F),Ls) )
     => ~ ! [L2: list(B),L5: list(B)] :
            ( ( Ls = aa(list(B),list(B),append(B,L2),L5) )
           => ( ( aa(list(B),list(A),map(B,A,F),L2) = Fl )
             => ( aa(list(B),list(A),map(B,A,F),L5) != Fl2 ) ) ) ) ).

% append_eq_mapE
tff(fact_6352_map__eq__appendE,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),Ls: list(B),Fl: list(A),Fl2: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F),Ls) = aa(list(A),list(A),append(A,Fl),Fl2) )
     => ~ ! [L2: list(B),L5: list(B)] :
            ( ( Ls = aa(list(B),list(B),append(B,L2),L5) )
           => ( ( aa(list(B),list(A),map(B,A,F),L2) = Fl )
             => ( aa(list(B),list(A),map(B,A,F),L5) != Fl2 ) ) ) ) ).

% map_eq_appendE
tff(fact_6353_Misc_Oappend__eq__map__conv,axiom,
    ! [B: $tType,A: $tType,Fl: list(A),Fl2: list(A),F: fun(B,A),Ls: list(B)] :
      ( ( aa(list(A),list(A),append(A,Fl),Fl2) = aa(list(B),list(A),map(B,A,F),Ls) )
    <=> ? [L4: list(B),L6: list(B)] :
          ( ( Ls = aa(list(B),list(B),append(B,L4),L6) )
          & ( aa(list(B),list(A),map(B,A,F),L4) = Fl )
          & ( aa(list(B),list(A),map(B,A,F),L6) = Fl2 ) ) ) ).

% Misc.append_eq_map_conv
tff(fact_6354_Misc_Omap__eq__append__conv,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),Ls: list(B),Fl: list(A),Fl2: list(A)] :
      ( ( aa(list(B),list(A),map(B,A,F),Ls) = aa(list(A),list(A),append(A,Fl),Fl2) )
    <=> ? [L4: list(B),L6: list(B)] :
          ( ( Ls = aa(list(B),list(B),append(B,L4),L6) )
          & ( aa(list(B),list(A),map(B,A,F),L4) = Fl )
          & ( aa(list(B),list(A),map(B,A,F),L6) = Fl2 ) ) ) ).

% Misc.map_eq_append_conv
tff(fact_6355_mergesort__by__rel__split_Osimps_I3_J,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A),X1: A,X23: A,Xs: 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)),Xs12),Xs23),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs))) = 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)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),Xs12)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs23)),Xs) ).

% mergesort_by_rel_split.simps(3)
tff(fact_6356_mergesort__by__rel__split_Osimps_I1_J,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: 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)),Xs12),Xs23),nil(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)),Xs12),Xs23) ).

% mergesort_by_rel_split.simps(1)
tff(fact_6357_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),Y5: A,Ys5: list(A)] :
            ( ( X2 != Y5 )
            & ( Xs = aa(list(A),list(A),append(A,Pre),aa(list(A),list(A),append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))),Xs5)) )
            & ( Ys = aa(list(A),list(A),append(A,Pre),aa(list(A),list(A),append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),nil(A))),Ys5)) ) ) ) ) ).

% same_length_different
tff(fact_6358_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,aa(list(A),list(A),append(A,Xs),Ys),I,Xb) = aa(list(A),list(A),append(A,list_update(A,Xs,I,Xb)),Ys) ) ) ).

% list_update_append1
tff(fact_6359_length__Suc__conv__rev,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,Nb) )
    <=> ? [Y3: A,Ys3: list(A)] :
          ( ( Xs = aa(list(A),list(A),append(A,Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = Nb ) ) ) ).

% length_Suc_conv_rev
tff(fact_6360_length__Suc__rev__conv,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,suc,Nb) )
    <=> ? [Ys3: list(A),Y3: A] :
          ( ( Xs = aa(list(A),list(A),append(A,Ys3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),nil(A))) )
          & ( aa(list(A),nat,size_size(list(A)),Ys3) = Nb ) ) ) ).

% length_Suc_rev_conv
tff(fact_6361_length__append__singleton,axiom,
    ! [A: $tType,Xs: list(A),Xb: A] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),append(A,Xs),aa(list(A),list(A),aa(A,fun(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_6362_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,aa(nat,fun(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),$o,P,aa(list(A),list(A),append(A,L2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A)))) )
       => aa(list(A),$o,P,L) ) ) ).

% length_compl_rev_induct
tff(fact_6363_nth__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat] :
      aa(nat,A,nth(A,aa(list(A),list(A),append(A,Xs),Ys)),Nb) = $ite(aa(nat,$o,ord_less(nat,Nb),aa(list(A),nat,size_size(list(A)),Xs)),aa(nat,A,nth(A,Xs),Nb),aa(nat,A,nth(A,Ys),aa(nat,nat,minus_minus(nat,Nb),aa(list(A),nat,size_size(list(A)),Xs)))) ).

% nth_append
tff(fact_6364_replicate__Suc__conv__snoc,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : replicate(A,aa(nat,nat,suc,Nb),Xb) = aa(list(A),list(A),append(A,replicate(A,Nb,Xb)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))) ).

% replicate_Suc_conv_snoc
tff(fact_6365_mergesort__by__rel__split_Oelims,axiom,
    ! [A: $tType,Xb: product_prod(list(A),list(A)),Xaa: list(A),Y: product_prod(list(A),list(A))] :
      ( ( merges295452479951948502_split(A,Xb,Xaa) = Y )
     => ( ! [Xs1: list(A),Xs22: list(A)] :
            ( ( Xb = 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) )
           => ( ( Xaa = nil(A) )
             => ( Y != 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) ) ) )
       => ( ! [Xs1: list(A),Xs22: list(A)] :
              ( ( Xb = 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) )
             => ! [X2: A] :
                  ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) )
                 => ( Y != 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),aa(A,fun(list(A),list(A)),cons(A),X2),Xs1)),Xs22) ) ) )
         => ~ ! [Xs1: list(A),Xs22: list(A)] :
                ( ( Xb = 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) )
               => ! [X15: A,X24: A,Xs2: list(A)] :
                    ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X24),Xs2)) )
                   => ( Y != 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)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),Xs1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X24),Xs22)),Xs2) ) ) ) ) ) ) ).

% mergesort_by_rel_split.elims
tff(fact_6366_mergesort__by__rel__split_Osimps_I2_J,axiom,
    ! [A: $tType,Xs12: list(A),Xs23: list(A),Xb: 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)),Xs12),Xs23),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(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),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs12)),Xs23) ).

% mergesort_by_rel_split.simps(2)
tff(fact_6367_list__update__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),Nb: nat,Xb: A] :
      list_update(A,aa(list(A),list(A),append(A,Xs),Ys),Nb,Xb) = $ite(aa(nat,$o,ord_less(nat,Nb),aa(list(A),nat,size_size(list(A)),Xs)),aa(list(A),list(A),append(A,list_update(A,Xs,Nb,Xb)),Ys),aa(list(A),list(A),append(A,Xs),list_update(A,Ys,aa(nat,nat,minus_minus(nat,Nb),aa(list(A),nat,size_size(list(A)),Xs)),Xb))) ).

% list_update_append
tff(fact_6368_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),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),Ys) = aa(list(product_prod(A,B)),list(product_prod(A,B)),append(product_prod(A,B),aa(list(B),list(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_6369_slice__prepend,axiom,
    ! [A: $tType,I: nat,K: nat,Xs: list(A),Ys: list(A)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K)
     => ( aa(nat,$o,aa(nat,fun(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),aa(list(A),list(A),append(A,Ys),Xs)) ) ) ) ).

% slice_prepend
tff(fact_6370_mergesort__by__rel_Opinduct,axiom,
    ! [A: $tType,A0: fun(A,fun(A,$o)),A1: list(A),P: fun(fun(A,fun(A,$o)),fun(list(A),$o))] :
      ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),A0),A1))
     => ( ! [R7: fun(A,fun(A,$o)),Xs2: list(A)] :
            ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),R7),Xs2))
           => ( ( ~ aa(nat,$o,ord_less(nat,aa(list(A),nat,size_size(list(A)),Xs2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))
               => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P,R7),aa(product_prod(list(A),list(A)),list(A),product_fst(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2))) )
             => ( ( ~ aa(nat,$o,ord_less(nat,aa(list(A),nat,size_size(list(A)),Xs2)),aa(num,nat,numeral_numeral(nat),bit0(one2)))
                 => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P,R7),aa(product_prod(list(A),list(A)),list(A),product_snd(list(A),list(A)),merges295452479951948502_split(A,aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),nil(A)),Xs2))) )
               => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P,R7),Xs2) ) ) )
       => aa(list(A),$o,aa(fun(A,fun(A,$o)),fun(list(A),$o),P,A0),A1) ) ) ).

% mergesort_by_rel.pinduct
tff(fact_6371_quickcheck__narrowing__samples_Onarrowing__samples_Opelims,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),Zero: A,Xb: code_integer,Y: list(A)] :
          ( ( code_T4080844693773952564amples(A,A_of_integer,Zero,Xb) = Y )
         => ( aa(code_integer,$o,accp(code_integer,code_T1710151556404007877es_rel(A,A_of_integer)),Xb)
           => ~ ( ( 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_age(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),aa(A,fun(list(A),list(A)),cons(A),Zero),nil(A))) )
               => ~ aa(code_integer,$o,accp(code_integer,code_T1710151556404007877es_rel(A,A_of_integer)),Xb) ) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.pelims
tff(fact_6372_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_age(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),aa(A,fun(list(A),list(A)),cons(A),Zero),nil(A))) ) ).

% quickcheck_narrowing_samples.narrowing_samples.simps
tff(fact_6373_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_age(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),aa(A,fun(list(A),list(A)),cons(A),Zero),nil(A))) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.elims
tff(fact_6374_quickcheck__narrowing__samples_Onarrowing__samples_Opsimps,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [A_of_integer: fun(code_integer,product_prod(A,A)),I: code_integer,Zero: A] :
          ( aa(code_integer,$o,accp(code_integer,code_T1710151556404007877es_rel(A,A_of_integer)),I)
         => ( 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(A,fun(A,fun(A,list(A))),aa(code_integer,fun(A,fun(A,fun(A,list(A)))),aTP_Lamp_agf(fun(code_integer,product_prod(A,A)),fun(code_integer,fun(A,fun(A,fun(A,list(A))))),A_of_integer),I),Zero)),aa(code_integer,product_prod(A,A),A_of_integer,I)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Zero),nil(A))) ) ) ) ).

% quickcheck_narrowing_samples.narrowing_samples.psimps
tff(fact_6375_mergesort__by__rel__split_Opelims,axiom,
    ! [A: $tType,Xb: product_prod(list(A),list(A)),Xaa: list(A),Y: product_prod(list(A),list(A))] :
      ( ( merges295452479951948502_split(A,Xb,Xaa) = Y )
     => ( aa(product_prod(product_prod(list(A),list(A)),list(A)),$o,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),Xb),Xaa))
       => ( ! [Xs1: list(A),Xs22: list(A)] :
              ( ( Xb = 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) )
             => ( ( Xaa = nil(A) )
               => ( ( Y = 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) )
                 => ~ aa(product_prod(product_prod(list(A),list(A)),list(A)),$o,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),list(A)),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22)),nil(A))) ) ) )
         => ( ! [Xs1: list(A),Xs22: list(A)] :
                ( ( Xb = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xs1),Xs22) )
               => ! [X2: A] :
                    ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)) )
                   => ( ( Y = aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs1)),Xs22) )
                     => ~ aa(product_prod(product_prod(list(A),list(A)),list(A)),$o,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),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)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A)))) ) ) )
           => ~ ! [Xs1: list(A),Xs22: list(A)] :
                  ( ( Xb = 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) )
                 => ! [X15: A,X24: A,Xs2: list(A)] :
                      ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X24),Xs2)) )
                     => ( ( Y = 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)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),Xs1)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X24),Xs22)),Xs2) )
                       => ~ aa(product_prod(product_prod(list(A),list(A)),list(A)),$o,accp(product_prod(product_prod(list(A),list(A)),list(A)),merges7066485432131860899it_rel(A)),aa(list(A),product_prod(product_prod(list(A),list(A)),list(A)),aa(product_prod(list(A),list(A)),fun(list(A),product_prod(product_prod(list(A),list(A)),list(A))),product_Pair(product_prod(list(A),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)),Xs1),Xs22)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X15),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X24),Xs2)))) ) ) ) ) ) ) ) ).

% mergesort_by_rel_split.pelims
tff(fact_6376_mergesort__by__rel_Opsimps,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xs: list(A)] :
      ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),R),Xs))
     => ( aa(list(A),list(A),mergesort_by_rel(A,R),Xs) = $ite(aa(nat,$o,ord_less(nat,aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xs,merges9089515139780605204_merge(A,R,aa(list(A),list(A),mergesort_by_rel(A,R),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)),nil(A)),nil(A)),Xs))),aa(list(A),list(A),mergesort_by_rel(A,R),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)),nil(A)),nil(A)),Xs))))) ) ) ).

% mergesort_by_rel.psimps
tff(fact_6377_mergesort__by__rel__simps_I1_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] : aa(list(A),list(A),mergesort_by_rel(A,R),nil(A)) = nil(A) ).

% mergesort_by_rel_simps(1)
tff(fact_6378_set__mergesort__by__rel,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),mergesort_by_rel(A,R),Xs)) = aa(list(A),set(A),set2(A),Xs) ).

% set_mergesort_by_rel
tff(fact_6379_mergesort__by__rel__merge__simps_I3_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Ys: list(A)] : merges9089515139780605204_merge(A,R,nil(A),Ys) = Ys ).

% mergesort_by_rel_merge_simps(3)
tff(fact_6380_mergesort__by__rel__simps_I2_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xb: A] : aa(list(A),list(A),mergesort_by_rel(A,R),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)) ).

% mergesort_by_rel_simps(2)
tff(fact_6381_mergesort__by__rel__simps_I3_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),X1: A,X23: A,Xs: list(A)] : aa(list(A),list(A),mergesort_by_rel(A,R),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),Xs))) = aa(product_prod(list(A),list(A)),list(A),product_case_prod(list(A),list(A),list(A),aTP_Lamp_agg(fun(A,fun(A,$o)),fun(list(A),fun(list(A),list(A))),R)),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)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X1),nil(A))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X23),nil(A))),Xs)) ).

% mergesort_by_rel_simps(3)
tff(fact_6382_mergesort__by__rel__merge__simps_I2_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xs: list(A)] : merges9089515139780605204_merge(A,R,Xs,nil(A)) = Xs ).

% mergesort_by_rel_merge_simps(2)
tff(fact_6383_mergesort__by__rel__merge__simps_I1_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xb: A,Xs: list(A),Y: A,Ys: list(A)] :
      merges9089515139780605204_merge(A,R,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)) = $ite(aa(A,$o,aa(A,fun(A,$o),R,Xb),Y),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),merges9089515139780605204_merge(A,R,Xs,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),merges9089515139780605204_merge(A,R,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),Ys))) ).

% mergesort_by_rel_merge_simps(1)
tff(fact_6384_mergesort__by__rel__merge_Oelims,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,$o)),Xaa: list(A),Xba: list(A),Y: list(A)] :
      ( ( merges9089515139780605204_merge(A,Xb,Xaa,Xba) = Y )
     => ( ! [X2: A,Xs2: list(A)] :
            ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
           => ! [Y5: A,Ys4: list(A)] :
                ( ( Xba = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) )
               => ( Y != $ite(aa(A,$o,aa(A,fun(A,$o),Xb,X2),Y5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),merges9089515139780605204_merge(A,Xb,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),merges9089515139780605204_merge(A,Xb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2),Ys4))) ) ) )
       => ( ( ( Xba = nil(A) )
           => ( Y != Xaa ) )
         => ~ ( ( Xaa = nil(A) )
             => ! [V3: A,Va2: list(A)] :
                  ( ( Xba = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va2) )
                 => ( Y != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va2) ) ) ) ) ) ) ).

% mergesort_by_rel_merge.elims
tff(fact_6385_mergesort__by__rel__merge_Osimps_I3_J,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),V: A,Vaa: list(A)] : merges9089515139780605204_merge(A,R,nil(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Vaa)) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V),Vaa) ).

% mergesort_by_rel_merge.simps(3)
tff(fact_6386_mergesort__by__rel_Osimps,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xs: list(A)] :
      aa(list(A),list(A),mergesort_by_rel(A,R),Xs) = $ite(aa(nat,$o,ord_less(nat,aa(list(A),nat,size_size(list(A)),Xs)),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xs,merges9089515139780605204_merge(A,R,aa(list(A),list(A),mergesort_by_rel(A,R),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)),nil(A)),nil(A)),Xs))),aa(list(A),list(A),mergesort_by_rel(A,R),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)),nil(A)),nil(A)),Xs))))) ).

% mergesort_by_rel.simps
tff(fact_6387_mergesort__by__rel_Oelims,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,$o)),Xaa: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),mergesort_by_rel(A,Xb),Xaa) = Y )
     => ( Y = $ite(aa(nat,$o,ord_less(nat,aa(list(A),nat,size_size(list(A)),Xaa)),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xaa,merges9089515139780605204_merge(A,Xb,aa(list(A),list(A),mergesort_by_rel(A,Xb),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)),nil(A)),nil(A)),Xaa))),aa(list(A),list(A),mergesort_by_rel(A,Xb),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)),nil(A)),nil(A)),Xaa))))) ) ) ).

% mergesort_by_rel.elims
tff(fact_6388_mergesort__by__rel_Opelims,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,$o)),Xaa: list(A),Y: list(A)] :
      ( ( aa(list(A),list(A),mergesort_by_rel(A,Xb),Xaa) = Y )
     => ( aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),Xb),Xaa))
       => ~ ( ( Y = $ite(aa(nat,$o,ord_less(nat,aa(list(A),nat,size_size(list(A)),Xaa)),aa(num,nat,numeral_numeral(nat),bit0(one2))),Xaa,merges9089515139780605204_merge(A,Xb,aa(list(A),list(A),mergesort_by_rel(A,Xb),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)),nil(A)),nil(A)),Xaa))),aa(list(A),list(A),mergesort_by_rel(A,Xb),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)),nil(A)),nil(A)),Xaa))))) )
           => ~ aa(product_prod(fun(A,fun(A,$o)),list(A)),$o,accp(product_prod(fun(A,fun(A,$o)),list(A)),mergesort_by_rel_rel(A)),aa(list(A),product_prod(fun(A,fun(A,$o)),list(A)),aa(fun(A,fun(A,$o)),fun(list(A),product_prod(fun(A,fun(A,$o)),list(A))),product_Pair(fun(A,fun(A,$o)),list(A)),Xb),Xaa)) ) ) ) ).

% mergesort_by_rel.pelims
tff(fact_6389_mergesort__by__rel__merge_Opelims,axiom,
    ! [A: $tType,Xb: fun(A,fun(A,$o)),Xaa: list(A),Xba: list(A),Y: list(A)] :
      ( ( merges9089515139780605204_merge(A,Xb,Xaa,Xba) = Y )
     => ( aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),Xb),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)),Xaa),Xba)))
       => ( ! [X2: A,Xs2: list(A)] :
              ( ( Xaa = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
             => ! [Y5: A,Ys4: list(A)] :
                  ( ( Xba = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4) )
                 => ( ( Y = $ite(aa(A,$o,aa(A,fun(A,$o),Xb,X2),Y5),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),merges9089515139780605204_merge(A,Xb,Xs2,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),merges9089515139780605204_merge(A,Xb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2),Ys4))) )
                   => ~ aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),Xb),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),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y5),Ys4)))) ) ) )
         => ( ( ( Xba = nil(A) )
             => ( ( Y = Xaa )
               => ~ aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),Xb),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xaa),nil(A)))) ) )
           => ~ ( ( Xaa = nil(A) )
               => ! [V3: A,Va2: list(A)] :
                    ( ( Xba = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va2) )
                   => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va2) )
                     => ~ aa(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),$o,accp(product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),merges2244889521215249637ge_rel(A)),aa(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A))),aa(fun(A,fun(A,$o)),fun(product_prod(list(A),list(A)),product_prod(fun(A,fun(A,$o)),product_prod(list(A),list(A)))),product_Pair(fun(A,fun(A,$o)),product_prod(list(A),list(A))),Xb),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)),nil(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),V3),Va2)))) ) ) ) ) ) ) ) ).

% mergesort_by_rel_merge.pelims
tff(fact_6390_dup__1,axiom,
    code_dup(one_one(code_integer)) = aa(num,code_integer,numeral_numeral(code_integer),bit0(one2)) ).

% dup_1
tff(fact_6391_dup_Oabs__eq,axiom,
    ! [Xb: int] : code_dup(code_integer_of_int(Xb)) = code_integer_of_int(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),Xb)) ).

% dup.abs_eq
tff(fact_6392_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_6393_mergesort__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ( mergesort(A) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).

% mergesort_def
tff(fact_6394_pow_Osimps_I1_J,axiom,
    ! [Xb: num] : pow(Xb,one2) = Xb ).

% pow.simps(1)
tff(fact_6395_timeFrame_Oelims,axiom,
    ! [A: $tType,Xb: nat,Xaa: option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Y: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
      ( ( heap_Time_timeFrame(A,Xb,Xaa) = Y )
     => ( ! [R2: A,H2: heap_ext(product_unit),N5: nat] :
            ( ( Xaa = 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),N5))) )
           => ( Y != 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),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xb),N5)))) ) )
       => ~ ( ( Xaa = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
           => ( Y != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ) ) ).

% timeFrame.elims
tff(fact_6396_return__def,axiom,
    ! [A: $tType,Xb: A] : aa(A,heap_Time_Heap(A),heap_Time_return(A),Xb) = heap_Time_heap(A,aTP_Lamp_agh(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Xb)) ).

% return_def
tff(fact_6397_timeFrame__assoc,axiom,
    ! [A: $tType,Nb: nat,N3: nat,F: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_timeFrame(A,Nb,heap_Time_timeFrame(A,N3,F)) = heap_Time_timeFrame(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),N3),F) ).

% timeFrame_assoc
tff(fact_6398_timeFrame_Osimps_I2_J,axiom,
    ! [A: $tType,Nb: nat] : heap_Time_timeFrame(A,Nb,none(product_prod(A,product_prod(heap_ext(product_unit),nat)))) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ).

% timeFrame.simps(2)
tff(fact_6399_effect__heapE,axiom,
    ! [A: $tType,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: A,Nb: nat] :
      ( heap_Time_effect(A,heap_Time_heap(A,F),H,H3,R3,Nb)
     => ~ ( ( H3 = 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)),F,H))) )
         => ( ( Nb = 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)),F,H))) )
           => ( R3 != 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)),F,H)) ) ) ) ) ).

% effect_heapE
tff(fact_6400_effect__heapI,axiom,
    ! [A: $tType,Nb: nat,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: A] :
      ( ( Nb = 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)),F,H))) )
     => ( ( H3 = 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)),F,H))) )
       => ( ( R3 = 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)),F,H)) )
         => heap_Time_effect(A,heap_Time_heap(A,F),H,H3,R3,Nb) ) ) ) ).

% effect_heapI
tff(fact_6401_timeFrame_Osimps_I1_J,axiom,
    ! [A: $tType,Nb: nat,R3: A,H: heap_ext(product_unit),N3: nat] : heap_Time_timeFrame(A,Nb,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)),R3),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),N3)))) = 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)),R3),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),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),N3)))) ).

% timeFrame.simps(1)
tff(fact_6402_timeFrame_Opelims,axiom,
    ! [A: $tType,Xb: nat,Xaa: option(product_prod(A,product_prod(heap_ext(product_unit),nat))),Y: option(product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
      ( ( heap_Time_timeFrame(A,Xb,Xaa) = Y )
     => ( aa(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),$o,accp(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(A)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Xb),Xaa))
       => ( ! [R2: A,H2: heap_ext(product_unit),N5: nat] :
              ( ( Xaa = 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),N5))) )
             => ( ( Y = 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),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Xb),N5)))) )
               => ~ aa(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),$o,accp(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(A)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Xb),aa(product_prod(A,product_prod(heap_ext(product_unit),nat)),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),some(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat)),aa(A,fun(product_prod(heap_ext(product_unit),nat),product_prod(A,product_prod(heap_ext(product_unit),nat))),product_Pair(A,product_prod(heap_ext(product_unit),nat)),R2),aa(nat,product_prod(heap_ext(product_unit),nat),aa(heap_ext(product_unit),fun(nat,product_prod(heap_ext(product_unit),nat)),product_Pair(heap_ext(product_unit),nat),H2),N5))))) ) )
         => ~ ( ( Xaa = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
             => ( ( Y = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
               => ~ aa(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),$o,accp(product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_T5500966940807335491me_rel(A)),aa(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),aa(nat,fun(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),product_prod(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat))))),product_Pair(nat,option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),Xb),none(product_prod(A,product_prod(heap_ext(product_unit),nat))))) ) ) ) ) ) ).

% timeFrame.pelims
tff(fact_6403_ureturn__def,axiom,
    ! [A: $tType,Xb: A] : aa(A,heap_Time_Heap(A),heap_Time_ureturn(A),Xb) = heap_Time_heap(A,aTP_Lamp_agi(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Xb)) ).

% ureturn_def
tff(fact_6404_bind__ureturn,axiom,
    ! [A: $tType,F: heap_Time_Heap(A)] : heap_Time_bind(A,A,F,heap_Time_ureturn(A)) = F ).

% bind_ureturn
tff(fact_6405_ureturn__bind,axiom,
    ! [A: $tType,B: $tType,Xb: B,F: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,aa(B,heap_Time_Heap(B),heap_Time_ureturn(B),Xb),F) = aa(B,heap_Time_Heap(A),F,Xb) ).

% ureturn_bind
tff(fact_6406_effect__guardI,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),$o),H: heap_ext(product_unit),H3: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Nb: nat,R3: A] :
      ( aa(heap_ext(product_unit),$o,P,H)
     => ( ( H3 = 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)),F,H))) )
       => ( ( Nb = 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)),F,H))) )
         => ( ( R3 = 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)),F,H)) )
           => heap_Time_effect(A,heap_Time_guard(A,P,F),H,H3,R3,Nb) ) ) ) ) ).

% effect_guardI
tff(fact_6407_effect__guardE,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),$o),F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: A,Nb: nat] :
      ( heap_Time_effect(A,heap_Time_guard(A,P,F),H,H3,R3,Nb)
     => ~ ( ( H3 = 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)),F,H))) )
         => ( ( Nb = 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)),F,H))) )
           => ( ( R3 = 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)),F,H)) )
             => ~ aa(heap_ext(product_unit),$o,P,H) ) ) ) ) ).

% effect_guardE
tff(fact_6408_guard__def,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),$o),F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_guard(A,P,F) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),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_agj(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),F)) ).

% guard_def
tff(fact_6409_len_H__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( semiring_1(A)
        & heap(B) )
     => ! [A2: array(B)] : array_len2(B,A,A2) = heap_Time_bind(nat,A,array_len(B,A2),aTP_Lamp_agk(nat,heap_Time_Heap(A))) ) ).

% len'_def
tff(fact_6410_cis__multiple__2pi,axiom,
    ! [Nb: real] :
      ( member(real,Nb,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)),Nb)) = one_one(complex) ) ) ).

% cis_multiple_2pi
tff(fact_6411_floor__add2,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,Y: A] :
          ( ( member(A,Xb,ring_1_Ints(A))
            | member(A,Y,ring_1_Ints(A)) )
         => ( archim6421214686448440834_floor(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),archim6421214686448440834_floor(A,Xb)),archim6421214686448440834_floor(A,Y)) ) ) ) ).

% floor_add2
tff(fact_6412_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_6413_Ints__add,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( member(A,B2,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2),ring_1_Ints(A)) ) ) ) ).

% Ints_add
tff(fact_6414_Ints__double__eq__0__iff,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),A2) = zero_zero(A) )
          <=> ( A2 = zero_zero(A) ) ) ) ) ).

% Ints_double_eq_0_iff
tff(fact_6415_Ints__mult,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( member(A,B2,ring_1_Ints(A))
           => member(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2),ring_1_Ints(A)) ) ) ) ).

% Ints_mult
tff(fact_6416_Ints__numeral,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Nb: num] : member(A,aa(num,A,numeral_numeral(A),Nb),ring_1_Ints(A)) ) ).

% Ints_numeral
tff(fact_6417_Ints__diff,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,B2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( member(A,B2,ring_1_Ints(A))
           => member(A,aa(A,A,minus_minus(A,A2),B2),ring_1_Ints(A)) ) ) ) ).

% Ints_diff
tff(fact_6418_Ints__power,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [A2: A,Nb: nat] :
          ( member(A,A2,ring_1_Ints(A))
         => member(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb),ring_1_Ints(A)) ) ) ).

% Ints_power
tff(fact_6419_Ints__1,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => member(A,one_one(A),ring_1_Ints(A)) ) ).

% Ints_1
tff(fact_6420_finite__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A,B2: A] : finite_finite2(A,collect(A,aa(A,fun(A,$o),aTP_Lamp_agl(A,fun(A,fun(A,$o)),A2),B2))) ) ).

% finite_int_segment
tff(fact_6421_Ints__odd__nonzero,axiom,
    ! [A: $tType] :
      ( ring_char_0(A)
     => ! [A2: A] :
          ( member(A,A2,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)),A2)),A2) != zero_zero(A) ) ) ) ).

% Ints_odd_nonzero
tff(fact_6422_of__int__divide__in__Ints,axiom,
    ! [A: $tType] :
      ( idom_divide(A)
     => ! [B2: int,A2: int] :
          ( aa(int,$o,dvd_dvd(int,B2),A2)
         => member(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),aa(int,A,ring_1_of_int(A),A2)),aa(int,A,ring_1_of_int(A),B2)),ring_1_Ints(A)) ) ) ).

% of_int_divide_in_Ints
tff(fact_6423_finite__abs__int__segment,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [A2: A] : finite_finite2(A,collect(A,aTP_Lamp_agm(A,fun(A,$o),A2))) ) ).

% finite_abs_int_segment
tff(fact_6424_Ints__odd__less__0,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [A2: A] :
          ( member(A,A2,ring_1_Ints(A))
         => ( aa(A,$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)),A2)),A2)),zero_zero(A))
          <=> aa(A,$o,ord_less(A,A2),zero_zero(A)) ) ) ) ).

% Ints_odd_less_0
tff(fact_6425_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,aa(A,fun(A,$o),ord_less_eq(A),one_one(A)),aa(A,A,abs_abs(A),Xb)) ) ) ) ).

% Ints_nonzero_abs_ge1
tff(fact_6426_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,aa(A,A,abs_abs(A),Xb)),one_one(A))
           => ( Xb = zero_zero(A) ) ) ) ) ).

% Ints_nonzero_abs_less1
tff(fact_6427_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,aa(A,A,abs_abs(A),aa(A,A,minus_minus(A,Xb),Y))),one_one(A)) ) ) ) ) ).

% Ints_eq_abs_less1
tff(fact_6428_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_6429_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_6430_upd__ureturn,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xb: A,A2: array(A)] : heap_Time_bind(array(A),array(A),array_upd(A,I,Xb,A2),aTP_Lamp_agn(array(A),fun(array(A),heap_Time_Heap(array(A))),A2)) = array_upd(A,I,Xb,A2) ) ).

% upd_ureturn
tff(fact_6431_le__mult__floor__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( member(A,A2,ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(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,A2)),archim6421214686448440834_floor(A,B2)))),aa(int,B,ring_1_of_int(B),archim6421214686448440834_floor(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)))) ) ) ) ).

% le_mult_floor_Ints
tff(fact_6432_mult__ceiling__le__Ints,axiom,
    ! [B: $tType,A: $tType] :
      ( ( archim2362893244070406136eiling(A)
        & linordered_idom(B) )
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
         => ( member(A,A2,ring_1_Ints(A))
           => aa(B,$o,aa(B,fun(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),A2),B2)))),aa(int,B,ring_1_of_int(B),aa(int,int,aa(int,fun(int,int),times_times(int),archimedean_ceiling(A,A2)),archimedean_ceiling(A,B2)))) ) ) ) ).

% mult_ceiling_le_Ints
tff(fact_6433_frac__unique__iff,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Xb: A,A2: A] :
          ( ( archimedean_frac(A,Xb) = A2 )
        <=> ( member(A,aa(A,A,minus_minus(A,Xb),A2),ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),zero_zero(A)),A2)
            & aa(A,$o,ord_less(A,A2),one_one(A)) ) ) ) ).

% frac_unique_iff
tff(fact_6434_sin__integer__2pi,axiom,
    ! [Nb: real] :
      ( member(real,Nb,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)),Nb)) = zero_zero(real) ) ) ).

% sin_integer_2pi
tff(fact_6435_cos__integer__2pi,axiom,
    ! [Nb: real] :
      ( member(real,Nb,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)),Nb)) = one_one(real) ) ) ).

% cos_integer_2pi
tff(fact_6436_ran__nth__set__encoding__conv,axiom,
    ! [A: $tType,L: list(A)] : ran(nat,A,aTP_Lamp_ago(list(A),fun(nat,option(A)),L)) = aa(list(A),set(A),set2(A),L) ).

% ran_nth_set_encoding_conv
tff(fact_6437_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_agp(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_inverse_code
tff(fact_6438_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_6439_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_6440_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_agq(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_6441_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_6442_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_6443_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_6444_divide__rat__def,axiom,
    ! [Q2: rat,R3: rat] : aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),Q2),R3) = aa(rat,rat,aa(rat,fun(rat,rat),times_times(rat),Q2),aa(rat,rat,inverse_inverse(rat),R3)) ).

% divide_rat_def
tff(fact_6445_quotient__of__div,axiom,
    ! [R3: rat,Nb: int,D2: int] :
      ( ( quotient_of(R3) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),Nb),D2) )
     => ( R3 = aa(rat,rat,aa(rat,fun(rat,rat),divide_divide(rat),aa(int,rat,ring_1_of_int(rat),Nb)),aa(int,rat,ring_1_of_int(rat),D2)) ) ) ).

% quotient_of_div
tff(fact_6446_quotient__of__denom__pos,axiom,
    ! [R3: rat,P2: int,Q2: int] :
      ( ( quotient_of(R3) = 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_6447_quotient__of__denom__pos_H,axiom,
    ! [R3: rat] : aa(int,$o,ord_less(int,zero_zero(int)),aa(product_prod(int,int),int,product_snd(int,int),quotient_of(R3))) ).

% quotient_of_denom_pos'
tff(fact_6448_rat__abs__code,axiom,
    ! [P2: rat] : quotient_of(aa(rat,rat,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_6449_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_ags(int,fun(int,product_prod(int,int)))),quotient_of(P2)) ).

% rat_uminus_code
tff(fact_6450_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,$o,aTP_Lamp_agu(rat,fun(int,fun(int,$o)),Q2)),quotient_of(P2)) ) ).

% rat_less_code
tff(fact_6451_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_6452_rat__less__eq__code,axiom,
    ! [P2: rat,Q2: rat] :
      ( aa(rat,$o,aa(rat,fun(rat,$o),ord_less_eq(rat),P2),Q2)
    <=> aa(product_prod(int,int),$o,product_case_prod(int,int,$o,aTP_Lamp_agw(rat,fun(int,fun(int,$o)),Q2)),quotient_of(P2)) ) ).

% rat_less_eq_code
tff(fact_6453_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_6454_ran__empty,axiom,
    ! [B: $tType,A: $tType] : ran(B,A,aTP_Lamp_agx(B,option(A))) = bot_bot(set(A)) ).

% ran_empty
tff(fact_6455_quotient__of__int,axiom,
    ! [A2: int] : quotient_of(of_int(A2)) = aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),one_one(int)) ).

% quotient_of_int
tff(fact_6456_ranI,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A)),A2: B,B2: A] :
      ( ( aa(B,option(A),M,A2) = aa(A,option(A),some(A),B2) )
     => member(A,B2,ran(B,A,M)) ) ).

% ranI
tff(fact_6457_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_6458_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_agz(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_minus_code
tff(fact_6459_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_6460_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_6461_normalize__denom__pos,axiom,
    ! [R3: product_prod(int,int),P2: int,Q2: int] :
      ( ( normalize(R3) = 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_6462_normalize__crossproduct,axiom,
    ! [Q2: int,S3: int,P2: int,R3: int] :
      ( ( Q2 != zero_zero(int) )
     => ( ( S3 != 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),R3),S3)) )
         => ( aa(int,int,aa(int,fun(int,int),times_times(int),P2),S3) = aa(int,int,aa(int,fun(int,int),times_times(int),R3),Q2) ) ) ) ) ).

% normalize_crossproduct
tff(fact_6463_Frct__code__post_I2_J,axiom,
    ! [A2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),zero_zero(int))) = zero_zero(rat) ).

% Frct_code_post(2)
tff(fact_6464_Frct__code__post_I1_J,axiom,
    ! [A2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),zero_zero(int)),A2)) = zero_zero(rat) ).

% Frct_code_post(1)
tff(fact_6465_Frct__code__post_I7_J,axiom,
    ! [A2: int,B2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),aa(int,int,uminus_uminus(int),A2)),B2)) = aa(rat,rat,uminus_uminus(rat),frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),B2))) ).

% Frct_code_post(7)
tff(fact_6466_Frct__code__post_I8_J,axiom,
    ! [A2: int,B2: int] : frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),aa(int,int,uminus_uminus(int),B2))) = aa(rat,rat,uminus_uminus(rat),frct(aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),product_Pair(int,int),A2),B2))) ).

% Frct_code_post(8)
tff(fact_6467_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_6468_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_6469_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_6470_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_ahb(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_times_code
tff(fact_6471_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_ahd(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_divide_code
tff(fact_6472_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_ahf(rat,fun(int,fun(int,product_prod(int,int))),Q2)),quotient_of(P2)) ).

% rat_plus_code
tff(fact_6473_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_6474_setceilmax,axiom,
    ! [S3: vEBT_VEBT,M: nat,Listy: list(vEBT_VEBT),Nb: nat] :
      ( vEBT_invar_vebt(S3,M)
     => ( ! [X2: vEBT_VEBT] :
            ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))
           => vEBT_invar_vebt(X2,Nb) )
       => ( ( M = aa(nat,nat,suc,Nb) )
         => ( ! [X2: vEBT_VEBT] :
                ( member(vEBT_VEBT,X2,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))
               => ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X2)) = archimedean_ceiling(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),Nb))) ) )
           => ( ( aa(nat,int,semiring_1_of_nat(int),aa(vEBT_VEBT,nat,vEBT_VEBT_height,S3)) = archimedean_ceiling(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,S3),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Listy))))) = archimedean_ceiling(real,log(aa(num,real,numeral_numeral(real),bit0(one2)),aa(nat,real,semiring_1_of_nat(real),M))) ) ) ) ) ) ) ).

% setceilmax
tff(fact_6475_image__ident,axiom,
    ! [A: $tType,Y7: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_dz(A,A)),Y7) = Y7 ).

% image_ident
tff(fact_6476_height__compose__list,axiom,
    ! [Ta: vEBT_VEBT,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] :
      ( member(vEBT_VEBT,Ta,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,Ta)),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,Summarya),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))))) ) ).

% height_compose_list
tff(fact_6477_gcd__add1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,Nb: A] : gcd_gcd(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),M),Nb),Nb) = gcd_gcd(A,M,Nb) ) ).

% gcd_add1
tff(fact_6478_gcd__add2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,Nb: A] : gcd_gcd(A,M,aa(A,A,aa(A,fun(A,A),plus_plus(A),M),Nb)) = gcd_gcd(A,M,Nb) ) ).

% gcd_add2
tff(fact_6479_gcd_Obottom__left__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : gcd_gcd(A,one_one(A),A2) = one_one(A) ) ).

% gcd.bottom_left_bottom
tff(fact_6480_gcd_Obottom__right__bottom,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A] : gcd_gcd(A,A2,one_one(A)) = one_one(A) ) ).

% gcd.bottom_right_bottom
tff(fact_6481_gcd__exp,axiom,
    ! [A: $tType] :
      ( semiri6843258321239162965malize(A)
     => ! [A2: A,Nb: nat,B2: A] : gcd_gcd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),Nb),aa(nat,A,aa(A,fun(nat,A),power_power(A),B2),Nb)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),gcd_gcd(A,A2,B2)),Nb) ) ).

% gcd_exp
tff(fact_6482_gcd__1__int,axiom,
    ! [M: int] : gcd_gcd(int,M,one_one(int)) = one_one(int) ).

% gcd_1_int
tff(fact_6483_max__ins__scaled,axiom,
    ! [Nb: nat,X14a: vEBT_VEBT,M: nat,X13a: list(vEBT_VEBT)] : aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14a))),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),lattic643756798349783984er_Max(nat,aa(set(nat),set(nat),insert(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14a)),aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13a))))))) ).

% max_ins_scaled
tff(fact_6484_height__i__max,axiom,
    ! [I: nat,X13a: list(vEBT_VEBT),Foo: nat] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13a))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13a),I))),aa(nat,nat,ord_max(nat,Foo),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13a))))) ) ).

% height_i_max
tff(fact_6485_image__add__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [S2: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A))),S2) = S2 ) ).

% image_add_0
tff(fact_6486_surj__plus,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_plus
tff(fact_6487_range__add,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% range_add
tff(fact_6488_range__diff,axiom,
    ! [A: $tType] :
      ( group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,minus_minus(A,A2)),top_top(set(A))) = top_top(set(A)) ) ).

% range_diff
tff(fact_6489_image__add__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(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_6490_image__diff__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D2: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,minus_minus(A,D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,D2),B2),aa(A,A,minus_minus(A,D2),A2)) ) ).

% image_diff_atLeastAtMost
tff(fact_6491_image__add__atLeastLessThan,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [K: A,I: A,J: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(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_6492_gcd__neg__numeral__1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Nb: num,A2: A] : gcd_gcd(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb)),A2) = gcd_gcd(A,aa(num,A,numeral_numeral(A),Nb),A2) ) ).

% gcd_neg_numeral_1
tff(fact_6493_gcd__neg__numeral__2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [A2: A,Nb: num] : gcd_gcd(A,A2,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),Nb))) = gcd_gcd(A,A2,aa(num,A,numeral_numeral(A),Nb)) ) ).

% gcd_neg_numeral_2
tff(fact_6494_list_Oset__map,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),V: list(B)] : aa(list(A),set(A),set2(A),aa(list(B),list(A),map(B,A,F),V)) = aa(set(B),set(A),image(B,A,F),aa(list(B),set(B),set2(B),V)) ).

% list.set_map
tff(fact_6495_is__unit__gcd__iff,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,dvd_dvd(A,gcd_gcd(A,A2,B2)),one_one(A))
        <=> ( gcd_gcd(A,A2,B2) = one_one(A) ) ) ) ).

% is_unit_gcd_iff
tff(fact_6496_image__add__atMost,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [C2: A,A2: A] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),C2)),set_ord_atMost(A,A2)) = set_ord_atMost(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),C2),A2)) ) ).

% image_add_atMost
tff(fact_6497_bij__betw__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A),B5: set(A)] :
          ( bij_betw(A,A,aa(A,fun(A,A),plus_plus(A),A2),A3,B5)
        <=> ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),A3) = B5 ) ) ) ).

% bij_betw_add
tff(fact_6498_max__idx__list,axiom,
    ! [I: nat,X13a: list(vEBT_VEBT),Nb: nat,X14a: vEBT_VEBT] :
      ( aa(nat,$o,ord_less(nat,I),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),X13a))
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(vEBT_VEBT,nat,vEBT_VEBT_height,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,X13a),I)))),aa(nat,nat,suc,aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Nb),aa(nat,nat,ord_max(nat,aa(vEBT_VEBT,nat,vEBT_VEBT_height,X14a)),lattic643756798349783984er_Max(nat,aa(set(vEBT_VEBT),set(nat),image(vEBT_VEBT,nat,vEBT_VEBT_height),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),X13a)))))))) ) ).

% max_idx_list
tff(fact_6499_gcd__neg__numeral__2__int,axiom,
    ! [Xb: int,Nb: num] : gcd_gcd(int,Xb,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = gcd_gcd(int,Xb,aa(num,int,numeral_numeral(int),Nb)) ).

% gcd_neg_numeral_2_int
tff(fact_6500_gcd__neg__numeral__1__int,axiom,
    ! [Nb: num,Xb: int] : gcd_gcd(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb)),Xb) = gcd_gcd(int,aa(num,int,numeral_numeral(int),Nb),Xb) ).

% gcd_neg_numeral_1_int
tff(fact_6501_surj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_ahg(A,fun(A,A),A2)),top_top(set(A))) = top_top(set(A)) ) ).

% surj_diff_right
tff(fact_6502_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_ahh(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_6503_image__minus__const__atLeastAtMost_H,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [D2: A,A2: A,B2: A] : aa(set(A),set(A),image(A,A,aTP_Lamp_ahi(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,A2),D2),aa(A,A,minus_minus(A,B2),D2)) ) ).

% image_minus_const_atLeastAtMost'
tff(fact_6504_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_ahh(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_6505_Max__divisors__self__nat,axiom,
    ! [Nb: nat] :
      ( ( Nb != zero_zero(nat) )
     => ( lattic643756798349783984er_Max(nat,collect(nat,aTP_Lamp_sn(nat,fun(nat,$o),Nb))) = Nb ) ) ).

% Max_divisors_self_nat
tff(fact_6506_Max_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,A3)),Xb)
            <=> ! [X: A] :
                  ( member(A,X,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),Xb) ) ) ) ) ) ).

% Max.bounded_iff
tff(fact_6507_Max__less__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,ord_less(A,lattic643756798349783984er_Max(A,A3)),Xb)
            <=> ! [X: A] :
                  ( member(A,X,A3)
                 => aa(A,$o,ord_less(A,X),Xb) ) ) ) ) ) ).

% Max_less_iff
tff(fact_6508_range__constant,axiom,
    ! [B: $tType,A: $tType,Xb: A] : aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_eq(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_6509_Max__const,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [A3: set(A),C2: B] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( lattic643756798349783984er_Max(B,aa(set(A),set(B),image(A,B,aTP_Lamp_ahj(B,fun(A,B),C2)),A3)) = C2 ) ) ) ) ).

% Max_const
tff(fact_6510_image__mult__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),D2)
         => ( aa(set(A),set(A),image(A,A,aa(A,fun(A,A),times_times(A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),times_times(A),D2),A2),aa(A,A,aa(A,fun(A,A),times_times(A),D2),B2)) ) ) ) ).

% image_mult_atLeastAtMost
tff(fact_6511_image__divide__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [D2: A,A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,zero_zero(A)),D2)
         => ( aa(set(A),set(A),image(A,A,aTP_Lamp_ahk(A,fun(A,A),D2)),set_or1337092689740270186AtMost(A,A2,B2)) = set_or1337092689740270186AtMost(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),D2),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),D2)) ) ) ) ).

% image_divide_atLeastAtMost
tff(fact_6512_translation__subtract__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,Ta: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_ahg(A,fun(A,A),A2)),aa(set(A),set(A),uminus_uminus(set(A)),Ta)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image(A,A,aTP_Lamp_ahg(A,fun(A,A),A2)),Ta)) ) ).

% translation_subtract_Compl
tff(fact_6513_translation__Compl,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,Ta: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),aa(set(A),set(A),uminus_uminus(set(A)),Ta)) = aa(set(A),set(A),uminus_uminus(set(A)),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),Ta)) ) ).

% translation_Compl
tff(fact_6514_Max__add__commute,axiom,
    ! [A: $tType,B: $tType] :
      ( linord4140545234300271783up_add(B)
     => ! [S2: set(A),F: fun(A,B),K: B] :
          ( finite_finite2(A,S2)
         => ( ( S2 != bot_bot(set(A)) )
           => ( lattic643756798349783984er_Max(B,aa(set(A),set(B),image(A,B,aa(B,fun(A,B),aTP_Lamp_ahl(fun(A,B),fun(B,fun(A,B)),F),K)),S2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),lattic643756798349783984er_Max(B,aa(set(A),set(B),image(A,B,F),S2))),K) ) ) ) ) ).

% Max_add_commute
tff(fact_6515_translation__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S3: set(A),Ta: set(A)] : aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),aa(set(A),set(A),minus_minus(set(A),S3),Ta)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),S3)),aa(set(A),set(A),image(A,A,aa(A,fun(A,A),plus_plus(A),A2)),Ta)) ) ).

% translation_diff
tff(fact_6516_prod_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [S2: set(A),H: fun(A,B),G: fun(A,C)] :
          ( finite_finite2(A,S2)
         => ( groups7121269368397514597t_prod(A,B,H,S2) = groups7121269368397514597t_prod(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ahn(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S2),H),G),aa(set(A),set(C),image(A,C,G),S2)) ) ) ) ).

% prod.image_gen
tff(fact_6517_gcd__diff1,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [M: A,Nb: A] : gcd_gcd(A,aa(A,A,minus_minus(A,M),Nb),Nb) = gcd_gcd(A,M,Nb) ) ).

% gcd_diff1
tff(fact_6518_gcd__diff2,axiom,
    ! [A: $tType] :
      ( ring_gcd(A)
     => ! [Nb: A,M: A] : gcd_gcd(A,aa(A,A,minus_minus(A,Nb),M),Nb) = gcd_gcd(A,M,Nb) ) ).

% gcd_diff2
tff(fact_6519_image__diff__subset,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: set(B),B5: set(B)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),aa(set(B),set(A),image(B,A,F),A3)),aa(set(B),set(A),image(B,A,F),B5))),aa(set(B),set(A),image(B,A,F),aa(set(B),set(B),minus_minus(set(B),A3),B5))) ).

% image_diff_subset
tff(fact_6520_Max_OcoboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( finite_finite2(A,A3)
         => ( member(A,A2,A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),lattic643756798349783984er_Max(A,A3)) ) ) ) ).

% Max.coboundedI
tff(fact_6521_Max__eq__if,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B5: set(A)] :
          ( finite_finite2(A,A3)
         => ( finite_finite2(A,B5)
           => ( ! [X2: A] :
                  ( member(A,X2,A3)
                 => ? [Xa: A] :
                      ( member(A,Xa,B5)
                      & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xa) ) )
             => ( ! [X2: A] :
                    ( member(A,X2,B5)
                   => ? [Xa: A] :
                        ( member(A,Xa,A3)
                        & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Xa) ) )
               => ( lattic643756798349783984er_Max(A,A3) = lattic643756798349783984er_Max(A,B5) ) ) ) ) ) ) ).

% Max_eq_if
tff(fact_6522_Max__eqI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ! [Y5: A] :
                ( member(A,Y5,A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y5),Xb) )
           => ( member(A,Xb,A3)
             => ( lattic643756798349783984er_Max(A,A3) = Xb ) ) ) ) ) ).

% Max_eqI
tff(fact_6523_Max__ge,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( member(A,Xb,A3)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),lattic643756798349783984er_Max(A,A3)) ) ) ) ).

% Max_ge
tff(fact_6524_finite__surj,axiom,
    ! [A: $tType,B: $tType,A3: set(A),B5: set(B),F: fun(A,B)] :
      ( finite_finite2(A,A3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B5),aa(set(A),set(B),image(A,B,F),A3))
       => finite_finite2(B,B5) ) ) ).

% finite_surj
tff(fact_6525_finite__subset__image,axiom,
    ! [A: $tType,B: $tType,B5: set(A),F: fun(B,A),A3: set(B)] :
      ( finite_finite2(A,B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(B),set(A),image(B,A,F),A3))
       => ? [C8: set(B)] :
            ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C8),A3)
            & finite_finite2(B,C8)
            & ( B5 = aa(set(B),set(A),image(B,A,F),C8) ) ) ) ) ).

% finite_subset_image
tff(fact_6526_ex__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: set(B),P: fun(set(A),$o)] :
      ( ? [B11: set(A)] :
          ( finite_finite2(A,B11)
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B11),aa(set(B),set(A),image(B,A,F),A3))
          & aa(set(A),$o,P,B11) )
    <=> ? [B11: set(B)] :
          ( finite_finite2(B,B11)
          & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B11),A3)
          & aa(set(A),$o,P,aa(set(B),set(A),image(B,A,F),B11)) ) ) ).

% ex_finite_subset_image
tff(fact_6527_all__finite__subset__image,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: set(B),P: fun(set(A),$o)] :
      ( ! [B11: set(A)] :
          ( ( finite_finite2(A,B11)
            & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B11),aa(set(B),set(A),image(B,A,F),A3)) )
         => aa(set(A),$o,P,B11) )
    <=> ! [B11: set(B)] :
          ( ( finite_finite2(B,B11)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B11),A3) )
         => aa(set(A),$o,P,aa(set(B),set(A),image(B,A,F),B11)) ) ) ).

% all_finite_subset_image
tff(fact_6528_range__composition,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: fun(C,A),G: fun(B,C)] : aa(set(B),set(A),image(B,A,aa(fun(B,C),fun(B,A),aTP_Lamp_aho(fun(C,A),fun(fun(B,C),fun(B,A)),F),G)),top_top(set(B))) = aa(set(C),set(A),image(C,A,F),aa(set(B),set(C),image(B,C,G),top_top(set(B)))) ).

% range_composition
tff(fact_6529_rangeE,axiom,
    ! [A: $tType,B: $tType,B2: A,F: fun(B,A)] :
      ( member(A,B2,aa(set(B),set(A),image(B,A,F),top_top(set(B))))
     => ~ ! [X2: B] : B2 != aa(B,A,F,X2) ) ).

% rangeE
tff(fact_6530_image__mono,axiom,
    ! [B: $tType,A: $tType,A3: set(A),B5: set(A),F: fun(A,B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),A3)),aa(set(A),set(B),image(A,B,F),B5)) ) ).

% image_mono
tff(fact_6531_image__subsetI,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F: fun(A,B),B5: set(B)] :
      ( ! [X2: A] :
          ( member(A,X2,A3)
         => member(B,aa(A,B,F,X2),B5) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),A3)),B5) ) ).

% image_subsetI
tff(fact_6532_subset__imageE,axiom,
    ! [A: $tType,B: $tType,B5: set(A),F: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(B),set(A),image(B,A,F),A3))
     => ~ ! [C8: set(B)] :
            ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),C8),A3)
           => ( B5 != aa(set(B),set(A),image(B,A,F),C8) ) ) ) ).

% subset_imageE
tff(fact_6533_image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A3: set(B),B5: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F),A3)),B5)
    <=> ! [X: B] :
          ( member(B,X,A3)
         => member(A,aa(B,A,F,X),B5) ) ) ).

% image_subset_iff
tff(fact_6534_subset__image__iff,axiom,
    ! [A: $tType,B: $tType,B5: set(A),F: fun(B,A),A3: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),aa(set(B),set(A),image(B,A,F),A3))
    <=> ? [AA: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),AA),A3)
          & ( B5 = aa(set(B),set(A),image(B,A,F),AA) ) ) ) ).

% subset_image_iff
tff(fact_6535_all__subset__image,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: set(B),P: fun(set(A),$o)] :
      ( ! [B11: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B11),aa(set(B),set(A),image(B,A,F),A3))
         => aa(set(A),$o,P,B11) )
    <=> ! [B11: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),B11),A3)
         => aa(set(A),$o,P,aa(set(B),set(A),image(B,A,F),B11)) ) ) ).

% all_subset_image
tff(fact_6536_range__subsetD,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),B5: set(A),I: B] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F),top_top(set(B)))),B5)
     => member(A,aa(B,A,F,I),B5) ) ).

% range_subsetD
tff(fact_6537_imageE,axiom,
    ! [A: $tType,B: $tType,B2: A,F: fun(B,A),A3: set(B)] :
      ( member(A,B2,aa(set(B),set(A),image(B,A,F),A3))
     => ~ ! [X2: B] :
            ( ( B2 = aa(B,A,F,X2) )
           => ~ member(B,X2,A3) ) ) ).

% imageE
tff(fact_6538_image__image,axiom,
    ! [B: $tType,A: $tType,C: $tType,F: fun(B,A),G: fun(C,B),A3: set(C)] : aa(set(B),set(A),image(B,A,F),aa(set(C),set(B),image(C,B,G),A3)) = aa(set(C),set(A),image(C,A,aa(fun(C,B),fun(C,A),aTP_Lamp_bk(fun(B,A),fun(fun(C,B),fun(C,A)),F),G)),A3) ).

% image_image
tff(fact_6539_Compr__image__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(B,A),A3: set(B),P: fun(A,$o)] : collect(A,aa(fun(A,$o),fun(A,$o),aa(set(B),fun(fun(A,$o),fun(A,$o)),aTP_Lamp_ahp(fun(B,A),fun(set(B),fun(fun(A,$o),fun(A,$o))),F),A3),P)) = aa(set(B),set(A),image(B,A,F),collect(B,aa(fun(A,$o),fun(B,$o),aa(set(B),fun(fun(A,$o),fun(B,$o)),aTP_Lamp_ahq(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),F),A3),P))) ).

% Compr_image_eq
tff(fact_6540_pigeonhole__infinite,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F: fun(A,B)] :
      ( ~ finite_finite2(A,A3)
     => ( finite_finite2(B,aa(set(A),set(B),image(A,B,F),A3))
       => ? [X2: A] :
            ( member(A,X2,A3)
            & ~ finite_finite2(A,collect(A,aa(A,fun(A,$o),aa(fun(A,B),fun(A,fun(A,$o)),aTP_Lamp_ahr(set(A),fun(fun(A,B),fun(A,fun(A,$o))),A3),F),X2))) ) ) ) ).

% pigeonhole_infinite
tff(fact_6541_gcd__ge__0__int,axiom,
    ! [Xb: int,Y: int] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),gcd_gcd(int,Xb,Y)) ).

% gcd_ge_0_int
tff(fact_6542_bij__betw__byWitness,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F4: fun(B,A),F: fun(A,B),A9: set(B)] :
      ( ! [X2: A] :
          ( member(A,X2,A3)
         => ( aa(B,A,F4,aa(A,B,F,X2)) = X2 ) )
     => ( ! [X2: B] :
            ( member(B,X2,A9)
           => ( aa(A,B,F,aa(B,A,F4,X2)) = X2 ) )
       => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),A3)),A9)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,F4),A9)),A3)
           => bij_betw(A,B,F,A3,A9) ) ) ) ) ).

% bij_betw_byWitness
tff(fact_6543_bij__betw__subset,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A3: set(A),A9: set(B),B5: set(A),B10: set(B)] :
      ( bij_betw(A,B,F,A3,A9)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
       => ( ( aa(set(A),set(B),image(A,B,F),B5) = B10 )
         => bij_betw(A,B,F,B5,B10) ) ) ) ).

% bij_betw_subset
tff(fact_6544_image__constant__conv,axiom,
    ! [B: $tType,A: $tType,C2: A,A3: set(B)] :
      aa(set(B),set(A),image(B,A,aa(A,fun(B,A),aTP_Lamp_eq(A,fun(B,A)),C2)),A3) = $ite(A3 = 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_6545_image__constant,axiom,
    ! [A: $tType,B: $tType,Xb: A,A3: set(A),C2: B] :
      ( member(A,Xb,A3)
     => ( aa(set(A),set(B),image(A,B,aTP_Lamp_ahs(B,fun(A,B),C2)),A3) = aa(set(B),set(B),insert(B,C2),bot_bot(set(B))) ) ) ).

% image_constant
tff(fact_6546_finite__range__imageI,axiom,
    ! [A: $tType,C: $tType,B: $tType,G: fun(B,A),F: 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_aht(fun(B,A),fun(fun(A,C),fun(B,C)),G),F)),top_top(set(B)))) ) ).

% finite_range_imageI
tff(fact_6547_image__set,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : aa(set(B),set(A),image(B,A,F),aa(list(B),set(B),set2(B),Xs)) = aa(list(A),set(A),set2(A),aa(list(B),list(A),map(B,A,F),Xs)) ).

% image_set
tff(fact_6548_translation__subtract__diff,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A,S3: set(A),Ta: set(A)] : aa(set(A),set(A),image(A,A,aTP_Lamp_ahg(A,fun(A,A),A2)),aa(set(A),set(A),minus_minus(set(A),S3),Ta)) = aa(set(A),set(A),minus_minus(set(A),aa(set(A),set(A),image(A,A,aTP_Lamp_ahg(A,fun(A,A),A2)),S3)),aa(set(A),set(A),image(A,A,aTP_Lamp_ahg(A,fun(A,A),A2)),Ta)) ) ).

% translation_subtract_diff
tff(fact_6549_sum_Oimage__gen,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),H: fun(A,B),G: fun(A,C)] :
          ( finite_finite2(A,S2)
         => ( aa(set(A),B,groups7311177749621191930dd_sum(A,B,H),S2) = 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_ahu(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),S2),H),G)),aa(set(A),set(C),image(A,C,G),S2)) ) ) ) ).

% sum.image_gen
tff(fact_6550_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,Nb: int] : aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,abs_abs(int),K)),gcd_gcd(int,M,Nb)) = gcd_gcd(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),Nb)) ).

% gcd_mult_distrib_int
tff(fact_6551_gcd__dvd__prod,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,K: A] : aa(A,$o,dvd_dvd(A,gcd_gcd(A,A2,B2)),aa(A,A,aa(A,fun(A,A),times_times(A),K),B2)) ) ).

% gcd_dvd_prod
tff(fact_6552_gcd__add__mult,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [M: A,K: A,Nb: A] : gcd_gcd(A,M,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),K),M)),Nb)) = gcd_gcd(A,M,Nb) ) ).

% gcd_add_mult
tff(fact_6553_bezout__int,axiom,
    ! [Xb: int,Y: int] :
    ? [U3: int,V3: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),U3),Xb)),aa(int,int,aa(int,fun(int,int),times_times(int),V3),Y)) = gcd_gcd(int,Xb,Y) ).

% bezout_int
tff(fact_6554_gcd__mult__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( gcd_gcd(A,B2,aa(A,A,aa(A,fun(A,A),times_times(A),C2),A2)) = gcd_gcd(A,B2,C2) ) ) ) ).

% gcd_mult_unit2
tff(fact_6555_gcd__mult__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( gcd_gcd(A,aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2),C2) = gcd_gcd(A,B2,C2) ) ) ) ).

% gcd_mult_unit1
tff(fact_6556_gcd__div__unit1,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( gcd_gcd(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),A2),C2) = gcd_gcd(A,B2,C2) ) ) ) ).

% gcd_div_unit1
tff(fact_6557_gcd__div__unit2,axiom,
    ! [A: $tType] :
      ( semiring_gcd(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,dvd_dvd(A,A2),one_one(A))
         => ( gcd_gcd(A,B2,aa(A,A,aa(A,fun(A,A),divide_divide(A),C2),A2)) = gcd_gcd(A,B2,C2) ) ) ) ).

% gcd_div_unit2
tff(fact_6558_gcd__le2__int,axiom,
    ! [B2: int,A2: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),gcd_gcd(int,A2,B2)),B2) ) ).

% gcd_le2_int
tff(fact_6559_gcd__le1__int,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,ord_less(int,zero_zero(int)),A2)
     => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),gcd_gcd(int,A2,B2)),A2) ) ).

% gcd_le1_int
tff(fact_6560_gcd__cases__int,axiom,
    ! [Xb: int,Y: int,P: fun(int,$o)] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
       => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Y)
         => aa(int,$o,P,gcd_gcd(int,Xb,Y)) ) )
     => ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),Xb)
         => ( aa(int,$o,aa(int,fun(int,$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,aa(int,fun(int,$o),ord_less_eq(int),Xb),zero_zero(int))
           => ( aa(int,$o,aa(int,fun(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,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),zero_zero(int))
             => ( aa(int,$o,aa(int,fun(int,$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_6561_gcd__unique__int,axiom,
    ! [D2: int,A2: int,B2: int] :
      ( ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),D2)
        & aa(int,$o,dvd_dvd(int,D2),A2)
        & aa(int,$o,dvd_dvd(int,D2),B2)
        & ! [E4: int] :
            ( ( aa(int,$o,dvd_dvd(int,E4),A2)
              & aa(int,$o,dvd_dvd(int,E4),B2) )
           => aa(int,$o,dvd_dvd(int,E4),D2) ) )
    <=> ( D2 = gcd_gcd(int,A2,B2) ) ) ).

% gcd_unique_int
tff(fact_6562_surj__Compl__image__subset,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: set(B)] :
      ( ( aa(set(B),set(A),image(B,A,F),top_top(set(B))) = top_top(set(A)) )
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),uminus_uminus(set(A)),aa(set(B),set(A),image(B,A,F),A3))),aa(set(B),set(A),image(B,A,F),aa(set(B),set(B),uminus_uminus(set(B)),A3))) ) ).

% surj_Compl_image_subset
tff(fact_6563_Max__eq__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( lattic643756798349783984er_Max(A,A3) = M )
            <=> ( member(A,M,A3)
                & ! [X: A] :
                    ( member(A,X,A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),M) ) ) ) ) ) ) ).

% Max_eq_iff
tff(fact_6564_Max__ge__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),lattic643756798349783984er_Max(A,A3))
            <=> ? [X: A] :
                  ( member(A,X,A3)
                  & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),X) ) ) ) ) ) ).

% Max_ge_iff
tff(fact_6565_eq__Max__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),M: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ( M = lattic643756798349783984er_Max(A,A3) )
            <=> ( member(A,M,A3)
                & ! [X: A] :
                    ( member(A,X,A3)
                   => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X),M) ) ) ) ) ) ) ).

% eq_Max_iff
tff(fact_6566_Max_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,A3)),Xb)
             => ! [A7: A] :
                  ( member(A,A7,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A7),Xb) ) ) ) ) ) ).

% Max.boundedE
tff(fact_6567_Max_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( ! [A4: A] :
                  ( member(A,A4,A3)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),Xb) )
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,A3)),Xb) ) ) ) ) ).

% Max.boundedI
tff(fact_6568_Max__gr__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( ( A3 != bot_bot(set(A)) )
           => ( aa(A,$o,ord_less(A,Xb),lattic643756798349783984er_Max(A,A3))
            <=> ? [X: A] :
                  ( member(A,X,A3)
                  & aa(A,$o,ord_less(A,Xb),X) ) ) ) ) ) ).

% Max_gr_iff
tff(fact_6569_Max__insert2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),A2: A] :
          ( finite_finite2(A,A3)
         => ( ! [B3: A] :
                ( member(A,B3,A3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B3),A2) )
           => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),insert(A,A2),A3)) = A2 ) ) ) ) ).

% Max_insert2
tff(fact_6570_VEBT__internal_Oheight_Osimps_I2_J,axiom,
    ! [Uu: option(product_prod(nat,nat)),Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_VEBT_height,vEBT_Node(Uu,Deg,TreeLista,Summarya)) = 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,Summarya),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))))) ).

% VEBT_internal.height.simps(2)
tff(fact_6571_Max_Oinfinite,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( ~ finite_finite2(A,A3)
         => ( lattic643756798349783984er_Max(A,A3) = aa(option(A),A,the2(A),none(A)) ) ) ) ).

% Max.infinite
tff(fact_6572_sum_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_add(C)
     => ! [S2: set(A),T5: set(B),G: fun(A,B),H: fun(A,C)] :
          ( finite_finite2(A,S2)
         => ( finite_finite2(B,T5)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),S2)),T5)
             => ( 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_ahw(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S2),G),H)),T5) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,H),S2) ) ) ) ) ) ).

% sum.group
tff(fact_6573_prod_Ogroup,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [S2: set(A),T5: set(B),G: fun(A,B),H: fun(A,C)] :
          ( finite_finite2(A,S2)
         => ( finite_finite2(B,T5)
           => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,G),S2)),T5)
             => ( groups7121269368397514597t_prod(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_ahx(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),S2),G),H),T5) = groups7121269368397514597t_prod(A,C,H,S2) ) ) ) ) ) ).

% prod.group
tff(fact_6574_set__image__eq__pointwiseI,axiom,
    ! [B: $tType,A: $tType,L: list(A),L3: list(A),F: fun(A,B)] :
      ( ( aa(list(A),nat,size_size(list(A)),L) = aa(list(A),nat,size_size(list(A)),L3) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,ord_less(nat,I3),aa(list(A),nat,size_size(list(A)),L))
           => ( aa(A,B,F,aa(nat,A,nth(A,L),I3)) = aa(A,B,F,aa(nat,A,nth(A,L3),I3)) ) )
       => ( aa(set(A),set(B),image(A,B,F),aa(list(A),set(A),set2(A),L)) = aa(set(A),set(B),image(A,B,F),aa(list(A),set(A),set2(A),L3)) ) ) ) ).

% set_image_eq_pointwiseI
tff(fact_6575_in__set__image__conv__nth,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Xb: B,L: list(B)] :
      ( member(A,aa(B,A,F,Xb),aa(set(B),set(A),image(B,A,F),aa(list(B),set(B),set2(B),L)))
    <=> ? [I2: nat] :
          ( aa(nat,$o,ord_less(nat,I2),aa(list(B),nat,size_size(list(B)),L))
          & ( aa(B,A,F,aa(nat,B,nth(B,L),I2)) = aa(B,A,F,Xb) ) ) ) ).

% in_set_image_conv_nth
tff(fact_6576_Max__mono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M8: set(A),N: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),M8),N)
         => ( ( M8 != bot_bot(set(A)) )
           => ( finite_finite2(A,N)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,M8)),lattic643756798349783984er_Max(A,N)) ) ) ) ) ).

% Max_mono
tff(fact_6577_Max_Osubset__imp,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B5: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
         => ( ( A3 != bot_bot(set(A)) )
           => ( finite_finite2(A,B5)
             => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),lattic643756798349783984er_Max(A,A3)),lattic643756798349783984er_Max(A,B5)) ) ) ) ) ).

% Max.subset_imp
tff(fact_6578_VEBT__internal_Oheight_Oelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,Xb) = Y )
     => ( ( ? [A4: $o,B3: $o] : Xb = vEBT_Leaf((A4),(B3))
         => ( Y != zero_zero(nat) ) )
       => ~ ! [Uu2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Uu2,Deg2,TreeList,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),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))))) ) ) ) ) ).

% VEBT_internal.height.elims
tff(fact_6579_Max_Osubset,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),B5: set(A)] :
          ( finite_finite2(A,A3)
         => ( ( B5 != bot_bot(set(A)) )
           => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
             => ( aa(A,A,ord_max(A,lattic643756798349783984er_Max(A,B5)),lattic643756798349783984er_Max(A,A3)) = lattic643756798349783984er_Max(A,A3) ) ) ) ) ) ).

% Max.subset
tff(fact_6580_divide__nat__def,axiom,
    ! [M: nat,Nb: nat] :
      aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),M),Nb) = $ite(Nb = zero_zero(nat),zero_zero(nat),lattic643756798349783984er_Max(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_ahy(nat,fun(nat,fun(nat,$o)),M),Nb)))) ).

% divide_nat_def
tff(fact_6581_Max_Oremove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( member(A,Xb,A3)
           => ( lattic643756798349783984er_Max(A,A3) = $ite(aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) = bot_bot(set(A)),Xb,aa(A,A,ord_max(A,Xb),lattic643756798349783984er_Max(A,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))))) ) ) ) ) ).

% Max.remove
tff(fact_6582_Max_Oinsert__remove,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A),Xb: A] :
          ( finite_finite2(A,A3)
         => ( lattic643756798349783984er_Max(A,aa(set(A),set(A),insert(A,Xb),A3)) = $ite(aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))) = bot_bot(set(A)),Xb,aa(A,A,ord_max(A,Xb),lattic643756798349783984er_Max(A,aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A))))))) ) ) ) ).

% Max.insert_remove
tff(fact_6583_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,aa(A,fun(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_6584_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_ahz(A,fun(A,A),C2)),set_or1337092689740270186AtMost(A,Xb,Y)) = $ite(
            aa(A,$o,aa(A,fun(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_6585_image__affinity__atLeastAtMost,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: 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,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(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),A2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),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),B2)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2))) ) ) ).

% image_affinity_atLeastAtMost
tff(fact_6586_image__affinity__atLeastAtMost__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: 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,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(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),A2)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),B2)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),M),A2)),C2))) ) ) ).

% image_affinity_atLeastAtMost_diff
tff(fact_6587_image__affinity__atLeastAtMost__div,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: 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,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(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),A2),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),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),B2),M)),C2),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div
tff(fact_6588_image__affinity__atLeastAtMost__div__diff,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [M: A,C2: A,A2: A,B2: A] :
          aa(set(A),set(A),image(A,A,aa(A,fun(A,A),aTP_Lamp_aid(A,fun(A,fun(A,A)),M),C2)),set_or1337092689740270186AtMost(A,A2,B2)) = $ite(
            set_or1337092689740270186AtMost(A,A2,B2) = bot_bot(set(A)),
            bot_bot(set(A)),
            $ite(aa(A,$o,aa(A,fun(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),A2),M)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2)),set_or1337092689740270186AtMost(A,aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),B2),M)),C2),aa(A,A,minus_minus(A,aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),M)),C2))) ) ) ).

% image_affinity_atLeastAtMost_div_diff
tff(fact_6589_VEBT__internal_Oheight_Opelims,axiom,
    ! [Xb: vEBT_VEBT,Y: nat] :
      ( ( aa(vEBT_VEBT,nat,vEBT_VEBT_height,Xb) = Y )
     => ( aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_height_rel),Xb)
       => ( ! [A4: $o,B3: $o] :
              ( ( Xb = vEBT_Leaf((A4),(B3)) )
             => ( ( Y = zero_zero(nat) )
               => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_height_rel),vEBT_Leaf((A4),(B3))) ) )
         => ~ ! [Uu2: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Uu2,Deg2,TreeList,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),aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))))) )
                 => ~ aa(vEBT_VEBT,$o,accp(vEBT_VEBT,vEBT_VEBT_height_rel),vEBT_Node(Uu2,Deg2,TreeList,Summary)) ) ) ) ) ) ).

% VEBT_internal.height.pelims
tff(fact_6590_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_6591_bij__betw__Suc,axiom,
    ! [M8: set(nat),N: set(nat)] :
      ( bij_betw(nat,nat,suc,M8,N)
    <=> ( aa(set(nat),set(nat),image(nat,nat,suc),M8) = N ) ) ).

% bij_betw_Suc
tff(fact_6592_gcd__1__nat,axiom,
    ! [M: nat] : gcd_gcd(nat,M,one_one(nat)) = one_one(nat) ).

% gcd_1_nat
tff(fact_6593_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_6594_gcd__pos__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),gcd_gcd(nat,M,Nb))
    <=> ( ( M != zero_zero(nat) )
        | ( Nb != zero_zero(nat) ) ) ) ).

% gcd_pos_nat
tff(fact_6595_img__fst,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,S2: 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),A2),B2),S2)
     => member(A,A2,aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),S2)) ) ).

% img_fst
tff(fact_6596_img__snd,axiom,
    ! [B: $tType,A: $tType,A2: A,B2: B,S2: 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),A2),B2),S2)
     => member(B,B2,aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),S2)) ) ).

% img_snd
tff(fact_6597_op__conc__empty__img__id,axiom,
    ! [A: $tType,L7: set(list(A))] : aa(set(list(A)),set(list(A)),image(list(A),list(A),append(A,nil(A))),L7) = L7 ).

% op_conc_empty_img_id
tff(fact_6598_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_or7035219750837199246ssThan(nat,I,J)) = set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastLessThan
tff(fact_6599_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_or1337092689740270186AtMost(nat,I,J)) = set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,I),aa(nat,nat,suc,J)) ).

% image_Suc_atLeastAtMost
tff(fact_6600_pair__imageI,axiom,
    ! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set(product_prod(A,B)),F: 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),A2),B2),A3)
     => member(C,aa(B,C,aa(A,fun(B,C),F,A2),B2),aa(set(product_prod(A,B)),set(C),image(product_prod(A,B),C,product_case_prod(A,B,C,F)),A3)) ) ).

% pair_imageI
tff(fact_6601_range__mult,axiom,
    ! [A2: real] :
      aa(set(real),set(real),image(real,real,aa(real,fun(real,real),times_times(real),A2)),top_top(set(real))) = $ite(A2 = 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_6602_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_6603_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_6604_Max__divisors__self__int,axiom,
    ! [Nb: int] :
      ( ( Nb != zero_zero(int) )
     => ( lattic643756798349783984er_Max(int,collect(int,aTP_Lamp_sz(int,fun(int,$o),Nb))) = aa(int,int,abs_abs(int),Nb) ) ) ).

% Max_divisors_self_int
tff(fact_6605_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))) = aa(list(A),set(A),set2(A),L) ).

% nth_image_indices
tff(fact_6606_gcd__diff2__nat,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
     => ( gcd_gcd(nat,aa(nat,nat,minus_minus(nat,Nb),M),Nb) = gcd_gcd(nat,M,Nb) ) ) ).

% gcd_diff2_nat
tff(fact_6607_gcd__diff1__nat,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
     => ( gcd_gcd(nat,aa(nat,nat,minus_minus(nat,M),Nb),Nb) = gcd_gcd(nat,M,Nb) ) ) ).

% gcd_diff1_nat
tff(fact_6608_None__notin__image__Some,axiom,
    ! [A: $tType,A3: set(A)] : ~ member(option(A),none(A),aa(set(A),set(option(A)),image(A,option(A),some(A)),A3)) ).

% None_notin_image_Some
tff(fact_6609_zero__notin__Suc__image,axiom,
    ! [A3: set(nat)] : ~ member(nat,zero_zero(nat),aa(set(nat),set(nat),image(nat,nat,suc),A3)) ).

% zero_notin_Suc_image
tff(fact_6610_gcd__le1__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),gcd_gcd(nat,A2,B2)),A2) ) ).

% gcd_le1_nat
tff(fact_6611_gcd__le2__nat,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2 != zero_zero(nat) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),gcd_gcd(nat,A2,B2)),B2) ) ).

% gcd_le2_nat
tff(fact_6612_nat__seg__image__imp__finite,axiom,
    ! [A: $tType,A3: set(A),F: fun(nat,A),Nb: nat] :
      ( ( A3 = aa(set(nat),set(A),image(nat,A,F),collect(nat,aTP_Lamp_sq(nat,fun(nat,$o),Nb))) )
     => finite_finite2(A,A3) ) ).

% nat_seg_image_imp_finite
tff(fact_6613_finite__conv__nat__seg__image,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
    <=> ? [N6: nat,F8: fun(nat,A)] : A3 = aa(set(nat),set(A),image(nat,A,F8),collect(nat,aTP_Lamp_sq(nat,fun(nat,$o),N6))) ) ).

% finite_conv_nat_seg_image
tff(fact_6614_in__fst__imageE,axiom,
    ! [B: $tType,A: $tType,Xb: A,S2: 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)),S2))
     => ~ ! [Y5: 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),Y5),S2) ) ).

% in_fst_imageE
tff(fact_6615_in__snd__imageE,axiom,
    ! [A: $tType,B: $tType,Y: A,S2: 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)),S2))
     => ~ ! [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),S2) ) ).

% in_snd_imageE
tff(fact_6616_finite__int__iff__bounded,axiom,
    ! [S2: set(int)] :
      ( finite_finite2(int,S2)
    <=> ? [K4: int] : aa(set(int),$o,aa(set(int),fun(set(int),$o),ord_less_eq(set(int)),aa(set(int),set(int),image(int,int,abs_abs(int)),S2)),set_ord_lessThan(int,K4)) ) ).

% finite_int_iff_bounded
tff(fact_6617_finite__int__iff__bounded__le,axiom,
    ! [S2: set(int)] :
      ( finite_finite2(int,S2)
    <=> ? [K4: int] : aa(set(int),$o,aa(set(int),fun(set(int),$o),ord_less_eq(set(int)),aa(set(int),set(int),image(int,int,abs_abs(int)),S2)),set_ord_atMost(int,K4)) ) ).

% finite_int_iff_bounded_le
tff(fact_6618_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),K),gcd_gcd(nat,M,Nb)) = gcd_gcd(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),Nb)) ).

% gcd_mult_distrib_nat
tff(fact_6619_bitval__bin__last__integer,axiom,
    ! [I: code_integer] : aa($o,code_integer,zero_neq_one_of_bool(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)) ).

% bitval_bin_last_integer
tff(fact_6620_bezout__nat,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2 != zero_zero(nat) )
     => ? [X2: nat,Y5: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y5)),gcd_gcd(nat,A2,B2)) ) ).

% bezout_nat
tff(fact_6621_bezout__gcd__nat_H,axiom,
    ! [B2: nat,A2: nat] :
    ? [X2: nat,Y5: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y5)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2))
        & ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),Y5)) = gcd_gcd(nat,A2,B2) ) )
      | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y5)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2))
        & ( aa(nat,nat,minus_minus(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),B2),X2)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),A2),Y5)) = gcd_gcd(nat,A2,B2) ) ) ) ).

% bezout_gcd_nat'
tff(fact_6622_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_6623_fst__image__mp,axiom,
    ! [B: $tType,A: $tType,A3: set(product_prod(A,B)),B5: set(A),Xb: A,Y: B] :
      ( aa(set(A),$o,aa(set(A),fun(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)),A3)),B5)
     => ( 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),A3)
       => member(A,Xb,B5) ) ) ).

% fst_image_mp
tff(fact_6624_snd__image__mp,axiom,
    ! [B: $tType,A: $tType,A3: set(product_prod(B,A)),B5: set(A),Xb: B,Y: A] :
      ( aa(set(A),$o,aa(set(A),fun(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)),A3)),B5)
     => ( 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),A3)
       => member(A,Y,B5) ) ) ).

% snd_image_mp
tff(fact_6625_in__image__insert__iff,axiom,
    ! [A: $tType,B5: set(set(A)),Xb: A,A3: set(A)] :
      ( ! [C8: set(A)] :
          ( member(set(A),C8,B5)
         => ~ member(A,Xb,C8) )
     => ( member(set(A),A3,aa(set(set(A)),set(set(A)),image(set(A),set(A),insert(A,Xb)),B5))
      <=> ( member(A,Xb,A3)
          & member(set(A),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,Xb),bot_bot(set(A)))),B5) ) ) ) ).

% in_image_insert_iff
tff(fact_6626_gcd__is__Max__divisors__int,axiom,
    ! [Nb: int,M: int] :
      ( ( Nb != zero_zero(int) )
     => ( gcd_gcd(int,M,Nb) = lattic643756798349783984er_Max(int,collect(int,aa(int,fun(int,$o),aTP_Lamp_aie(int,fun(int,fun(int,$o)),Nb),M))) ) ) ).

% gcd_is_Max_divisors_int
tff(fact_6627_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_6628_image__Suc__lessThan,axiom,
    ! [Nb: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_ord_lessThan(nat,Nb)) = set_or1337092689740270186AtMost(nat,one_one(nat),Nb) ).

% image_Suc_lessThan
tff(fact_6629_image__Suc__atMost,axiom,
    ! [Nb: nat] : aa(set(nat),set(nat),image(nat,nat,suc),set_ord_atMost(nat,Nb)) = set_or1337092689740270186AtMost(nat,one_one(nat),aa(nat,nat,suc,Nb)) ).

% image_Suc_atMost
tff(fact_6630_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = 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),Nb))) ).

% atLeast0_lessThan_Suc_eq_insert_0
tff(fact_6631_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb)) = 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),Nb))) ).

% atLeast0_atMost_Suc_eq_insert_0
tff(fact_6632_lessThan__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_ord_lessThan(nat,aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_ord_lessThan(nat,Nb))) ).

% lessThan_Suc_eq_insert_0
tff(fact_6633_atMost__Suc__eq__insert__0,axiom,
    ! [Nb: nat] : set_ord_atMost(nat,aa(nat,nat,suc,Nb)) = aa(set(nat),set(nat),insert(nat,zero_zero(nat)),aa(set(nat),set(nat),image(nat,nat,suc),set_ord_atMost(nat,Nb))) ).

% atMost_Suc_eq_insert_0
tff(fact_6634_Collect__split__mono__strong,axiom,
    ! [B: $tType,A: $tType,X5: set(A),A3: set(product_prod(A,B)),Y7: set(B),P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o))] :
      ( ( X5 = aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),A3) )
     => ( ( Y7 = aa(set(product_prod(A,B)),set(B),image(product_prod(A,B),B,product_snd(A,B)),A3) )
       => ( ! [X2: A] :
              ( member(A,X2,X5)
             => ! [Xa2: B] :
                  ( member(B,Xa2,Y7)
                 => ( aa(B,$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,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A3),collect(product_prod(A,B),product_case_prod(A,B,$o,P)))
           => aa(set(product_prod(A,B)),$o,aa(set(product_prod(A,B)),fun(set(product_prod(A,B)),$o),ord_less_eq(set(product_prod(A,B))),A3),collect(product_prod(A,B),product_case_prod(A,B,$o,Q))) ) ) ) ) ).

% Collect_split_mono_strong
tff(fact_6635_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_6636_range__mod,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_aif(nat,fun(nat,nat),Nb)),top_top(set(nat))) = set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb) ) ) ).

% range_mod
tff(fact_6637_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_6638_gcd__is__Max__divisors__nat,axiom,
    ! [Nb: nat,M: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( gcd_gcd(nat,M,Nb) = lattic643756798349783984er_Max(nat,collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_aig(nat,fun(nat,fun(nat,$o)),Nb),M))) ) ) ).

% gcd_is_Max_divisors_nat
tff(fact_6639_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] : aa(set(int),set(int),image(int,int,aTP_Lamp_aih(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_6640_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_aii(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_6641_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( aa(int,$o,aa(int,fun(int,$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_6642_bezw__aux,axiom,
    ! [Xb: nat,Y: nat] : aa(nat,int,semiring_1_of_nat(int),gcd_gcd(nat,Xb,Y)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_fst(int,int),bezw(Xb,Y))),aa(nat,int,semiring_1_of_nat(int),Xb))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(product_prod(int,int),int,product_snd(int,int),bezw(Xb,Y))),aa(nat,int,semiring_1_of_nat(int),Y))) ).

% bezw_aux
tff(fact_6643_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Xb: nat,Y: nat] :
      aa(set(nat),set(nat),image(nat,nat,aTP_Lamp_aij(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_6644_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_6645_gcd__nat_Opelims,axiom,
    ! [Xb: nat,Xaa: nat,Y: nat] :
      ( ( gcd_gcd(nat,Xb,Xaa) = Y )
     => ( aa(product_prod(nat,nat),$o,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 = $ite(Xaa = zero_zero(nat),Xb,gcd_gcd(nat,Xaa,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xb),Xaa))) )
           => ~ aa(product_prod(nat,nat),$o,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_6646_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_6647_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_6648_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_6649_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_6650_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_6651_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_6652_set__Cons__sing__Nil,axiom,
    ! [A: $tType,A3: set(A)] : set_Cons(A,A3,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_aik(A,list(A))),A3) ).

% set_Cons_sing_Nil
tff(fact_6653_flat__lub__def,axiom,
    ! [A: $tType,B2: A,A3: set(A)] :
      aa(set(A),A,partial_flat_lub(A,B2),A3) = $ite(aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),insert(A,B2),bot_bot(set(A)))),B2,the(A,aa(set(A),fun(A,$o),aTP_Lamp_ail(A,fun(set(A),fun(A,$o)),B2),A3))) ).

% flat_lub_def
tff(fact_6654_bind__return,axiom,
    ! [A: $tType,F: heap_Time_Heap(A)] : heap_Time_bind(A,A,F,heap_Time_return(A)) = heap_Time_bind(product_unit,A,heap_Time_wait(one_one(nat)),aTP_Lamp_cy(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),F)) ).

% bind_return
tff(fact_6655_wait__bind__decon,axiom,
    ! [A: $tType,P: assn,M: heap_Time_Heap(A),Q: fun(A,assn),Nb: nat] :
      ( hoare_hoare_triple(A,P,M,Q)
     => hoare_hoare_triple(A,P,heap_Time_bind(product_unit,A,heap_Time_wait(Nb),aTP_Lamp_cy(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),M)),Q) ) ).

% wait_bind_decon
tff(fact_6656_wait__rule,axiom,
    ! [Nb: nat] : hoare_hoare_triple(product_unit,one_one(assn),heap_Time_wait(Nb),aTP_Lamp_aim(product_unit,assn)) ).

% wait_rule
tff(fact_6657_return__bind,axiom,
    ! [A: $tType,B: $tType,Xb: B,F: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,aa(B,heap_Time_Heap(B),heap_Time_return(B),Xb),F) = 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_ain(B,fun(fun(B,heap_Time_Heap(A)),fun(product_unit,heap_Time_Heap(A))),Xb),F)) ).

% return_bind
tff(fact_6658_tailrec_Ofixp__induct__uc,axiom,
    ! [A: $tType,B: $tType,C: $tType,U2: fun(C,fun(A,B)),F5: fun(C,C),C5: fun(fun(A,B),C),F: C,P: fun(fun(A,B),$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_aio(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))
     => ( ( F = 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_aip(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))) )
       => ( ! [F2: fun(A,B)] : aa(C,fun(A,B),U2,aa(fun(A,B),C,C5,F2)) = F2
         => ( 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),$o,P,aTP_Lamp_aiq(A,B))
             => ( ! [F2: C] :
                    ( aa(fun(A,B),$o,P,aa(C,fun(A,B),U2,F2))
                   => aa(fun(A,B),$o,P,aa(C,fun(A,B),U2,aa(C,C,F5,F2))) )
               => aa(fun(A,B),$o,P,aa(C,fun(A,B),U2,F)) ) ) ) ) ) ) ).

% tailrec.fixp_induct_uc
tff(fact_6659_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_6660_drop__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat,A2: 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),Nb),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)),A2) ) ).

% drop_bit_drop_bit
tff(fact_6661_drop__bit__Suc__bit0,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),bit0(K))) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit0
tff(fact_6662_drop__bit__Suc__bit1,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),aa(num,num,bit1,K))) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),aa(num,A,numeral_numeral(A),K)) ) ).

% drop_bit_Suc_bit1
tff(fact_6663_drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),one_one(A)) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% drop_bit_of_1
tff(fact_6664_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_6665_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_6666_fixp__induct__option,axiom,
    ! [C: $tType,A: $tType,B: $tType,U2: fun(C,fun(A,option(B))),F5: fun(C,C),C5: fun(fun(A,option(B)),C),F: C,P: fun(A,fun(B,$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_air(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))
     => ( ( F = 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_ais(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))) )
       => ( ! [F2: fun(A,option(B))] : aa(C,fun(A,option(B)),U2,aa(fun(A,option(B)),C,C5,F2)) = F2
         => ( ! [F2: C,X2: A,Y5: B] :
                ( ! [Xa: A,Ya2: B] :
                    ( ( aa(A,option(B),aa(C,fun(A,option(B)),U2,F2),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,F2)),X2) = aa(B,option(B),some(B),Y5) )
                 => aa(B,$o,aa(A,fun(B,$o),P,X2),Y5) ) )
           => ( ( aa(A,option(B),aa(C,fun(A,option(B)),U2,F),Xb) = aa(B,option(B),some(B),Y) )
             => aa(B,$o,aa(A,fun(B,$o),P,Xb),Y) ) ) ) ) ) ).

% fixp_induct_option
tff(fact_6667_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_ait(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_6668_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),F: 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_air(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))
     => ( ( F = 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_ais(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))) )
       => ( ! [F2: C] : aa(fun(A,option(B)),C,C5,aa(C,fun(A,option(B)),U2,F2)) = F2
         => ( F = aa(C,C,F5,F) ) ) ) ) ).

% option.fixp_rule_uc
tff(fact_6669_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_6670_option_Ofixp__induct__uc,axiom,
    ! [A: $tType,B: $tType,C: $tType,U2: fun(C,fun(A,option(B))),F5: fun(C,C),C5: fun(fun(A,option(B)),C),F: C,P: fun(fun(A,option(B)),$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_air(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))
     => ( ( F = 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_ais(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))) )
       => ( ! [F2: fun(A,option(B))] : aa(C,fun(A,option(B)),U2,aa(fun(A,option(B)),C,C5,F2)) = F2
         => ( 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)),$o,P,aTP_Lamp_di(A,option(B)))
             => ( ! [F2: C] :
                    ( aa(fun(A,option(B)),$o,P,aa(C,fun(A,option(B)),U2,F2))
                   => aa(fun(A,option(B)),$o,P,aa(C,fun(A,option(B)),U2,aa(C,C,F5,F2))) )
               => aa(fun(A,option(B)),$o,P,aa(C,fun(A,option(B)),U2,F)) ) ) ) ) ) ) ).

% option.fixp_induct_uc
tff(fact_6671_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_aiu(option(B),fun(A,option(B)),C2)) ).

% option.const_mono
tff(fact_6672_drop__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,Nb),M)),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),A2)) ) ).

% drop_bit_take_bit
tff(fact_6673_drop__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: nat,Nb: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),bit_se2239418461657761734s_mask(A,Nb)) = bit_se2239418461657761734s_mask(A,aa(nat,nat,minus_minus(nat,Nb),M)) ) ).

% drop_bit_mask_eq
tff(fact_6674_take__bit__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)),A2)) ) ).

% take_bit_drop_bit
tff(fact_6675_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_6676_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_6677_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_6678_bit__word__iff__drop__bit__and,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A2),Nb)
        <=> ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),Nb),A2)),one_one(word(A))) = one_one(word(A)) ) ) ) ).

% bit_word_iff_drop_bit_and
tff(fact_6679_bit__iff__and__drop__bit__eq__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Nb)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),A2)),one_one(A)) = one_one(A) ) ) ) ).

% bit_iff_and_drop_bit_eq_1
tff(fact_6680_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_ahs(B,fun(A,B),C2)) ).

% tailrec.const_mono
tff(fact_6681_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_aiv(fun(A,fun(B,$o)),fun(fun(A,option(B)),$o),P)) ).

% option_admissible
tff(fact_6682_drop__bit__half,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),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),Nb),A2)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% drop_bit_half
tff(fact_6683_stable__imp__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( ( aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))) = A2 )
         => ( aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),A2) = A2 ) ) ) ).

% stable_imp_drop_bit_eq
tff(fact_6684_drop__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,Nb)),A2) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% drop_bit_Suc
tff(fact_6685_drop__bit__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),A2) = aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) ) ).

% drop_bit_eq_div
tff(fact_6686_even__drop__bit__iff__not__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),A2))
        <=> ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Nb) ) ) ).

% even_drop_bit_iff_not_bit
tff(fact_6687_bit__iff__odd__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Nb)
        <=> ~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),A2)) ) ) ).

% bit_iff_odd_drop_bit
tff(fact_6688_heap__step__admissible,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,P: fun(E,fun(A,fun(C,fun(B,fun(D,$o))))),Xb: E] : comple1908693960933563346ssible(fun(A,option(product_prod(B,product_prod(C,D)))),partial_fun_lub(option(product_prod(B,product_prod(C,D))),option(product_prod(B,product_prod(C,D))),A,partial_flat_lub(option(product_prod(B,product_prod(C,D))),none(product_prod(B,product_prod(C,D))))),partial_fun_ord(option(product_prod(B,product_prod(C,D))),option(product_prod(B,product_prod(C,D))),A,partial_flat_ord(option(product_prod(B,product_prod(C,D))),none(product_prod(B,product_prod(C,D))))),aa(E,fun(fun(A,option(product_prod(B,product_prod(C,D)))),$o),aTP_Lamp_aiw(fun(E,fun(A,fun(C,fun(B,fun(D,$o))))),fun(E,fun(fun(A,option(product_prod(B,product_prod(C,D)))),$o)),P),Xb)) ).

% heap_step_admissible
tff(fact_6689_drop__bit__rec,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),A2) = $ite(Nb = zero_zero(nat),A2,aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,minus_minus(nat,Nb),one_one(nat))),aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))))) ) ).

% drop_bit_rec
tff(fact_6690_fixp__induct__tailrec,axiom,
    ! [C: $tType,A: $tType,B: $tType,C2: B,U2: fun(C,fun(A,B)),F5: fun(C,C),C5: fun(fun(A,B),C),F: C,P: fun(A,fun(B,$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_aio(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))
     => ( ( F = 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_aip(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))) )
       => ( ! [F2: fun(A,B)] : aa(C,fun(A,B),U2,aa(fun(A,B),C,C5,F2)) = F2
         => ( ! [F2: C,X2: A,Y5: B] :
                ( ! [Xa: A,Ya2: B] :
                    ( ( aa(A,B,aa(C,fun(A,B),U2,F2),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,F2)),X2) = Y5 )
                 => ( ( Y5 != C2 )
                   => aa(B,$o,aa(A,fun(B,$o),P,X2),Y5) ) ) )
           => ( ( aa(A,B,aa(C,fun(A,B),U2,F),Xb) = Y )
             => ( ( Y != C2 )
               => aa(B,$o,aa(A,fun(B,$o),P,Xb),Y) ) ) ) ) ) ) ).

% fixp_induct_tailrec
tff(fact_6691_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_aix(B,fun(fun(A,fun(B,$o)),fun(fun(A,B),$o)),C2),P)) ).

% tailrec_admissible
tff(fact_6692_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),F: 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_aio(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))
     => ( ( F = 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_aip(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))) )
       => ( ! [F2: C] : aa(fun(A,B),C,C5,aa(C,fun(A,B),U2,F2)) = F2
         => ( F = aa(C,C,F5,F) ) ) ) ) ).

% tailrec.fixp_rule_uc
tff(fact_6693_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_aiy(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_6694_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,aa(word(A),fun(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_6695_take__bit__num__def,axiom,
    ! [Nb: nat,M: num] :
      bit_take_bit_num(Nb,M) = $ite(aa(nat,nat,bit_se2584673776208193580ke_bit(nat,Nb),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,Nb),aa(num,nat,numeral_numeral(nat),M))))) ).

% take_bit_num_def
tff(fact_6696_drop__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% drop_bit_nonnegative_int_iff
tff(fact_6697_drop__bit__minus__one,axiom,
    ! [Nb: nat] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Nb),aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% drop_bit_minus_one
tff(fact_6698_push__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat,A2: 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),Nb),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)),A2) ) ).

% push_bit_push_bit
tff(fact_6699_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_6700_drop__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),Nb),aa(nat,nat,suc,zero_zero(nat))) = aa($o,nat,zero_neq_one_of_bool(nat),Nb = zero_zero(nat)) ).

% drop_bit_of_Suc_0
tff(fact_6701_push__bit__Suc__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),K)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),aa(num,A,numeral_numeral(A),bit0(K))) ) ).

% push_bit_Suc_numeral
tff(fact_6702_drop__bit__Suc__minus__bit0,axiom,
    ! [Nb: nat,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,Nb)),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),Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K))) ).

% drop_bit_Suc_minus_bit0
tff(fact_6703_push__bit__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,K: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,Nb)),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),Nb),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(K)))) ) ).

% push_bit_Suc_minus_numeral
tff(fact_6704_push__bit__minus__one__eq__not__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),aa(A,A,uminus_uminus(A),one_one(A))) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb)) ) ).

% push_bit_minus_one_eq_not_mask
tff(fact_6705_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_6706_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_6707_drop__bit__Suc__minus__bit1,axiom,
    ! [Nb: nat,K: num] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,Nb)),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),Nb),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),inc(K)))) ).

% drop_bit_Suc_minus_bit1
tff(fact_6708_push__bit__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,Nb)),A2) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),bit0(one2)))) ) ).

% push_bit_Suc
tff(fact_6709_push__bit__of__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),one_one(A)) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) ) ).

% push_bit_of_1
tff(fact_6710_even__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),A2))
        <=> ( ( Nb != zero_zero(nat) )
            | aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),A2) ) ) ) ).

% even_push_bit_iff
tff(fact_6711_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_6712_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_6713_push__bit__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(num,nat,numeral_numeral(nat),Nb)),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),Nb))) ) ).

% push_bit_numeral_minus_1
tff(fact_6714_drop__bit__int__code_I2_J,axiom,
    ! [Nb: nat] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,suc,Nb)),zero_zero(int)) = zero_zero(int) ).

% drop_bit_int_code(2)
tff(fact_6715_push__bit__add,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A,B2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),A2)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),B2)) ) ).

% push_bit_add
tff(fact_6716_push__bit__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) = aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),A2)) ) ).

% push_bit_take_bit
tff(fact_6717_take__bit__push__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),A2)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,minus_minus(nat,M),Nb)),A2)) ) ).

% take_bit_push_bit
tff(fact_6718_num__of__nat_Osimps_I1_J,axiom,
    num_of_nat(zero_zero(nat)) = one2 ).

% num_of_nat.simps(1)
tff(fact_6719_div__push__bit__of__1__eq__drop__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] : aa(A,A,aa(A,fun(A,A),divide_divide(A),A2),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),one_one(A))) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),A2) ) ).

% div_push_bit_of_1_eq_drop_bit
tff(fact_6720_bits__ident,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),A2))),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2)) = A2 ) ).

% bits_ident
tff(fact_6721_set__bit__eq__or,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se5668285175392031749et_bit(A,Nb,A2) = aa(A,A,aa(A,fun(A,A),bit_se1065995026697491101ons_or(A),A2),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),one_one(A))) ) ).

% set_bit_eq_or
tff(fact_6722_flip__bit__eq__xor,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : bit_se8732182000553998342ip_bit(A,Nb,A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344971392196577ns_xor(A),A2),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),one_one(A))) ) ).

% flip_bit_eq_xor
tff(fact_6723_push__bit__double,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),aa(A,A,aa(A,fun(A,A),times_times(A),A2),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),Nb),A2)),aa(num,A,numeral_numeral(A),bit0(one2))) ) ).

% push_bit_double
tff(fact_6724_bit__iff__and__push__bit__not__eq__0,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Nb)
        <=> ( aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),one_one(A))) != zero_zero(A) ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
tff(fact_6725_numeral__num__of__nat,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(num,nat,numeral_numeral(nat),num_of_nat(Nb)) = Nb ) ) ).

% numeral_num_of_nat
tff(fact_6726_num__of__nat__One,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),one_one(nat))
     => ( num_of_nat(Nb) = one2 ) ) ).

% num_of_nat_One
tff(fact_6727_push__bit__mask__eq,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Nb: nat] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),bit_se2239418461657761734s_mask(A,Nb)) = 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),Nb),M))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,M))) ) ).

% push_bit_mask_eq
tff(fact_6728_unset__bit__eq__and__not,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] : bit_se2638667681897837118et_bit(A,Nb,A2) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),aa(A,A,bit_ri4277139882892585799ns_not(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),one_one(A)))) ) ).

% unset_bit_eq_and_not
tff(fact_6729_shiftr__integer__conv__div__pow2,axiom,
    ! [Nb: nat,Xb: code_integer] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4197421643247451524op_bit(code_integer),Nb),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))),Nb)) ).

% shiftr_integer_conv_div_pow2
tff(fact_6730_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_6731_drop__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Nb),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))),Nb)) ).

% drop_bit_int_def
tff(fact_6732_push__bit__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),A2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) ) ).

% push_bit_eq_mult
tff(fact_6733_exp__dvdE,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] :
          ( aa(A,$o,dvd_dvd(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)),A2)
         => ~ ! [B3: A] : A2 != aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),B3) ) ) ).

% exp_dvdE
tff(fact_6734_drop__bit__nat__def,axiom,
    ! [Nb: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4197421643247451524op_bit(nat),Nb),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))),Nb)) ).

% drop_bit_nat_def
tff(fact_6735_numeral__num__of__nat__unfold,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Nb: nat] :
          aa(num,A,numeral_numeral(A),num_of_nat(Nb)) = $ite(Nb = zero_zero(nat),one_one(A),aa(nat,A,semiring_1_of_nat(A),Nb)) ) ).

% numeral_num_of_nat_unfold
tff(fact_6736_num__of__nat__double,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),Nb)) = bit0(num_of_nat(Nb)) ) ) ).

% num_of_nat_double
tff(fact_6737_num__of__nat__plus__distrib,axiom,
    ! [M: nat,Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),M)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
       => ( num_of_nat(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)) = aa(num,num,aa(num,fun(num,num),plus_plus(num),num_of_nat(M)),num_of_nat(Nb)) ) ) ) ).

% num_of_nat_plus_distrib
tff(fact_6738_slice__eq__mask,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,M: nat,A2: A] : aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Nb),aa(A,A,bit_se2584673776208193580ke_bit(A,M),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),A2))) = aa(A,A,aa(A,fun(A,A),bit_se5824344872417868541ns_and(A),A2),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),Nb))),aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,Nb)))) ) ).

% slice_eq_mask
tff(fact_6739_take__bit__sum,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,A2: A] : aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aTP_Lamp_aiz(A,fun(nat,A),A2)),set_or7035219750837199246ssThan(nat,zero_zero(nat),Nb)) ) ).

% take_bit_sum
tff(fact_6740_num__of__nat_Osimps_I2_J,axiom,
    ! [Nb: nat] :
      num_of_nat(aa(nat,nat,suc,Nb)) = $ite(aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb),inc(num_of_nat(Nb)),one2) ).

% num_of_nat.simps(2)
tff(fact_6741_word__and__mask__or__conv__and__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),Index: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Nb),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)),Nb),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)),Nb),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_6742_signed__take__bit__code,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat,A2: A] :
          aa(A,A,bit_ri4674362597316999326ke_bit(A,Nb),A2) = $let(
            l: A,
            l:= aa(A,A,bit_se2584673776208193580ke_bit(A,aa(nat,nat,suc,Nb)),A2),
            $ite(aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),l),Nb),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,Nb)),aa(A,A,uminus_uminus(A),one_one(A)))),l) ) ) ).

% signed_take_bit_code
tff(fact_6743_set__bits__aux__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,$o),Nb: nat,W: word(A)] :
          code_T2661198915054445665ts_aux(A,F,aa(nat,nat,suc,Nb),W) = code_T2661198915054445665ts_aux(A,F,Nb,
            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,F,Nb),one_one(word(A)),zero_zero(word(A))))) ) ).

% set_bits_aux_Suc
tff(fact_6744_set__bits__aux__rec,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,$o),Nb: nat,W: word(A)] :
          code_T2661198915054445665ts_aux(A,F,Nb,W) = $ite(
            Nb = zero_zero(nat),
            W,
            $let(
              n: nat,
              n:= aa(nat,nat,minus_minus(nat,Nb),one_one(nat)),
              code_T2661198915054445665ts_aux(A,F,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,F,n),one_one(word(A)),zero_zero(word(A))))) ) ) ) ).

% set_bits_aux_rec
tff(fact_6745_push__bit__nonnegative__int__iff,axiom,
    ! [Nb: nat,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Nb),K))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),K) ) ).

% push_bit_nonnegative_int_iff
tff(fact_6746_push__bit__of__Suc__0,axiom,
    ! [Nb: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),Nb),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))),Nb) ).

% push_bit_of_Suc_0
tff(fact_6747_drop__bit__push__bit__int,axiom,
    ! [M: nat,Nb: 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),Nb),K)) = aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,minus_minus(nat,M),Nb)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),aa(nat,nat,minus_minus(nat,Nb),M)),K)) ).

% drop_bit_push_bit_int
tff(fact_6748_set__bit__nat__def,axiom,
    ! [M: nat,Nb: nat] : bit_se5668285175392031749et_bit(nat,M,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se1065995026697491101ons_or(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),M),one_one(nat))) ).

% set_bit_nat_def
tff(fact_6749_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_6750_flip__bit__nat__def,axiom,
    ! [M: nat,Nb: nat] : bit_se8732182000553998342ip_bit(nat,M,Nb) = aa(nat,nat,aa(nat,fun(nat,nat),bit_se5824344971392196577ns_xor(nat),Nb),aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),M),one_one(nat))) ).

% flip_bit_nat_def
tff(fact_6751_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,Nb: 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)),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
        & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),aa(nat,nat,minus_minus(nat,Nb),M)) ) ) ).

% bit_push_bit_iff_int
tff(fact_6752_Bit__Operations_Oset__bit__int__def,axiom,
    ! [Nb: nat,K: int] : bit_se5668285175392031749et_bit(int,Nb,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),Nb),one_one(int))) ).

% Bit_Operations.set_bit_int_def
tff(fact_6753_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q2: nat,Nb: 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)),Nb)
    <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
        & aa(nat,$o,aa(nat,fun(nat,$o),bit_se5641148757651400278ts_bit(nat),Q2),aa(nat,nat,minus_minus(nat,Nb),M)) ) ) ).

% bit_push_bit_iff_nat
tff(fact_6754_flip__bit__int__def,axiom,
    ! [Nb: nat,K: int] : bit_se8732182000553998342ip_bit(int,Nb,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),Nb),one_one(int))) ).

% flip_bit_int_def
tff(fact_6755_shiftl__integer__conv__mult__pow2,axiom,
    ! [Nb: nat,Xb: code_integer] : aa(code_integer,code_integer,aa(nat,fun(code_integer,code_integer),bit_se4730199178511100633sh_bit(code_integer),Nb),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))),Nb)) ).

% shiftl_integer_conv_mult_pow2
tff(fact_6756_unset__bit__int__def,axiom,
    ! [Nb: nat,K: int] : bit_se2638667681897837118et_bit(int,Nb,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),Nb),one_one(int)))) ).

% unset_bit_int_def
tff(fact_6757_push__bit__int__def,axiom,
    ! [Nb: nat,K: int] : aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Nb),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))),Nb)) ).

% push_bit_int_def
tff(fact_6758_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_6759_push__bit__nat__def,axiom,
    ! [Nb: nat,M: nat] : aa(nat,nat,aa(nat,fun(nat,nat),bit_se4730199178511100633sh_bit(nat),Nb),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))),Nb)) ).

% push_bit_nat_def
tff(fact_6760_push__bit__minus__one,axiom,
    ! [Nb: nat] : aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Nb),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))),Nb)) ).

% push_bit_minus_one
tff(fact_6761_test__bit__split,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A2: word(A),B2: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A2),B2) )
         => ( ! [N8: nat] :
                ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),B2),N8)
              <=> ( aa(nat,$o,ord_less(nat,N8),aa(word(B),nat,size_size(word(B)),B2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),N8) ) )
            & ! [M3: nat] :
                ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A2),M3)
              <=> ( aa(nat,$o,ord_less(nat,M3),aa(word(A),nat,size_size(word(A)),A2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M3),aa(word(B),nat,size_size(word(B)),B2))) ) ) ) ) ) ).

% test_bit_split
tff(fact_6762_test__bit__split_H,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A2: word(A),B2: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A2),B2) )
         => ! [N8: nat,M3: nat] :
              ( ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),B2),N8)
              <=> ( aa(nat,$o,ord_less(nat,N8),aa(word(B),nat,size_size(word(B)),B2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),N8) ) )
              & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A2),M3)
              <=> ( aa(nat,$o,ord_less(nat,M3),aa(word(A),nat,size_size(word(A)),A2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M3),aa(word(B),nat,size_size(word(B)),B2))) ) ) ) ) ) ).

% test_bit_split'
tff(fact_6763_test__bit__split__eq,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [C2: word(C),A2: word(A),B2: word(B)] :
          ( ( word_split(C,A,B,C2) = aa(word(B),product_prod(word(A),word(B)),aa(word(A),fun(word(B),product_prod(word(A),word(B))),product_Pair(word(A),word(B)),A2),B2) )
        <=> ( ! [N6: nat] :
                ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),B2),N6)
              <=> ( aa(nat,$o,ord_less(nat,N6),aa(word(B),nat,size_size(word(B)),B2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),N6) ) )
            & ! [M7: nat] :
                ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A2),M7)
              <=> ( aa(nat,$o,ord_less(nat,M7),aa(word(A),nat,size_size(word(A)),A2))
                  & aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),C2),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M7),aa(word(B),nat,size_size(word(B)),B2))) ) ) ) ) ) ).

% test_bit_split_eq
tff(fact_6764_set__bit__integer__conv__masks,axiom,
    ! [Xb: code_integer,I: nat,B2: $o] :
      aa($o,code_integer,aa(nat,fun($o,code_integer),aa(code_integer,fun(nat,fun($o,code_integer)),generi7602027413899671122et_bit(code_integer),Xb),I),(B2)) = $ite((B2),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),I),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),I),one_one(code_integer))))) ).

% set_bit_integer_conv_masks
tff(fact_6765_admissible__all,axiom,
    ! [A: $tType,B: $tType,Lub: fun(set(B),B),Ord: fun(B,fun(B,$o)),P: fun(B,fun(A,$o))] :
      ( ! [Y5: A] : comple1908693960933563346ssible(B,Lub,Ord,aa(A,fun(B,$o),aTP_Lamp_aja(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),Y5))
     => comple1908693960933563346ssible(B,Lub,Ord,aTP_Lamp_ajb(fun(B,fun(A,$o)),fun(B,$o),P)) ) ).

% admissible_all
tff(fact_6766_admissible__True,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,$o))] : comple1908693960933563346ssible(A,Lub,Ord,aTP_Lamp_bn(A,$o)) ).

% admissible_True
tff(fact_6767_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_st(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)) ) ) ).

% admissible_conj
tff(fact_6768_admissible__const,axiom,
    ! [A: $tType,Lub: fun(set(A),A),Ord: fun(A,fun(A,$o)),Ta: $o] : comple1908693960933563346ssible(A,Lub,Ord,aTP_Lamp_ex($o,fun(A,$o),(Ta))) ).

% admissible_const
tff(fact_6769_bit__set__bit__iff__2n,axiom,
    ! [A: $tType] :
      ( generic_set_set_bit(A)
     => ! [A2: A,M: nat,B2: $o,Nb: nat] :
          ( aa(nat,$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),A2),M),(B2))),Nb)
        <=> ( $ite(M = Nb,(B2),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),Nb))
            & ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) != zero_zero(A) ) ) ) ) ).

% bit_set_bit_iff_2n
tff(fact_6770_Code__Numeral_Opositive__def,axiom,
    code_positive = numeral_numeral(code_integer) ).

% Code_Numeral.positive_def
tff(fact_6771_test__bit__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: 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),Nb),(Xb))),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),aa(word(A),nat,size_size(word(A)),W))
            & (Xb) ) ) ) ).

% test_bit_set
tff(fact_6772_test__bit__set__gen,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: 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),Nb),(Xb))),M)
        <=> $ite(
              M = Nb,
              ( aa(nat,$o,ord_less(nat,Nb),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_6773_word__set__nth__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat,B2: $o] :
          ( ( aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),W),Nb),(B2)) = W )
        <=> ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Nb)
            <=> (B2) )
            | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),W)),Nb) ) ) ) ).

% word_set_nth_iff
tff(fact_6774_set__bit__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: nat,B2: $o] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),Xb)),Nb)
         => ( aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),Xb),Nb),(B2)) = Xb ) ) ) ).

% set_bit_beyond
tff(fact_6775_word__clr__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),W),Nb),$false)),W) ) ).

% word_clr_le
tff(fact_6776_word__set__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),W),Nb),$true)) ) ).

% word_set_ge
tff(fact_6777_one__bit__pow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : aa($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))),Nb),$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))),Nb) ) ).

% one_bit_pow
tff(fact_6778_int__set__bit__True__conv__OR,axiom,
    ! [I: int,Nb: nat] : aa($o,int,aa(nat,fun($o,int),aa(int,fun(nat,fun($o,int)),generi7602027413899671122et_bit(int),I),Nb),$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),Nb),one_one(int))) ).

% int_set_bit_True_conv_OR
tff(fact_6779_int__set__bit__False__conv__NAND,axiom,
    ! [I: int,Nb: nat] : aa($o,int,aa(nat,fun($o,int),aa(int,fun(nat,fun($o,int)),generi7602027413899671122et_bit(int),I),Nb),$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),Nb),one_one(int)))) ).

% int_set_bit_False_conv_NAND
tff(fact_6780_int__set__bit__conv__ops,axiom,
    ! [I: int,Nb: nat,B2: $o] :
      aa($o,int,aa(nat,fun($o,int),aa(int,fun(nat,fun($o,int)),generi7602027413899671122et_bit(int),I),Nb),(B2)) = $ite((B2),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),Nb),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),Nb),one_one(int))))) ).

% int_set_bit_conv_ops
tff(fact_6781_case__prod__app,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F: fun(B,fun(C,fun(D,A))),Xb: product_prod(B,C),Y: D] : aa(D,A,aa(product_prod(B,C),fun(D,A),product_case_prod(B,C,fun(D,A),F),Xb),Y) = aa(product_prod(B,C),A,product_case_prod(B,C,A,aa(D,fun(B,fun(C,A)),aTP_Lamp_el(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),F),Y)),Xb) ).

% case_prod_app
tff(fact_6782_the__elem__def,axiom,
    ! [A: $tType,X5: set(A)] : the_elem(A,X5) = the(A,aTP_Lamp_ajc(set(A),fun(A,$o),X5)) ).

% the_elem_def
tff(fact_6783_predicate2D__conj,axiom,
    ! [A: $tType,B: $tType,P: fun(A,fun(B,$o)),Q: fun(A,fun(B,$o)),R: $o,Xb: A,Y: B] :
      ( ( aa(fun(A,fun(B,$o)),$o,aa(fun(A,fun(B,$o)),fun(fun(A,fun(B,$o)),$o),ord_less_eq(fun(A,fun(B,$o))),P),Q)
        & (R) )
     => ( (R)
        & ( aa(B,$o,aa(A,fun(B,$o),P,Xb),Y)
         => aa(B,$o,aa(A,fun(B,$o),Q,Xb),Y) ) ) ) ).

% predicate2D_conj
tff(fact_6784_fun__cong__unused__0,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( zero(B)
     => ! [F: fun(fun(A,B),C),G: C] :
          ( ! [X2: fun(A,B)] : aa(fun(A,B),C,F,X2) = G
         => ( aa(fun(A,B),C,F,aTP_Lamp_ajd(A,B)) = G ) ) ) ).

% fun_cong_unused_0
tff(fact_6785_eq__subset,axiom,
    ! [A: $tType,P: fun(A,fun(A,$o))] : aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),aTP_Lamp_aje(fun(A,fun(A,$o)),fun(A,fun(A,$o)),P)) ).

% eq_subset
tff(fact_6786_the__elem__set,axiom,
    ! [A: $tType,Xb: A] : the_elem(A,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A)))) = Xb ).

% the_elem_set
tff(fact_6787_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_6788_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),aa(int,fun(list(int),list(int)),cons(int),J),Js))) ).

% upto_aux_rec
tff(fact_6789_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_6790_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_6791_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_6792_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_6793_word__lsb__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),A2)
        <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),aa(word(A),int,semiring_1_unsigned(A,int),A2)) ) ) ).

% word_lsb_def
tff(fact_6794_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_6795_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_6796_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_6797_upto_Opsimps,axiom,
    ! [I: int,J: int] :
      ( aa(product_prod(int,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),I),J))
     => ( upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),aa(int,fun(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_6798_upto_Opelims,axiom,
    ! [Xb: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xb,Xaa) = Y )
     => ( aa(product_prod(int,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),Xb),Xaa))
       => ~ ( ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),Xaa),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),Xb),upto(aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),one_one(int)),Xaa)),nil(int)) )
           => ~ aa(product_prod(int,int),$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),Xb),Xaa)) ) ) ) ).

% upto.pelims
tff(fact_6799_int__lsb__numeral_I2_J,axiom,
    aa(int,$o,least_8051144512741203767sb_lsb(int),one_one(int)) ).

% int_lsb_numeral(2)
tff(fact_6800_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_6801_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_6802_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_6803_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_6804_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_6805_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_6806_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_6807_nth__upto,axiom,
    ! [I: int,K: nat,J: int] :
      ( aa(int,$o,aa(int,fun(int,$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_6808_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_6809_upto__rec__numeral_I1_J,axiom,
    ! [M: num,Nb: num] :
      upto(aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),Nb)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Nb)),aa(list(int),list(int),aa(int,fun(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),Nb))),nil(int)) ).

% upto_rec_numeral(1)
tff(fact_6810_upto__rec__numeral_I4_J,axiom,
    ! [M: num,Nb: 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),Nb))) = $ite(aa(int,$o,aa(int,fun(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),Nb))),aa(list(int),list(int),aa(int,fun(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),Nb)))),nil(int)) ).

% upto_rec_numeral(4)
tff(fact_6811_upto__rec__numeral_I3_J,axiom,
    ! [M: num,Nb: num] :
      upto(aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Nb)) = $ite(aa(int,$o,aa(int,fun(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),Nb)),aa(list(int),list(int),aa(int,fun(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),Nb))),nil(int)) ).

% upto_rec_numeral(3)
tff(fact_6812_upto__rec__numeral_I2_J,axiom,
    ! [M: num,Nb: num] :
      upto(aa(num,int,numeral_numeral(int),M),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) = $ite(aa(int,$o,aa(int,fun(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),Nb))),aa(list(int),list(int),aa(int,fun(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),Nb)))),nil(int)) ).

% upto_rec_numeral(2)
tff(fact_6813_atLeastAtMost__upto,axiom,
    ! [I: int,J: int] : set_or1337092689740270186AtMost(int,I,J) = aa(list(int),set(int),set2(int),upto(I,J)) ).

% atLeastAtMost_upto
tff(fact_6814_upto__split2,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = aa(list(int),list(int),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_6815_upto__split1,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = aa(list(int),list(int),append(int,upto(I,aa(int,int,minus_minus(int,J),one_one(int)))),upto(J,K)) ) ) ) ).

% upto_split1
tff(fact_6816_atLeastLessThan__upto,axiom,
    ! [I: int,J: int] : set_or7035219750837199246ssThan(int,I,J) = aa(list(int),set(int),set2(int),upto(I,aa(int,int,minus_minus(int,J),one_one(int)))) ).

% atLeastLessThan_upto
tff(fact_6817_upto_Oelims,axiom,
    ! [Xb: int,Xaa: int,Y: list(int)] :
      ( ( upto(Xb,Xaa) = Y )
     => ( Y = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Xb),Xaa),aa(list(int),list(int),aa(int,fun(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_6818_upto_Osimps,axiom,
    ! [I: int,J: int] :
      upto(I,J) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J),aa(list(int),list(int),aa(int,fun(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_6819_upto__rec1,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = aa(list(int),list(int),aa(int,fun(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_6820_upto__rec2,axiom,
    ! [I: int,J: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( upto(I,J) = aa(list(int),list(int),append(int,upto(I,aa(int,int,minus_minus(int,J),one_one(int)))),aa(list(int),list(int),aa(int,fun(list(int),list(int)),cons(int),J),nil(int))) ) ) ).

% upto_rec2
tff(fact_6821_bin__last__conv__lsb,axiom,
    aTP_Lamp_ajf(int,$o) = least_8051144512741203767sb_lsb(int) ).

% bin_last_conv_lsb
tff(fact_6822_upto__split3,axiom,
    ! [I: int,J: int,K: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),I),J)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),J),K)
       => ( upto(I,K) = aa(list(int),list(int),append(int,upto(I,aa(int,int,minus_minus(int,J),one_one(int)))),aa(list(int),list(int),aa(int,fun(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_6823_concat__bit__Suc,axiom,
    ! [Nb: nat,K: int,L: int] : bit_concat_bit(aa(nat,nat,suc,Nb),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))),bit_concat_bit(Nb,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_6824_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,aa(nat,fun(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_6825_concat__bit__nonnegative__iff,axiom,
    ! [Nb: nat,K: int,L: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),bit_concat_bit(Nb,K,L))
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),L) ) ).

% concat_bit_nonnegative_iff
tff(fact_6826_concat__bit__assoc,axiom,
    ! [Nb: nat,K: int,M: nat,L: int,R3: int] : bit_concat_bit(Nb,K,bit_concat_bit(M,L,R3)) = bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb),bit_concat_bit(Nb,K,L),R3) ).

% concat_bit_assoc
tff(fact_6827_concat__bit__eq,axiom,
    ! [Nb: nat,K: int,L: int] : bit_concat_bit(Nb,K,L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),K)),aa(int,int,aa(nat,fun(int,int),bit_se4730199178511100633sh_bit(int),Nb),L)) ).

% concat_bit_eq
tff(fact_6828_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,Nb: nat] :
      ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),bit_concat_bit(M,K,L)),Nb)
    <=> ( ( aa(nat,$o,ord_less(nat,Nb),M)
          & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Nb) )
        | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
          & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),L),aa(nat,nat,minus_minus(nat,Nb),M)) ) ) ) ).

% bit_concat_bit_iff
tff(fact_6829_test__bit__cat,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [A2: word(B),B2: word(C),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_cat(B,C,A,A2,B2)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),aa(word(A),nat,size_size(word(A)),word_cat(B,C,A,A2,B2)))
            & $ite(aa(nat,$o,ord_less(nat,Nb),aa(word(C),nat,size_size(word(C)),B2)),aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),B2),Nb),aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),A2),aa(nat,nat,minus_minus(nat,Nb),aa(word(C),nat,size_size(word(C)),B2)))) ) ) ) ).

% test_bit_cat
tff(fact_6830_word__cat__split__size,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(A)
        & type_len(B) )
     => ! [Ta: word(A),U: word(B),V: word(C)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(word(A),nat,size_size(word(A)),Ta)),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,Ta) )
           => ( Ta = word_cat(B,C,A,U,V) ) ) ) ) ).

% word_cat_split_size
tff(fact_6831_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,aa(nat,fun(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_6832_cat__slices,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A)
        & type_len(C) )
     => ! [A2: word(A),Nb: nat,C2: word(B),B2: word(C)] :
          ( ( A2 = aa(word(B),word(A),slice2(B,A,Nb),C2) )
         => ( ( B2 = aa(word(B),word(C),slice2(B,C,zero_zero(nat)),C2) )
           => ( ( Nb = aa(word(C),nat,size_size(word(C)),B2) )
             => ( aa(nat,$o,aa(nat,fun(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)),A2)),aa(word(C),nat,size_size(word(C)),B2)))
               => ( word_cat(A,C,B,A2,B2) = C2 ) ) ) ) ) ) ).

% cat_slices
tff(fact_6833_mult_Osafe__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Xb: A,Y: A,A2: A,B2: A] :
          ( syntax7388354845996824322omatch(A,A,aa(A,A,aa(A,fun(A,A),times_times(A),Xb),Y),A2)
         => ( aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2) = aa(A,A,aa(A,fun(A,A),times_times(A),B2),A2) ) ) ) ).

% mult.safe_commute
tff(fact_6834_assn__aci_I11_J,axiom,
    ! [Xb: assn,Y: assn,A2: assn,B2: assn] :
      ( syntax7388354845996824322omatch(assn,assn,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Xb),Y),A2)
     => ( aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),A2),B2) = aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),B2),A2) ) ) ).

% assn_aci(11)
tff(fact_6835_add_Oright__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A2: A,B2: 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),C2)),B2) ) ).

% add.right_commute
tff(fact_6836_add_Oright__assoc,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [A2: A,B2: 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),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),C2)) ) ).

% add.right_assoc
tff(fact_6837_mult_Oright__assoc,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(A,A,aa(A,fun(A,A),times_times(A),B2),C2)) ) ).

% mult.right_assoc
tff(fact_6838_mult_Oright__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [A2: A,B2: A,C2: A] : aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),B2)),C2) = aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,aa(A,fun(A,A),times_times(A),A2),C2)),B2) ) ).

% mult.right_commute
tff(fact_6839_slice__cat1,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(C)
        & type_len(B) )
     => ! [A2: word(A),B2: word(B)] :
          ( aa(nat,$o,aa(nat,fun(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)),A2)),aa(word(B),nat,size_size(word(B)),B2))),aa(word(C),nat,size_size(word(C)),word_cat(A,B,C,A2,B2)))
         => ( aa(word(C),word(A),slice2(C,A,aa(word(B),nat,size_size(word(B)),B2)),word_cat(A,B,C,A2,B2)) = A2 ) ) ) ).

% slice_cat1
tff(fact_6840_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 = aa(word(C),word(A),slice2(C,A,aa(word(B),nat,size_size(word(B)),V)),W) )
            & ( V = aa(word(C),word(B),slice2(C,B,zero_zero(nat)),W) ) ) ) ) ).

% split_slices
tff(fact_6841_add_Osafe__commute,axiom,
    ! [A: $tType] :
      ( ab_semigroup_add(A)
     => ! [Xb: A,Y: A,A2: A,B2: A] :
          ( syntax7388354845996824322omatch(A,A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y),A2)
         => ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A2),B2) = aa(A,A,aa(A,fun(A,A),plus_plus(A),B2),A2) ) ) ) ).

% add.safe_commute
tff(fact_6842_sum__diff1_H__aux,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [F5: set(A),I5: set(A),F: fun(A,B),I: A] :
          ( finite_finite2(A,F5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajg(set(A),fun(fun(A,B),fun(A,$o)),I5),F))),F5)
           => ( groups1027152243600224163dd_sum(A,B,F,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,F,I5)),aa(A,B,F,I)),groups1027152243600224163dd_sum(A,B,F,I5)) ) ) ) ) ).

% sum_diff1'_aux
tff(fact_6843_bit__horner__sum__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Bs2: list($o),Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),Bs2)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),aa(list($o),nat,size_size(list($o)),Bs2))
            & aa(nat,$o,nth($o,Bs2),Nb) ) ) ) ).

% bit_horner_sum_bit_iff
tff(fact_6844_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_tm(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_6845_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_ajh(fun(B,A),fun(set(B),fun(B,$o)),G),I5))) = groups1027152243600224163dd_sum(B,A,G,I5) ) ).

% sum.non_neutral'
tff(fact_6846_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_aji(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_6847_sum_Omono__neutral__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T5: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
         => ( ! [X2: A] :
                ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T5),S2))
               => ( aa(A,B,G,X2) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,S2) = groups1027152243600224163dd_sum(A,B,G,T5) ) ) ) ) ).

% sum.mono_neutral_left'
tff(fact_6848_sum_Omono__neutral__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T5: set(A),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
         => ( ! [X2: A] :
                ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T5),S2))
               => ( aa(A,B,G,X2) = zero_zero(B) ) )
           => ( groups1027152243600224163dd_sum(A,B,G,T5) = groups1027152243600224163dd_sum(A,B,G,S2) ) ) ) ) ).

% sum.mono_neutral_right'
tff(fact_6849_sum_Omono__neutral__cong__left_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T5: set(A),H: fun(A,B),G: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
         => ( ! [I3: A] :
                ( member(A,I3,aa(set(A),set(A),minus_minus(set(A),T5),S2))
               => ( aa(A,B,H,I3) = zero_zero(B) ) )
           => ( ! [X2: A] :
                  ( member(A,X2,S2)
                 => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,S2) = groups1027152243600224163dd_sum(A,B,H,T5) ) ) ) ) ) ).

% sum.mono_neutral_cong_left'
tff(fact_6850_sum_Omono__neutral__cong__right_H,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [S2: set(A),T5: set(A),G: fun(A,B),H: fun(A,B)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),T5)
         => ( ! [X2: A] :
                ( member(A,X2,aa(set(A),set(A),minus_minus(set(A),T5),S2))
               => ( aa(A,B,G,X2) = zero_zero(B) ) )
           => ( ! [X2: A] :
                  ( member(A,X2,S2)
                 => ( aa(A,B,G,X2) = aa(A,B,H,X2) ) )
             => ( groups1027152243600224163dd_sum(A,B,G,T5) = groups1027152243600224163dd_sum(A,B,H,S2) ) ) ) ) ) ).

% sum.mono_neutral_cong_right'
tff(fact_6851_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_tm(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_tm(set(A),fun(fun(A,B),fun(A,$o)),I5),H)))
           => ( groups1027152243600224163dd_sum(A,B,aa(fun(A,B),fun(A,B),aTP_Lamp_aji(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_6852_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_ajh(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_ajh(fun(B,A),fun(set(B),fun(B,$o)),P2),I5))),zero_zero(A)) ) ).

% sum.G_def
tff(fact_6853_sum__diff1_H,axiom,
    ! [B: $tType,A: $tType] :
      ( ab_group_add(B)
     => ! [I5: set(A),F: fun(A,B),I: A] :
          ( finite_finite2(A,collect(A,aa(fun(A,B),fun(A,$o),aTP_Lamp_ajg(set(A),fun(fun(A,B),fun(A,$o)),I5),F)))
         => ( groups1027152243600224163dd_sum(A,B,F,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,F,I5)),aa(A,B,F,I)),groups1027152243600224163dd_sum(A,B,F,I5)) ) ) ) ).

% sum_diff1'
tff(fact_6854_horner__sum__of__bool__2__less,axiom,
    ! [Bs2: list($o)] : aa(int,$o,ord_less(int,groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),Bs2)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),aa(list($o),nat,size_size(list($o)),Bs2))) ).

% horner_sum_of_bool_2_less
tff(fact_6855_horner__sum__simps_I2_J,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_0(A)
     => ! [F: fun(B,A),A2: A,Xb: B,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F,A2,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xb),Xs)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,F,Xb)),aa(A,A,aa(A,fun(A,A),times_times(A),A2),groups4207007520872428315er_sum(B,A,F,A2,Xs))) ) ).

% horner_sum_simps(2)
tff(fact_6856_horner__sum__eq__sum,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F,A2,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_ajj(fun(B,A),fun(A,fun(list(B),fun(nat,A))),F),A2),Xs)),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(list(B),nat,size_size(list(B)),Xs))) ) ).

% horner_sum_eq_sum
tff(fact_6857_horner__sum__foldr,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_semiring_0(A)
     => ! [F: fun(B,A),A2: A,Xs: list(B)] : groups4207007520872428315er_sum(B,A,F,A2,Xs) = foldr(B,A,aa(A,fun(B,fun(A,A)),aTP_Lamp_ajk(fun(B,A),fun(A,fun(B,fun(A,A))),F),A2),Xs,zero_zero(A)) ) ).

% horner_sum_foldr
tff(fact_6858_horner__sum__append,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [F: fun(B,A),A2: A,Xs: list(B),Ys: list(B)] : groups4207007520872428315er_sum(B,A,F,A2,aa(list(B),list(B),append(B,Xs),Ys)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),groups4207007520872428315er_sum(B,A,F,A2,Xs)),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),A2),aa(list(B),nat,size_size(list(B)),Xs))),groups4207007520872428315er_sum(B,A,F,A2,Ys))) ) ).

% horner_sum_append
tff(fact_6859_horner__sum__bit__eq__take__bit,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] : groups4207007520872428315er_sum($o,A,zero_neq_one_of_bool(A),aa(num,A,numeral_numeral(A),bit0(one2)),aa(list(nat),list($o),map(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2)),upt(zero_zero(nat),Nb))) = aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),A2) ) ).

% horner_sum_bit_eq_take_bit
tff(fact_6860_sdiv__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,aa(int,fun(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)),A2)),one_one(nat))))),signed7115095781618012415divide(int,ring_1_signed(A,int,A2),ring_1_signed(A,int,B2))) ) ).

% sdiv_word_min
tff(fact_6861_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_6862_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
     => ( upt(I,J) = nil(nat) ) ) ).

% upt_conv_Nil
tff(fact_6863_upt__merge,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),K) )
     => ( aa(list(nat),list(nat),append(nat,upt(I,J)),upt(J,K)) = upt(I,K) ) ) ).

% upt_merge
tff(fact_6864_int__sdiv__simps_I1_J,axiom,
    ! [A2: int] : signed7115095781618012415divide(int,A2,one_one(int)) = A2 ).

% int_sdiv_simps(1)
tff(fact_6865_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( upt(I,J) = nil(nat) )
    <=> ( ( J = zero_zero(nat) )
        | aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I) ) ) ).

% upt_eq_Nil_conv
tff(fact_6866_int__sdiv__same__is__1,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero(int) )
     => ( ( signed7115095781618012415divide(int,A2,B2) = A2 )
      <=> ( B2 = one_one(int) ) ) ) ).

% int_sdiv_same_is_1
tff(fact_6867_int__sdiv__simps_I3_J,axiom,
    ! [A2: int] : signed7115095781618012415divide(int,A2,aa(int,int,uminus_uminus(int),one_one(int))) = aa(int,int,uminus_uminus(int),A2) ).

% int_sdiv_simps(3)
tff(fact_6868_sdiv__int__numeral__numeral,axiom,
    ! [M: num,Nb: num] : signed7115095781618012415divide(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),Nb)) = aa(int,int,aa(int,fun(int,int),divide_divide(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Nb)) ).

% sdiv_int_numeral_numeral
tff(fact_6869_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_6870_upt__rec__numeral,axiom,
    ! [M: num,Nb: num] :
      upt(aa(num,nat,numeral_numeral(nat),M),aa(num,nat,numeral_numeral(nat),Nb)) = $ite(aa(nat,$o,ord_less(nat,aa(num,nat,numeral_numeral(nat),M)),aa(num,nat,numeral_numeral(nat),Nb)),aa(list(nat),list(nat),aa(nat,fun(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),Nb))),nil(nat)) ).

% upt_rec_numeral
tff(fact_6871_int__sdiv__negated__is__minus1,axiom,
    ! [A2: int,B2: int] :
      ( ( A2 != zero_zero(int) )
     => ( ( signed7115095781618012415divide(int,A2,B2) = aa(int,int,uminus_uminus(int),A2) )
      <=> ( B2 = aa(int,int,uminus_uminus(int),one_one(int)) ) ) ) ).

% int_sdiv_negated_is_minus1
tff(fact_6872_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_6873_map__fst__enumerate,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(product_prod(nat,A)),list(nat),map(product_prod(nat,A),nat,product_fst(nat,A)),enumerate(A,Nb,Xs)) = upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),aa(list(A),nat,size_size(list(A)),Xs))) ).

% map_fst_enumerate
tff(fact_6874_upt__eq__lel__conv,axiom,
    ! [L: nat,H: nat,Is1: list(nat),I: nat,Is2: list(nat)] :
      ( ( upt(L,H) = aa(list(nat),list(nat),append(nat,Is1),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),Is2)) )
    <=> ( ( Is1 = upt(L,I) )
        & ( Is2 = upt(aa(nat,nat,suc,I),H) )
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),L),I)
        & aa(nat,$o,ord_less(nat,I),H) ) ) ).

% upt_eq_lel_conv
tff(fact_6875_upt__rec,axiom,
    ! [I: nat,J: nat] :
      upt(I,J) = $ite(aa(nat,$o,ord_less(nat,I),J),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),upt(aa(nat,nat,suc,I),J)),nil(nat)) ).

% upt_rec
tff(fact_6876_map__add__upt,axiom,
    ! [Nb: nat,M: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_ajl(nat,fun(nat,nat),Nb)),upt(zero_zero(nat),M)) = upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)) ).

% map_add_upt
tff(fact_6877_map__add__upt_H,axiom,
    ! [Ofs: nat,A2: nat,B2: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_ajl(nat,fun(nat,nat),Ofs)),upt(A2,B2)) = upt(aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),A2),Ofs),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),B2),Ofs)) ).

% map_add_upt'
tff(fact_6878_map__Suc__upt,axiom,
    ! [M: nat,Nb: nat] : aa(list(nat),list(nat),map(nat,nat,suc),upt(M,Nb)) = upt(aa(nat,nat,suc,M),aa(nat,nat,suc,Nb)) ).

% map_Suc_upt
tff(fact_6879_upt__conv__Cons__Cons,axiom,
    ! [M: nat,Nb: nat,Ns: list(nat),Q2: nat] :
      ( ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),M),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Nb),Ns)) = upt(M,Q2) )
    <=> ( aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),Nb),Ns) = upt(aa(nat,nat,suc,M),Q2) ) ) ).

% upt_conv_Cons_Cons
tff(fact_6880_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),I),upt(aa(nat,nat,suc,I),J)) ) ) ).

% upt_conv_Cons
tff(fact_6881_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( upt(I,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),J),K)) = aa(list(nat),list(nat),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_6882_upt__eq__append__conv,axiom,
    ! [I: nat,J: nat,Xs: list(nat),Ys: list(nat)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( ( upt(I,J) = aa(list(nat),list(nat),append(nat,Xs),Ys) )
      <=> ? [K4: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),K4)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K4),J)
            & ( upt(I,K4) = Xs )
            & ( upt(K4,J) = Ys ) ) ) ) ).

% upt_eq_append_conv
tff(fact_6883_atLeastLessThan__upt,axiom,
    ! [I: nat,J: nat] : set_or7035219750837199246ssThan(nat,I,J) = aa(list(nat),set(nat),set2(nat),upt(I,J)) ).

% atLeastLessThan_upt
tff(fact_6884_upt__append,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,ord_less(nat,I),J)
     => ( aa(list(nat),list(nat),append(nat,upt(zero_zero(nat),I)),upt(I,J)) = upt(zero_zero(nat),J) ) ) ).

% upt_append
tff(fact_6885_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      upt(I,aa(nat,nat,suc,J)) = $ite(aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J),aa(list(nat),list(nat),append(nat,upt(I,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))),nil(nat)) ).

% upt_Suc
tff(fact_6886_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),I),J)
     => ( upt(I,aa(nat,nat,suc,J)) = aa(list(nat),list(nat),append(nat,upt(I,J)),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) ) ) ).

% upt_Suc_append
tff(fact_6887_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,Xb: nat,Xs: list(nat)] :
      ( ( upt(I,J) = aa(list(nat),list(nat),aa(nat,fun(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_6888_map__decr__upt,axiom,
    ! [M: nat,Nb: nat] : aa(list(nat),list(nat),map(nat,nat,aTP_Lamp_ajm(nat,nat)),upt(aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = upt(M,Nb) ).

% map_decr_upt
tff(fact_6889_atLeastAtMost__upt,axiom,
    ! [Nb: nat,M: nat] : set_or1337092689740270186AtMost(nat,Nb,M) = aa(list(nat),set(nat),set2(nat),upt(Nb,aa(nat,nat,suc,M))) ).

% atLeastAtMost_upt
tff(fact_6890_atLeast__upt,axiom,
    ! [Nb: nat] : set_ord_lessThan(nat,Nb) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),Nb)) ).

% atLeast_upt
tff(fact_6891_map__replicate__trivial,axiom,
    ! [A: $tType,Xb: A,I: nat] : aa(list(nat),list(A),map(nat,A,aTP_Lamp_ajn(A,fun(nat,A),Xb)),upt(zero_zero(nat),I)) = replicate(A,I,Xb) ).

% map_replicate_trivial
tff(fact_6892_enumerate__map__upt,axiom,
    ! [A: $tType,Nb: nat,F: fun(nat,A),M: nat] : enumerate(A,Nb,aa(list(nat),list(A),map(nat,A,F),upt(Nb,M))) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_ajo(fun(nat,A),fun(nat,product_prod(nat,A)),F)),upt(Nb,M)) ).

% enumerate_map_upt
tff(fact_6893_sgn__sdiv__eq__sgn__mult,axiom,
    ! [A2: int,B2: int] :
      ( ( signed7115095781618012415divide(int,A2,B2) != zero_zero(int) )
     => ( aa(int,int,sgn_sgn(int),signed7115095781618012415divide(int,A2,B2)) = aa(int,int,sgn_sgn(int),aa(int,int,aa(int,fun(int,int),times_times(int),A2),B2)) ) ) ).

% sgn_sdiv_eq_sgn_mult
tff(fact_6894_atMost__upto,axiom,
    ! [Nb: nat] : set_ord_atMost(nat,Nb) = aa(list(nat),set(nat),set2(nat),upt(zero_zero(nat),aa(nat,nat,suc,Nb))) ).

% atMost_upto
tff(fact_6895_map__upt__Suc,axiom,
    ! [A: $tType,F: fun(nat,A),Nb: nat] : aa(list(nat),list(A),map(nat,A,F),upt(zero_zero(nat),aa(nat,nat,suc,Nb))) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(nat,A,F,zero_zero(nat))),aa(list(nat),list(A),map(nat,A,aTP_Lamp_ajp(fun(nat,A),fun(nat,A),F)),upt(zero_zero(nat),Nb))) ).

% map_upt_Suc
tff(fact_6896_map__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(nat),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_6897_enumerate__replicate__eq,axiom,
    ! [A: $tType,Nb: nat,M: nat,A2: A] : enumerate(A,Nb,replicate(A,M,A2)) = aa(list(nat),list(product_prod(nat,A)),map(nat,product_prod(nat,A),aTP_Lamp_ajq(A,fun(nat,product_prod(nat,A)),A2)),upt(Nb,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),M))) ).

% enumerate_replicate_eq
tff(fact_6898_nth__map__upt,axiom,
    ! [A: $tType,I: nat,Nb: nat,M: nat,F: fun(nat,A)] :
      ( aa(nat,$o,ord_less(nat,I),aa(nat,nat,minus_minus(nat,Nb),M))
     => ( aa(nat,A,nth(A,aa(list(nat),list(A),map(nat,A,F),upt(M,Nb))),I) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I)) ) ) ).

% nth_map_upt
tff(fact_6899_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),aa(int,int,abs_abs(int),K)),aa(int,int,abs_abs(int),L))) ).

% signed_divide_int_def
tff(fact_6900_map__upt__eqI,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat,M: nat,F: fun(nat,A)] :
      ( ( aa(list(A),nat,size_size(list(A)),Xs) = aa(nat,nat,minus_minus(nat,Nb),M) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,ord_less(nat,I3),aa(list(A),nat,size_size(list(A)),Xs))
           => ( aa(nat,A,nth(A,Xs),I3) = aa(nat,A,F,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),I3)) ) )
       => ( aa(list(nat),list(A),map(nat,A,F),upt(M,Nb)) = Xs ) ) ) ).

% map_upt_eqI
tff(fact_6901_sdiv__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),signed7115095781618012415divide(int,ring_1_signed(A,int,A2),ring_1_signed(A,int,B2))),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)),A2)),aa(nat,nat,suc,zero_zero(nat))))) ) ).

% sdiv_word_max
tff(fact_6902_make__rule,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,F: fun(nat,A)] : hoare_hoare_triple(array(A),one_one(assn),array_make(A,Nb,F),aa(fun(nat,A),fun(array(A),assn),aTP_Lamp_ajr(nat,fun(fun(nat,A),fun(array(A),assn)),Nb),F)) ) ).

% make_rule
tff(fact_6903_Cauchy__iff2,axiom,
    ! [X5: fun(nat,real)] :
      ( topolo3814608138187158403Cauchy(real,X5)
    <=> ! [J3: nat] :
        ? [M11: nat] :
        ! [M7: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),M7)
         => ! [N6: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),N6)
             => aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,aa(nat,real,X5,M7)),aa(nat,real,X5,N6)))),aa(real,real,inverse_inverse(real),aa(nat,real,semiring_1_of_nat(real),aa(nat,nat,suc,J3)))) ) ) ) ).

% Cauchy_iff2
tff(fact_6904_array__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,Xb: A] : array_new(A,Nb,Xb) = array_make(A,Nb,aTP_Lamp_ajs(A,fun(nat,A),Xb)) ) ).

% array_make
tff(fact_6905_time__array__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,F: fun(nat,A),H: heap_ext(product_unit)] : time_time(array(A),array_make(A,Nb,F),H) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat)) ) ).

% time_array_make
tff(fact_6906_TBOUND__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,F: fun(nat,A)] : time_TBOUND(array(A),array_make(A,Nb,F),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))) ) ).

% TBOUND_make
tff(fact_6907_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_6908_Cauchy__iff,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [X5: fun(nat,A)] :
          ( topolo3814608138187158403Cauchy(A,X5)
        <=> ! [E4: real] :
              ( aa(real,$o,ord_less(real,zero_zero(real)),E4)
             => ? [M11: nat] :
                ! [M7: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),M7)
                 => ! [N6: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M11),N6)
                     => aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,aa(A,A,minus_minus(A,aa(nat,A,X5,M7)),aa(nat,A,X5,N6)))),E4) ) ) ) ) ) ).

% Cauchy_iff
tff(fact_6909_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] :
                ! [M2: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M13),M2)
                 => ! [N2: nat] :
                      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M13),N2)
                     => 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,N2)))),E2) ) ) )
         => topolo3814608138187158403Cauchy(A,X5) ) ) ).

% CauchyI
tff(fact_6910_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] :
              ! [M3: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),M3)
               => ! [N8: nat] :
                    ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M10),N8)
                   => 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,N8)))),E3) ) ) ) ) ) ).

% CauchyD
tff(fact_6911_Array__Time_Omake__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,F: fun(nat,A)] : array_make(A,Nb,F) = 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)))),Nb),F)) ) ).

% Array_Time.make_def
tff(fact_6912_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] : linord4507533701916653071of_set(nat,set_ord_atMost(nat,aa(nat,nat,suc,K))) = aa(list(nat),list(nat),append(nat,linord4507533701916653071of_set(nat,set_ord_atMost(nat,K))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),aa(nat,nat,suc,K)),nil(nat))) ).

% sorted_list_of_set_atMost_Suc
tff(fact_6913_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] :
          ( finite_finite2(A,A3)
         => ( aa(list(A),set(A),set2(A),linord4507533701916653071of_set(A,A3)) = A3 ) ) ) ).

% sorted_list_of_set.set_sorted_key_list_of_set
tff(fact_6914_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] : linord4507533701916653071of_set(nat,set_ord_lessThan(nat,aa(nat,nat,suc,K))) = aa(list(nat),list(nat),append(nat,linord4507533701916653071of_set(nat,set_ord_lessThan(nat,K))),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),K),nil(nat))) ).

% sorted_list_of_set_lessThan_Suc
tff(fact_6915_effect__of__listI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A2: array(A),H3: 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)),A2),H3) = array_alloc(A,Xs,H) )
         => heap_Time_effect(array(A),array_of_list(A,Xs),H,H3,A2,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_6916_effect__newI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A2: array(A),H3: heap_ext(product_unit),Nb: 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)),A2),H3) = array_alloc(A,replicate(A,Nb,Xb),H) )
         => heap_Time_effect(array(A),array_new(A,Nb,Xb),H,H3,A2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))) ) ) ).

% effect_newI
tff(fact_6917_of__list__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Xs: list(A)] : array_of_list(A,Xs) = heap_Time_heap(array(A),aTP_Lamp_ajw(list(A),fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat))),Xs)) ) ).

% of_list_def
tff(fact_6918_new__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,Xb: A] : array_new(A,Nb,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_ajx(nat,fun(A,fun(heap_ext(product_unit),product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),Nb),Xb)) ) ).

% new_def
tff(fact_6919_effect__makeI,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A2: array(A),H3: heap_ext(product_unit),F: fun(nat,A),Nb: 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)),A2),H3) = array_alloc(A,aa(list(nat),list(A),map(nat,A,F),upt(zero_zero(nat),Nb)),H) )
         => heap_Time_effect(array(A),array_make(A,Nb,F),H,H3,A2,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),one_one(nat))) ) ) ).

% effect_makeI
tff(fact_6920_execute__make,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: nat,F: 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))),aa(heap_Time_Heap(array(A)),fun(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(array(A)),array_make(A,Nb,F)),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)))),Nb)),array_alloc(A,aa(list(nat),list(A),map(nat,A,F),upt(zero_zero(nat),Nb)),H))) ) ).

% execute_make
tff(fact_6921_execute__new,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Nb: 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))),aa(heap_Time_Heap(array(A)),fun(heap_ext(product_unit),option(product_prod(array(A),product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(array(A)),array_new(A,Nb,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)))),Nb)),array_alloc(A,replicate(A,Nb,Xb),H))) ) ).

% execute_new
tff(fact_6922_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_ajz(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))),aa(heap_Time_Heap(A),fun(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_6923_execute__bind_I2_J,axiom,
    ! [A: $tType,B: $tType,F: 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))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),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))),aa(heap_Time_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(B),heap_Time_bind(A,B,F,G)),H) = none(product_prod(B,product_prod(heap_ext(product_unit),nat))) ) ) ).

% execute_bind(2)
tff(fact_6924_Heap__ordE,axiom,
    ! [A: $tType,Xb: heap_Time_Heap(A),Y: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( aa(heap_Time_Heap(A),$o,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),$o),heap_Time_Heap_ord(A),Xb),Y)
     => ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),Xb),H) != none(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))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),Xb),H) = aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),Y),H) ) ) ) ).

% Heap_ordE
tff(fact_6925_Heap__ordI,axiom,
    ! [A: $tType,Xb: heap_Time_Heap(A),Y: heap_Time_Heap(A)] :
      ( ! [H2: heap_ext(product_unit)] :
          ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),Xb),H2) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
          | ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),Xb),H2) = aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),Y),H2) ) )
     => aa(heap_Time_Heap(A),$o,aa(heap_Time_Heap(A),fun(heap_Time_Heap(A),$o),heap_Time_Heap_ord(A),Xb),Y) ) ).

% Heap_ordI
tff(fact_6926_execute__guard_I2_J,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),$o),H: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
      ( aa(heap_ext(product_unit),$o,P,H)
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),heap_Time_guard(A,P,F)),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(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)),F,H)) ) ) ).

% execute_guard(2)
tff(fact_6927_execute__guard_I1_J,axiom,
    ! [A: $tType,P: fun(heap_ext(product_unit),$o),H: heap_ext(product_unit),F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] :
      ( ~ aa(heap_ext(product_unit),$o,P,H)
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),heap_Time_guard(A,P,F)),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% execute_guard(1)
tff(fact_6928_Heap__cases,axiom,
    ! [A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( ! [X2: A,H5: product_prod(heap_ext(product_unit),nat)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),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)),X2),H5))
     => ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% Heap_cases
tff(fact_6929_execute__bind__case,axiom,
    ! [A: $tType,B: $tType,F: 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))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),heap_Time_bind(B,A,F,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_akb(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))),aa(heap_Time_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(B),F),H)) ).

% execute_bind_case
tff(fact_6930_effectI,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),R3: A,H3: heap_ext(product_unit),Nb: nat] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(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)),R3),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),H3),Nb))) )
     => heap_Time_effect(A,C2,H,H3,R3,Nb) ) ).

% effectI
tff(fact_6931_effect__def,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: A,Nb: nat] :
      ( heap_Time_effect(A,C2,H,H3,R3,Nb)
    <=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(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)),R3),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),H3),Nb))) ) ) ).

% effect_def
tff(fact_6932_refines__def,axiom,
    ! [A: $tType,P2: heap_Time_Heap(A),Q2: heap_Time_Heap(A)] :
      ( refine_Imp_refines(A,P2,Q2)
    <=> ! [H4: heap_ext(product_unit)] : case_option($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),$o)),aTP_Lamp_akd(heap_Time_Heap(A),fun(heap_ext(product_unit),fun(A,fun(product_prod(heap_ext(product_unit),nat),$o))),P2),H4)),aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),Q2),H4)) ) ).

% refines_def
tff(fact_6933_Heap__Time__Monad_Obind__def,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap(B),G: fun(B,heap_Time_Heap(A))] : heap_Time_bind(B,A,F,G) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),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_ake(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))))),F),G)) ).

% Heap_Time_Monad.bind_def
tff(fact_6934_execute__bind__eq__SomeI,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),Xb: A,H3: heap_ext(product_unit),Nb: nat,G: fun(A,heap_Time_Heap(B)),Y: B,H8: heap_ext(product_unit),N3: nat] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),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),H3),Nb))) )
     => ( ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(B),fun(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)),H3) = 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),H8),N3))) )
       => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(B),heap_Time_bind(A,B,F,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),H8),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Nb),N3)))) ) ) ) ).

% execute_bind_eq_SomeI
tff(fact_6935_execute__bind_I1_J,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),Xb: A,H3: heap_ext(product_unit),Nb: nat,G: fun(A,heap_Time_Heap(B))] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),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),H3),Nb))) )
     => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(B),heap_Time_bind(A,B,F,G)),H) = heap_Time_timeFrame(B,Nb,aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(B),fun(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)),H3)) ) ) ).

% execute_bind(1)
tff(fact_6936_execute__tap,axiom,
    ! [A: $tType,F: 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))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),heap_Time_tap(A,F)),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,F,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_6937_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))),aa(heap_Time_Heap(array(A)),fun(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_ajv(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_6938_execute__bind__success,axiom,
    ! [B: $tType,A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),G: fun(A,heap_Time_Heap(B))] :
      ( heap_Time_success(A,F,H)
     => ( aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(B),heap_Time_bind(A,B,F,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))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H)))),aa(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(B),fun(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))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),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))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H)))))) ) ) ).

% execute_bind_success
tff(fact_6939_Heap__lub__def,axiom,
    ! [A: $tType] : heap_Time_Heap_lub(A) = partial_img_lub(heap_Time_Heap(A),fun(heap_ext(product_unit),option(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)))),heap_Time_Heap(A),heap_Time_execute(A),heap_Time_Heap2(A),partial_fun_lub(option(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),partial_flat_lub(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))))) ).

% Heap_lub_def
tff(fact_6940_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_6941_success__LetI,axiom,
    ! [A: $tType,B: $tType,Xb: A,Ta: A,F: fun(A,heap_Time_Heap(B)),H: heap_ext(product_unit)] :
      ( ( Xb = Ta )
     => ( heap_Time_success(B,aa(A,heap_Time_Heap(B),F,Xb),H)
       => heap_Time_success(B,aa(A,heap_Time_Heap(B),F,Ta),H) ) ) ).

% success_LetI
tff(fact_6942_success__bind__effectI,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),Xb: A,Nb: nat,G: fun(A,heap_Time_Heap(B))] :
      ( heap_Time_effect(A,F,H,H3,Xb,Nb)
     => ( heap_Time_success(B,aa(A,heap_Time_Heap(B),G,Xb),H3)
       => heap_Time_success(B,heap_Time_bind(A,B,F,G),H) ) ) ).

% success_bind_effectI
tff(fact_6943_success__def,axiom,
    ! [A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( heap_Time_success(A,F,H)
    <=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H) != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% success_def
tff(fact_6944_successI,axiom,
    ! [A: $tType,F: 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))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),H) != none(product_prod(A,product_prod(heap_ext(product_unit),nat))) )
     => heap_Time_success(A,F,H) ) ).

% successI
tff(fact_6945_hoare__triple__success,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))
       => heap_Time_success(A,C2,H) ) ) ).

% hoare_triple_success
tff(fact_6946_successE,axiom,
    ! [A: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( heap_Time_success(A,F,H)
     => ~ ! [R2: A,H5: product_prod(heap_ext(product_unit),nat)] : aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),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),H5)) ) ).

% successE
tff(fact_6947_success__bind__executeI,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap(A),H: heap_ext(product_unit),Xb: A,H3: heap_ext(product_unit),Nb: nat,G: fun(A,heap_Time_Heap(B))] :
      ( ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),F),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),H3),Nb))) )
     => ( heap_Time_success(B,aa(A,heap_Time_Heap(B),G,Xb),H3)
       => heap_Time_success(B,heap_Time_bind(A,B,F,G),H) ) ) ).

% success_bind_executeI
tff(fact_6948_effectE,axiom,
    ! [A: $tType,C2: heap_Time_Heap(A),H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: A,Nb: nat] :
      ( heap_Time_effect(A,C2,H,H3,R3,Nb)
     => ~ ( ( R3 = 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))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),C2),H))) )
         => ( ( H3 = 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))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),C2),H)))) )
           => ( ( Nb = 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))),aa(heap_Time_Heap(A),fun(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_6949_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_akg(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))),aa(heap_Time_Heap(A),fun(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_6950_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))),aa(heap_Time_Heap(A),fun(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_6951_assert__cong,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),P3: fun(A,$o),F: fun(A,heap_Time_Heap(B)),F4: fun(A,heap_Time_Heap(B)),Xb: A] :
      ( ( P = P3 )
     => ( ! [X2: A] :
            ( aa(A,$o,P3,X2)
           => ( aa(A,heap_Time_Heap(B),F,X2) = aa(A,heap_Time_Heap(B),F4,X2) ) )
       => ( heap_Time_bind(A,B,heap_Time_assert(A,P,Xb),F) = heap_Time_bind(A,B,heap_Time_assert(A,P3,Xb),F4) ) ) ) ).

% assert_cong
tff(fact_6952_effect__assertI,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,H3: heap_ext(product_unit),H: heap_ext(product_unit),R3: A,Nb: nat] :
      ( aa(A,$o,P,Xb)
     => ( ( H3 = H )
       => ( ( R3 = Xb )
         => ( ( Nb = one_one(nat) )
           => heap_Time_effect(A,heap_Time_assert(A,P,Xb),H,H3,R3,Nb) ) ) ) ) ).

% effect_assertI
tff(fact_6953_effect__assertE,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,H: heap_ext(product_unit),H3: heap_ext(product_unit),R3: A,Nb: nat] :
      ( heap_Time_effect(A,heap_Time_assert(A,P,Xb),H,H3,R3,Nb)
     => ~ ( aa(A,$o,P,Xb)
         => ( ( R3 = Xb )
           => ( ( H3 = H )
             => ( Nb != one_one(nat) ) ) ) ) ) ).

% effect_assertE
tff(fact_6954_execute__assert_I2_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))),aa(heap_Time_Heap(A),fun(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) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% execute_assert(2)
tff(fact_6955_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_aki(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))),aa(heap_Time_Heap(A),fun(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_6956_Heap__ord__def,axiom,
    ! [A: $tType] : heap_Time_Heap_ord(A) = partial_img_ord(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),$o,heap_Time_execute(A),partial_fun_ord(option(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),partial_flat_ord(option(product_prod(A,product_prod(heap_ext(product_unit),nat))),none(product_prod(A,product_prod(heap_ext(product_unit),nat)))))) ).

% Heap_ord_def
tff(fact_6957_img__ord__def,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,C),Ord: fun(C,fun(C,B)),X3: A,Xa: A] : aa(A,B,aa(A,fun(A,B),partial_img_ord(A,C,B,F,Ord),X3),Xa) = aa(C,B,aa(C,fun(C,B),Ord,aa(A,C,F,X3)),aa(A,C,F,Xa)) ).

% img_ord_def
tff(fact_6958_time__bind,axiom,
    ! [B: $tType,A: $tType,M: heap_Time_Heap(B),F: fun(B,heap_Time_Heap(A)),H: heap_ext(product_unit)] :
      time_time(A,heap_Time_bind(B,A,M,F),H) = $ite(
        ( ~ time_fails(B,M,H)
        & ~ time_fails(A,aa(B,heap_Time_Heap(A),F,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),F,time_the_res(B,M,H)),time_the_heap(B,M,H))),
        zero_zero(nat) ) ).

% time_bind
tff(fact_6959_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,$o),aTP_Lamp_akj(fun(A,$o),fun(list(A),fun(nat,$o)),P),L),upt(zero_zero(nat),aa(list(A),nat,size_size(list(A)),L))) = aa(list(nat),list(nat),append(nat,Js),aa(list(nat),list(nat),aa(nat,fun(list(nat),list(nat)),cons(nat),J),nil(nat))) )
     => ( aa(nat,$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_6960_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,$o),aTP_Lamp_akk(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q),Xs) ).

% filter_filter
tff(fact_6961_filter__True,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X2) )
     => ( filter2(A,P,Xs) = Xs ) ) ).

% filter_True
tff(fact_6962_set__filter,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(list(A),set(A),set2(A),filter2(A,P,Xs)) = collect(A,aa(list(A),fun(A,$o),aTP_Lamp_akl(fun(A,$o),fun(list(A),fun(A,$o)),P),Xs)) ).

% set_filter
tff(fact_6963_filter__False,axiom,
    ! [A: $tType,Xs: list(A),P: fun(A,$o)] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P,X2) )
     => ( filter2(A,P,Xs) = nil(A) ) ) ).

% filter_False
tff(fact_6964_length__filter__less,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o)] :
      ( member(A,Xb,aa(list(A),set(A),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_6965_length__filter__le,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(nat,$o,aa(nat,fun(nat,$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_6966_replicate__length__filter,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] : replicate(A,aa(list(A),nat,size_size(list(A)),filter2(A,aa(A,fun(A,$o),fequal(A),Xb),Xs)),Xb) = filter2(A,aa(A,fun(A,$o),fequal(A),Xb),Xs) ).

% replicate_length_filter
tff(fact_6967_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_ss(fun(A,$o),fun(A,$o),P),Xs))) = aa(list(A),nat,size_size(list(A)),Xs) ).

% sum_length_filter_compl
tff(fact_6968_filter__cong,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( ( Xs = Ys )
     => ( ! [X2: A] :
            ( member(A,X2,aa(list(A),set(A),set2(A),Ys))
           => ( aa(A,$o,P,X2)
            <=> aa(A,$o,Q,X2) ) )
       => ( filter2(A,P,Xs) = filter2(A,Q,Ys) ) ) ) ).

% filter_cong
tff(fact_6969_filter__id__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( filter2(A,P,Xs) = Xs )
    <=> ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => aa(A,$o,P,X) ) ) ).

% filter_id_conv
tff(fact_6970_filter__is__subset,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),filter2(A,P,Xs))),aa(list(A),set(A),set2(A),Xs)) ).

% filter_is_subset
tff(fact_6971_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_6972_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_6973_filter__eq__Cons__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Ys: list(A),Xb: A,Xs: list(A)] :
      ( ( filter2(A,P,Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) )
    <=> ? [Us2: list(A),Vs2: list(A)] :
          ( ( Ys = aa(list(A),list(A),append(A,Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Vs2)) )
          & ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Us2))
             => ~ aa(A,$o,P,X) )
          & aa(A,$o,P,Xb)
          & ( Xs = filter2(A,P,Vs2) ) ) ) ).

% filter_eq_Cons_iff
tff(fact_6974_Cons__eq__filter__iff,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) = filter2(A,P,Ys) )
    <=> ? [Us2: list(A),Vs2: list(A)] :
          ( ( Ys = aa(list(A),list(A),append(A,Us2),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Vs2)) )
          & ! [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Us2))
             => ~ aa(A,$o,P,X) )
          & aa(A,$o,P,Xb)
          & ( Xs = filter2(A,P,Vs2) ) ) ) ).

% Cons_eq_filter_iff
tff(fact_6975_filter__eq__ConsD,axiom,
    ! [A: $tType,P: fun(A,$o),Ys: list(A),Xb: A,Xs: list(A)] :
      ( ( filter2(A,P,Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) )
     => ? [Us3: list(A),Vs3: list(A)] :
          ( ( Ys = aa(list(A),list(A),append(A,Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Vs3)) )
          & ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Us3))
             => ~ aa(A,$o,P,X3) )
          & aa(A,$o,P,Xb)
          & ( Xs = filter2(A,P,Vs3) ) ) ) ).

% filter_eq_ConsD
tff(fact_6976_Cons__eq__filterD,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),P: fun(A,$o),Ys: list(A)] :
      ( ( aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs) = filter2(A,P,Ys) )
     => ? [Us3: list(A),Vs3: list(A)] :
          ( ( Ys = aa(list(A),list(A),append(A,Us3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Vs3)) )
          & ! [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Us3))
             => ~ aa(A,$o,P,X3) )
          & aa(A,$o,P,Xb)
          & ( Xs = filter2(A,P,Vs3) ) ) ) ).

% Cons_eq_filterD
tff(fact_6977_empty__filter__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( nil(A) = filter2(A,P,Xs) )
    <=> ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P,X) ) ) ).

% empty_filter_conv
tff(fact_6978_filter__empty__conv,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( filter2(A,P,Xs) = nil(A) )
    <=> ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),Xs))
         => ~ aa(A,$o,P,X) ) ) ).

% filter_empty_conv
tff(fact_6979_filter__eq__snocD,axiom,
    ! [A: $tType,P: fun(A,$o),L: list(A),L3: list(A),Xb: A] :
      ( ( filter2(A,P,L) = aa(list(A),list(A),append(A,L3),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),L))
        & aa(A,$o,P,Xb) ) ) ).

% filter_eq_snocD
tff(fact_6980_filter__conv__foldr,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : filter2(A,P,Xs) = foldr(A,list(A),aTP_Lamp_akm(fun(A,$o),fun(A,fun(list(A),list(A))),P),Xs,nil(A)) ).

% filter_conv_foldr
tff(fact_6981_fails__def,axiom,
    ! [A: $tType,M: heap_Time_Heap(A),H: heap_ext(product_unit)] :
      ( time_fails(A,M,H)
    <=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),M),H) = none(product_prod(A,product_prod(heap_ext(product_unit),nat))) ) ) ).

% fails_def
tff(fact_6982_set__minus__filter__out,axiom,
    ! [A: $tType,Xs: list(A),Y: A] : aa(set(A),set(A),minus_minus(set(A),aa(list(A),set(A),set2(A),Xs)),aa(set(A),set(A),insert(A,Y),bot_bot(set(A)))) = aa(list(A),set(A),set2(A),filter2(A,aTP_Lamp_akn(A,fun(A,$o),Y),Xs)) ).

% set_minus_filter_out
tff(fact_6983_upt__filter__extend,axiom,
    ! [U: nat,U4: nat,P: fun(nat,$o)] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),U),U4)
     => ( ! [I3: nat] :
            ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),U),I3)
              & aa(nat,$o,ord_less(nat,I3),U4) )
           => ~ aa(nat,$o,P,I3) )
       => ( filter2(nat,P,upt(zero_zero(nat),U)) = filter2(nat,P,upt(zero_zero(nat),U4)) ) ) ) ).

% upt_filter_extend
tff(fact_6984_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,aa(nat,fun(nat,$o),ord_less_eq(nat),time_time(A,C2,H)),time_time(A,C6,H)) ) ) ).

% time_refines
tff(fact_6985_TBOUND__bind__weak,axiom,
    ! [A: $tType,B: $tType,M: heap_Time_Heap(A),T_1: nat,F: fun(A,heap_Time_Heap(B)),T_2: nat] :
      ( time_TBOUND(A,M,T_1)
     => ( ! [X2: A,H2: heap_ext(product_unit)] :
            ( ~ time_fails(A,M,H2)
           => time_TBOUND(B,aa(A,heap_Time_Heap(B),F,X2),T_2) )
       => time_TBOUND(B,heap_Time_bind(A,B,M,F),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),T_1),T_2)) ) ) ).

% TBOUND_bind_weak
tff(fact_6986_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_6987_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_6988_fails__bind,axiom,
    ! [A: $tType,B: $tType,M: heap_Time_Heap(B),F: fun(B,heap_Time_Heap(A)),H: heap_ext(product_unit)] :
      ( time_fails(A,heap_Time_bind(B,A,M,F),H)
    <=> ( ~ time_fails(B,M,H)
       => time_fails(A,aa(B,heap_Time_Heap(A),F,time_the_res(B,M,H)),time_the_heap(B,M,H)) ) ) ).

% fails_bind
tff(fact_6989_TBOUND__bind,axiom,
    ! [A: $tType,B: $tType,M: heap_Time_Heap(A),T_1: nat,F: fun(A,heap_Time_Heap(B)),T_2: nat] :
      ( time_TBOUND(A,M,T_1)
     => ( ! [X2: A,H2: heap_ext(product_unit)] :
            ( ( X2 = time_the_res(A,M,H2) )
           => ( ~ time_fails(A,M,H2)
             => time_TBOUND(B,aa(A,heap_Time_Heap(B),F,X2),T_2) ) )
       => time_TBOUND(B,heap_Time_bind(A,B,M,F),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),T_1),T_2)) ) ) ).

% TBOUND_bind
tff(fact_6990_sort__key__by__quicksort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,B,F),Xs) = aa(list(A),list(A),append(A,aa(list(A),list(A),linorder_sort_key(A,B,F),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_ako(fun(A,B),fun(list(A),fun(A,$o)),F),Xs),Xs))),aa(list(A),list(A),append(A,filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_akp(fun(A,B),fun(list(A),fun(A,$o)),F),Xs),Xs)),aa(list(A),list(A),linorder_sort_key(A,B,F),filter2(A,aa(list(A),fun(A,$o),aTP_Lamp_akq(fun(A,B),fun(list(A),fun(A,$o)),F),Xs),Xs)))) ) ).

% sort_key_by_quicksort
tff(fact_6991_sort__upt,axiom,
    ! [M: nat,Nb: nat] : aa(list(nat),list(nat),linorder_sort_key(nat,nat,aTP_Lamp_vc(nat,nat)),upt(M,Nb)) = upt(M,Nb) ).

% sort_upt
tff(fact_6992_sort__upto,axiom,
    ! [I: int,J: int] : aa(list(int),list(int),linorder_sort_key(int,int,aTP_Lamp_vx(int,int)),upto(I,J)) = upto(I,J) ).

% sort_upto
tff(fact_6993_length__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Xs: list(A)] : aa(list(A),nat,size_size(list(A)),aa(list(A),list(A),linorder_sort_key(A,B,F),Xs)) = aa(list(A),nat,size_size(list(A)),Xs) ) ).

% length_sort
tff(fact_6994_set__sort,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),linorder_sort_key(A,B,F),Xs)) = aa(list(A),set(A),set2(A),Xs) ) ).

% set_sort
tff(fact_6995_sort__key__stable,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),K: B,Xs: list(A)] : filter2(A,aa(B,fun(A,$o),aTP_Lamp_akr(fun(A,B),fun(B,fun(A,$o)),F),K),aa(list(A),list(A),linorder_sort_key(A,B,F),Xs)) = filter2(A,aa(B,fun(A,$o),aTP_Lamp_akr(fun(A,B),fun(B,fun(A,$o)),F),K),Xs) ) ).

% sort_key_stable
tff(fact_6996_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_ahj(B,fun(A,B),C2)),Xs) = Xs ) ).

% sort_key_const
tff(fact_6997_sort__mergesort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_aks(A,A)) = mergesort(A) ) ) ).

% sort_mergesort
tff(fact_6998_sort__mergesort__by__rel,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_aks(A,A)) = mergesort_by_rel(A,ord_less_eq(A)) ) ) ).

% sort_mergesort_by_rel
tff(fact_6999_transpose__aux__max,axiom,
    ! [A: $tType,B: $tType,Xs: list(A),Xss: list(list(B))] : aa(nat,nat,ord_max(nat,aa(nat,nat,suc,aa(list(A),nat,size_size(list(A)),Xs))),foldr(list(B),nat,aTP_Lamp_akt(list(B),fun(nat,nat)),Xss,zero_zero(nat))) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,aa(list(A),nat,size_size(list(A)),Xs)),foldr(list(B),nat,aTP_Lamp_aku(list(B),fun(nat,nat)),filter2(list(B),aTP_Lamp_akv(list(B),$o),Xss),zero_zero(nat)))) ).

% transpose_aux_max
tff(fact_7000_sort__by__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_aks(A,A)),Xs) = aa(list(A),list(A),append(A,aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_aks(A,A)),filter2(A,aTP_Lamp_akw(list(A),fun(A,$o),Xs),Xs))),aa(list(A),list(A),append(A,filter2(A,aTP_Lamp_akx(list(A),fun(A,$o),Xs),Xs)),aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_aks(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_7001_time__array__map__entry,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,F: fun(A,A),P2: array(A),H: heap_ext(product_unit)] :
          time_time(array(A),array_map_entry(A,I,F,P2),H) = $ite(time_fails(array(A),array_map_entry(A,I,F,P2),H),zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% time_array_map_entry
tff(fact_7002_sort__key__by__quicksort__code,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Xs: list(A)] : aa(list(A),list(A),linorder_sort_key(A,B,F),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_alc(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),F),Xs)),Xs) ) ).

% sort_key_by_quicksort_code
tff(fact_7003_list_Ocase__distrib,axiom,
    ! [B: $tType,A: $tType,C: $tType,H: fun(B,A),F1: B,F22: fun(C,fun(list(C),B)),List: list(C)] : aa(B,A,H,aa(list(C),B,case_list(B,C,F1,F22),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_ald(fun(B,A),fun(fun(C,fun(list(C),B)),fun(C,fun(list(C),A))),H),F22)),List) ).

% list.case_distrib
tff(fact_7004_part__code_I1_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Pivot: B] : linorder_part(A,B,F,Pivot,nil(A)) = 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))),nil(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)),nil(A)),nil(A))) ) ).

% part_code(1)
tff(fact_7005_TBOUND__map__entry,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xb: fun(A,A),A2: array(A)] : time_TBOUND(array(A),array_map_entry(A,I,Xb,A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% TBOUND_map_entry
tff(fact_7006_part__code_I2_J,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Pivot: B,Xb: A,Xs: list(A)] : linorder_part(A,B,F,Pivot,aa(list(A),list(A),aa(A,fun(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_alf(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))))))),F),Pivot),Xb)),linorder_part(A,B,F,Pivot,Xs)) ) ).

% part_code(2)
tff(fact_7007_part__def,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [F: fun(A,B),Pivot: B,Xs: list(A)] : linorder_part(A,B,F,Pivot,Xs) = aa(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A))),aa(list(A),fun(product_prod(list(A),list(A)),product_prod(list(A),product_prod(list(A),list(A)))),product_Pair(list(A),product_prod(list(A),list(A))),filter2(A,aa(B,fun(A,$o),aTP_Lamp_alg(fun(A,B),fun(B,fun(A,$o)),F),Pivot),Xs)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),filter2(A,aa(B,fun(A,$o),aTP_Lamp_akr(fun(A,B),fun(B,fun(A,$o)),F),Pivot),Xs)),filter2(A,aa(B,fun(A,$o),aTP_Lamp_alh(fun(A,B),fun(B,fun(A,$o)),F),Pivot),Xs))) ) ).

% part_def
tff(fact_7008_time__array__swap,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xb: A,P2: array(A),H: heap_ext(product_unit),F: fun(A,A)] :
          time_time(A,array_swap(A,I,Xb,P2),H) = $ite(time_fails(array(A),array_map_entry(A,I,F,P2),H),zero_zero(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% time_array_swap
tff(fact_7009_signed_Ostable__sort__key__def,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [Sk: fun(fun(A,word(B)),fun(list(A),list(A)))] :
          ( stable_sort_key(A,word(B),Sk)
        <=> ! [F8: fun(A,word(B)),Xs3: list(A),K4: word(B)] : filter2(A,aa(word(B),fun(A,$o),aTP_Lamp_ali(fun(A,word(B)),fun(word(B),fun(A,$o)),F8),K4),aa(list(A),list(A),aa(fun(A,word(B)),fun(list(A),list(A)),Sk,F8),Xs3)) = filter2(A,aa(word(B),fun(A,$o),aTP_Lamp_ali(fun(A,word(B)),fun(word(B),fun(A,$o)),F8),K4),Xs3) ) ) ).

% signed.stable_sort_key_def
tff(fact_7010_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_alj(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(2)
tff(fact_7011_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_alk(A,fun(list(A),$o))),List) ) ).

% list.disc_eq_case(1)
tff(fact_7012_case__list__rule,axiom,
    ! [A: $tType,B: $tType,L: list(A),P: assn,Fn: heap_Time_Heap(B),Q: fun(B,assn),Fc: fun(A,fun(list(A),heap_Time_Heap(B)))] :
      ( ( ( L = nil(A) )
       => hoare_hoare_triple(B,P,Fn,Q) )
     => ( ! [X2: A,Xs2: list(A)] :
            ( ( L = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
           => hoare_hoare_triple(B,P,aa(list(A),heap_Time_Heap(B),aa(A,fun(list(A),heap_Time_Heap(B)),Fc,X2),Xs2),Q) )
       => hoare_hoare_triple(B,P,aa(list(A),heap_Time_Heap(B),case_list(heap_Time_Heap(B),A,Fn,Fc),L),Q) ) ) ).

% case_list_rule
tff(fact_7013_TBOUND__swap,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [I: nat,Xb: A,A2: array(A)] : time_TBOUND(A,array_swap(A,I,Xb,A2),aa(num,nat,numeral_numeral(nat),bit0(one2))) ) ).

% TBOUND_swap
tff(fact_7014_transpose__max__length,axiom,
    ! [A: $tType,Xs: list(list(A))] : foldr(list(A),nat,aTP_Lamp_all(list(A),fun(nat,nat)),transpose(A,Xs),zero_zero(nat)) = aa(list(list(A)),nat,size_size(list(list(A))),filter2(list(A),aTP_Lamp_alm(list(A),$o),Xs)) ).

% transpose_max_length
tff(fact_7015_Bleast__code,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A),P: fun(A,$o)] : bleast(A,aa(list(A),set(A),set2(A),Xs),P) = aa(list(A),A,case_list(A,A,abort_Bleast(A,aa(list(A),set(A),set2(A),Xs),P),aTP_Lamp_aln(A,fun(list(A),A))),filter2(A,P,aa(list(A),list(A),linorder_sort_key(A,A,aTP_Lamp_aks(A,A)),Xs))) ) ).

% Bleast_code
tff(fact_7016_transpose__empty,axiom,
    ! [A: $tType,Xs: list(list(A))] :
      ( ( transpose(A,Xs) = nil(list(A)) )
    <=> ! [X: list(A)] :
          ( member(list(A),X,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
         => ( X = nil(A) ) ) ) ).

% transpose_empty
tff(fact_7017_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) = aa(list(list(A)),list(A),map(list(A),A,aTP_Lamp_alo(nat,fun(list(A),A),I)),filter2(list(A),aTP_Lamp_alp(nat,fun(list(A),$o),I),Xs)) ) ) ).

% nth_transpose
tff(fact_7018_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_all(list(A),fun(nat,nat)),Xs,zero_zero(nat)) ).

% length_transpose
tff(fact_7019_transpose__rectangle,axiom,
    ! [A: $tType,Xs: list(list(A)),Nb: nat] :
      ( ( ( Xs = nil(list(A)) )
       => ( Nb = zero_zero(nat) ) )
     => ( ! [I3: nat] :
            ( aa(nat,$o,ord_less(nat,I3),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),I3)) = Nb ) )
       => ( transpose(A,Xs) = aa(list(nat),list(list(A)),map(nat,list(A),aTP_Lamp_alr(list(list(A)),fun(nat,list(A)),Xs)),upt(zero_zero(nat),Nb)) ) ) ) ).

% transpose_rectangle
tff(fact_7020_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),aa(list(list(A)),list(nat),map(list(A),nat,size_size(list(A))),Xss),one_one(nat)) ).

% length_product_lists
tff(fact_7021_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_7022_Cons__in__subseqsD,axiom,
    ! [A: $tType,Y: A,Ys: list(A),Xs: list(A)] :
      ( member(list(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))
     => member(list(A),Ys,aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))) ) ).

% Cons_in_subseqsD
tff(fact_7023_subseqs_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      subseqs(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $let(
        xss: list(list(A)),
        xss:= subseqs(A,Xs),
        aa(list(list(A)),list(list(A)),append(list(A),aa(list(list(A)),list(list(A)),map(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb)),xss)),xss) ) ).

% subseqs.simps(2)
tff(fact_7024_subseqs__refl,axiom,
    ! [A: $tType,Xs: list(A)] : member(list(A),Xs,aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs))) ).

% subseqs_refl
tff(fact_7025_merge__list_Ocases,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: product_prod(list(list(A)),list(list(A)))] :
          ( ( Xb != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),nil(list(A))) )
         => ( ! [L2: list(A)] : Xb != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),nil(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A))))
           => ( ! [La: list(A),Acc22: list(list(A))] : Xb != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),nil(list(A)))
             => ( ! [La: list(A),Acc22: list(list(A)),L2: list(A)] : Xb != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),La),Acc22)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L2),nil(list(A))))
               => ~ ! [Acc22: list(list(A)),L1: list(A),L22: list(A),Ls2: list(list(A))] : Xb != aa(list(list(A)),product_prod(list(list(A)),list(list(A))),aa(list(list(A)),fun(list(list(A)),product_prod(list(list(A)),list(list(A)))),product_Pair(list(list(A)),list(list(A))),Acc22),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L1),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),L22),Ls2))) ) ) ) ) ) ).

% merge_list.cases
tff(fact_7026_subset__subseqs,axiom,
    ! [A: $tType,X5: set(A),Xs: list(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),aa(list(A),set(A),set2(A),Xs))
     => member(set(A),X5,aa(set(list(A)),set(set(A)),image(list(A),set(A),set2(A)),aa(list(list(A)),set(list(A)),set2(list(A)),subseqs(A,Xs)))) ) ).

% subset_subseqs
tff(fact_7027_in__set__product__lists__length,axiom,
    ! [A: $tType,Xs: list(A),Xss: list(list(A))] :
      ( member(list(A),Xs,aa(list(list(A)),set(list(A)),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_7028_length__mul__elem,axiom,
    ! [A: $tType,Xs: list(list(A)),Nb: nat] :
      ( ! [X2: list(A)] :
          ( member(list(A),X2,aa(list(list(A)),set(list(A)),set2(list(A)),Xs))
         => ( aa(list(A),nat,size_size(list(A)),X2) = Nb ) )
     => ( 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)),Nb) ) ) ).

% length_mul_elem
tff(fact_7029_set__n__lists,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,Nb,Xs)) = collect(list(A),aa(list(A),fun(list(A),$o),aTP_Lamp_als(nat,fun(list(A),fun(list(A),$o)),Nb),Xs)) ).

% set_n_lists
tff(fact_7030_Nil__eq__concat__conv,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ( nil(A) = concat(A,Xss) )
    <=> ! [X: list(A)] :
          ( member(list(A),X,aa(list(list(A)),set(list(A)),set2(list(A)),Xss))
         => ( X = nil(A) ) ) ) ).

% Nil_eq_concat_conv
tff(fact_7031_concat__eq__Nil__conv,axiom,
    ! [A: $tType,Xss: list(list(A))] :
      ( ( concat(A,Xss) = nil(A) )
    <=> ! [X: list(A)] :
          ( member(list(A),X,aa(list(list(A)),set(list(A)),set2(list(A)),Xss))
         => ( X = nil(A) ) ) ) ).

% concat_eq_Nil_conv
tff(fact_7032_concat__map__singleton,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Xs: list(B)] : concat(A,aa(list(B),list(list(A)),map(B,list(A),aTP_Lamp_alt(fun(B,A),fun(B,list(A)),F)),Xs)) = aa(list(B),list(A),map(B,A,F),Xs) ).

% concat_map_singleton
tff(fact_7033_n__lists_Osimps_I2_J,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : n_lists(A,aa(nat,nat,suc,Nb),Xs) = concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),aTP_Lamp_alv(list(A),fun(list(A),list(list(A))),Xs)),n_lists(A,Nb,Xs))) ).

% n_lists.simps(2)
tff(fact_7034_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)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),Xss)) = concat(list(A),aa(list(A),list(list(list(A))),map(A,list(list(A)),aTP_Lamp_alw(list(list(A)),fun(A,list(list(A))),Xss)),Xs)) ).

% product_lists.simps(2)
tff(fact_7035_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)),aa(list(A),fun(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)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Xss2) )
               => ( Y != aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_alx(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aly(A,fun(list(A),list(list(A)))))),Xss2))))) ) ) ) ) ) ).

% transpose.elims
tff(fact_7036_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)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Xss)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_alx(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aly(A,fun(list(A),list(list(A)))))),Xss))))) ).

% transpose.simps(3)
tff(fact_7037_concat__filter__neq__Nil,axiom,
    ! [A: $tType,Xs: list(list(A))] : concat(A,filter2(list(A),aTP_Lamp_alm(list(A),$o),Xs)) = concat(A,Xs) ).

% concat_filter_neq_Nil
tff(fact_7038_product__concat__map,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product(A,B,Xs,Ys) = concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_alz(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs)) ).

% product_concat_map
tff(fact_7039_comm__append__is__replicate,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( Ys != nil(A) )
       => ( ( aa(list(A),list(A),append(A,Xs),Ys) = aa(list(A),list(A),append(A,Ys),Xs) )
         => ? [N2: nat,Zs2: list(A)] :
              ( aa(nat,$o,ord_less(nat,one_one(nat)),N2)
              & ( concat(A,replicate(list(A),N2,Zs2)) = aa(list(A),list(A),append(A,Xs),Ys) ) ) ) ) ) ).

% comm_append_is_replicate
tff(fact_7040_length__n__lists__elem,axiom,
    ! [A: $tType,Ys: list(A),Nb: nat,Xs: list(A)] :
      ( member(list(A),Ys,aa(list(list(A)),set(list(A)),set2(list(A)),n_lists(A,Nb,Xs)))
     => ( aa(list(A),nat,size_size(list(A)),Ys) = Nb ) ) ).

% length_n_lists_elem
tff(fact_7041_length__n__lists,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] : aa(list(list(A)),nat,size_size(list(list(A))),n_lists(A,Nb,Xs)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(list(A),nat,size_size(list(A)),Xs)),Nb) ).

% length_n_lists
tff(fact_7042_transpose_Opelims,axiom,
    ! [A: $tType,Xb: list(list(A)),Y: list(list(A))] :
      ( ( transpose(A,Xb) = Y )
     => ( aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),Xb)
       => ( ( ( Xb = nil(list(A)) )
           => ( ( Y = nil(list(A)) )
             => ~ aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),nil(list(A))) ) )
         => ( ! [Xss2: list(list(A))] :
                ( ( Xb = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2) )
               => ( ( Y = transpose(A,Xss2) )
                 => ~ aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(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)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Xss2) )
                 => ( ( Y = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_alx(A,fun(list(A),list(A))))),Xss2)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aly(A,fun(list(A),list(list(A)))))),Xss2))))) )
                   => ~ aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Xss2)) ) ) ) ) ) ) ).

% transpose.pelims
tff(fact_7043_transpose_Opsimps_I3_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Xss: list(list(A))] :
      ( aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Xss))
     => ( transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Xss)) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_alx(A,fun(list(A),list(A))))),Xss)))),transpose(A,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aly(A,fun(list(A),list(list(A)))))),Xss))))) ) ) ).

% transpose.psimps(3)
tff(fact_7044_transpose_Opinduct,axiom,
    ! [A: $tType,A0: list(list(A)),P: fun(list(list(A)),$o)] :
      ( aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),A0)
     => ( ( aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),nil(list(A)))
         => aa(list(list(A)),$o,P,nil(list(A))) )
       => ( ! [Xss2: list(list(A))] :
              ( aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(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)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),nil(A)),Xss2)) ) )
         => ( ! [X2: A,Xs2: list(A),Xss2: list(list(A))] :
                ( aa(list(list(A)),$o,accp(list(list(A)),transpose_rel(A)),aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Xss2))
               => ( aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Xs2),concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aly(A,fun(list(A),list(list(A)))))),Xss2))))
                 => aa(list(list(A)),$o,P,aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Xss2)) ) )
           => aa(list(list(A)),$o,P,A0) ) ) ) ) ).

% transpose.pinduct
tff(fact_7045_product__code,axiom,
    ! [B: $tType,A: $tType,Xs: list(A),Ys: list(B)] : product_product(A,B,aa(list(A),set(A),set2(A),Xs),aa(list(B),set(B),set2(B),Ys)) = aa(list(product_prod(A,B)),set(product_prod(A,B)),set2(product_prod(A,B)),concat(product_prod(A,B),aa(list(A),list(list(product_prod(A,B))),map(A,list(product_prod(A,B)),aTP_Lamp_alz(list(B),fun(A,list(product_prod(A,B))),Ys)),Xs))) ).

% product_code
tff(fact_7046_transpose__aux__filter__tail,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(list(A),aa(list(list(A)),list(list(list(A))),map(list(A),list(list(A)),case_list(list(list(A)),A,nil(list(A)),aTP_Lamp_aly(A,fun(list(A),list(list(A)))))),Xss)) = aa(list(list(A)),list(list(A)),map(list(A),list(A),tl(A)),filter2(list(A),aTP_Lamp_alm(list(A),$o),Xss)) ).

% transpose_aux_filter_tail
tff(fact_7047_tl__upt,axiom,
    ! [M: nat,Nb: nat] : aa(list(nat),list(nat),tl(nat),upt(M,Nb)) = upt(aa(nat,nat,suc,M),Nb) ).

% tl_upt
tff(fact_7048_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_7049_tl__replicate,axiom,
    ! [A: $tType,Nb: nat,Xb: A] : aa(list(A),list(A),tl(A),replicate(A,Nb,Xb)) = replicate(A,aa(nat,nat,minus_minus(nat,Nb),one_one(nat)),Xb) ).

% tl_replicate
tff(fact_7050_tl__append,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),list(A),tl(A),aa(list(A),list(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_ama(list(A),fun(A,fun(list(A),list(A))),Ys)),Xs) ).

% tl_append
tff(fact_7051_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_amb(A,fun(list(A),list(A)))),List) ).

% tl_def
tff(fact_7052_list_Oset__sel_I2_J,axiom,
    ! [A: $tType,A2: list(A),Xb: A] :
      ( ( A2 != nil(A) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),A2)))
       => member(A,Xb,aa(list(A),set(A),set2(A),A2)) ) ) ).

% list.set_sel(2)
tff(fact_7053_in__set__tlD,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xs)))
     => member(A,Xb,aa(list(A),set(A),set2(A),Xs)) ) ).

% in_set_tlD
tff(fact_7054_tl__obtain__elem,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => ( ( aa(list(A),list(A),tl(A),Xs) = nil(A) )
       => ~ ! [E2: A] : Xs != aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),E2),nil(A)) ) ) ).

% tl_obtain_elem
tff(fact_7055_Misc_Onth__tl,axiom,
    ! [A: $tType,Xs: list(A),Nb: nat] :
      ( ( Xs != nil(A) )
     => ( aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),Nb) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,Nb)) ) ) ).

% Misc.nth_tl
tff(fact_7056_tl__subset,axiom,
    ! [A: $tType,Xs: list(A),A3: set(A)] :
      ( ( Xs != nil(A) )
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Xs)),A3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xs))),A3) ) ) ).

% tl_subset
tff(fact_7057_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) )
     => ( ! [N2: nat] :
            ( aa(nat,$o,ord_less(nat,N2),aa(list(A),nat,size_size(list(A)),Xs))
           => aa(A,$o,aa(B,fun(A,$o),P,aa(nat,B,nth(B,Ys),N2)),aa(nat,A,nth(A,Xs),N2)) )
       => ! [N8: nat] :
            ( aa(nat,$o,ord_less(nat,N8),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)),N8)),aa(nat,A,nth(A,aa(list(A),list(A),tl(A),Xs)),N8)) ) ) ) ).

% list_take_induct_tl2
tff(fact_7058_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_7059_List_Onth__tl,axiom,
    ! [A: $tType,Nb: nat,Xs: list(A)] :
      ( aa(nat,$o,ord_less(nat,Nb),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)),Nb) = aa(nat,A,nth(A,Xs),aa(nat,nat,suc,Nb)) ) ) ).

% List.nth_tl
tff(fact_7060_extract__Some__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A),Y: A,Zs: list(A)] :
      ( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),Ys),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Y),Zs))) )
    <=> ( ( Xs = aa(list(A),list(A),append(A,Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) )
        & aa(A,$o,P,Y)
        & ~ ? [X: A] :
              ( member(A,X,aa(list(A),set(A),set2(A),Ys))
              & aa(A,$o,P,X) ) ) ) ).

% extract_Some_iff
tff(fact_7061_extract__SomeE,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A),Ys: list(A),Y: A,Zs: list(A)] :
      ( ( extract(A,P,Xs) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),Ys),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Y),Zs))) )
     => ( ( Xs = aa(list(A),list(A),append(A,Ys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)) )
        & aa(A,$o,P,Y)
        & ~ ? [X3: A] :
              ( member(A,X3,aa(list(A),set(A),set2(A),Ys))
              & aa(A,$o,P,X3) ) ) ) ).

% extract_SomeE
tff(fact_7062_extract__None__iff,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] :
      ( ( extract(A,P,Xs) = none(product_prod(list(A),product_prod(A,list(A)))) )
    <=> ~ ? [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),Xs))
            & aa(A,$o,P,X) ) ) ).

% extract_None_iff
tff(fact_7063_extract__Nil__code,axiom,
    ! [A: $tType,P: fun(A,$o)] : extract(A,P,nil(A)) = none(product_prod(list(A),product_prod(A,list(A)))) ).

% extract_Nil_code
tff(fact_7064_extract__Cons__code,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A,Xs: list(A)] :
      extract(A,P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = $ite(aa(A,$o,P,Xb),aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),nil(A)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Xb),Xs))),case_option(option(product_prod(list(A),product_prod(A,list(A)))),product_prod(list(A),product_prod(A,list(A))),none(product_prod(list(A),product_prod(A,list(A)))),product_case_prod(list(A),product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))),aTP_Lamp_amd(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Xb)),extract(A,P,Xs))) ).

% extract_Cons_code
tff(fact_7065_map__filter__map__filter,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),P: fun(B,$o),Xs: list(B)] : aa(list(B),list(A),map(B,A,F),filter2(B,P,Xs)) = map_filter(B,A,aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_ame(fun(B,A),fun(fun(B,$o),fun(B,option(A))),F),P),Xs) ).

% map_filter_map_filter
tff(fact_7066_subset__eq__mset__impl_Oelims,axiom,
    ! [A: $tType,Xb: list(A),Xaa: list(A),Y: option($o)] :
      ( ( subset_eq_mset_impl(A,Xb,Xaa) = Y )
     => ( ( ( Xb = nil(A) )
         => ( Y != aa($o,option($o),some($o),Xaa != nil(A)) ) )
       => ~ ! [X2: A,Xs2: list(A)] :
              ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
             => ( Y != case_option(option($o),product_prod(list(A),product_prod(A,list(A))),none($o),product_case_prod(list(A),product_prod(A,list(A)),option($o),aTP_Lamp_amg(list(A),fun(list(A),fun(product_prod(A,list(A)),option($o))),Xs2)),extract(A,aa(A,fun(A,$o),fequal(A),X2),Xaa)) ) ) ) ) ).

% subset_eq_mset_impl.elims
tff(fact_7067_subset__eq__mset__impl_Osimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] : subset_eq_mset_impl(A,nil(A),Ys) = aa($o,option($o),some($o),Ys != nil(A)) ).

% subset_eq_mset_impl.simps(1)
tff(fact_7068_map__filter__simps_I1_J,axiom,
    ! [A: $tType,B: $tType,F: fun(B,option(A)),Xb: B,Xs: list(B)] : map_filter(B,A,F,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Xb),Xs)) = case_option(list(A),A,map_filter(B,A,F,Xs),aa(list(B),fun(A,list(A)),aTP_Lamp_amh(fun(B,option(A)),fun(list(B),fun(A,list(A))),F),Xs),aa(B,option(A),F,Xb)) ).

% map_filter_simps(1)
tff(fact_7069_subset__eq__mset__impl_Osimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A)] : subset_eq_mset_impl(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),Ys) = case_option(option($o),product_prod(list(A),product_prod(A,list(A))),none($o),product_case_prod(list(A),product_prod(A,list(A)),option($o),aTP_Lamp_amg(list(A),fun(list(A),fun(product_prod(A,list(A)),option($o))),Xs)),extract(A,aa(A,fun(A,$o),fequal(A),Xb),Ys)) ).

% subset_eq_mset_impl.simps(2)
tff(fact_7070_subset__eq__mset__impl_Opelims,axiom,
    ! [A: $tType,Xb: list(A),Xaa: list(A),Y: option($o)] :
      ( ( subset_eq_mset_impl(A,Xb,Xaa) = Y )
     => ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Xb),Xaa))
       => ( ( ( Xb = nil(A) )
           => ( ( Y = aa($o,option($o),some($o),Xaa != nil(A)) )
             => ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xaa)) ) )
         => ~ ! [X2: A,Xs2: list(A)] :
                ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
               => ( ( Y = case_option(option($o),product_prod(list(A),product_prod(A,list(A))),none($o),product_case_prod(list(A),product_prod(A,list(A)),option($o),aTP_Lamp_amg(list(A),fun(list(A),fun(product_prod(A,list(A)),option($o))),Xs2)),extract(A,aa(A,fun(A,$o),fequal(A),X2),Xaa)) )
                 => ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),subset751672762298770561pl_rel(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),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Xaa)) ) ) ) ) ) ).

% subset_eq_mset_impl.pelims
tff(fact_7071_quicksort_Osimps_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Xs: list(A)] : aa(list(A),list(A),linorder_quicksort(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)) = aa(list(A),list(A),append(A,aa(list(A),list(A),linorder_quicksort(A),filter2(A,aTP_Lamp_ami(A,fun(A,$o),Xb),Xs))),aa(list(A),list(A),append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))),aa(list(A),list(A),linorder_quicksort(A),filter2(A,aa(A,fun(A,$o),ord_less_eq(A),Xb),Xs)))) ) ).

% quicksort.simps(2)
tff(fact_7072_set__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xs: list(A)] : aa(list(A),set(A),set2(A),aa(list(A),list(A),linorder_quicksort(A),Xs)) = aa(list(A),set(A),set2(A),Xs) ) ).

% set_quicksort
tff(fact_7073_sort__quicksort,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ( linorder_sort_key(A,A,aTP_Lamp_aks(A,A)) = linorder_quicksort(A) ) ) ).

% sort_quicksort
tff(fact_7074_quicksort_Oelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: list(A),Y: list(A)] :
          ( ( aa(list(A),list(A),linorder_quicksort(A),Xb) = Y )
         => ( ( ( Xb = nil(A) )
             => ( Y != nil(A) ) )
           => ~ ! [X2: A,Xs2: list(A)] :
                  ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
                 => ( Y != aa(list(A),list(A),append(A,aa(list(A),list(A),linorder_quicksort(A),filter2(A,aTP_Lamp_ami(A,fun(A,$o),X2),Xs2))),aa(list(A),list(A),append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))),aa(list(A),list(A),linorder_quicksort(A),filter2(A,aa(A,fun(A,$o),ord_less_eq(A),X2),Xs2)))) ) ) ) ) ) ).

% quicksort.elims
tff(fact_7075_quicksort_Opelims,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: list(A),Y: list(A)] :
          ( ( aa(list(A),list(A),linorder_quicksort(A),Xb) = Y )
         => ( aa(list(A),$o,accp(list(A),linord6200660962353139674rt_rel(A)),Xb)
           => ( ( ( Xb = nil(A) )
               => ( ( Y = nil(A) )
                 => ~ aa(list(A),$o,accp(list(A),linord6200660962353139674rt_rel(A)),nil(A)) ) )
             => ~ ! [X2: A,Xs2: list(A)] :
                    ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
                   => ( ( Y = aa(list(A),list(A),append(A,aa(list(A),list(A),linorder_quicksort(A),filter2(A,aTP_Lamp_ami(A,fun(A,$o),X2),Xs2))),aa(list(A),list(A),append(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),nil(A))),aa(list(A),list(A),linorder_quicksort(A),filter2(A,aa(A,fun(A,$o),ord_less_eq(A),X2),Xs2)))) )
                     => ~ aa(list(A),$o,accp(list(A),linord6200660962353139674rt_rel(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)) ) ) ) ) ) ) ).

% quicksort.pelims
tff(fact_7076_transpose__aux__filter__head,axiom,
    ! [A: $tType,Xss: list(list(A))] : concat(A,aa(list(list(A)),list(list(A)),map(list(A),list(A),case_list(list(A),A,nil(A),aTP_Lamp_alx(A,fun(list(A),list(A))))),Xss)) = aa(list(list(A)),list(A),map(list(A),A,hd(A)),filter2(list(A),aTP_Lamp_alm(list(A),$o),Xss)) ).

% transpose_aux_filter_head
tff(fact_7077_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_7078_in__hd__or__tl__conv,axiom,
    ! [A: $tType,L: list(A),Xb: A] :
      ( ( L != nil(A) )
     => ( ( ( Xb = aa(list(A),A,hd(A),L) )
          | member(A,Xb,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),L))) )
      <=> member(A,Xb,aa(list(A),set(A),set2(A),L)) ) ) ).

% in_hd_or_tl_conv
tff(fact_7079_not__hd__in__tl,axiom,
    ! [A: $tType,Xb: A,Xs: list(A)] :
      ( ( Xb != aa(list(A),A,hd(A),Xs) )
     => ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
       => member(A,Xb,aa(list(A),set(A),set2(A),aa(list(A),list(A),tl(A),Xs))) ) ) ).

% not_hd_in_tl
tff(fact_7080_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_amj(A,fun(list(A),A))),List) ).

% hd_def
tff(fact_7081_list_Oset__sel_I1_J,axiom,
    ! [A: $tType,A2: list(A)] :
      ( ( A2 != nil(A) )
     => member(A,aa(list(A),A,hd(A),A2),aa(list(A),set(A),set2(A),A2)) ) ).

% list.set_sel(1)
tff(fact_7082_hd__in__set,axiom,
    ! [A: $tType,Xs: list(A)] :
      ( ( Xs != nil(A) )
     => member(A,aa(list(A),A,hd(A),Xs),aa(list(A),set(A),set2(A),Xs)) ) ).

% hd_in_set
tff(fact_7083_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_7084_slice__head,axiom,
    ! [A: $tType,From: nat,To: nat,Xs: list(A)] :
      ( aa(nat,$o,ord_less(nat,From),To)
     => ( aa(nat,$o,aa(nat,fun(nat,$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_7085_Nitpick_Osize__list__simp_I1_J,axiom,
    ! [A: $tType,F: fun(A,nat),Xs: list(A)] :
      size_list(A,F,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,F,aa(list(A),A,hd(A),Xs))),size_list(A,F,aa(list(A),list(A),tl(A),Xs))))) ).

% Nitpick.size_list_simp(1)
tff(fact_7086_splice_Opinduct,axiom,
    ! [A: $tType,A0: list(A),A1: list(A),P: fun(list(A),fun(list(A),$o))] :
      ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),A0),A1))
     => ( ! [Ys4: list(A)] :
            ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(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)),nil(A)),Ys4))
           => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,nil(A)),Ys4) )
       => ( ! [X2: A,Xs2: list(A),Ys4: list(A)] :
              ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Ys4))
             => ( aa(list(A),$o,aa(list(A),fun(list(A),$o),P,Ys4),Xs2)
               => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Ys4) ) )
         => aa(list(A),$o,aa(list(A),fun(list(A),$o),P,A0),A1) ) ) ) ).

% splice.pinduct
tff(fact_7087_size__list__append,axiom,
    ! [A: $tType,F: fun(A,nat),Xs: list(A),Ys: list(A)] : size_list(A,F,aa(list(A),list(A),append(A,Xs),Ys)) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),size_list(A,F,Xs)),size_list(A,F,Ys)) ).

% size_list_append
tff(fact_7088_size__list__estimation,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: nat,F: fun(A,nat)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,ord_less(nat,Y),aa(A,nat,F,Xb))
       => aa(nat,$o,ord_less(nat,Y),size_list(A,F,Xs)) ) ) ).

% size_list_estimation
tff(fact_7089_size__list__pointwise,axiom,
    ! [A: $tType,Xs: list(A),F: fun(A,nat),G: fun(A,nat)] :
      ( ! [X2: A] :
          ( member(A,X2,aa(list(A),set(A),set2(A),Xs))
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(A,nat,F,X2)),aa(A,nat,G,X2)) )
     => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),size_list(A,F,Xs)),size_list(A,G,Xs)) ) ).

% size_list_pointwise
tff(fact_7090_size__list__estimation_H,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: nat,F: fun(A,nat)] :
      ( member(A,Xb,aa(list(A),set(A),set2(A),Xs))
     => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),aa(A,nat,F,Xb))
       => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),size_list(A,F,Xs)) ) ) ).

% size_list_estimation'
tff(fact_7091_list_Osize__gen_I2_J,axiom,
    ! [A: $tType,Xb: fun(A,nat),X21: A,X22: list(A)] : size_list(A,Xb,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X21),X22)) = 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,X22))),aa(nat,nat,suc,zero_zero(nat))) ).

% list.size_gen(2)
tff(fact_7092_VEBT_Osize_I3_J,axiom,
    ! [X11a: option(product_prod(nat,nat)),X12a: nat,X13a: list(vEBT_VEBT),X14a: vEBT_VEBT] : aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),vEBT_Node(X11a,X12a,X13a,X14a)) = 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),X13a)),aa(vEBT_VEBT,nat,size_size(vEBT_VEBT),X14a))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size(3)
tff(fact_7093_VEBT_Osize__gen_I1_J,axiom,
    ! [X11a: option(product_prod(nat,nat)),X12a: nat,X13a: list(vEBT_VEBT),X14a: vEBT_VEBT] : aa(vEBT_VEBT,nat,vEBT_size_VEBT,vEBT_Node(X11a,X12a,X13a,X14a)) = 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,X13a)),aa(vEBT_VEBT,nat,vEBT_size_VEBT,X14a))),aa(nat,nat,suc,zero_zero(nat))) ).

% VEBT.size_gen(1)
tff(fact_7094_splice_Opelims,axiom,
    ! [A: $tType,Xb: list(A),Xaa: list(A),Y: list(A)] :
      ( ( splice(A,Xb,Xaa) = Y )
     => ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(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)),Xb),Xaa))
       => ( ( ( Xb = nil(A) )
           => ( ( Y = Xaa )
             => ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),nil(A)),Xaa)) ) )
         => ~ ! [X2: A,Xs2: list(A)] :
                ( ( Xb = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2) )
               => ( ( Y = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),splice(A,Xaa,Xs2)) )
                 => ~ aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(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),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)),Xaa)) ) ) ) ) ) ).

% splice.pelims
tff(fact_7095_length__splice,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A)] : aa(list(A),nat,size_size(list(A)),splice(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_splice
tff(fact_7096_splice__replicate,axiom,
    ! [A: $tType,M: nat,Xb: A,Nb: nat] : splice(A,replicate(A,M,Xb),replicate(A,Nb,Xb)) = replicate(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb),Xb) ).

% splice_replicate
tff(fact_7097_splice_Opsimps_I2_J,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Ys: list(A)] :
      ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(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),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),Ys))
     => ( splice(A,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs),Ys) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),splice(A,Ys,Xs)) ) ) ).

% splice.psimps(2)
tff(fact_7098_splice_Opsimps_I1_J,axiom,
    ! [A: $tType,Ys: list(A)] :
      ( aa(product_prod(list(A),list(A)),$o,accp(product_prod(list(A),list(A)),splice_rel(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)),nil(A)),Ys))
     => ( splice(A,nil(A),Ys) = Ys ) ) ).

% splice.psimps(1)
tff(fact_7099_entails__solve__init_I1_J,axiom,
    ! [P: assn,Q: assn] :
      ( fI_QUERY(P,Q,top_top(assn))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),top_top(assn))) ) ).

% entails_solve_init(1)
tff(fact_7100_image__Collect__subsetI,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F: fun(A,B),B5: set(B)] :
      ( ! [X2: A] :
          ( aa(A,$o,P,X2)
         => member(B,aa(A,B,F,X2),B5) )
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),collect(A,P))),B5) ) ).

% image_Collect_subsetI
tff(fact_7101_ssubst__Pair__rhs,axiom,
    ! [B: $tType,A: $tType,R3: A,S3: 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),R3),S3),R)
     => ( ( S4 = S3 )
       => member(product_prod(A,B),aa(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),R3),S4),R) ) ) ).

% ssubst_Pair_rhs
tff(fact_7102_prop__restrict,axiom,
    ! [A: $tType,Xb: A,Z6: set(A),X5: set(A),P: fun(A,$o)] :
      ( member(A,Xb,Z6)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Z6),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cv(set(A),fun(fun(A,$o),fun(A,$o)),X5),P)))
       => aa(A,$o,P,Xb) ) ) ).

% prop_restrict
tff(fact_7103_Collect__restrict,axiom,
    ! [A: $tType,X5: set(A),P: fun(A,$o)] : aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cv(set(A),fun(fun(A,$o),fun(A,$o)),X5),P))),X5) ).

% Collect_restrict
tff(fact_7104_FI__QUERY__def,axiom,
    ! [P: assn,Q: assn,F5: assn] :
      ( fI_QUERY(P,Q,F5)
    <=> aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),F5)) ) ).

% FI_QUERY_def
tff(fact_7105_frame__inference__init,axiom,
    ! [P: assn,Q: assn,F5: assn] :
      ( fI_QUERY(P,Q,F5)
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,P),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q),F5)) ) ).

% frame_inference_init
tff(fact_7106_entails__solve__init_I2_J,axiom,
    ! [P: assn,Q: assn] :
      ( fI_QUERY(P,Q,one_one(assn))
     => aa(assn,$o,aa(assn,fun(assn,$o),entails,P),Q) ) ).

% entails_solve_init(2)
tff(fact_7107_subset__emptyI,axiom,
    ! [A: $tType,A3: set(A)] :
      ( ! [X2: A] : ~ member(A,X2,A3)
     => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),bot_bot(set(A))) ) ).

% subset_emptyI
tff(fact_7108_insert__subsetI,axiom,
    ! [A: $tType,Xb: A,A3: set(A),X5: set(A)] :
      ( member(A,Xb,A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),X5),A3)
       => aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),insert(A,Xb),X5)),A3) ) ) ).

% insert_subsetI
tff(fact_7109_smod__int__range,axiom,
    ! [B2: int,A2: int] :
      ( ( B2 != zero_zero(int) )
     => member(int,signed6721504322012087516modulo(int,A2,B2),set_or1337092689740270186AtMost(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,uminus_uminus(int),aa(int,int,abs_abs(int),B2))),one_one(int)),aa(int,int,minus_minus(int,aa(int,int,abs_abs(int),B2)),one_one(int)))) ) ).

% smod_int_range
tff(fact_7110_mod__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2))) = 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),B2))))) ) ).

% mod_word_minus_1_minus_numeral
tff(fact_7111_smod__int__numeral__numeral,axiom,
    ! [M: num,Nb: num] : signed6721504322012087516modulo(int,aa(num,int,numeral_numeral(int),M),aa(num,int,numeral_numeral(int),Nb)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),M)),aa(num,int,numeral_numeral(int),Nb)) ).

% smod_int_numeral_numeral
tff(fact_7112_drop__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),Nb),W) = zero_zero(word(A)) ) ) ) ).

% drop_bit_word_beyond
tff(fact_7113_push__bit__word__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),Nb),W) = zero_zero(word(A)) ) ) ) ).

% push_bit_word_beyond
tff(fact_7114_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_7115_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_7116_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_7117_less__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(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),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% less_eq_word_numeral_numeral
tff(fact_7118_less__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> 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),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% less_word_numeral_numeral
tff(fact_7119_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_7120_bit__numeral__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num,Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(num,word(A),numeral_numeral(word(A)),W)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),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)),Nb) ) ) ) ).

% bit_numeral_word_iff
tff(fact_7121_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_7122_unsigned__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [Nb: num] : aa(word(B),A,semiring_1_unsigned(B,A),aa(num,word(B),numeral_numeral(word(B)),Nb)) = 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),Nb))) ) ).

% unsigned_numeral
tff(fact_7123_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_7124_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_7125_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_7126_exp__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( ( aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb) = zero_zero(word(A)) )
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb) ) ) ).

% exp_eq_zero_iff
tff(fact_7127_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_7128_signed__take__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,K: num] : aa(word(A),word(A),bit_ri4674362597316999326ke_bit(word(A),aa(nat,nat,suc,Nb)),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,Nb)),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_7129_signed__take__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num,K: num] : aa(word(A),word(A),bit_ri4674362597316999326ke_bit(word(A),aa(num,nat,numeral_numeral(nat),Nb)),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),Nb)),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_7130_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_7131_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_7132_div__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% div_word_numeral_numeral
tff(fact_7133_mod__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% mod_word_numeral_numeral
tff(fact_7134_signed__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Nb: num] : ring_1_signed(B,A,aa(num,word(B),numeral_numeral(word(B)),Nb)) = 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),Nb))) ) ).

% signed_numeral
tff(fact_7135_less__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(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),A2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% less_eq_word_minus_numeral_numeral
tff(fact_7136_less__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(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),A2))),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),B2)))) ) ) ).

% less_eq_word_numeral_minus_numeral
tff(fact_7137_less__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(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),A2)))),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),B2)))) ) ) ).

% less_eq_word_minus_numeral_minus_numeral
tff(fact_7138_less__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> 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),A2)))),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),B2)))) ) ) ).

% less_word_minus_numeral_minus_numeral
tff(fact_7139_less__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> 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),A2))),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),B2)))) ) ) ).

% less_word_numeral_minus_numeral
tff(fact_7140_less__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> 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),A2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2))) ) ) ).

% less_word_minus_numeral_numeral
tff(fact_7141_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_7142_bit__neg__numeral__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num,Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W))),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),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))),Nb) ) ) ) ).

% bit_neg_numeral_word_iff
tff(fact_7143_drop__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,suc,Nb)),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,Nb)),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_7144_drop__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: 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),Nb)),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),Nb)),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_7145_unat__power__lower,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,ord_less(nat,Nb),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))),Nb)) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ) ) ) ).

% unat_power_lower
tff(fact_7146_unsigned__neg__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & semiring_1(A) )
     => ! [Nb: 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)),Nb))) = 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),Nb))))) ) ).

% unsigned_neg_numeral
tff(fact_7147_signed__take__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,K: num] : aa(word(A),word(A),bit_ri4674362597316999326ke_bit(word(A),aa(nat,nat,suc,Nb)),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,Nb)),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_7148_signed__take__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num,K: num] : aa(word(A),word(A),bit_ri4674362597316999326ke_bit(word(A),aa(num,nat,numeral_numeral(nat),Nb)),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),Nb)),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_7149_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_7150_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_7151_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_7152_div__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% div_word_minus_numeral_numeral
tff(fact_7153_div__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2))),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),B2))))) ) ).

% div_word_numeral_minus_numeral
tff(fact_7154_div__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2)))),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),B2))))) ) ).

% div_word_minus_numeral_minus_numeral
tff(fact_7155_mod__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2)))),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),B2))))) ) ).

% mod_word_minus_numeral_minus_numeral
tff(fact_7156_mod__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2))),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),B2))))) ) ).

% mod_word_numeral_minus_numeral
tff(fact_7157_mod__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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)),A2))),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),B2)))) ) ).

% mod_word_minus_numeral_numeral
tff(fact_7158_word__less__sub__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$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))),Nb)),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))),Nb)) ) ) ) ).

% word_less_sub_le
tff(fact_7159_signed__neg__numeral,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Nb: num] : ring_1_signed(B,A,aa(word(B),word(B),uminus_uminus(word(B)),aa(num,word(B),numeral_numeral(word(B)),Nb))) = 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),Nb)))) ) ).

% signed_neg_numeral
tff(fact_7160_drop__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,K: num] : aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4197421643247451524op_bit(word(A)),aa(nat,nat,suc,Nb)),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,Nb)),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_7161_drop__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: 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),Nb)),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),Nb)),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_7162_less__word__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(num,word(A),numeral_numeral(word(A)),A2)),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),A2))),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_7163_less__word__minus__numeral__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num] :
          ( aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2))),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),A2)))),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_7164_div__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2)) = 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),B2)))) ) ).

% div_word_minus_1_numeral
tff(fact_7165_mod__word__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2)) = 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),B2)))) ) ).

% mod_word_minus_1_numeral
tff(fact_7166_div__word__minus__1__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2))) = 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),B2))))) ) ).

% div_word_minus_1_minus_numeral
tff(fact_7167_word__split__bin_H,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [W: word(C)] : 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)),aa(word(C),word(A),semiring_1_unsigned(C,word(A)),aa(word(C),word(C),aa(nat,fun(word(C),word(C)),bit_se4197421643247451524op_bit(word(C)),type_len0_len_of(B,type2(B))),W))),aa(word(C),word(B),semiring_1_unsigned(C,word(B)),W)) ) ).

% word_split_bin'
tff(fact_7168_bit__word__cat__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(B)
        & type_len(A) )
     => ! [V: word(B),W: word(C),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_cat(B,C,A,V,W)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
            & $ite(aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(C,type2(C))),aa(nat,$o,aa(word(C),fun(nat,$o),bit_se5641148757651400278ts_bit(word(C)),W),Nb),aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),V),aa(nat,nat,minus_minus(nat,Nb),type_len0_len_of(C,type2(C))))) ) ) ) ).

% bit_word_cat_iff
tff(fact_7169_word__cat__inj,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(C)
        & type_len(A) )
     => ! [A2: word(A),B2: word(B),C2: word(A),D2: word(B)] :
          ( aa(nat,$o,aa(nat,fun(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,A2,B2) = word_cat(A,B,C,C2,D2) )
          <=> ( ( A2 = C2 )
              & ( B2 = D2 ) ) ) ) ) ).

% word_cat_inj
tff(fact_7170_word__of__int__bin__cat__eq__iff,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(C)
        & type_len(B) )
     => ! [B2: word(B),A2: word(A),D2: word(B),C2: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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)),bit_concat_bit(type_len0_len_of(B,type2(B)),aa(word(B),int,semiring_1_unsigned(B,int),B2),aa(word(A),int,semiring_1_unsigned(A,int),A2))) = aa(int,word(C),ring_1_of_int(word(C)),bit_concat_bit(type_len0_len_of(B,type2(B)),aa(word(B),int,semiring_1_unsigned(B,int),D2),aa(word(A),int,semiring_1_unsigned(A,int),C2))) )
          <=> ( ( B2 = D2 )
              & ( A2 = C2 ) ) ) ) ) ).

% word_of_int_bin_cat_eq_iff
tff(fact_7171_word__cat__bin_H,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( ( type_len(C)
        & type_len(A)
        & type_len(B) )
     => ! [V: word(B),W: word(C)] : word_cat(B,C,A,V,W) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),type_len0_len_of(C,type2(C))),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),V))),aa(word(C),word(A),semiring_1_unsigned(C,word(A)),W)) ) ).

% word_cat_bin'
tff(fact_7172_mask__exceed,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xb: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( 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),Nb))) = zero_zero(word(A)) ) ) ) ).

% mask_exceed
tff(fact_7173_ucast__ucast__add,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xb: word(A),Y: word(B)] :
          ( aa(nat,$o,aa(nat,fun(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_7174_ucast__sub__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Y: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),Xb)
         => ( aa(nat,$o,aa(nat,fun(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_7175_ucast__mask__drop,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Nb: nat,Xb: word(B)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( 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),Nb))) = aa(word(B),word(A),semiring_1_unsigned(B,word(A)),Xb) ) ) ) ).

% ucast_mask_drop
tff(fact_7176_bintr__uint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),aa(word(A),int,semiring_1_unsigned(A,int),W)) = aa(word(A),int,semiring_1_unsigned(A,int),W) ) ) ) ).

% bintr_uint
tff(fact_7177_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,aa(nat,fun(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_7178_ucast__le__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(B),$o,aa(word(B),fun(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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y) ) ) ) ).

% ucast_le_ucast
tff(fact_7179_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,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(B,type2(B))),type_len0_len_of(C,type2(C)))
           => ( aa(nat,$o,aa(nat,fun(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_7180_up__ucast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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_7181_ucast__up__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y)
           => aa(word(B),$o,aa(word(B),fun(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_7182_eq__ucast__ucast__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(B),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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_7183_unat__ucast__up__simp,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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_7184_ucast__up__mono,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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_7185_ucast__less__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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_7186_less__ucast__ucast__less,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(B),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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_7187_word__of__nat__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),semiring_1_of_nat(word(A)),M)),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb))
        <=> aa(nat,$o,aa(nat,fun(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))),Nb)) ) ) ).

% word_of_nat_less_eq_iff
tff(fact_7188_take__bit__word__beyond__length__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),Nb),W) = W ) ) ) ).

% take_bit_word_beyond_length_eq
tff(fact_7189_wi__bintr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: int] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,bit_se2584673776208193580ke_bit(int,Nb),W)) = aa(int,word(A),ring_1_of_int(word(A)),W) ) ) ) ).

% wi_bintr
tff(fact_7190_word__of__int__less__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,L: int] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(int,fun(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_7191_up__scast__inj__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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_7192_mask__over__length,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( bit_se2239418461657761734s_mask(word(A),Nb) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ) ) ).

% mask_over_length
tff(fact_7193_bin__nth__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),ring_1_signed(A,int,W)),Nb)
          <=> aa(nat,$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_7194_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_7195_neg__mask__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),bit_se2239418461657761734s_mask(word(A),Nb))),M)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
            & aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) ) ) ) ).

% neg_mask_test_bit
tff(fact_7196_uint__word__arith__bintrs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: 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)),A2),B2)) = aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% uint_word_arith_bintrs(3)
tff(fact_7197_num__of__bintr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( ( aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),aa(num,int,numeral_numeral(int),A2)) = aa(num,int,numeral_numeral(int),B2) )
         => ( aa(num,word(A),numeral_numeral(word(A)),A2) = aa(num,word(A),numeral_numeral(word(A)),B2) ) ) ) ).

% num_of_bintr'
tff(fact_7198_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_7199_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_7200_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_7201_neg__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xb)),Nb)
        <=> ( ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),Nb)
            & aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A))) ) ) ) ).

% neg_test_bit
tff(fact_7202_bit__uint__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Nb) ) ) ) ).

% bit_uint_iff
tff(fact_7203_bit__ucast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [A2: word(B),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),A2)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),A2),Nb) ) ) ) ).

% bit_ucast_iff
tff(fact_7204_bin__nth__uint__imp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),aa(word(A),int,semiring_1_unsigned(A,int),W)),Nb)
         => aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A))) ) ) ).

% bin_nth_uint_imp
tff(fact_7205_bit__word__ucast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(B,type2(B)))
            & aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),Nb) ) ) ) ).

% bit_word_ucast_iff
tff(fact_7206_test__bit__bin,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),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)),Nb) ) ) ) ).

% test_bit_bin
tff(fact_7207_nth__ucast,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [W: word(B),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W)),Nb)
        <=> ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),Nb)
            & aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_ucast
tff(fact_7208_word__eq__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( ( Xb = Y )
        <=> ! [N6: nat] :
              ( aa(nat,$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)),Xb),N6)
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),N6) ) ) ) ) ).

% word_eq_iff
tff(fact_7209_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_7210_bit__imp__le__length,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Nb)
         => aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A))) ) ) ).

% bit_imp_le_length
tff(fact_7211_bit__word__eqI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( ! [N2: nat] :
              ( aa(nat,$o,ord_less(nat,N2),type_len0_len_of(A,type2(A)))
             => ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),A2),N2)
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),B2),N2) ) )
         => ( A2 = B2 ) ) ) ).

% bit_word_eqI
tff(fact_7212_bit__word__of__int__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),ring_1_of_int(word(A)),K)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),K),Nb) ) ) ) ).

% bit_word_of_int_iff
tff(fact_7213_test__bit__wi,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int,Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),ring_1_of_int(word(A)),Xb)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),Xb),Nb) ) ) ) ).

% test_bit_wi
tff(fact_7214_max__test__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))),Nb)
        <=> aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A))) ) ) ).

% max_test_bit
tff(fact_7215_test__bit__1_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),one_one(word(A))),Nb)
        <=> ( aa(nat,$o,ord_less(nat,zero_zero(nat)),type_len0_len_of(A,type2(A)))
            & ( Nb = zero_zero(nat) ) ) ) ) ).

% test_bit_1'
tff(fact_7216_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_7217_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_7218_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_7219_max__word__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) = bit_se2239418461657761734s_mask(word(A),type_len0_len_of(A,type2(A))) ) ) ).

% max_word_mask
tff(fact_7220_unsigned__minus__1__eq__mask,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_se359711467146920520ations(A) )
     => ( aa(word(B),A,semiring_1_unsigned(B,A),aa(word(B),word(B),uminus_uminus(word(B)),one_one(word(B)))) = bit_se2239418461657761734s_mask(A,type_len0_len_of(B,type2(B))) ) ) ).

% unsigned_minus_1_eq_mask
tff(fact_7221_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_7222_signed__take__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [Nb: nat,W: word(B)] :
          ring_1_signed(B,A,aa(word(B),word(B),bit_se2584673776208193580ke_bit(word(B),Nb),W)) = $ite(aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(B,type2(B))),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),ring_1_signed(B,A,W)),ring_1_signed(B,A,W)) ) ).

% signed_take_bit_eq
tff(fact_7223_word__of__nat__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Nb: nat] :
          ( aa(word(A),$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)),Nb))
        <=> 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))),Nb)) ) ) ).

% word_of_nat_less_iff
tff(fact_7224_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_7225_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_7226_uint__word__arith__bintrs_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),A2),B2)) = 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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% uint_word_arith_bintrs(2)
tff(fact_7227_uint__word__arith__bintrs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: 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)),A2),B2)) = 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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))) ) ).

% uint_word_arith_bintrs(1)
tff(fact_7228_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_7229_num__of__sbintr_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: 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),A2)) = aa(num,int,numeral_numeral(int),B2) )
         => ( aa(num,word(A),numeral_numeral(word(A)),A2) = aa(num,word(A),numeral_numeral(word(A)),B2) ) ) ) ).

% num_of_sbintr'
tff(fact_7230_bit__set__bit__aux,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,$o),Nb: nat,W: word(A),M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),code_T2661198915054445665ts_aux(A,F,Nb,W)),M)
        <=> ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
            & $ite(aa(nat,$o,ord_less(nat,M),Nb),aa(nat,$o,F,M),aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),aa(nat,nat,minus_minus(nat,M),Nb))) ) ) ) ).

% bit_set_bit_aux
tff(fact_7231_two__power__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$o,ord_less(nat,Nb),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))),Nb) = 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) )
            <=> ( Nb = M ) ) ) ) ) ).

% two_power_eq
tff(fact_7232_ucast__drop__bit__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(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)),Nb),W)) = aa(word(B),word(B),aa(nat,fun(word(B),word(B)),bit_se4197421643247451524op_bit(word(B)),Nb),aa(word(A),word(B),semiring_1_unsigned(A,word(B)),W)) ) ) ) ).

% ucast_drop_bit_eq
tff(fact_7233_bit__set__bit__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: nat,B2: $o,Nb: nat] :
          ( aa(nat,$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),(B2))),Nb)
        <=> $ite(
              M = Nb,
              ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
              & (B2) ),
              aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),Nb) ) ) ) ).

% bit_set_bit_word_iff
tff(fact_7234_nth__slice,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Nb: nat,W: word(B),M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),slice2(B,A,Nb),W)),M)
        <=> ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb))
            & aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_slice
tff(fact_7235_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_7236_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_7237_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,aa(nat,fun(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_7238_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_7239_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_7240_smod__word__max,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,ord_less(int,signed6721504322012087516modulo(int,ring_1_signed(A,int,A2),ring_1_signed(A,int,B2))),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_7241_word__nchotomy,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W3: word(A)] :
        ? [N2: nat] :
          ( ( W3 = aa(nat,word(A),semiring_1_of_nat(word(A)),N2) )
          & aa(nat,$o,ord_less(nat,N2),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_7242_word__nat__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ~ ! [N2: nat] :
              ( ( Xb = aa(nat,word(A),semiring_1_of_nat(word(A)),N2) )
             => ~ aa(nat,$o,ord_less(nat,N2),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_7243_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_7244_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_7245_More__Word_Opower__not__zero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,ord_less(nat,Nb),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))),Nb) != zero_zero(word(A)) ) ) ) ).

% More_Word.power_not_zero
tff(fact_7246_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_7247_power__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( 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))),Nb) = zero_zero(word(A)) ) ) ) ).

% power_overflow
tff(fact_7248_nth__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)),M)
        <=> ( ( M = Nb )
            & aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) ) ) ) ).

% nth_w2p
tff(fact_7249_nth__w2p__same,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)),Nb)
        <=> aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A))) ) ) ).

% nth_w2p_same
tff(fact_7250_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_7251_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_7252_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_7253_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_7254_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_7255_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_7256_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_7257_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)),aa(word(A),fun(list(word(A)),list(word(A))),cons(word(A)),W),Ws)) = 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_7258_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_7259_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_7260_sint__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : ring_1_signed(A,int,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),B2)) = 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,A2)),ring_1_signed(A,int,B2))) ) ).

% sint_word_ariths(1)
tff(fact_7261_nth__sint,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),ring_1_signed(A,int,W)),Nb)
        <=> $ite(aa(nat,$o,ord_less(nat,Nb),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),Nb),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_7262_bit__sint__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),ring_1_signed(A,int,W)),Nb)
        <=> ( ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
              & aa(nat,$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),Nb) ) ) ) ).

% bit_sint_iff
tff(fact_7263_sint__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ring_1_signed(A,int,aa(word(A),word(A),uminus_uminus(word(A)),A2)) = 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,A2))) ) ).

% sint_word_ariths(4)
tff(fact_7264_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_7265_sint__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : ring_1_signed(A,int,aa(word(A),word(A),minus_minus(word(A),A2),B2)) = 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,A2)),ring_1_signed(A,int,B2))) ) ).

% sint_word_ariths(2)
tff(fact_7266_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_7267_sint__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : ring_1_signed(A,int,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)) = 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,A2)),ring_1_signed(A,int,B2))) ) ).

% sint_word_ariths(3)
tff(fact_7268_smod__word__min,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(int,$o,aa(int,fun(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,A2),ring_1_signed(A,int,B2))) ) ).

% smod_word_min
tff(fact_7269_smod__int__compares_I1_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
       => aa(int,$o,ord_less(int,signed6721504322012087516modulo(int,A2,B2)),B2) ) ) ).

% smod_int_compares(1)
tff(fact_7270_smod__int__compares_I2_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),signed6721504322012087516modulo(int,A2,B2)) ) ) ).

% smod_int_compares(2)
tff(fact_7271_smod__int__compares_I4_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),signed6721504322012087516modulo(int,A2,B2)),zero_zero(int)) ) ) ).

% smod_int_compares(4)
tff(fact_7272_smod__int__compares_I6_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,ord_less(int,B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),signed6721504322012087516modulo(int,A2,B2)) ) ) ).

% smod_int_compares(6)
tff(fact_7273_smod__int__compares_I7_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,ord_less(int,B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),signed6721504322012087516modulo(int,A2,B2)),zero_zero(int)) ) ) ).

% smod_int_compares(7)
tff(fact_7274_smod__int__compares_I8_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,ord_less(int,B2),zero_zero(int))
       => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),B2),signed6721504322012087516modulo(int,A2,B2)) ) ) ).

% smod_int_compares(8)
tff(fact_7275_less__Suc__unat__less__bound,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),Xb)))
         => aa(nat,$o,ord_less(nat,Nb),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_7276_smod__mod__positive,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),B2)
       => ( signed6721504322012087516modulo(int,A2,B2) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),A2),B2) ) ) ) ).

% smod_mod_positive
tff(fact_7277_uint__2__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( aa(nat,$o,aa(nat,fun(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_7278_lt2p__lem,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => aa(int,$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))),Nb)) ) ) ).

% lt2p_lem
tff(fact_7279_power__le__mono,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),M))
         => ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
           => ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M) ) ) ) ) ).

% power_le_mono
tff(fact_7280_two__power__increasing,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),M)
         => ( aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A)))
           => aa(word(A),$o,aa(word(A),fun(word(A),$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))),Nb)),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_7281_unat__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(word(A),nat,semiring_1_unsigned(A,nat),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(num,nat,numeral_numeral(nat),B2)),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_7282_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_7283_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_7284_unat__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),Xb: word(A)] :
          ( aa(nat,$o,P,aa(word(A),nat,semiring_1_unsigned(A,nat),Xb))
        <=> ! [N6: nat] :
              ( ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N6) = Xb )
                & aa(nat,$o,ord_less(nat,N6),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,N6) ) ) ) ).

% unat_split
tff(fact_7285_of__nat__inverse,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [R3: nat,A2: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),R3) = A2 )
         => ( aa(nat,$o,ord_less(nat,R3),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),A2) = R3 ) ) ) ) ).

% of_nat_inverse
tff(fact_7286_unat__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),Xb: word(A)] :
          ( aa(nat,$o,P,aa(word(A),nat,semiring_1_unsigned(A,nat),Xb))
        <=> ~ ? [N6: nat] :
                ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),N6) = Xb )
                & aa(nat,$o,ord_less(nat,N6),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,N6) ) ) ) ).

% unat_split_asm
tff(fact_7287_unat__eq__of__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),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) = Nb )
          <=> ( Xb = aa(nat,word(A),semiring_1_of_nat(word(A)),Nb) ) ) ) ) ).

% unat_eq_of_nat
tff(fact_7288_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_7289_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_7290_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_7291_test__bit__2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(int,word(A),ring_1_of_int(word(A)),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb))),M)
        <=> ( ( M = Nb )
            & aa(nat,$o,ord_less(nat,M),type_len0_len_of(A,type2(A))) ) ) ) ).

% test_bit_2p
tff(fact_7292_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_7293_word__1__le__power,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,aa(word(A),fun(word(A),$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))),Nb)) ) ) ).

% word_1_le_power
tff(fact_7294_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_7295_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_7296_uint__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : aa(word(A),int,semiring_1_unsigned(A,int),aa(num,word(A),numeral_numeral(word(A)),B2)) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(num,int,numeral_numeral(int),B2)),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_7297_p2__gt__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(word(A),$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))),Nb))
        <=> aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A))) ) ) ).

% p2_gt_0
tff(fact_7298_word__of__nat__eq__0__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Nb) = zero_zero(word(A)) )
        <=> aa(nat,$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)))),Nb) ) ) ).

% word_of_nat_eq_0_iff
tff(fact_7299_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_7300_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_7301_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_7302_mask__lt__2pn,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
         => aa(word(A),$o,ord_less(word(A),bit_se2239418461657761734s_mask(word(A),Nb)),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))),Nb)) ) ) ).

% mask_lt_2pn
tff(fact_7303_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_7304_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_7305_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_7306_signed__of__int,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Nb: int] : ring_1_signed(B,A,aa(int,word(B),ring_1_of_int(word(B)),Nb)) = 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)))),Nb)) ) ).

% signed_of_int
tff(fact_7307_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_7308_of__nat__n__less__equal__power__2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,ord_less(nat,Nb),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)),Nb)),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))),Nb)) ) ) ).

% of_nat_n_less_equal_power_2
tff(fact_7309_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_7310_complement__nth__w2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [N3: nat,Nb: nat] :
          ( aa(nat,$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)),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))),Nb))),N3)
          <=> ( N3 != Nb ) ) ) ) ).

% complement_nth_w2p
tff(fact_7311_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_7312_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_7313_bit__word__scast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),ring_1_signed(B,word(A),W)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
            & ( aa(nat,$o,aa(word(B),fun(nat,$o),bit_se5641148757651400278ts_bit(word(B)),W),Nb)
              | ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(B,type2(B))),Nb)
                & 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_7314_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_7315_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_7316_word__power__less__diff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,Q2: word(A),M: nat] :
          ( aa(word(A),$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))),Nb)),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),$o,ord_less(word(A),Q2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),Nb)))
           => aa(word(A),$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),Nb))) ) ) ) ).

% word_power_less_diff
tff(fact_7317_ucast__mono__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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),$o,aa(word(B),fun(word(B),$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_7318_unat__word__ariths_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: 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)),A2),B2)) = 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),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),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_7319_take__bit__word__eq__self__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( ( aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),Nb),W) = W )
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
            | aa(word(A),$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))),Nb)) ) ) ) ).

% take_bit_word_eq_self_iff
tff(fact_7320_signed__modulo__int__def,axiom,
    ! [K: int,L: int] : signed6721504322012087516modulo(int,K,L) = aa(int,int,minus_minus(int,K),aa(int,int,aa(int,fun(int,int),times_times(int),signed7115095781618012415divide(int,K,L)),L)) ).

% signed_modulo_int_def
tff(fact_7321_signed__push__bit__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [Nb: 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)),Nb),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),Nb),ring_1_signed(B,A,W))) ) ).

% signed_push_bit_eq
tff(fact_7322_msb0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A),I: nat] :
          ( ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se1065995026697491101ons_or(word(A)),Xb),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              | aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),Xb),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              & aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344971392196577ns_xor(word(A)),Xb),Y)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ~ ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
              <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Y),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I))) ) )
          & ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),bit_ri4277139882892585799ns_not(word(A)),Xb)),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,I)))
          <=> ~ aa(nat,$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,I))) ) ) ) ).

% msb0
tff(fact_7323_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_amk(word(B),$o)) ) ) ) ).

% ucast_range_less
tff(fact_7324_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_7325_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_7326_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_7327_More__Word_Oof__nat__0,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Nb) = zero_zero(word(A)) )
         => ( aa(nat,$o,ord_less(nat,Nb),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))))
           => ( Nb = zero_zero(nat) ) ) ) ) ).

% More_Word.of_nat_0
tff(fact_7328_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,aa(nat,fun(nat,$o),ord_less_eq(nat),Y),Xb)
            <=> aa(word(A),$o,aa(word(A),fun(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_7329_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_7330_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_7331_unat__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: 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)),A2),B2)) = 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),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),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_7332_uint__range_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ( aa(int,$o,aa(int,fun(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_7333_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_7334_word__of__int__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int,Y: int] :
          ( ( aa(int,$o,aa(int,fun(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,$o,aa(int,fun(int,$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_7335_word__int__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] :
          ~ ! [N2: int] :
              ( ( Xb = aa(int,word(A),ring_1_of_int(word(A)),N2) )
             => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N2)
               => ~ aa(int,$o,ord_less(int,N2),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_7336_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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y)
             => aa(word(B),$o,aa(word(B),fun(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_7337_of__nat__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( ( aa(nat,word(A),semiring_1_of_nat(word(A)),Nb) = W )
        <=> ? [Q4: nat] : Nb = 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_7338_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_7339_unat__ucast__no__overflow__le,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [B2: word(A),F: word(B)] :
          ( aa(nat,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),B2)),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)),F)),B2)
            <=> aa(nat,$o,ord_less(nat,aa(word(B),nat,semiring_1_unsigned(B,nat),F)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)) ) ) ) ) ).

% unat_ucast_no_overflow_le
tff(fact_7340_uint__m2p__not__non__neg,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : ~ aa(int,$o,aa(int,fun(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_7341_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_7342_unat__ucast__less__no__overflow,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,F: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),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,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),F)),Nb)
           => aa(word(A),$o,ord_less(word(A),F),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb)) ) ) ) ).

% unat_ucast_less_no_overflow
tff(fact_7343_unat__ucast__less__no__overflow__simp,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,F: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),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,$o,ord_less(nat,aa(word(A),nat,semiring_1_unsigned(A,nat),F)),Nb)
          <=> aa(word(A),$o,ord_less(word(A),F),aa(nat,word(A),semiring_1_of_nat(word(A)),Nb)) ) ) ) ).

% unat_ucast_less_no_overflow_simp
tff(fact_7344_uint__power__lower,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,ord_less(nat,Nb),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))),Nb)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),Nb) ) ) ) ).

% uint_power_lower
tff(fact_7345_less__2p__is__upper__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Nb: nat] :
          ( aa(word(A),$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))),Nb))
        <=> ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
            & ! [N11: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N11)
               => ( aa(nat,$o,ord_less(nat,N11),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P2),N11) ) ) ) ) ) ).

% less_2p_is_upper_bits_unset
tff(fact_7346_upper__bits__unset__is__l2p,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,P2: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
         => ( ! [N11: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N11)
               => ( aa(nat,$o,ord_less(nat,N11),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P2),N11) ) )
          <=> aa(word(A),$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))),Nb)) ) ) ) ).

% upper_bits_unset_is_l2p
tff(fact_7347_nth__bounded,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: nat,M: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Xb),Nb)
         => ( aa(word(A),$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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),type_len0_len_of(A,type2(A)))
             => aa(nat,$o,ord_less(nat,Nb),M) ) ) ) ) ).

% nth_bounded
tff(fact_7348_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_7349_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_7350_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_7351_wi__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: int,M: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),aa(int,word(A),ring_1_of_int(word(A)),Nb)),aa(int,word(A),ring_1_of_int(word(A)),M))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Nb),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_7352_uint__word__ariths_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: 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)),A2),B2)) = 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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),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_7353_word__2p__mult__inc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(word(A),$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))),Nb))),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,$o,ord_less(nat,aa(nat,nat,suc,Nb)),type_len0_len_of(A,type2(A)))
           => aa(word(A),$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))),Nb)),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_7354_wi__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: int,M: int] :
          ( aa(word(A),$o,ord_less(word(A),aa(int,word(A),ring_1_of_int(word(A)),Nb)),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),Nb),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_7355_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_7356_power__2__ge__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(word(A),$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))),Nb)),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))),Nb))
        <=> aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A))) ) ) ).

% power_2_ge_iff
tff(fact_7357_unat__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: 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)),A2),B2)) = 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),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),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_7358_uint__word__ariths_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),uminus_uminus(word(A)),A2)) = 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),A2))),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_7359_unat__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: 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)),A2),B2)) = 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),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),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_7360_le__mask__iff__lt__2n,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb))
          <=> 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))),Nb)) ) ) ) ).

% le_mask_iff_lt_2n
tff(fact_7361_and__mask__less_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),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),Nb))),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))),Nb)) ) ) ).

% and_mask_less'
tff(fact_7362_eq__mask__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),Nb: nat] :
          ( ( W = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),bit_se5824344872417868541ns_and(word(A)),W),bit_se2239418461657761734s_mask(word(A),Nb)) )
         => ( aa(nat,$o,ord_less(nat,Nb),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))),Nb)) ) ) ) ).

% eq_mask_less
tff(fact_7363_sint__1__cases,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
           => ( ( A2 = zero_zero(word(A)) )
             => ( ring_1_signed(A,int,A2) != zero_zero(int) ) ) )
         => ( ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
             => ( ( A2 = one_one(word(A)) )
               => ( ring_1_signed(A,int,one_one(word(A))) != aa(int,int,uminus_uminus(int),one_one(int)) ) ) )
           => ~ ( aa(nat,$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_7364_uint__word__ariths_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),A2),B2)) = 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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),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_7365_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_7366_uint__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: 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)),A2),B2)) = 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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),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_7367_signed__of__nat,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Nb: nat] : ring_1_signed(B,A,aa(nat,word(B),semiring_1_of_nat(word(B)),Nb)) = 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),Nb))) ) ).

% signed_of_nat
tff(fact_7368_word__power__mod__div,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat,Xb: word(A)] :
          ( aa(nat,$o,ord_less(nat,Nb),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))),Nb))),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,Nb),M))) ) ) ) ) ).

% word_power_mod_div
tff(fact_7369_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_7370_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_7371_unat__plus__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: 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)),A2),B2)) = $ite(aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(word(A),nat,semiring_1_unsigned(A,nat),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),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),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2)),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),A2)),aa(word(A),nat,semiring_1_unsigned(A,nat),B2))),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_7372_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_7373_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,aa(nat,fun(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_7374_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_7375_no__olen__add__nat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_7376_word__add__le__dest,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),J: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ).

% word_add_le_dest
tff(fact_7377_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,aa(word(A),fun(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,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ).

% word_add_le_iff
tff(fact_7378_word__add__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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),$o,aa(word(A),fun(word(A),$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_7379_word__add__le__mono2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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),$o,aa(word(A),fun(word(A),$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_7380_smod__int__compares_I3_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),A2),zero_zero(int))
     => ( aa(int,$o,ord_less(int,zero_zero(int)),B2)
       => aa(int,$o,ord_less(int,aa(int,int,uminus_uminus(int),B2)),signed6721504322012087516modulo(int,A2,B2)) ) ) ).

% smod_int_compares(3)
tff(fact_7381_smod__int__compares_I5_J,axiom,
    ! [A2: int,B2: int] :
      ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),A2)
     => ( aa(int,$o,ord_less(int,B2),zero_zero(int))
       => aa(int,$o,ord_less(int,signed6721504322012087516modulo(int,A2,B2)),aa(int,int,uminus_uminus(int),B2)) ) ) ).

% smod_int_compares(5)
tff(fact_7382_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_7383_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_7384_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_7385_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_7386_sint__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A)] : aa(int,$o,aa(int,fun(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_7387_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_7388_word__of__int__inverse,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [R3: int,A2: word(A)] :
          ( ( aa(int,word(A),ring_1_of_int(word(A)),R3) = A2 )
         => ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),R3)
           => ( aa(int,$o,ord_less(int,R3),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),A2) = R3 ) ) ) ) ) ).

% word_of_int_inverse
tff(fact_7389_uint__split__asm,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(int,$o),Xb: word(A)] :
          ( aa(int,$o,P,aa(word(A),int,semiring_1_unsigned(A,int),Xb))
        <=> ~ ? [I2: int] :
                ( ( aa(int,word(A),ring_1_of_int(word(A)),I2) = Xb )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I2)
                & aa(int,$o,ord_less(int,I2),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,I2) ) ) ) ).

% uint_split_asm
tff(fact_7390_uint__split,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(int,$o),Xb: word(A)] :
          ( aa(int,$o,P,aa(word(A),int,semiring_1_unsigned(A,int),Xb))
        <=> ! [I2: int] :
              ( ( ( aa(int,word(A),ring_1_of_int(word(A)),I2) = Xb )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I2)
                & aa(int,$o,ord_less(int,I2),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,I2) ) ) ) ).

% uint_split
tff(fact_7391_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_7392_div__lt_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_7393_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_7394_double__eq__zero__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: 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) = zero_zero(word(A)) )
        <=> ( ( A2 = zero_zero(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)))) ) ) ) ) ).

% double_eq_zero_iff
tff(fact_7395_no__olen__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_7396_no__olen__add_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_7397_word__le__exists_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y)
         => ? [Z3: word(A)] :
              ( ( Y = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),Xb),Z3) )
              & aa(int,$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),Z3))),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_7398_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_7399_uint__plus__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: 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)),A2),B2)) = $ite(aa(int,$o,ord_less(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)),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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),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_7400_word__less__power__trans2,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),M: nat,K: nat] :
          ( aa(word(A),$o,ord_less(word(A),Nb),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,$o,aa(nat,fun(nat,$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)),Nb),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_7401_word__less__power__trans,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),M: nat,K: nat] :
          ( aa(word(A),$o,ord_less(word(A),Nb),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,$o,aa(nat,fun(nat,$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)),Nb)),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_7402_uint__sub__if_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),minus_minus(word(A),A2),B2)) = $ite(aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(word(A),int,semiring_1_unsigned(A,int),B2)),aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(int,int,minus_minus(int,aa(word(A),int,semiring_1_unsigned(A,int),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2)),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),A2)),aa(word(A),int,semiring_1_unsigned(A,int),B2))),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_7403_word__less__two__pow__divI,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: nat,M: nat] :
          ( aa(word(A),$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,Nb),M)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
           => ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
             => aa(word(A),$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))),Nb)),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_7404_uint__neg__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: 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)),B2))) = aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),B2))),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_7405_word__power__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Nb: nat] :
          ( aa(word(A),$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))),Nb)))
         => ( aa(nat,$o,ord_less(nat,Nb),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))),Nb)) != zero_zero(word(A)) ) ) ) ) ) ).

% word_power_nonzero
tff(fact_7406_div__lt__uint_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),K: word(A),Xb: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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_7407_mult__pow2__inj,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Nb: nat,Xb: word(A),Y: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)),type_len0_len_of(A,type2(A)))
         => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),bit_se2239418461657761734s_mask(word(A),M))
           => ( aa(word(A),$o,aa(word(A),fun(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))),Nb)) = 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))),Nb)) )
               => ( Xb = Y ) ) ) ) ) ) ).

% mult_pow2_inj
tff(fact_7408_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_7409_push__bit__word__eq__nonzero,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: nat,Nb: nat] :
          ( aa(word(A),$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),Nb)),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)),Nb),W) != zero_zero(word(A)) ) ) ) ) ) ).

% push_bit_word_eq_nonzero
tff(fact_7410_smod__int__alt__def,axiom,
    ! [A2: int,B2: int] : signed6721504322012087516modulo(int,A2,B2) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(int,int,sgn_sgn(int),A2)),aa(int,int,aa(int,fun(int,int),modulo_modulo(int),aa(int,int,abs_abs(int),A2)),aa(int,int,abs_abs(int),B2))) ).

% smod_int_alt_def
tff(fact_7411_uint__and__mask__or__full,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: word(A),Mask1: word(A),Mask2: int] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),Nb),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)),Nb),Mask1))),Mask2) = aa(word(A),int,semiring_1_unsigned(A,int),Nb) ) ) ) ) ) ).

% uint_and_mask_or_full
tff(fact_7412_sint__greater__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : aa(int,$o,aa(int,fun(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_7413_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_7414_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_7415_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_7416_le__2p__upper__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)),one_one(word(A))))
         => ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
           => ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N4)
               => ( aa(nat,$o,ord_less(nat,N4),type_len0_len_of(A,type2(A)))
                 => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P2),N4) ) ) ) ) ) ).

% le_2p_upper_bits
tff(fact_7417_le2p__bits__unset,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P2: word(A),Nb: nat] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),P2),aa(word(A),word(A),minus_minus(word(A),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)),one_one(word(A))))
         => ! [N4: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N4)
             => ( aa(nat,$o,ord_less(nat,N4),type_len0_len_of(A,type2(A)))
               => ~ aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),P2),N4) ) ) ) ) ).

% le2p_bits_unset
tff(fact_7418_word__add__offset__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A),Nb: nat,Xb: word(A),M: nat,Sz: nat] :
          ( aa(word(A),$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))),Nb))
         => ( 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),$o,ord_less(word(A),Xb),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),Nb)))
               => ( ( Sz = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb) )
                 => aa(word(A),$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))),Nb))),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_7419_bit__horner__sum__uint__exp__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Ws: list(word(A)),Nb: nat] :
          ( aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),groups4207007520872428315er_sum(word(A),int,semiring_1_unsigned(A,int),aa(nat,int,aa(int,fun(nat,int),power_power(int),aa(num,int,numeral_numeral(int),bit0(one2))),type_len0_len_of(A,type2(A))),Ws)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Nb),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),Nb),type_len0_len_of(A,type2(A))))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Nb),type_len0_len_of(A,type2(A)))) ) ) ) ).

% bit_horner_sum_uint_exp_iff
tff(fact_7420_div__power__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: nat,Y: nat] :
          ( aa(nat,$o,aa(nat,fun(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_7421_even__mult__exp__div__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),M: nat,Nb: nat] :
          ( aa(word(A),$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)),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))),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))),Nb)))
        <=> ~ ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
              & aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
              & ~ aa(word(A),$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)),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,Nb),M)))) ) ) ) ).

% even_mult_exp_div_word_iff
tff(fact_7422_Suc__2p__unat__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,K: nat] :
          ( aa(nat,$o,ord_less(nat,Nb),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))),Nb)),K)),aa(word(A),nat,semiring_1_unsigned(A,nat),bit_se2239418461657761734s_mask(word(A),Nb)))) = 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))),Nb)),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),K),one_one(nat))) ) ) ) ).

% Suc_2p_unat_mask
tff(fact_7423_sint__of__nat__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: nat,A2: nat] :
          ( aa(nat,$o,ord_less(nat,B2),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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),A2),B2)
           => aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),A2))),ring_1_signed(A,int,aa(nat,word(A),semiring_1_of_nat(word(A)),B2))) ) ) ) ).

% sint_of_nat_le
tff(fact_7424_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,$o,aa(int,fun(int,$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_7425_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_7426_sint__of__int__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int] :
          ( aa(int,$o,aa(int,fun(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_7427_sint__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [B2: num] : ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),B2)),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_7428_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),$o,aa(word(A),fun(word(A),$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),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),I),J) ) ) ) ) ) ).

% word_mult_le_iff
tff(fact_7429_word__mult__le__mono1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [I: word(A),J: word(A),K: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(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,aa(word(A),fun(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_7430_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_7431_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_7432_of__nat__less__two__pow__div__set,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat] :
          ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
         => ( collect(word(A),aa(nat,fun(word(A),$o),aTP_Lamp_aml(nat,fun(nat,fun(word(A),$o)),Nb),M)) = aa(set(nat),set(word(A)),image(nat,word(A),semiring_1_of_nat(word(A))),collect(nat,aa(nat,fun(nat,$o),aTP_Lamp_amm(nat,fun(nat,fun(nat,$o)),Nb),M))) ) ) ) ).

% of_nat_less_two_pow_div_set
tff(fact_7433_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_7434_word__bit__induct,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(word(A),$o),A2: word(A)] :
          ( aa(word(A),$o,P,zero_zero(word(A)))
         => ( ! [A4: word(A)] :
                ( aa(word(A),$o,P,A4)
               => ( aa(word(A),$o,ord_less(word(A),zero_zero(word(A))),A4)
                 => ( aa(word(A),$o,ord_less(word(A),A4),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),$o,P,aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),A4)) ) ) )
           => ( ! [A4: word(A)] :
                  ( aa(word(A),$o,P,A4)
                 => ( aa(word(A),$o,ord_less(word(A),A4),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))))
                   => aa(word(A),$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))),A4))) ) )
             => aa(word(A),$o,P,A2) ) ) ) ) ).

% word_bit_induct
tff(fact_7435_word__less__power__trans__ofnat,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,M: nat,K: nat] :
          ( aa(nat,$o,ord_less(nat,Nb),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,$o,aa(nat,fun(nat,$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)),Nb)),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_7436_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_7437_bit__word__half__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: $o] :
          ( aa(word(A),$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),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)),(B2))),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))))),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))) = A2 ) ) ) ).

% bit_word_half_eq
tff(fact_7438_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,$o,aa(int,fun(int,$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,$o,aa(int,fun(int,$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_7439_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,$o,aa(nat,fun(nat,$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_7440_alignUp__div__helper,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: nat,Nb: nat,Xb: word(A),A2: word(A)] :
          ( aa(nat,$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))),Nb)))
         => ( ( 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))),Nb)),aa(nat,word(A),semiring_1_of_nat(word(A)),K)) )
           => ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),A2),Xb)
             => ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
               => ( ( aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),modulo_modulo(word(A)),A2),aa(nat,word(A),aa(word(A),fun(nat,word(A)),power_power(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2))),Nb)) != zero_zero(word(A)) )
                 => aa(word(A),$o,ord_less(word(A),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),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))),Nb))),aa(nat,word(A),semiring_1_of_nat(word(A)),K)) ) ) ) ) ) ) ).

% alignUp_div_helper
tff(fact_7441_less__eq__decr__length__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat))))
        <=> aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A))) ) ) ).

% less_eq_decr_length_iff
tff(fact_7442_decr__length__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,ord_less(nat,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Nb)
        <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb) ) ) ).

% decr_length_less_iff
tff(fact_7443_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_7444_length__not__greater__eq__2__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( ~ aa(nat,$o,aa(nat,fun(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_7445_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_7446_len__num1,axiom,
    ! [Uu: itself(numeral_num1)] : type_len0_len_of(numeral_num1,Uu) = one_one(nat) ).

% len_num1
tff(fact_7447_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_7448_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_7449_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_7450_two__less__eq__exp__length,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & linordered_idom(A) )
     => aa(A,$o,aa(A,fun(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_7451_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_7452_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,aa(word(A),fun(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,aa(word(A),fun(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_7453_signed__drop__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num,K: num] : signed_drop_bit(A,aa(num,nat,numeral_numeral(nat),Nb),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),Nb)),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_7454_signed__drop__bit__signed__drop__bit,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Nb: nat,W: word(A)] : signed_drop_bit(A,M,signed_drop_bit(A,Nb,W)) = signed_drop_bit(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb),W) ) ).

% signed_drop_bit_signed_drop_bit
tff(fact_7455_word__sdiv__div1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : signed7115095781618012415divide(word(A),A2,one_one(word(A))) = A2 ) ).

% word_sdiv_div1
tff(fact_7456_signed__drop__bit__of__minus__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : signed_drop_bit(A,Nb,aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ).

% signed_drop_bit_of_minus_1
tff(fact_7457_word__sdiv__div__minus1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : signed7115095781618012415divide(word(A),A2,aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) = aa(word(A),word(A),uminus_uminus(word(A)),A2) ) ).

% word_sdiv_div_minus1
tff(fact_7458_minus__one__sdiv__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed7115095781618012415divide(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),W) = aa(word(A),word(A),uminus_uminus(word(A)),signed7115095781618012415divide(word(A),one_one(word(A)),W)) ) ).

% minus_one_sdiv_word_eq
tff(fact_7459_one__sdiv__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          signed7115095781618012415divide(word(A),one_one(word(A)),W) = aa(word(A),word(A),
            aa(word(A),fun(word(A),word(A)),times_times(word(A)),
              aa($o,word(A),zero_neq_one_of_bool(word(A)),
                ( ( W = one_one(word(A)) )
                | ( W = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ))),
            W) ) ).

% one_sdiv_word_eq
tff(fact_7460_signed__drop__bit__of__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          signed_drop_bit(A,Nb,one_one(word(A))) = aa($o,word(A),zero_neq_one_of_bool(word(A)),
            ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
            | ( Nb = zero_zero(nat) ) )) ) ).

% signed_drop_bit_of_1
tff(fact_7461_sdiv__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2)),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),B2)))) ) ).

% sdiv_word_numeral_numeral
tff(fact_7462_signed__drop__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,K: num] : signed_drop_bit(A,aa(nat,nat,suc,Nb),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,Nb)),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_7463_signed__drop__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num,K: num] : signed_drop_bit(A,aa(num,nat,numeral_numeral(nat),Nb),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),Nb)),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_7464_sdiv__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : signed7115095781618012415divide(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2))),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),B2))))) ) ).

% sdiv_word_minus_numeral_minus_numeral
tff(fact_7465_sdiv__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),A2),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2)),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),B2))))) ) ).

% sdiv_word_numeral_minus_numeral
tff(fact_7466_sdiv__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : signed7115095781618012415divide(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2))),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),B2)))) ) ).

% sdiv_word_minus_numeral_numeral
tff(fact_7467_signed__drop__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,K: num] : signed_drop_bit(A,aa(nat,nat,suc,Nb),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,Nb)),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_7468_word__sdiv__numerals__lhs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: num,W: word(A)] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),Xb),W) = 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,W))) ) ).

% word_sdiv_numerals_lhs(1)
tff(fact_7469_word__sdiv__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: num,Y: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),Xb),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,aa(num,word(A),numeral_numeral(word(A)),Xb)),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_sdiv_numerals(1)
tff(fact_7470_word__sdiv__numerals__lhs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed7115095781618012415divide(word(A),one_one(word(A)),W) = 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,W))) ) ).

% word_sdiv_numerals_lhs(3)
tff(fact_7471_word__sdiv__numerals_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed7115095781618012415divide(word(A),one_one(word(A)),one_one(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,one_one(word(A))))) ) ) ).

% word_sdiv_numerals(9)
tff(fact_7472_word__sdiv__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] : signed7115095781618012415divide(word(A),one_one(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,one_one(word(A))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_sdiv_numerals(3)
tff(fact_7473_word__sdiv__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: num] : signed7115095781618012415divide(word(A),aa(num,word(A),numeral_numeral(word(A)),Xb),one_one(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,one_one(word(A))))) ) ).

% word_sdiv_numerals(7)
tff(fact_7474_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_7475_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_7476_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_7477_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_7478_signed__div__arith,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : ring_1_signed(A,int,signed7115095781618012415divide(word(A),A2,B2)) = 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,A2),ring_1_signed(A,int,B2))) ) ).

% signed_div_arith
tff(fact_7479_bit__signed__drop__bit__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),signed_drop_bit(A,M,W)),Nb)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),
              $ite(
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M)),Nb)
                & aa(nat,$o,ord_less(nat,Nb),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),Nb) )) ) ) ).

% bit_signed_drop_bit_iff
tff(fact_7480_signed__drop__bit__beyond,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),type_len0_len_of(A,type2(A))),Nb)
         => ( signed_drop_bit(A,Nb,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_7481_word__int__split__asm,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P: fun(A,$o),F: fun(int,A),Xb: word(B)] :
          ( aa(A,$o,P,word_int_case(A,B,F,Xb))
        <=> ~ ? [N6: int] :
                ( ( Xb = aa(int,word(B),ring_1_of_int(word(B)),N6) )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),N6)
                & aa(int,$o,ord_less(int,N6),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,F,N6)) ) ) ) ).

% word_int_split_asm
tff(fact_7482_word__int__split,axiom,
    ! [B: $tType,A: $tType] :
      ( type_len(B)
     => ! [P: fun(A,$o),F: fun(int,A),Xb: word(B)] :
          ( aa(A,$o,P,word_int_case(A,B,F,Xb))
        <=> ! [I2: int] :
              ( ( ( Xb = aa(int,word(B),ring_1_of_int(word(B)),I2) )
                & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I2)
                & aa(int,$o,ord_less(int,I2),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,F,I2)) ) ) ) ).

% word_int_split
tff(fact_7483_word__int__case__wi,axiom,
    ! [A: $tType,B: $tType] :
      ( type_len(B)
     => ! [F: fun(int,A),I: int] : word_int_case(A,B,F,aa(int,word(B),ring_1_of_int(word(B)),I)) = aa(int,A,F,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_7484_smod__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : signed6721504322012087516modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2))),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),B2))))) ) ).

% smod_word_minus_numeral_minus_numeral
tff(fact_7485_smod__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),A2),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2))) = 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),A2)),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),B2))))) ) ).

% smod_word_numeral_minus_numeral
tff(fact_7486_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_7487_minus__one__smod__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))),W) = aa(word(A),word(A),uminus_uminus(word(A)),signed6721504322012087516modulo(word(A),one_one(word(A)),W)) ) ).

% minus_one_smod_word_eq
tff(fact_7488_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_7489_one__smod__word__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] :
          signed6721504322012087516modulo(word(A),one_one(word(A)),W) = aa(word(A),word(A),minus_minus(word(A),one_one(word(A))),
            aa($o,word(A),zero_neq_one_of_bool(word(A)),
              ( ( W = one_one(word(A)) )
              | ( W = aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A))) ) ))) ) ).

% one_smod_word_eq
tff(fact_7490_smod__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2)),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),B2)))) ) ).

% smod_word_numeral_numeral
tff(fact_7491_smod__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] : signed6721504322012087516modulo(word(A),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2)) = 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),A2))),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),B2)))) ) ).

% smod_word_minus_numeral_numeral
tff(fact_7492_sdiv__smod__id,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: 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)),signed7115095781618012415divide(word(A),A2,B2)),B2)),signed6721504322012087516modulo(word(A),A2,B2)) = A2 ) ).

% sdiv_smod_id
tff(fact_7493_smod__word__alt__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : signed6721504322012087516modulo(word(A),A2,B2) = aa(word(A),word(A),minus_minus(word(A),A2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),signed7115095781618012415divide(word(A),A2,B2)),B2)) ) ).

% smod_word_alt_def
tff(fact_7494_word__smod__numerals_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: num,Y: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),Xb),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,aa(num,word(A),numeral_numeral(word(A)),Xb)),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_smod_numerals(1)
tff(fact_7495_word__smod__numerals__lhs_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: num,W: word(A)] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),Xb),W) = 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,W))) ) ).

% word_smod_numerals_lhs(1)
tff(fact_7496_word__smod__numerals_I9_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( signed6721504322012087516modulo(word(A),one_one(word(A)),one_one(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,one_one(word(A))))) ) ) ).

% word_smod_numerals(9)
tff(fact_7497_word__smod__numerals__lhs_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A)] : signed6721504322012087516modulo(word(A),one_one(word(A)),W) = 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,W))) ) ).

% word_smod_numerals_lhs(3)
tff(fact_7498_word__smod__numerals_I7_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: num] : signed6721504322012087516modulo(word(A),aa(num,word(A),numeral_numeral(word(A)),Xb),one_one(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,one_one(word(A))))) ) ).

% word_smod_numerals(7)
tff(fact_7499_word__smod__numerals_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: num] : signed6721504322012087516modulo(word(A),one_one(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,one_one(word(A))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Y)))) ) ).

% word_smod_numerals(3)
tff(fact_7500_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_7501_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_7502_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_7503_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_7504_signed__mod__arith,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : ring_1_signed(A,int,signed6721504322012087516modulo(word(A),A2,B2)) = 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,A2),ring_1_signed(A,int,B2))) ) ).

% signed_mod_arith
tff(fact_7505_uint__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_pred(A,A2)) = 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),A2)),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_7506_slice1__def,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Nb: nat,W: word(B)] :
          aa(word(B),word(A),slice1(B,A,Nb),W) = $ite(aa(nat,$o,ord_less(nat,Nb),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))),Nb)),W)),aa(word(A),word(A),aa(nat,fun(word(A),word(A)),bit_se4730199178511100633sh_bit(word(A)),aa(nat,nat,minus_minus(nat,Nb),type_len0_len_of(B,type2(B)))),aa(word(B),word(A),semiring_1_unsigned(B,word(A)),W))) ) ).

% slice1_def
tff(fact_7507_succ__pred__no_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] : word_pred(A,aa(num,word(A),numeral_numeral(word(A)),W)) = aa(word(A),word(A),minus_minus(word(A),aa(num,word(A),numeral_numeral(word(A)),W)),one_one(word(A))) ) ).

% succ_pred_no(2)
tff(fact_7508_succ__pred__no_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] : word_pred(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W))) = 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)),W))),one_one(word(A))) ) ).

% succ_pred_no(4)
tff(fact_7509_word__m1__ge,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Y: word(A)] : aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Y),word_pred(A,zero_zero(word(A)))) ) ).

% word_m1_ge
tff(fact_7510_word__pred__m1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : word_pred(A,A2) = aa(word(A),word(A),minus_minus(word(A),A2),one_one(word(A))) ) ).

% word_pred_m1
tff(fact_7511_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) = aa(word(B),word(A),slice1(B,A,aa(word(B),nat,size_size(word(B)),W)),W) ) ).

% ucast_slice1
tff(fact_7512_wi__hom__pred,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int] : word_pred(A,aa(int,word(A),ring_1_of_int(word(A)),A2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,minus_minus(int,A2),one_one(int))) ) ).

% wi_hom_pred
tff(fact_7513_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_7514_word__pred__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : word_pred(A,A2) = 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),A2)),one_one(int))) ) ).

% word_pred_alt
tff(fact_7515_Word_Oslice__def,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Nb: nat] : slice2(A,B,Nb) = slice1(A,B,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),Nb)) ) ).

% Word.slice_def
tff(fact_7516_uint__word__arith__bintrs_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_pred(A,A2)) = 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),A2)),one_one(int))) ) ).

% uint_word_arith_bintrs(6)
tff(fact_7517_sint__word__ariths_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ring_1_signed(A,int,word_pred(A,A2)) = 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,A2)),one_one(int))) ) ).

% sint_word_ariths(6)
tff(fact_7518_uint__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_succ(A,A2)) = 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),A2)),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_7519_bit__word__roti__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [K: int,W: word(A),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word_roti(A,K,W)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),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),Nb)),K)),aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))) ) ) ) ).

% bit_word_roti_iff
tff(fact_7520_succ__pred__no_I1_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] : word_succ(A,aa(num,word(A),numeral_numeral(word(A)),W)) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W)),one_one(word(A))) ) ).

% succ_pred_no(1)
tff(fact_7521_succ__pred__no_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: num] : word_succ(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W))) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),W))),one_one(word(A))) ) ).

% succ_pred_no(3)
tff(fact_7522_Abs__fnat__hom__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: nat] : word_succ(A,aa(nat,word(A),semiring_1_of_nat(word(A)),A2)) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,suc,A2)) ) ).

% Abs_fnat_hom_Suc
tff(fact_7523_word__succ__p1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : word_succ(A,A2) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),A2),one_one(word(A))) ) ).

% word_succ_p1
tff(fact_7524_word__roti__add,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: int,Nb: int,W: word(A)] : word_roti(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),M),Nb),W) = word_roti(A,M,word_roti(A,Nb,W)) ) ).

% word_roti_add
tff(fact_7525_word__mult__succ,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),word_succ(A,A2)),B2) = aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),B2),aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),A2),B2)) ) ).

% word_mult_succ
tff(fact_7526_wi__hom__succ,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: int] : word_succ(A,aa(int,word(A),ring_1_of_int(word(A)),A2)) = aa(int,word(A),ring_1_of_int(word(A)),aa(int,int,aa(int,fun(int,int),plus_plus(int),A2),one_one(int))) ) ).

% wi_hom_succ
tff(fact_7527_word__arith__nat__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : word_succ(A,A2) = aa(nat,word(A),semiring_1_of_nat(word(A)),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),A2))) ) ).

% word_arith_nat_Suc
tff(fact_7528_word__roti__conv__mod_H,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: int,W: word(A)] : word_roti(A,Nb,W) = word_roti(A,aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Nb),aa(nat,int,semiring_1_of_nat(int),aa(word(A),nat,size_size(word(A)),W))),W) ) ).

% word_roti_conv_mod'
tff(fact_7529_word__succ__alt,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : word_succ(A,A2) = 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),A2)),one_one(int))) ) ).

% word_succ_alt
tff(fact_7530_word__sp__01,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType] :
      ( ( type_len(D)
        & 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(D,one_one(word(D))) = zero_zero(word(D)) ) ) ) ).

% word_sp_01
tff(fact_7531_uint__word__arith__bintrs_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),word_succ(A,A2)) = 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),A2)),one_one(int))) ) ).

% uint_word_arith_bintrs(5)
tff(fact_7532_sint__word__ariths_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : ring_1_signed(A,int,word_succ(A,A2)) = 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,A2)),one_one(int))) ) ).

% sint_word_ariths(5)
tff(fact_7533_unat__word__ariths_I3_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] : aa(word(A),nat,semiring_1_unsigned(A,nat),word_succ(A,A2)) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),aa(nat,nat,suc,aa(word(A),nat,semiring_1_unsigned(A,nat),A2))),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_7534_sless__eq__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sle(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(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),A2)))),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),B2)))) ) ) ).

% sless_eq_word_minus_numeral_minus_numeral
tff(fact_7535_sless__eq__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),A2),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$o,aa(int,fun(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),A2))),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),B2)))) ) ) ).

% sless_eq_word_numeral_minus_numeral
tff(fact_7536_word__sle__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),A2))),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),B2))) ) ) ).

% word_sle_no
tff(fact_7537_extra__sle__sless__unfolds_I6_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),Nb),one_one(word(A)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Nb))),ring_1_signed(A,int,one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(6)
tff(fact_7538_extra__sle__sless__unfolds_I4_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num] :
          ( word_sle(A,one_one(word(A)),aa(num,word(A),numeral_numeral(word(A)),Nb))
        <=> aa(int,$o,aa(int,fun(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)),Nb))) ) ) ).

% extra_sle_sless_unfolds(4)
tff(fact_7539_extra__sle__sless__unfolds_I5_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),Nb),zero_zero(word(A)))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Nb))),ring_1_signed(A,int,zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(5)
tff(fact_7540_extra__sle__sless__unfolds_I2_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num] :
          ( word_sle(A,zero_zero(word(A)),aa(num,word(A),numeral_numeral(word(A)),Nb))
        <=> aa(int,$o,aa(int,fun(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)),Nb))) ) ) ).

% extra_sle_sless_unfolds(2)
tff(fact_7541_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,aa(int,fun(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_7542_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,aa(int,fun(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_7543_sless__eq__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(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),A2))),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),B2))) ) ) ).

% sless_eq_word_numeral_numeral
tff(fact_7544_sless__eq__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sle(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$o,aa(int,fun(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),A2)))),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),B2))) ) ) ).

% sless_eq_word_minus_numeral_numeral
tff(fact_7545_word__sle__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A),B2: word(A)] :
          ( word_sle(A,A2,B2)
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),ring_1_signed(A,int,A2)),ring_1_signed(A,int,B2)) ) ) ).

% word_sle_eq
tff(fact_7546_signed_Olift__Suc__mono__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,word(A)),Nb: nat,N3: nat] :
          ( ! [N2: nat] : word_sle(A,aa(nat,word(A),F,N2),aa(nat,word(A),F,aa(nat,nat,suc,N2)))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
           => word_sle(A,aa(nat,word(A),F,Nb),aa(nat,word(A),F,N3)) ) ) ) ).

% signed.lift_Suc_mono_le
tff(fact_7547_signed_Olift__Suc__antimono__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,word(A)),Nb: nat,N3: nat] :
          ( ! [N2: nat] : word_sle(A,aa(nat,word(A),F,aa(nat,nat,suc,N2)),aa(nat,word(A),F,N2))
         => ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N3)
           => word_sle(A,aa(nat,word(A),F,N3),aa(nat,word(A),F,Nb)) ) ) ) ).

% signed.lift_Suc_antimono_le
tff(fact_7548_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_7549_sless__word__minus__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sless(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$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),A2)))),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),B2)))) ) ) ).

% sless_word_minus_numeral_minus_numeral
tff(fact_7550_sless__word__numeral__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),A2),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),B2)))
        <=> aa(int,$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),A2))),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),B2)))) ) ) ).

% sless_word_numeral_minus_numeral
tff(fact_7551_word__sless__no,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> ( word_sle(A,aa(num,word(A),numeral_numeral(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),B2))
            & ( aa(num,word(A),numeral_numeral(word(A)),A2) != aa(num,word(A),numeral_numeral(word(A)),B2) ) ) ) ) ).

% word_sless_no
tff(fact_7552_extra__sle__sless__unfolds_I10_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num] :
          ( word_sless(A,one_one(word(A)),aa(num,word(A),numeral_numeral(word(A)),Nb))
        <=> aa(int,$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)),Nb))) ) ) ).

% extra_sle_sless_unfolds(10)
tff(fact_7553_extra__sle__sless__unfolds_I12_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),Nb),one_one(word(A)))
        <=> aa(int,$o,ord_less(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Nb))),ring_1_signed(A,int,one_one(word(A)))) ) ) ).

% extra_sle_sless_unfolds(12)
tff(fact_7554_extra__sle__sless__unfolds_I8_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num] :
          ( word_sless(A,zero_zero(word(A)),aa(num,word(A),numeral_numeral(word(A)),Nb))
        <=> aa(int,$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)),Nb))) ) ) ).

% extra_sle_sless_unfolds(8)
tff(fact_7555_extra__sle__sless__unfolds_I11_J,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),Nb),zero_zero(word(A)))
        <=> aa(int,$o,ord_less(int,ring_1_signed(A,int,aa(num,word(A),numeral_numeral(word(A)),Nb))),ring_1_signed(A,int,zero_zero(word(A)))) ) ) ).

% extra_sle_sless_unfolds(11)
tff(fact_7556_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_7557_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_7558_sless__word__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sless(A,aa(num,word(A),numeral_numeral(word(A)),A2),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$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),A2))),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),B2))) ) ) ).

% sless_word_numeral_numeral
tff(fact_7559_sless__word__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: num,B2: num] :
          ( word_sless(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),A2)),aa(num,word(A),numeral_numeral(word(A)),B2))
        <=> aa(int,$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),A2)))),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),B2))) ) ) ).

% sless_word_minus_numeral_numeral
tff(fact_7560_signed_Olift__Suc__mono__less,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,word(A)),Nb: nat,N3: nat] :
          ( ! [N2: nat] : word_sless(A,aa(nat,word(A),F,N2),aa(nat,word(A),F,aa(nat,nat,suc,N2)))
         => ( aa(nat,$o,ord_less(nat,Nb),N3)
           => word_sless(A,aa(nat,word(A),F,Nb),aa(nat,word(A),F,N3)) ) ) ) ).

% signed.lift_Suc_mono_less
tff(fact_7561_signed_Olift__Suc__mono__less__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,word(A)),Nb: nat,M: nat] :
          ( ! [N2: nat] : word_sless(A,aa(nat,word(A),F,N2),aa(nat,word(A),F,aa(nat,nat,suc,N2)))
         => ( word_sless(A,aa(nat,word(A),F,Nb),aa(nat,word(A),F,M))
          <=> aa(nat,$o,ord_less(nat,Nb),M) ) ) ) ).

% signed.lift_Suc_mono_less_iff
tff(fact_7562_word__sless__sint__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( word_sless(A,Xb,Y)
         => aa(int,$o,aa(int,fun(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_7563_take__bit__word__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num,K: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),Nb)),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),Nb))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))) ) ).

% take_bit_word_minus_numeral
tff(fact_7564_take__bit__word__Suc__minus__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,K: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(nat,nat,suc,Nb)),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,Nb))),aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),K)))) ) ).

% take_bit_word_Suc_minus_numeral
tff(fact_7565_min_Obounded__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),ord_min(A,B2,C2))
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% min.bounded_iff
tff(fact_7566_min_Oabsorb2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
         => ( ord_min(A,A2,B2) = B2 ) ) ) ).

% min.absorb2
tff(fact_7567_min_Oabsorb1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( ord_min(A,A2,B2) = A2 ) ) ) ).

% min.absorb1
tff(fact_7568_min__eq__arg_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Nb: A] :
          ( ( ord_min(A,M,Nb) = M )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),Nb) ) ) ).

% min_eq_arg(1)
tff(fact_7569_min__eq__arg_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Nb: A] :
          ( ( ord_min(A,M,Nb) = Nb )
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),M) ) ) ).

% min_eq_arg(2)
tff(fact_7570_min__arg__le_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Nb: A,M: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Nb),ord_min(A,M,Nb))
        <=> ( ord_min(A,M,Nb) = Nb ) ) ) ).

% min_arg_le(1)
tff(fact_7571_min__arg__le_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [M: A,Nb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),M),ord_min(A,M,Nb))
        <=> ( ord_min(A,M,Nb) = M ) ) ) ).

% min_arg_le(2)
tff(fact_7572_min__simps_I2_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
         => ( ord_min(A,A2,B2) = B2 ) ) ) ).

% min_simps(2)
tff(fact_7573_min__simps_I1_J,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( ord_min(A,A2,B2) = A2 ) ) ) ).

% min_simps(1)
tff(fact_7574_min__less__self__conv_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,ord_min(A,A2,B2)),B2)
        <=> aa(A,$o,ord_less(A,A2),B2) ) ) ).

% min_less_self_conv(2)
tff(fact_7575_min__less__self__conv_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,ord_min(A,A2,B2)),A2)
        <=> aa(A,$o,ord_less(A,B2),A2) ) ) ).

% min_less_self_conv(1)
tff(fact_7576_min__arg__not__ge_I2_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Nb: A] :
          ( ~ aa(A,$o,ord_less(A,ord_min(A,M,Nb)),Nb)
        <=> ( ord_min(A,M,Nb) = Nb ) ) ) ).

% min_arg_not_ge(2)
tff(fact_7577_min__arg__not__ge_I1_J,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [M: A,Nb: A] :
          ( ~ aa(A,$o,ord_less(A,ord_min(A,M,Nb)),M)
        <=> ( ord_min(A,M,Nb) = M ) ) ) ).

% min_arg_not_ge(1)
tff(fact_7578_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_7579_min_Oabsorb4,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,ord_less(A,B2),A2)
         => ( ord_min(A,A2,B2) = B2 ) ) ) ).

% min.absorb4
tff(fact_7580_min_Oabsorb3,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( ord_min(A,A2,B2) = A2 ) ) ) ).

% min.absorb3
tff(fact_7581_min__Suc__Suc,axiom,
    ! [M: nat,Nb: nat] : ord_min(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,ord_min(nat,M,Nb)) ).

% min_Suc_Suc
tff(fact_7582_min__0R,axiom,
    ! [Nb: nat] : ord_min(nat,Nb,zero_zero(nat)) = zero_zero(nat) ).

% min_0R
tff(fact_7583_min__0L,axiom,
    ! [Nb: nat] : ord_min(nat,zero_zero(nat),Nb) = zero_zero(nat) ).

% min_0L
tff(fact_7584_min__minus_H,axiom,
    ! [M: nat,K: nat] : ord_min(nat,aa(nat,nat,minus_minus(nat,M),K),M) = aa(nat,nat,minus_minus(nat,M),K) ).

% min_minus'
tff(fact_7585_min__minus,axiom,
    ! [M: nat,K: nat] : ord_min(nat,M,aa(nat,nat,minus_minus(nat,M),K)) = aa(nat,nat,minus_minus(nat,M),K) ).

% min_minus
tff(fact_7586_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_7587_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_7588_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,aa(A,fun(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_7589_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_7590_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_7591_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_7592_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_7593_min__Suc__gt_I2_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,ord_less(nat,A2),B2)
     => ( ord_min(nat,B2,aa(nat,nat,suc,A2)) = aa(nat,nat,suc,A2) ) ) ).

% min_Suc_gt(2)
tff(fact_7594_min__Suc__gt_I1_J,axiom,
    ! [A2: nat,B2: nat] :
      ( aa(nat,$o,ord_less(nat,A2),B2)
     => ( ord_min(nat,aa(nat,nat,suc,A2),B2) = aa(nat,nat,suc,A2) ) ) ).

% min_Suc_gt(1)
tff(fact_7595_min__pm,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),ord_min(nat,A2,B2)),aa(nat,nat,minus_minus(nat,A2),B2)) = A2 ).

% min_pm
tff(fact_7596_min__pm1,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,A2),B2)),ord_min(nat,A2,B2)) = A2 ).

% min_pm1
tff(fact_7597_rev__min__pm,axiom,
    ! [B2: nat,A2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),ord_min(nat,B2,A2)),aa(nat,nat,minus_minus(nat,A2),B2)) = A2 ).

% rev_min_pm
tff(fact_7598_rev__min__pm1,axiom,
    ! [A2: nat,B2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa(nat,nat,minus_minus(nat,A2),B2)),ord_min(nat,B2,A2)) = A2 ).

% rev_min_pm1
tff(fact_7599_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,aa(A,fun(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_7600_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,aa(A,fun(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_7601_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,aa(A,fun(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_7602_min__Suc__numeral,axiom,
    ! [Nb: nat,K: num] : ord_min(nat,aa(nat,nat,suc,Nb),aa(num,nat,numeral_numeral(nat),K)) = aa(nat,nat,suc,ord_min(nat,Nb,pred_numeral(K))) ).

% min_Suc_numeral
tff(fact_7603_min__numeral__Suc,axiom,
    ! [K: num,Nb: nat] : ord_min(nat,aa(num,nat,numeral_numeral(nat),K),aa(nat,nat,suc,Nb)) = aa(nat,nat,suc,ord_min(nat,pred_numeral(K),Nb)) ).

% min_numeral_Suc
tff(fact_7604_take__bit__word__Suc__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,K: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(nat,nat,suc,Nb)),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,Nb))),aa(num,int,numeral_numeral(int),K))) ) ).

% take_bit_word_Suc_numeral
tff(fact_7605_take__bit__word__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: num,K: num] : aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(num,nat,numeral_numeral(nat),Nb)),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),Nb))),aa(num,int,numeral_numeral(int),K))) ) ).

% take_bit_word_numeral
tff(fact_7606_min__add__distrib__right,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),ord_min(A,Y,Z)) = ord_min(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Y),aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z)) ) ).

% min_add_distrib_right
tff(fact_7607_min__add__distrib__left,axiom,
    ! [A: $tType] :
      ( ordere2412721322843649153imp_le(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),ord_min(A,Xb,Y)),Z) = ord_min(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Xb),Z),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) ) ).

% min_add_distrib_left
tff(fact_7608_min__diff,axiom,
    ! [M: nat,I: nat,Nb: nat] : ord_min(nat,aa(nat,nat,minus_minus(nat,M),I),aa(nat,nat,minus_minus(nat,Nb),I)) = aa(nat,nat,minus_minus(nat,ord_min(nat,M,Nb)),I) ).

% min_diff
tff(fact_7609_min__diff__distrib__left,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Xb: A,Y: A,Z: A] : aa(A,A,minus_minus(A,ord_min(A,Xb,Y)),Z) = ord_min(A,aa(A,A,minus_minus(A,Xb),Z),aa(A,A,minus_minus(A,Y),Z)) ) ).

% min_diff_distrib_left
tff(fact_7610_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_7611_min_Ostrict__boundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,ord_less(A,A2),ord_min(A,B2,C2))
         => ~ ( aa(A,$o,ord_less(A,A2),B2)
             => ~ aa(A,$o,ord_less(A,A2),C2) ) ) ) ).

% min.strict_boundedE
tff(fact_7612_min_Ostrict__order__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
        <=> ( ( A2 = ord_min(A,A2,B2) )
            & ( A2 != B2 ) ) ) ) ).

% min.strict_order_iff
tff(fact_7613_min_Ostrict__coboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),C2)
         => aa(A,$o,ord_less(A,ord_min(A,A2,B2)),C2) ) ) ).

% min.strict_coboundedI1
tff(fact_7614_min_Ostrict__coboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,ord_less(A,B2),C2)
         => aa(A,$o,ord_less(A,ord_min(A,A2,B2)),C2) ) ) ).

% min.strict_coboundedI2
tff(fact_7615_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_7616_nat__mult__min__right,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),ord_min(nat,Nb,Q2)) = ord_min(nat,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Nb),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),M),Q2)) ).

% nat_mult_min_right
tff(fact_7617_nat__mult__min__left,axiom,
    ! [M: nat,Nb: nat,Q2: nat] : aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),ord_min(nat,M,Nb)),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),Nb),Q2)) ).

% nat_mult_min_left
tff(fact_7618_min__absorb2,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Y: A,Xb: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Xb)
         => ( ord_min(A,Xb,Y) = Y ) ) ) ).

% min_absorb2
tff(fact_7619_min__absorb1,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Y: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y)
         => ( ord_min(A,Xb,Y) = Xb ) ) ) ).

% min_absorb1
tff(fact_7620_min__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [A2: A,B2: A] :
          ord_min(A,A2,B2) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2),A2,B2) ) ).

% min_def
tff(fact_7621_min_Omono,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A,D2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),D2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_min(A,A2,B2)),ord_min(A,C2,D2)) ) ) ) ).

% min.mono
tff(fact_7622_min_OorderE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( A2 = ord_min(A,A2,B2) ) ) ) ).

% min.orderE
tff(fact_7623_min_OorderI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( ( A2 = ord_min(A,A2,B2) )
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2) ) ) ).

% min.orderI
tff(fact_7624_min_OboundedE,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),ord_min(A,B2,C2))
         => ~ ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
             => ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2) ) ) ) ).

% min.boundedE
tff(fact_7625_min_OboundedI,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A,C2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
         => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),ord_min(A,B2,C2)) ) ) ) ).

% min.boundedI
tff(fact_7626_min_Oorder__iff,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( A2 = ord_min(A,A2,B2) ) ) ) ).

% min.order_iff
tff(fact_7627_min_Ocobounded1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_min(A,A2,B2)),A2) ) ).

% min.cobounded1
tff(fact_7628_min_Ocobounded2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] : aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_min(A,A2,B2)),B2) ) ).

% min.cobounded2
tff(fact_7629_min_Oabsorb__iff1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),B2)
        <=> ( ord_min(A,A2,B2) = A2 ) ) ) ).

% min.absorb_iff1
tff(fact_7630_min_Oabsorb__iff2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),A2)
        <=> ( ord_min(A,A2,B2) = B2 ) ) ) ).

% min.absorb_iff2
tff(fact_7631_min_OcoboundedI1,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A2: A,C2: A,B2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_min(A,A2,B2)),C2) ) ) ).

% min.coboundedI1
tff(fact_7632_min_OcoboundedI2,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [B2: A,C2: A,A2: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),B2),C2)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_min(A,A2,B2)),C2) ) ) ).

% min.coboundedI2
tff(fact_7633_min__le__iff__disj,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Xb: A,Y: A,Z: A] :
          ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_min(A,Xb,Y)),Z)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Z)
            | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y),Z) ) ) ) ).

% min_le_iff_disj
tff(fact_7634_min__def__raw,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X3: A,Xa: A] :
          ord_min(A,X3,Xa) = $ite(aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Xa),X3,Xa) ) ).

% min_def_raw
tff(fact_7635_concat__bit__assoc__sym,axiom,
    ! [M: nat,Nb: nat,K: int,L: int,R3: int] : bit_concat_bit(M,bit_concat_bit(Nb,K,L),R3) = bit_concat_bit(ord_min(nat,M,Nb),K,bit_concat_bit(aa(nat,nat,minus_minus(nat,M),Nb),L,R3)) ).

% concat_bit_assoc_sym
tff(fact_7636_take__bit__concat__bit__eq,axiom,
    ! [M: nat,Nb: nat,K: int,L: int] : aa(int,int,bit_se2584673776208193580ke_bit(int,M),bit_concat_bit(Nb,K,L)) = bit_concat_bit(ord_min(nat,M,Nb),K,aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,minus_minus(nat,M),Nb)),L)) ).

% take_bit_concat_bit_eq
tff(fact_7637_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,aa(A,fun(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_7638_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,aa(A,fun(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_7639_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,aa(A,fun(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_7640_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,aa(A,fun(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_7641_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,aa(A,fun(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_7642_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,aa(A,fun(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_7643_min__Suc2,axiom,
    ! [M: nat,Nb: nat] : ord_min(nat,M,aa(nat,nat,suc,Nb)) = case_nat(nat,zero_zero(nat),aTP_Lamp_amn(nat,fun(nat,nat),Nb),M) ).

% min_Suc2
tff(fact_7644_min__Suc1,axiom,
    ! [Nb: nat,M: nat] : ord_min(nat,aa(nat,nat,suc,Nb),M) = case_nat(nat,zero_zero(nat),aTP_Lamp_amo(nat,fun(nat,nat),Nb),M) ).

% min_Suc1
tff(fact_7645_mod__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,M: nat,Nb: nat] : aa(A,A,aa(A,fun(A,A),modulo_modulo(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),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))),Nb)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),ord_min(nat,M,Nb))) ) ).

% mod_exp_eq
tff(fact_7646_mod__mod__power,axiom,
    ! [K: nat,M: nat,Nb: 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))),Nb)) = 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,Nb))) ).

% mod_mod_power
tff(fact_7647_Word_Obit__mask__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_se2239418461657761734s_mask(word(A),M)),Nb)
        <=> aa(nat,$o,ord_less(nat,Nb),ord_min(nat,type_len0_len_of(A,type2(A)),M)) ) ) ).

% Word.bit_mask_iff
tff(fact_7648_mask__mod__exp,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Nb: 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))),Nb)),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,Nb))),one_one(A)) ) ).

% mask_mod_exp
tff(fact_7649_bit__slice__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [M: nat,W: word(B),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),slice2(B,A,M),W)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),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),Nb),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_7650_bit__slice1__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [M: nat,W: word(B),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(B),word(A),slice1(B,A,M),W)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,M),type_len0_len_of(B,type2(B)))),Nb)
            & aa(nat,$o,ord_less(nat,Nb),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),Nb),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_7651_unat__mask,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : aa(word(A),nat,semiring_1_unsigned(A,nat),bit_se2239418461657761734s_mask(word(A),Nb)) = 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,Nb,type_len0_len_of(A,type2(A))))),one_one(nat)) ) ).

% unat_mask
tff(fact_7652_bit__horner__sum__bit__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Bs2: list($o),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),groups4207007520872428315er_sum($o,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)),Bs2)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),ord_min(nat,type_len0_len_of(A,type2(A)),aa(list($o),nat,size_size(list($o)),Bs2)))
            & aa(nat,$o,nth($o,Bs2),Nb) ) ) ) ).

% bit_horner_sum_bit_word_iff
tff(fact_7653_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),aa(A,fun(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_amp(A,fun(list(A),fun(A,fun(list(A),A))),X2),Xs2)),Xs2) ) )
           => ~ ( ( Xb = nil(A) )
               => ( Y != undefined(A) ) ) ) ) ) ).

% min_list.elims
tff(fact_7654_drop__bit__exp__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat] :
          aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),M),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)) = aa(A,A,
            aa(A,fun(A,A),times_times(A),
              aa($o,A,zero_neq_one_of_bool(A),
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
                & bit_se6407376104438227557le_bit(A,type2(A),Nb) ))),
            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,Nb),M))) ) ).

% drop_bit_exp_eq
tff(fact_7655_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_7656_bit__minus__1__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),one_one(A))),Nb)
        <=> bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ).

% bit_minus_1_iff
tff(fact_7657_bit__minus__2__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),bit0(one2)))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb) ) ) ) ).

% bit_minus_2_iff
tff(fact_7658_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,aa(nat,fun(nat,$o),ord_less_eq(nat),J),I)
           => bit_se6407376104438227557le_bit(A,Tyrep,J) ) ) ) ).

% possible_bit_less_imp
tff(fact_7659_bit__mask__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),bit_se2239418461657761734s_mask(A,M)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,ord_less(nat,Nb),M) ) ) ) ).

% bit_mask_iff
tff(fact_7660_possible__bit__def,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Tyrep: itself(A),Nb: nat] :
          ( bit_se6407376104438227557le_bit(A,Tyrep,Nb)
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) != zero_zero(A) ) ) ) ).

% possible_bit_def
tff(fact_7661_bit__minus__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),A2)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & ~ aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,minus_minus(A,A2),one_one(A))),Nb) ) ) ) ).

% bit_minus_iff
tff(fact_7662_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_7663_bit__push__bit__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: nat,A2: A,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),M),A2)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
            & bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,nat,minus_minus(nat,Nb),M)) ) ) ) ).

% bit_push_bit_iff
tff(fact_7664_fold__possible__bit,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] :
          ( ( aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) = zero_zero(A) )
        <=> ~ bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ).

% fold_possible_bit
tff(fact_7665_bit__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$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)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & ( M = Nb ) ) ) ) ).

% bit_exp_iff
tff(fact_7666_bit__2__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),one_one(nat))
            & ( Nb = one_one(nat) ) ) ) ) ).

% bit_2_iff
tff(fact_7667_min__list_Osimps,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: A,Xs: list(A)] : min_list(A,aa(list(A),list(A),aa(A,fun(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_amp(A,fun(list(A),fun(A,fun(list(A),A))),Xb),Xs)),Xs) ) ).

% min_list.simps
tff(fact_7668_bit__not__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$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))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & ( Nb != M ) ) ) ) ).

% bit_not_exp_iff
tff(fact_7669_bit__minus__exp__iff,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,uminus_uminus(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb) ) ) ) ).

% bit_minus_exp_iff
tff(fact_7670_bit__mask__sub__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [M: nat,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,minus_minus(A,aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),M)),one_one(A))),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,ord_less(nat,Nb),M) ) ) ) ).

% bit_mask_sub_iff
tff(fact_7671_bit__double__iff,axiom,
    ! [A: $tType] :
      ( bit_semiring_bits(A)
     => ! [A2: A,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(A,fun(A,A),times_times(A),aa(num,A,numeral_numeral(A),bit0(one2))),A2)),Nb)
        <=> ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,nat,minus_minus(nat,Nb),one_one(nat)))
            & ( Nb != zero_zero(nat) )
            & bit_se6407376104438227557le_bit(A,type2(A),Nb) ) ) ) ).

% bit_double_iff
tff(fact_7672_bit__signed__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & bit_ri3973907225187159222ations(A) )
     => ! [W: word(B),Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),ring_1_signed(B,A,W)),Nb)
        <=> ( bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$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))),Nb)) ) ) ) ).

% bit_signed_iff
tff(fact_7673_min__list_Opelims,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Xb: list(A),Y: A] :
          ( ( min_list(A,Xb) = Y )
         => ( aa(list(A),$o,accp(list(A),min_list_rel(A)),Xb)
           => ( ! [X2: A,Xs2: list(A)] :
                  ( ( Xb = aa(list(A),list(A),aa(A,fun(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_amp(A,fun(list(A),fun(A,fun(list(A),A))),X2),Xs2)),Xs2) )
                   => ~ aa(list(A),$o,accp(list(A),min_list_rel(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X2),Xs2)) ) )
             => ~ ( ( Xb = nil(A) )
                 => ( ( Y = undefined(A) )
                   => ~ aa(list(A),$o,accp(list(A),min_list_rel(A)),nil(A)) ) ) ) ) ) ) ).

% min_list.pelims
tff(fact_7674_word__set__bits__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o)] : bit_bi4170147762399595738t_bits(word(A),P) = groups4207007520872428315er_sum($o,word(A),zero_neq_one_of_bool(word(A)),aa(num,word(A),numeral_numeral(word(A)),bit0(one2)),aa(list(nat),list($o),map(nat,$o,P),upt(zero_zero(nat),type_len0_len_of(A,type2(A))))) ) ).

% word_set_bits_def
tff(fact_7675_set__bits__False__eq,axiom,
    ! [A: $tType] :
      ( bit_bi6583157726757044596ension(A)
     => ( bit_bi4170147762399595738t_bits(A,aTP_Lamp_afc(nat,$o)) = zero_zero(A) ) ) ).

% set_bits_False_eq
tff(fact_7676_set__bits__K__False,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( bit_bi4170147762399595738t_bits(word(A),aTP_Lamp_afc(nat,$o)) = zero_zero(word(A)) ) ) ).

% set_bits_K_False
tff(fact_7677_bit__set__bits__word__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [P: fun(nat,$o),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_bi4170147762399595738t_bits(word(A),P)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,P,Nb) ) ) ) ).

% bit_set_bits_word_iff
tff(fact_7678_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_7679_word__test__bit__set__bits,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [F: fun(nat,$o),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_bi4170147762399595738t_bits(word(A),F)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,F,Nb) ) ) ) ).

% word_test_bit_set_bits
tff(fact_7680_uint__word__rotr__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat,W: word(A)] : aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),word_rotr(A,Nb),W)) = bit_concat_bit(aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Nb),type_len0_len_of(A,type2(A)))),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Nb),type_len0_len_of(A,type2(A)))),aa(word(A),int,semiring_1_unsigned(A,int),W)),aa(word(A),int,semiring_1_unsigned(A,int),aa(word(A),word(A),bit_se2584673776208193580ke_bit(word(A),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Nb),type_len0_len_of(A,type2(A)))),W))) ) ).

% uint_word_rotr_eq
tff(fact_7681_word__roti_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xb: int] : word_roti(A,Xaa,word2(A,Xb)) = word2(A,bit_concat_bit(aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),nat2(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xaa),aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),nat2(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xaa),aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xb)),aa(int,int,bit_se2584673776208193580ke_bit(int,nat2(aa(int,int,aa(int,fun(int,int),modulo_modulo(int),Xaa),aa(nat,int,semiring_1_of_nat(int),type_len0_len_of(A,type2(A)))))),Xb))) ) ).

% word_roti.abs_eq
tff(fact_7682_int__set__bits__K__False,axiom,
    bit_bi4170147762399595738t_bits(int,aTP_Lamp_afc(nat,$o)) = zero_zero(int) ).

% int_set_bits_K_False
tff(fact_7683_int__set__bits__K__True,axiom,
    bit_bi4170147762399595738t_bits(int,aTP_Lamp_afb(nat,$o)) = aa(int,int,uminus_uminus(int),one_one(int)) ).

% int_set_bits_K_True
tff(fact_7684_size__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int] : aa(word(A),nat,size_size(word(A)),word2(A,Xb)) = type_len0_len_of(A,type2(A)) ) ).

% size_word.abs_eq
tff(fact_7685_word__pred_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int] : word_pred(A,word2(A,Xb)) = word2(A,aa(int,int,minus_minus(int,Xb),one_one(int))) ) ).

% word_pred.abs_eq
tff(fact_7686_times__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xb: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),times_times(word(A)),word2(A,Xaa)),word2(A,Xb)) = word2(A,aa(int,int,aa(int,fun(int,int),times_times(int),Xaa),Xb)) ) ).

% times_word.abs_eq
tff(fact_7687_one__word__def,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ( one_one(word(A)) = word2(A,one_one(int)) ) ) ).

% one_word_def
tff(fact_7688_minus__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xb: int] : aa(word(A),word(A),minus_minus(word(A),word2(A,Xaa)),word2(A,Xb)) = word2(A,aa(int,int,minus_minus(int,Xaa),Xb)) ) ).

% minus_word.abs_eq
tff(fact_7689_word__rotr__word__rotr__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,Nb: nat,W: word(A)] : aa(word(A),word(A),word_rotr(A,M),aa(word(A),word(A),word_rotr(A,Nb),W)) = aa(word(A),word(A),word_rotr(A,aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),M),Nb)),W) ) ).

% word_rotr_word_rotr_eq
tff(fact_7690_plus__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xb: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),plus_plus(word(A)),word2(A,Xaa)),word2(A,Xb)) = word2(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xaa),Xb)) ) ).

% plus_word.abs_eq
tff(fact_7691_word__succ_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int] : word_succ(A,word2(A,Xb)) = word2(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),Xb),one_one(int))) ) ).

% word_succ.abs_eq
tff(fact_7692_word__rotr_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: nat,Xb: int] : aa(word(A),word(A),word_rotr(A,Xaa),word2(A,Xb)) = word2(A,bit_concat_bit(aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xaa),type_len0_len_of(A,type2(A)))),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xaa),type_len0_len_of(A,type2(A)))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xb)),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xaa),type_len0_len_of(A,type2(A)))),Xb))) ) ).

% word_rotr.abs_eq
tff(fact_7693_less__eq__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xb: int] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),word2(A,Xaa)),word2(A,Xb))
        <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xaa)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xb)) ) ) ).

% less_eq_word.abs_eq
tff(fact_7694_bit__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int,X3: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),word2(A,Xb)),X3)
        <=> ( aa(nat,$o,ord_less(nat,X3),type_len0_len_of(A,type2(A)))
            & aa(nat,$o,aa(int,fun(nat,$o),bit_se5641148757651400278ts_bit(int),Xb),X3) ) ) ) ).

% bit_word.abs_eq
tff(fact_7695_divide__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xb: int] : aa(word(A),word(A),aa(word(A),fun(word(A),word(A)),divide_divide(word(A)),word2(A,Xaa)),word2(A,Xb)) = word2(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))),Xaa)),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xb))) ) ).

% divide_word.abs_eq
tff(fact_7696_word__sle_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xb: int] :
          ( word_sle(A,word2(A,Xaa),word2(A,Xb))
        <=> aa(int,$o,aa(int,fun(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))),aa(nat,nat,suc,zero_zero(nat)))),Xaa)),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)))),Xb)) ) ) ).

% word_sle.abs_eq
tff(fact_7697_word__sless_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xb: int] :
          ( word_sless(A,word2(A,Xaa),word2(A,Xb))
        <=> aa(int,$o,ord_less(int,aa(int,int,bit_ri4674362597316999326ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,suc,zero_zero(nat)))),Xaa)),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)))),Xb)) ) ) ).

% word_sless.abs_eq
tff(fact_7698_signed_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & ring_1(A) )
     => ! [Xb: int] : ring_1_signed(B,A,word2(B,Xb)) = 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)))),Xb)) ) ).

% signed.abs_eq
tff(fact_7699_signed__drop__bit_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: nat,Xb: int] : signed_drop_bit(A,Xaa,word2(A,Xb)) = word2(A,aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),Xaa),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)))),Xb))) ) ).

% signed_drop_bit.abs_eq
tff(fact_7700_bit__word__rotr__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),word_rotr(A,M),W)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),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),Nb),M)),type_len0_len_of(A,type2(A)))) ) ) ) ).

% bit_word_rotr_iff
tff(fact_7701_signed__modulo__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xb: int] : signed6721504322012087516modulo(word(A),word2(A,Xaa),word2(A,Xb)) = word2(A,signed6721504322012087516modulo(int,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)))),Xaa),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)))),Xb))) ) ).

% signed_modulo_word.abs_eq
tff(fact_7702_signed__divide__word_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: int,Xb: int] : signed7115095781618012415divide(word(A),word2(A,Xaa),word2(A,Xb)) = word2(A,signed7115095781618012415divide(int,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)))),Xaa),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)))),Xb))) ) ).

% signed_divide_word.abs_eq
tff(fact_7703_signed__cast_Oabs__eq,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [Xb: int] : signed_cast(B,A,word2(B,Xb)) = word2(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)))),Xb)) ) ).

% signed_cast.abs_eq
tff(fact_7704_the__signed__int_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: int] : the_signed_int(A,word2(A,Xb)) = 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)))),Xb) ) ).

% the_signed_int.abs_eq
tff(fact_7705_bin__last__set__bits,axiom,
    ! [F: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F)
     => ( ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),bit_bi4170147762399595738t_bits(int,F))
      <=> aa(nat,$o,F,zero_zero(nat)) ) ) ).

% bin_last_set_bits
tff(fact_7706_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,aa(int,fun(int,$o),ord_less_eq(int),zero_zero(int)),I),aa(word(A),word(A),word_rotr(A,nat2(I)),W),aa(word(A),word(A),word_rotl(A,nat2(aa(int,int,uminus_uminus(int),I))),W)) ) ).

% word_roti_eq_word_rotr_word_rotl
tff(fact_7707_wf__set__bits__int__Suc,axiom,
    ! [F: fun(nat,$o)] :
      ( bit_wf_set_bits_int(aTP_Lamp_amq(fun(nat,$o),fun(nat,$o),F))
    <=> bit_wf_set_bits_int(F) ) ).

% wf_set_bits_int_Suc
tff(fact_7708_wf__set__bits__int__const,axiom,
    ! [B2: $o] : bit_wf_set_bits_int(aTP_Lamp_amr($o,fun(nat,$o),(B2))) ).

% wf_set_bits_int_const
tff(fact_7709_ones,axiom,
    ! [Nb: nat,F: fun(nat,$o)] :
      ( ! [N5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N5)
         => aa(nat,$o,F,N5) )
     => bit_wf_set_bits_int(F) ) ).

% ones
tff(fact_7710_wf__set__bits__int_Ocases,axiom,
    ! [F: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F)
     => ( ! [N2: nat] :
            ~ ! [N4: nat] :
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N4)
               => ~ aa(nat,$o,F,N4) )
       => ~ ! [N2: nat] :
              ~ ! [N4: nat] :
                  ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N2),N4)
                 => aa(nat,$o,F,N4) ) ) ) ).

% wf_set_bits_int.cases
tff(fact_7711_wf__set__bits__int_Osimps,axiom,
    ! [F: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F)
    <=> ( ? [N6: nat] :
          ! [N11: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N11)
           => ~ aa(nat,$o,F,N11) )
        | ? [N6: nat] :
          ! [N11: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N11)
           => aa(nat,$o,F,N11) ) ) ) ).

% wf_set_bits_int.simps
tff(fact_7712_zeros,axiom,
    ! [Nb: nat,F: fun(nat,$o)] :
      ( ! [N5: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Nb),N5)
         => ~ aa(nat,$o,F,N5) )
     => bit_wf_set_bits_int(F) ) ).

% zeros
tff(fact_7713_wf__set__bits__int__simps,axiom,
    ! [F: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F)
    <=> ? [N6: nat] :
          ( ! [N11: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N11)
             => ~ aa(nat,$o,F,N11) )
          | ! [N11: nat] :
              ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N11)
             => aa(nat,$o,F,N11) ) ) ) ).

% wf_set_bits_int_simps
tff(fact_7714_word__rotl__eq__word__rotr,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : word_rotl(A,Nb) = word_rotr(A,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Nb),type_len0_len_of(A,type2(A))))) ) ).

% word_rotl_eq_word_rotr
tff(fact_7715_bit__word__rotl__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: nat,W: word(A),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),aa(word(A),word(A),word_rotl(A,M),W)),Nb)
        <=> ( aa(nat,$o,ord_less(nat,Nb),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),Nb),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_7716_word__rotl_Oabs__eq,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xaa: nat,Xb: int] : aa(word(A),word(A),word_rotl(A,Xaa),word2(A,Xb)) = word2(A,bit_concat_bit(aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xaa),type_len0_len_of(A,type2(A))),aa(int,int,aa(nat,fun(int,int),bit_se4197421643247451524op_bit(int),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xaa),type_len0_len_of(A,type2(A))))),aa(int,int,bit_se2584673776208193580ke_bit(int,type_len0_len_of(A,type2(A))),Xb)),aa(int,int,bit_se2584673776208193580ke_bit(int,aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Xaa),type_len0_len_of(A,type2(A))))),Xb))) ) ).

% word_rotl.abs_eq
tff(fact_7717_int__set__bits__unfold__BIT,axiom,
    ! [F: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F)
     => ( bit_bi4170147762399595738t_bits(int,F) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa($o,int,zero_neq_one_of_bool(int),aa(nat,$o,F,zero_zero(nat)))),aa(int,int,aa(int,fun(int,int),times_times(int),aa(num,int,numeral_numeral(int),bit0(one2))),bit_bi4170147762399595738t_bits(int,comp(nat,$o,nat,F,suc)))) ) ) ).

% int_set_bits_unfold_BIT
tff(fact_7718_CHAR__eq0__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( ( semiri4206861660011772517g_char(A,type2(A)) = zero_zero(nat) )
      <=> ! [N6: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N6)
           => ( aa(nat,A,semiring_1_of_nat(A),N6) != zero_zero(A) ) ) ) ) ).

% CHAR_eq0_iff
tff(fact_7719_length__filter__map,axiom,
    ! [A: $tType,B: $tType,P: fun(A,$o),F: fun(B,A),Xs: list(B)] : aa(list(A),nat,size_size(list(A)),filter2(A,P,aa(list(B),list(A),map(B,A,F),Xs))) = aa(list(B),nat,size_size(list(B)),filter2(B,comp(A,$o,B,P,F),Xs)) ).

% length_filter_map
tff(fact_7720_bin__rest__set__bits,axiom,
    ! [F: fun(nat,$o)] :
      ( bit_wf_set_bits_int(F)
     => ( aa(int,int,aa(int,fun(int,int),divide_divide(int),bit_bi4170147762399595738t_bits(int,F)),aa(num,int,numeral_numeral(int),bit0(one2))) = bit_bi4170147762399595738t_bits(int,comp(nat,$o,nat,F,suc)) ) ) ).

% bin_rest_set_bits
tff(fact_7721_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),A3: set(C)] :
          ( ( aa(B,A,H,zero_zero(B)) = zero_zero(A) )
         => ( ! [X2: B,Y5: B] : aa(B,A,H,aa(B,B,aa(B,fun(B,B),plus_plus(B),X2),Y5)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(B,A,H,X2)),aa(B,A,H,Y5))
           => ( aa(set(C),A,groups7311177749621191930dd_sum(C,A,comp(B,A,C,H,G)),A3) = aa(B,A,H,aa(set(C),B,groups7311177749621191930dd_sum(C,B,G),A3)) ) ) ) ) ).

% sum_comp_morphism
tff(fact_7722_fun__comp__eq__conv,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: fun(C,B),G: fun(A,C),Fg: fun(A,B)] :
      ( ( comp(C,B,A,F,G) = Fg )
    <=> ! [X: A] : aa(C,B,F,aa(A,C,G,X)) = aa(A,B,Fg,X) ) ).

% fun_comp_eq_conv
tff(fact_7723_comp__cong__right,axiom,
    ! [C: $tType,B: $tType,A: $tType,Xb: fun(A,B),Y: fun(A,B),F: fun(B,C)] :
      ( ( Xb = Y )
     => ( comp(B,C,A,F,Xb) = comp(B,C,A,F,Y) ) ) ).

% comp_cong_right
tff(fact_7724_comp__cong__left,axiom,
    ! [B: $tType,A: $tType,C: $tType,Xb: fun(A,B),Y: fun(A,B),F: fun(C,A)] :
      ( ( Xb = Y )
     => ( comp(A,B,C,Xb,F) = comp(A,B,C,Y,F) ) ) ).

% comp_cong_left
tff(fact_7725_prod_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Nb: 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),Nb),K))) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),K)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% prod.atLeastAtMost_shift_bounds
tff(fact_7726_prod_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Nb: 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),Nb),K))) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),K)),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% prod.atLeastLessThan_shift_bounds
tff(fact_7727_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,suc),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% prod.atLeast_Suc_atMost_Suc_shift
tff(fact_7728_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,suc),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% prod.atLeast_Suc_lessThan_Suc_shift
tff(fact_7729_sum_OatLeastAtMost__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Nb: 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),Nb),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% sum.atLeastAtMost_shift_bounds
tff(fact_7730_sum_OatLeastLessThan__shift__bounds,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,K: nat,Nb: 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),Nb),K))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aa(nat,fun(nat,nat),plus_plus(nat),K))),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% sum.atLeastLessThan_shift_bounds
tff(fact_7731_bit__drop__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat,A2: A] : aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),Nb),A2)) = comp(nat,$o,nat,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,fun(nat,nat),plus_plus(nat),Nb)) ) ).

% bit_drop_bit_eq
tff(fact_7732_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,suc)),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% sum.atLeast_Suc_atMost_Suc_shift
tff(fact_7733_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,suc)),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% sum.atLeast_Suc_lessThan_Suc_shift
tff(fact_7734_bij__betw__comp__iff2,axiom,
    ! [C: $tType,A: $tType,B: $tType,F4: fun(A,B),A9: set(A),A10: set(B),F: fun(C,A),A3: set(C)] :
      ( bij_betw(A,B,F4,A9,A10)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(C),set(A),image(C,A,F),A3)),A9)
       => ( bij_betw(C,A,F,A3,A9)
        <=> bij_betw(C,B,comp(A,B,C,F4,F),A3,A10) ) ) ) ).

% bij_betw_comp_iff2
tff(fact_7735_summable__inverse__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(nat,A),C2: A] :
          ( summable(A,comp(A,A,nat,inverse_inverse(A),F))
         => summable(A,aa(A,fun(nat,A),aTP_Lamp_ams(fun(nat,A),fun(A,fun(nat,A)),F),C2)) ) ) ).

% summable_inverse_divide
tff(fact_7736_prod_Oreindex__nontrivial,axiom,
    ! [B: $tType,C: $tType,A: $tType] :
      ( comm_monoid_mult(C)
     => ! [A3: set(A),H: fun(A,B),G: fun(B,C)] :
          ( finite_finite2(A,A3)
         => ( ! [X2: A,Y5: A] :
                ( member(A,X2,A3)
               => ( member(A,Y5,A3)
                 => ( ( X2 != Y5 )
                   => ( ( aa(A,B,H,X2) = aa(A,B,H,Y5) )
                     => ( 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),A3)) = groups7121269368397514597t_prod(A,C,comp(B,C,A,G,H),A3) ) ) ) ) ).

% prod.reindex_nontrivial
tff(fact_7737_sum__image__le,axiom,
    ! [C: $tType,B: $tType,A: $tType] :
      ( ordere6911136660526730532id_add(B)
     => ! [I5: set(A),G: fun(C,B),F: fun(A,C)] :
          ( finite_finite2(A,I5)
         => ( ! [I3: A] :
                ( member(A,I3,I5)
               => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),zero_zero(B)),aa(C,B,G,aa(A,C,F,I3))) )
           => aa(B,$o,aa(B,fun(B,$o),ord_less_eq(B),aa(set(C),B,groups7311177749621191930dd_sum(C,B,G),aa(set(A),set(C),image(A,C,F),I5))),aa(set(A),B,groups7311177749621191930dd_sum(A,B,comp(C,B,A,G,F)),I5)) ) ) ) ).

% sum_image_le
tff(fact_7738_sum_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = 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),Nb))) ) ).

% sum.atLeast0_lessThan_Suc_shift
tff(fact_7739_sum_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = 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),Nb))) ) ).

% sum.atLeast0_atMost_Suc_shift
tff(fact_7740_prod_OatLeast0__lessThan__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = 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),Nb))) ) ).

% prod.atLeast0_lessThan_Suc_shift
tff(fact_7741_prod_OatLeast0__atMost__Suc__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),Nb: nat] : groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,zero_zero(nat),aa(nat,nat,suc,Nb))) = 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),Nb))) ) ).

% prod.atLeast0_atMost_Suc_shift
tff(fact_7742_sum_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,Nb)) = 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,Nb),M))) ) ).

% sum.atLeastLessThan_shift_0
tff(fact_7743_prod_OatLeastLessThan__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Nb)) = 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,Nb),M))) ) ).

% prod.atLeastLessThan_shift_0
tff(fact_7744_sum_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aTP_Lamp_ajm(nat,nat))),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% sum.atLeast_lessThan_pred_shift
tff(fact_7745_sum_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,comp(nat,A,nat,G,aTP_Lamp_ajm(nat,nat))),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% sum.atLeast_atMost_pred_shift
tff(fact_7746_prod_OatLeast__lessThan__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aTP_Lamp_ajm(nat,nat)),set_or7035219750837199246ssThan(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = groups7121269368397514597t_prod(nat,A,G,set_or7035219750837199246ssThan(nat,M,Nb)) ) ).

% prod.atLeast_lessThan_pred_shift
tff(fact_7747_prod_OatLeast__atMost__pred__shift,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(nat,A),M: nat,Nb: nat] : groups7121269368397514597t_prod(nat,A,comp(nat,A,nat,G,aTP_Lamp_ajm(nat,nat)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,M),aa(nat,nat,suc,Nb))) = groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Nb)) ) ).

% prod.atLeast_atMost_pred_shift
tff(fact_7748_sum_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,G),set_or1337092689740270186AtMost(nat,M,Nb)) = 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,Nb),M))) ) ) ) ).

% sum.atLeastAtMost_shift_0
tff(fact_7749_prod_OatLeastAtMost__shift__0,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [M: nat,Nb: nat,G: fun(nat,A)] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
         => ( groups7121269368397514597t_prod(nat,A,G,set_or1337092689740270186AtMost(nat,M,Nb)) = 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,Nb),M))) ) ) ) ).

% prod.atLeastAtMost_shift_0
tff(fact_7750_CHAR__pos__iff,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ( aa(nat,$o,ord_less(nat,zero_zero(nat)),semiri4206861660011772517g_char(A,type2(A)))
      <=> ? [N6: nat] :
            ( aa(nat,$o,ord_less(nat,zero_zero(nat)),N6)
            & ( aa(nat,A,semiring_1_of_nat(A),N6) = zero_zero(A) ) ) ) ) ).

% CHAR_pos_iff
tff(fact_7751_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_7752_bit__sshiftr__iff,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [W: word(A),M: nat,Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),bit_Sh8784991116023147202shiftr(A,W,M)),Nb)
        <=> aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),W),
              $ite(
                ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),M)),Nb)
                & aa(nat,$o,ord_less(nat,Nb),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),Nb) )) ) ) ).

% bit_sshiftr_iff
tff(fact_7753_sshiftr__1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] :
          bit_Sh8784991116023147202shiftr(A,one_one(word(A)),Nb) = aa($o,word(A),zero_neq_one_of_bool(word(A)),
            ( ( type_len0_len_of(A,type2(A)) = one_one(nat) )
            | ( Nb = zero_zero(nat) ) )) ) ).

% sshiftr_1
tff(fact_7754_sshiftr__numeral__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Nb: nat] : bit_Sh8784991116023147202shiftr(A,aa(num,word(A),numeral_numeral(word(A)),M),aa(nat,nat,suc,Nb)) = signed_drop_bit(A,aa(nat,nat,suc,Nb),aa(num,word(A),numeral_numeral(word(A)),M)) ) ).

% sshiftr_numeral_Suc
tff(fact_7755_sshiftr__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Nb: num] : bit_Sh8784991116023147202shiftr(A,aa(num,word(A),numeral_numeral(word(A)),M),aa(num,nat,numeral_numeral(nat),Nb)) = signed_drop_bit(A,aa(num,nat,numeral_numeral(nat),Nb),aa(num,word(A),numeral_numeral(word(A)),M)) ) ).

% sshiftr_numeral_numeral
tff(fact_7756_sshiftr__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Nb: nat] : bit_Sh8784991116023147202shiftr(A,aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M)),aa(nat,nat,suc,Nb)) = signed_drop_bit(A,aa(nat,nat,suc,Nb),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M))) ) ).

% sshiftr_minus_numeral_Suc
tff(fact_7757_sshiftr__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [M: num,Nb: 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),Nb)) = signed_drop_bit(A,aa(num,nat,numeral_numeral(nat),Nb),aa(word(A),word(A),uminus_uminus(word(A)),aa(num,word(A),numeral_numeral(word(A)),M))) ) ).

% sshiftr_minus_numeral_numeral
tff(fact_7758_card_Ocomp__fun__commute__on,axiom,
    comp(nat,nat,nat,suc,suc) = comp(nat,nat,nat,suc,suc) ).

% card.comp_fun_commute_on
tff(fact_7759_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_amt(B,product_prod(B,B)),product_snd(A,B))) = product_snd(A,B) ).

% fst_diag_snd
tff(fact_7760_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_amu(A,product_prod(A,A)),product_fst(A,B))) = product_fst(A,B) ).

% snd_diag_fst
tff(fact_7761_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_amu(A,product_prod(A,A)),product_fst(A,B))) = product_fst(A,B) ).

% fst_diag_fst
tff(fact_7762_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_amt(B,product_prod(B,B)),product_snd(A,B))) = product_snd(A,B) ).

% snd_diag_snd
tff(fact_7763_set__oo__map__alt,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),X3: list(A)] : aa(list(A),set(B),comp(list(B),set(B),list(A),set2(B),map(A,B,F)),X3) = aa(set(A),set(B),image(A,B,F),aa(list(A),set(A),set2(A),X3)) ).

% set_oo_map_alt
tff(fact_7764_case__prod__comp,axiom,
    ! [D: $tType,A: $tType,C: $tType,B: $tType,F: fun(D,fun(C,A)),G: fun(B,D),Xb: product_prod(B,C)] : aa(product_prod(B,C),A,product_case_prod(B,C,A,comp(D,fun(C,A),B,F,G)),Xb) = aa(C,A,aa(D,fun(C,A),F,aa(B,D,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_7765_execute__heap,axiom,
    ! [A: $tType,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] : aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(A),heap_Time_heap(A,F)) = 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))),F) ).

% execute_heap
tff(fact_7766_heap__def,axiom,
    ! [A: $tType,F: fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat)))] : heap_Time_heap(A,F) = aa(fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(A),heap_Time_Heap2(A),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))),F)) ).

% heap_def
tff(fact_7767_sum_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(A)
     => ! [G: fun(B,A),A3: set(B)] : aa(set(B),A,groups7311177749621191930dd_sum(B,A,G),A3) = finite_fold(B,A,comp(A,fun(A,A),B,plus_plus(A),G),zero_zero(A),A3) ) ).

% sum.eq_fold
tff(fact_7768_prod_Oeq__fold,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(A)
     => ! [G: fun(B,A),A3: set(B)] : groups7121269368397514597t_prod(B,A,G,A3) = finite_fold(B,A,comp(A,fun(A,A),B,times_times(A),G),one_one(A),A3) ) ).

% prod.eq_fold
tff(fact_7769_finite__range__prod,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: 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),F)),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),F)),top_top(set(B))))
       => finite_finite2(product_prod(A,C),aa(set(B),set(product_prod(A,C)),image(B,product_prod(A,C),F),top_top(set(B)))) ) ) ).

% finite_range_prod
tff(fact_7770_map__filter__def,axiom,
    ! [A: $tType,B: $tType,F: fun(B,option(A)),Xs: list(B)] : map_filter(B,A,F,Xs) = aa(list(B),list(A),map(B,A,comp(option(A),A,B,the2(A),F)),filter2(B,aTP_Lamp_amv(fun(B,option(A)),fun(B,$o),F),Xs)) ).

% map_filter_def
tff(fact_7771_execute__return,axiom,
    ! [A: $tType,Xb: A] : aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),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_agh(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Xb)) ).

% execute_return
tff(fact_7772_execute__ureturn,axiom,
    ! [A: $tType,Xb: A] : aa(heap_Time_Heap(A),fun(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat)))),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_agi(A,fun(heap_ext(product_unit),product_prod(A,product_prod(heap_ext(product_unit),nat))),Xb)) ).

% execute_ureturn
tff(fact_7773_Code__Numeral_Onegative__def,axiom,
    code_negative = comp(code_integer,code_integer,num,uminus_uminus(code_integer),numeral_numeral(code_integer)) ).

% Code_Numeral.negative_def
tff(fact_7774_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_ti(A,fun(B,product_prod(B,A))))),Xy) ).

% fst_snd_flip
tff(fact_7775_ge__eq__refl,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o)),Xb: A] :
      ( aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),R)
     => aa(A,$o,aa(A,fun(A,$o),R,Xb),Xb) ) ).

% ge_eq_refl
tff(fact_7776_refl__ge__eq,axiom,
    ! [A: $tType,R: fun(A,fun(A,$o))] :
      ( ! [X2: A] : aa(A,$o,aa(A,fun(A,$o),R,X2),X2)
     => aa(fun(A,fun(A,$o)),$o,aa(fun(A,fun(A,$o)),fun(fun(A,fun(A,$o)),$o),ord_less_eq(fun(A,fun(A,$o))),fequal(A)),R) ) ).

% refl_ge_eq
tff(fact_7777_image__fold__insert,axiom,
    ! [B: $tType,A: $tType,A3: set(A),F: fun(A,B)] :
      ( finite_finite2(A,A3)
     => ( aa(set(A),set(B),image(A,B,F),A3) = finite_fold(A,set(B),aTP_Lamp_amw(fun(A,B),fun(A,fun(set(B),set(B))),F),bot_bot(set(B)),A3) ) ) ).

% image_fold_insert
tff(fact_7778_Max_Oeq__fold_H,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [A3: set(A)] : lattic643756798349783984er_Max(A,A3) = aa(option(A),A,the2(A),finite_fold(A,option(A),aTP_Lamp_amx(A,fun(option(A),option(A))),none(A),A3)) ) ).

% Max.eq_fold'
tff(fact_7779_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_7780_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_7781_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_ek(B,fun(A,product_prod(A,B))))),Xy) ).

% snd_fst_flip
tff(fact_7782_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_7783_shiftl__Suc__0,axiom,
    ! [Nb: nat] : bit_Sh4282982442137083160shiftl(nat,aa(nat,nat,suc,zero_zero(nat)),Nb) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb) ).

% shiftl_Suc_0
tff(fact_7784_shiftl__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,Nb: nat] : bit_Sh4282982442137083160shiftl(A,aa(num,A,numeral_numeral(A),M),aa(nat,nat,suc,Nb)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftl_numeral_Suc
tff(fact_7785_shiftl__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,Nb: num] : bit_Sh4282982442137083160shiftl(A,aa(num,A,numeral_numeral(A),M),aa(num,nat,numeral_numeral(nat),Nb)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(num,nat,numeral_numeral(nat),Nb)),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftl_numeral_numeral
tff(fact_7786_shiftl__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,Nb: nat] : bit_Sh4282982442137083160shiftl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(nat,nat,suc,Nb)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(nat,nat,suc,Nb)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% shiftl_minus_numeral_Suc
tff(fact_7787_shiftl__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,Nb: num] : bit_Sh4282982442137083160shiftl(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(num,nat,numeral_numeral(nat),Nb)) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),aa(num,nat,numeral_numeral(nat),Nb)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% shiftl_minus_numeral_numeral
tff(fact_7788_shiftl__of__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] : bit_Sh4282982442137083160shiftl(A,A2,aa(nat,nat,suc,Nb)) = bit_Sh4282982442137083160shiftl(A,aa(A,A,aa(A,fun(A,A),times_times(A),A2),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) ) ).

% shiftl_of_Suc
tff(fact_7789_shiftl__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_Sh4282982442137083160shiftl(A,one_one(A),Nb) = aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(num,A,numeral_numeral(A),bit0(one2))),Nb) ) ).

% shiftl_1
tff(fact_7790_shiftl__minus__1__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [Nb: num] : bit_Sh4282982442137083160shiftl(A,aa(A,A,uminus_uminus(A),one_one(A)),aa(num,nat,numeral_numeral(nat),Nb)) = aa(A,A,bit_ri4277139882892585799ns_not(A),bit_se2239418461657761734s_mask(A,aa(num,nat,numeral_numeral(nat),Nb))) ) ).

% shiftl_minus_1_numeral
tff(fact_7791_shiftl__eq__mult,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Xb: A,Nb: nat] : bit_Sh4282982442137083160shiftl(A,Xb,Nb) = 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))),Nb)) ) ).

% shiftl_eq_mult
tff(fact_7792_bit__shiftl__iff,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,M: nat,Nb: nat] :
          ( aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),bit_Sh4282982442137083160shiftl(A,A2,M)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),M),Nb)
            & bit_se6407376104438227557le_bit(A,type2(A),Nb)
            & aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,nat,minus_minus(nat,Nb),M)) ) ) ) ).

% bit_shiftl_iff
tff(fact_7793_empty__natural,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F: fun(A,C),G: fun(D,B)] : comp(C,set(B),A,aTP_Lamp_amy(C,set(B)),F) = comp(set(D),set(B),A,image(D,B,G),aTP_Lamp_amz(A,set(D))) ).

% empty_natural
tff(fact_7794_shiftr__minus__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,Nb: num] : bit_Sh4282982442137083166shiftr(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(num,nat,numeral_numeral(nat),Nb)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(num,nat,numeral_numeral(nat),Nb)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% shiftr_minus_numeral_numeral
tff(fact_7795_shiftr__Suc__0,axiom,
    ! [Nb: nat] : bit_Sh4282982442137083166shiftr(nat,aa(nat,nat,suc,zero_zero(nat)),Nb) = aa($o,nat,zero_neq_one_of_bool(nat),Nb = zero_zero(nat)) ).

% shiftr_Suc_0
tff(fact_7796_shiftr__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,Nb: nat] : bit_Sh4282982442137083166shiftr(A,aa(num,A,numeral_numeral(A),M),aa(nat,nat,suc,Nb)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,Nb)),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftr_numeral_Suc
tff(fact_7797_shiftr__numeral__numeral,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [M: num,Nb: num] : bit_Sh4282982442137083166shiftr(A,aa(num,A,numeral_numeral(A),M),aa(num,nat,numeral_numeral(nat),Nb)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(num,nat,numeral_numeral(nat),Nb)),aa(num,A,numeral_numeral(A),M)) ) ).

% shiftr_numeral_numeral
tff(fact_7798_shiftr__1,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Nb: nat] : bit_Sh4282982442137083166shiftr(A,one_one(A),Nb) = aa($o,A,zero_neq_one_of_bool(A),Nb = zero_zero(nat)) ) ).

% shiftr_1
tff(fact_7799_shiftr__minus__numeral__Suc,axiom,
    ! [A: $tType] :
      ( bit_ri3973907225187159222ations(A)
     => ! [M: num,Nb: nat] : bit_Sh4282982442137083166shiftr(A,aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M)),aa(nat,nat,suc,Nb)) = aa(A,A,aa(nat,fun(A,A),bit_se4197421643247451524op_bit(A),aa(nat,nat,suc,Nb)),aa(A,A,uminus_uminus(A),aa(num,A,numeral_numeral(A),M))) ) ).

% shiftr_minus_numeral_Suc
tff(fact_7800_conj__subset__def,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o),Q: fun(A,$o)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),collect(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_st(fun(A,$o),fun(fun(A,$o),fun(A,$o)),P),Q)))
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),collect(A,P))
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),collect(A,Q)) ) ) ).

% conj_subset_def
tff(fact_7801_one__bit__shiftl,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Nb: nat] : aa($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))),Nb),$true) = bit_Sh4282982442137083160shiftl(word(A),one_one(word(A)),Nb) ) ).

% one_bit_shiftl
tff(fact_7802_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,$o),aTP_Lamp_ana(fun(B,$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_7803_bit__shiftr__eq,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [A2: A,Nb: nat] : aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),bit_Sh4282982442137083166shiftr(A,A2,Nb)) = comp(nat,$o,nat,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),A2),aa(nat,fun(nat,nat),plus_plus(nat),Nb)) ) ).

% bit_shiftr_eq
tff(fact_7804_shiftr__eq__div,axiom,
    ! [A: $tType] :
      ( bit_se359711467146920520ations(A)
     => ! [Xb: A,Nb: nat] : bit_Sh4282982442137083166shiftr(A,Xb,Nb) = 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))),Nb)) ) ).

% shiftr_eq_div
tff(fact_7805_bit__revcast__iff,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B),Nb: nat] :
          ( aa(nat,$o,aa(word(A),fun(nat,$o),bit_se5641148757651400278ts_bit(word(A)),revcast(B,A,W)),Nb)
        <=> ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,minus_minus(nat,type_len0_len_of(A,type2(A))),type_len0_len_of(B,type2(B)))),Nb)
            & aa(nat,$o,ord_less(nat,Nb),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),Nb),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_7806_Set__filter__fold,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] :
      ( finite_finite2(A,A3)
     => ( filter3(A,P,A3) = finite_fold(A,set(A),aTP_Lamp_anb(fun(A,$o),fun(A,fun(set(A),set(A))),P),bot_bot(set(A)),A3) ) ) ).

% Set_filter_fold
tff(fact_7807_Set_Ofilter__def,axiom,
    ! [A: $tType,P: fun(A,$o),A3: set(A)] : filter3(A,P,A3) = collect(A,aa(set(A),fun(A,$o),aTP_Lamp_anc(fun(A,$o),fun(set(A),fun(A,$o)),P),A3)) ).

% Set.filter_def
tff(fact_7808_revcast__slice1,axiom,
    ! [A: $tType,B: $tType] :
      ( ( type_len(B)
        & type_len(A) )
     => ! [W: word(B)] : aa(word(B),word(A),slice1(B,A,aa(word(A),nat,size_size(word(A)),revcast(B,A,W))),W) = revcast(B,A,W) ) ).

% revcast_slice1
tff(fact_7809_filter__set,axiom,
    ! [A: $tType,P: fun(A,$o),Xs: list(A)] : filter3(A,P,aa(list(A),set(A),set2(A),Xs)) = aa(list(A),set(A),set2(A),filter2(A,P,Xs)) ).

% filter_set
tff(fact_7810_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_7811_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_7812_inj__mult__left,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [A2: A] :
          ( inj_on(A,A,aa(A,fun(A,A),times_times(A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_mult_left
tff(fact_7813_inj__divide__right,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [A2: A] :
          ( inj_on(A,A,aTP_Lamp_and(A,fun(A,A),A2),top_top(set(A)))
        <=> ( A2 != zero_zero(A) ) ) ) ).

% inj_divide_right
tff(fact_7814_inj__on__insert,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A2: A,A3: set(A)] :
      ( inj_on(A,B,F,aa(set(A),set(A),insert(A,A2),A3))
    <=> ( inj_on(A,B,F,A3)
        & ~ member(B,aa(A,B,F,A2),aa(set(A),set(B),image(A,B,F),aa(set(A),set(A),minus_minus(set(A),A3),aa(set(A),set(A),insert(A,A2),bot_bot(set(A)))))) ) ) ).

% inj_on_insert
tff(fact_7815_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_7816_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_7817_inj__on__filter__key__eq,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),Y: A,Xs: list(A)] :
      ( inj_on(A,B,F,aa(set(A),set(A),insert(A,Y),aa(list(A),set(A),set2(A),Xs)))
     => ( filter2(A,aa(A,fun(A,$o),aTP_Lamp_ane(fun(A,B),fun(A,fun(A,$o)),F),Y),Xs) = filter2(A,aa(A,fun(A,$o),fequal(A),Y),Xs) ) ) ).

% inj_on_filter_key_eq
tff(fact_7818_inj__on__add,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A)] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A2),A3) ) ).

% inj_on_add
tff(fact_7819_inj__on__add_H,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A,A3: set(A)] : inj_on(A,A,aTP_Lamp_anf(A,fun(A,A),A2),A3) ) ).

% inj_on_add'
tff(fact_7820_inj__on__diff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: set(A),B5: set(A)] :
      ( inj_on(A,B,F,A3)
     => inj_on(A,B,F,aa(set(A),set(A),minus_minus(set(A),A3),B5)) ) ).

% inj_on_diff
tff(fact_7821_word__msb__n1,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => most_s684356279273892711sb_msb(word(A),aa(word(A),word(A),uminus_uminus(word(A)),one_one(word(A)))) ) ).

% word_msb_n1
tff(fact_7822_inj__on__id2,axiom,
    ! [A: $tType,A3: set(A)] : inj_on(A,A,aTP_Lamp_dz(A,A),A3) ).

% inj_on_id2
tff(fact_7823_inj__on__mult,axiom,
    ! [A: $tType] :
      ( semidom_divide(A)
     => ! [A2: A,A3: set(A)] :
          ( ( A2 != zero_zero(A) )
         => inj_on(A,A,aa(A,fun(A,A),times_times(A),A2),A3) ) ) ).

% inj_on_mult
tff(fact_7824_linorder__inj__onI,axiom,
    ! [B: $tType,A: $tType] :
      ( order(A)
     => ! [A3: set(A),F: fun(A,B)] :
          ( ! [X2: A,Y5: A] :
              ( aa(A,$o,ord_less(A,X2),Y5)
             => ( member(A,X2,A3)
               => ( member(A,Y5,A3)
                 => ( aa(A,B,F,X2) != aa(A,B,F,Y5) ) ) ) )
         => ( ! [X2: A,Y5: A] :
                ( member(A,X2,A3)
               => ( member(A,Y5,A3)
                 => ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X2),Y5)
                    | aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y5),X2) ) ) )
           => inj_on(A,B,F,A3) ) ) ) ).

% linorder_inj_onI
tff(fact_7825_inj__on__subset,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: set(A),B5: set(A)] :
      ( inj_on(A,B,F,A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),A3)
       => inj_on(A,B,F,B5) ) ) ).

% inj_on_subset
tff(fact_7826_subset__inj__on,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),B5: set(A),A3: set(A)] :
      ( inj_on(A,B,F,B5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
       => inj_on(A,B,F,A3) ) ) ).

% subset_inj_on
tff(fact_7827_inj__add__left,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [A2: A] : inj_on(A,A,aa(A,fun(A,A),plus_plus(A),A2),top_top(set(A))) ) ).

% inj_add_left
tff(fact_7828_inj__diff__right,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [A2: A] : inj_on(A,A,aTP_Lamp_ahg(A,fun(A,A),A2),top_top(set(A))) ) ).

% inj_diff_right
tff(fact_7829_sorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => inj_on(A,A,aTP_Lamp_aks(A,A),top_top(set(A))) ) ).

% sorted_list_of_set.inj_on
tff(fact_7830_linorder__injI,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(A)
     => ! [F: fun(A,B)] :
          ( ! [X2: A,Y5: A] :
              ( aa(A,$o,ord_less(A,X2),Y5)
             => ( aa(A,B,F,X2) != aa(A,B,F,Y5) ) )
         => inj_on(A,B,F,top_top(set(A))) ) ) ).

% linorder_injI
tff(fact_7831_inj__fun,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: fun(A,B)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => inj_on(A,fun(C,B),aTP_Lamp_ang(fun(A,B),fun(A,fun(C,B)),F),top_top(set(A))) ) ).

% inj_fun
tff(fact_7832_word__sle__msb__le,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Xb: word(A),Y: word(A)] :
          ( word_sle(A,Xb,Y)
        <=> ( ( most_s684356279273892711sb_msb(word(A),Y)
             => most_s684356279273892711sb_msb(word(A),Xb) )
            & ( ( most_s684356279273892711sb_msb(word(A),Xb)
                & ~ most_s684356279273892711sb_msb(word(A),Y) )
              | aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Xb),Y) ) ) ) ) ).

% word_sle_msb_le
tff(fact_7833_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_7834_inj__on__image__eq__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),C5: set(A),A3: set(A),B5: set(A)] :
      ( inj_on(A,B,F,C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),C5)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
         => ( ( aa(set(A),set(B),image(A,B,F),A3) = aa(set(A),set(B),image(A,B,F),B5) )
          <=> ( A3 = B5 ) ) ) ) ) ).

% inj_on_image_eq_iff
tff(fact_7835_inj__on__image__mem__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),B5: set(A),A2: A,A3: set(A)] :
      ( inj_on(A,B,F,B5)
     => ( member(A,A2,B5)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
         => ( member(B,aa(A,B,F,A2),aa(set(A),set(B),image(A,B,F),A3))
          <=> member(A,A2,A3) ) ) ) ) ).

% inj_on_image_mem_iff
tff(fact_7836_sum_Oimage__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(B)
     => ! [G: fun(A,B),A3: set(A)] :
          ( inj_on(A,B,G,A3)
         => ( aa(set(B),B,groups7311177749621191930dd_sum(B,B,aTP_Lamp_anh(B,B)),aa(set(A),set(B),image(A,B,G),A3)) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,G),A3) ) ) ) ).

% sum.image_eq
tff(fact_7837_prod_Oimage__eq,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(B)
     => ! [G: fun(A,B),A3: set(A)] :
          ( inj_on(A,B,G,A3)
         => ( groups7121269368397514597t_prod(B,B,aTP_Lamp_ani(B,B),aa(set(A),set(B),image(A,B,G),A3)) = groups7121269368397514597t_prod(A,B,G,A3) ) ) ) ).

% prod.image_eq
tff(fact_7838_subset__image__inj,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F: fun(B,A),T5: set(B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S2),aa(set(B),set(A),image(B,A,F),T5))
    <=> ? [U5: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),U5),T5)
          & inj_on(B,A,F,U5)
          & ( S2 = aa(set(B),set(A),image(B,A,F),U5) ) ) ) ).

% subset_image_inj
tff(fact_7839_finite__surj__inj,axiom,
    ! [A: $tType,A3: set(A),F: fun(A,A)] :
      ( finite_finite2(A,A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),aa(set(A),set(A),image(A,A,F),A3))
       => inj_on(A,A,F,A3) ) ) ).

% finite_surj_inj
tff(fact_7840_inj__on__finite,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: set(A),B5: set(B)] :
      ( inj_on(A,B,F,A3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),A3)),B5)
       => ( finite_finite2(B,B5)
         => finite_finite2(A,A3) ) ) ) ).

% inj_on_finite
tff(fact_7841_endo__inj__surj,axiom,
    ! [A: $tType,A3: set(A),F: fun(A,A)] :
      ( finite_finite2(A,A3)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),image(A,A,F),A3)),A3)
       => ( inj_on(A,A,F,A3)
         => ( aa(set(A),set(A),image(A,A,F),A3) = A3 ) ) ) ) ).

% endo_inj_surj
tff(fact_7842_inj__image__subset__iff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: set(A),B5: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),A3)),aa(set(A),set(B),image(A,B,F),B5))
      <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5) ) ) ).

% inj_image_subset_iff
tff(fact_7843_inj__on__iff__surj,axiom,
    ! [B: $tType,A: $tType,A3: set(A),A9: set(B)] :
      ( ( A3 != bot_bot(set(A)) )
     => ( ? [F8: fun(A,B)] :
            ( inj_on(A,B,F8,A3)
            & aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F8),A3)),A9) )
      <=> ? [G5: fun(B,A)] : aa(set(B),set(A),image(B,A,G5),A9) = A3 ) ) ).

% inj_on_iff_surj
tff(fact_7844_image__set__diff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: set(A),B5: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => ( aa(set(A),set(B),image(A,B,F),aa(set(A),set(A),minus_minus(set(A),A3),B5)) = aa(set(B),set(B),minus_minus(set(B),aa(set(A),set(B),image(A,B,F),A3)),aa(set(A),set(B),image(A,B,F),B5)) ) ) ).

% image_set_diff
tff(fact_7845_inj__on__image__set__diff,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),C5: set(A),A3: set(A),B5: set(A)] :
      ( inj_on(A,B,F,C5)
     => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(A),set(A),minus_minus(set(A),A3),B5)),C5)
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),B5),C5)
         => ( aa(set(A),set(B),image(A,B,F),aa(set(A),set(A),minus_minus(set(A),A3),B5)) = aa(set(B),set(B),minus_minus(set(B),aa(set(A),set(B),image(A,B,F),A3)),aa(set(A),set(B),image(A,B,F),B5)) ) ) ) ) ).

% inj_on_image_set_diff
tff(fact_7846_finite__inverse__image,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F: fun(B,A)] :
      ( finite_finite2(A,A3)
     => ( inj_on(B,A,F,top_top(set(B)))
       => finite_finite2(B,collect(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_anj(set(A),fun(fun(B,A),fun(B,$o)),A3),F))) ) ) ).

% finite_inverse_image
tff(fact_7847_finite__Collect,axiom,
    ! [A: $tType,B: $tType,S2: set(A),F: fun(B,A)] :
      ( finite_finite2(A,S2)
     => ( inj_on(B,A,F,top_top(set(B)))
       => finite_finite2(B,collect(B,aa(fun(B,A),fun(B,$o),aTP_Lamp_anj(set(A),fun(fun(B,A),fun(B,$o)),S2),F))) ) ) ).

% finite_Collect
tff(fact_7848_the__inv__into__into,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: set(A),Xb: B,B5: set(A)] :
      ( inj_on(A,B,F,A3)
     => ( member(B,Xb,aa(set(A),set(B),image(A,B,F),A3))
       => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),B5)
         => member(A,the_inv_into(A,B,A3,F,Xb),B5) ) ) ) ).

% the_inv_into_into
tff(fact_7849_finite__inverse__image__gen,axiom,
    ! [A: $tType,B: $tType,A3: set(A),F: fun(B,A),D6: set(B)] :
      ( finite_finite2(A,A3)
     => ( inj_on(B,A,F,D6)
       => finite_finite2(B,collect(B,aa(set(B),fun(B,$o),aa(fun(B,A),fun(set(B),fun(B,$o)),aTP_Lamp_ank(set(A),fun(fun(B,A),fun(set(B),fun(B,$o))),A3),F),D6))) ) ) ).

% finite_inverse_image_gen
tff(fact_7850_inj__image__Compl__subset,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: set(A)] :
      ( inj_on(A,B,F,top_top(set(A)))
     => aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),aa(set(A),set(A),uminus_uminus(set(A)),A3))),aa(set(B),set(B),uminus_uminus(set(B)),aa(set(A),set(B),image(A,B,F),A3))) ) ).

% inj_image_Compl_subset
tff(fact_7851_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_7852_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_7853_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_7854_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_7855_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_7856_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_7857_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_7858_msb__big,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [A2: word(A)] :
          ( most_s684356279273892711sb_msb(word(A),A2)
        <=> aa(word(A),$o,aa(word(A),fun(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))))),A2) ) ) ).

% msb_big
tff(fact_7859_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)) )
         => ( ! [P6: B] :
                ( most_s684356279273892711sb_msb(B,P6)
              <=> most_s684356279273892711sb_msb(word(A),aa(B,word(A),Word_of,P6)) )
           => ( ! [P6: B] :
                  ( aa(B,$o,least_8051144512741203767sb_lsb(B),P6)
                <=> aa(word(A),$o,least_8051144512741203767sb_lsb(word(A)),aa(B,word(A),Word_of,P6)) )
             => ( ! [P6: B] : aa(B,nat,size_size(B),P6) = aa(word(A),nat,size_size(word(A)),aa(B,word(A),Word_of,P6))
               => ( ! [P6: B,N2: nat,B3: $o] : aa(B,word(A),Word_of,aa($o,B,aa(nat,fun($o,B),aa(B,fun(nat,fun($o,B)),generi7602027413899671122et_bit(B),P6),N2),(B3))) = aa($o,word(A),aa(nat,fun($o,word(A)),aa(word(A),fun(nat,fun($o,word(A))),generi7602027413899671122et_bit(word(A)),aa(B,word(A),Word_of,P6)),N2),(B3))
                 => ( ! [P4: fun(nat,$o)] : aa(B,word(A),Word_of,bit_bi4170147762399595738t_bits(B,P4)) = bit_bi4170147762399595738t_bits(word(A),P4)
                   => ( ! [P4: fun(nat,$o),N2: nat,P6: B] : aa(B,word(A),Word_of,aa(B,B,aa(nat,fun(B,B),aa(fun(nat,$o),fun(nat,fun(B,B)),Set_bits_aux,P4),N2),P6)) = code_T2661198915054445665ts_aux(A,P4,N2,aa(B,word(A),Word_of,P6))
                     => word_T8964210463127689547axioms(B,A,Word_of,Size,Set_bits_aux) ) ) ) ) ) ) ) ) ).

% word_type_copy_misc_axioms.intro
tff(fact_7860_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,N6: nat,B6: $o] : aa(A,word(B),Word_of,aa($o,A,aa(nat,fun($o,A),aa(A,fun(nat,fun($o,A)),generi7602027413899671122et_bit(A),P7),N6),(B6))) = aa($o,word(B),aa(nat,fun($o,word(B)),aa(word(B),fun(nat,fun($o,word(B))),generi7602027413899671122et_bit(word(B)),aa(A,word(B),Word_of,P7)),N6),(B6))
            & ! [P8: fun(nat,$o)] : aa(A,word(B),Word_of,bit_bi4170147762399595738t_bits(A,P8)) = bit_bi4170147762399595738t_bits(word(B),P8)
            & ! [P8: fun(nat,$o),N6: nat,P7: A] : aa(A,word(B),Word_of,aa(A,A,aa(nat,fun(A,A),aa(fun(nat,$o),fun(nat,fun(A,A)),Set_bits_aux,P8),N6),P7)) = code_T2661198915054445665ts_aux(B,P8,N6,aa(A,word(B),Word_of,P7)) ) ) ) ).

% word_type_copy_misc_axioms_def
tff(fact_7861_msb__numeral_I1_J,axiom,
    ! [Nb: num] : ~ most_s684356279273892711sb_msb(int,aa(num,int,numeral_numeral(int),Nb)) ).

% msb_numeral(1)
tff(fact_7862_msb__1,axiom,
    ~ most_s684356279273892711sb_msb(int,one_one(int)) ).

% msb_1
tff(fact_7863_msb__numeral_I2_J,axiom,
    ! [Nb: num] : most_s684356279273892711sb_msb(int,aa(int,int,uminus_uminus(int),aa(num,int,numeral_numeral(int),Nb))) ).

% msb_numeral(2)
tff(fact_7864_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_7865_swap__inj__on,axiom,
    ! [B: $tType,A: $tType,A3: 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_ti(A,fun(B,product_prod(B,A)))),A3) ).

% swap_inj_on
tff(fact_7866_inj__Some,axiom,
    ! [A: $tType,A3: set(A)] : inj_on(A,option(A),some(A),A3) ).

% inj_Some
tff(fact_7867_inj__singleton,axiom,
    ! [A: $tType,A3: set(A)] : inj_on(A,set(A),aTP_Lamp_anl(A,set(A)),A3) ).

% inj_singleton
tff(fact_7868_inj__Suc,axiom,
    ! [N: set(nat)] : inj_on(nat,nat,suc,N) ).

% inj_Suc
tff(fact_7869_inj__on__diff__nat,axiom,
    ! [N: set(nat),K: nat] :
      ( ! [N2: nat] :
          ( member(nat,N2,N)
         => aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),K),N2) )
     => inj_on(nat,nat,aTP_Lamp_aij(nat,fun(nat,nat),K),N) ) ).

% inj_on_diff_nat
tff(fact_7870_inj__Pair_I2_J,axiom,
    ! [B: $tType,A: $tType,C2: fun(A,B),S2: set(A)] : inj_on(A,product_prod(B,A),aTP_Lamp_anm(fun(A,B),fun(A,product_prod(B,A)),C2),S2) ).

% inj_Pair(2)
tff(fact_7871_inj__Pair_I1_J,axiom,
    ! [B: $tType,A: $tType,C2: fun(A,B),S2: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_ann(fun(A,B),fun(A,product_prod(A,B)),C2),S2) ).

% inj_Pair(1)
tff(fact_7872_inj__on__convol__ident,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),X5: set(A)] : inj_on(A,product_prod(A,B),aTP_Lamp_ann(fun(A,B),fun(A,product_prod(A,B)),F),X5) ).

% inj_on_convol_ident
tff(fact_7873_signed_Osorted__list__of__set_Oinj__on,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => inj_on(word(A),word(A),aTP_Lamp_ano(word(A),word(A)),top_top(set(word(A)))) ) ).

% signed.sorted_list_of_set.inj_on
tff(fact_7874_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_7875_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_alu(list(A),fun(A,list(A)))),X5) ).

% inj_split_Cons
tff(fact_7876_inj__graph,axiom,
    ! [B: $tType,A: $tType] : inj_on(fun(A,B),set(product_prod(A,B)),aTP_Lamp_anq(fun(A,B),set(product_prod(A,B))),top_top(set(fun(A,B)))) ).

% inj_graph
tff(fact_7877_finite__imp__nat__seg__image__inj__on,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => ? [N2: nat,F2: fun(nat,A)] :
          ( ( A3 = aa(set(nat),set(A),image(nat,A,F2),collect(nat,aTP_Lamp_sq(nat,fun(nat,$o),N2))) )
          & inj_on(nat,A,F2,collect(nat,aTP_Lamp_sq(nat,fun(nat,$o),N2))) ) ) ).

% finite_imp_nat_seg_image_inj_on
tff(fact_7878_finite__imp__inj__to__nat__seg,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => ? [F2: fun(A,nat),N2: nat] :
          ( ( aa(set(A),set(nat),image(A,nat,F2),A3) = collect(nat,aTP_Lamp_sq(nat,fun(nat,$o),N2)) )
          & inj_on(A,nat,F2,A3) ) ) ).

% finite_imp_inj_to_nat_seg
tff(fact_7879_finite__imp__inj__to__nat__seg_H,axiom,
    ! [A: $tType,A3: set(A)] :
      ( finite_finite2(A,A3)
     => ~ ! [F2: fun(A,nat)] :
            ( ? [N2: nat] : aa(set(A),set(nat),image(A,nat,F2),A3) = collect(nat,aTP_Lamp_sq(nat,fun(nat,$o),N2))
           => ~ inj_on(A,nat,F2,A3) ) ) ).

% finite_imp_inj_to_nat_seg'
tff(fact_7880_infinite__iff__countable__subset,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ~ finite_finite2(A,S2)
    <=> ? [F8: fun(nat,A)] :
          ( inj_on(nat,A,F8,top_top(set(nat)))
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(nat),set(A),image(nat,A,F8),top_top(set(nat)))),S2) ) ) ).

% infinite_iff_countable_subset
tff(fact_7881_infinite__countable__subset,axiom,
    ! [A: $tType,S2: set(A)] :
      ( ~ finite_finite2(A,S2)
     => ? [F2: fun(nat,A)] :
          ( inj_on(nat,A,F2,top_top(set(nat)))
          & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(nat),set(A),image(nat,A,F2),top_top(set(nat)))),S2) ) ) ).

% infinite_countable_subset
tff(fact_7882_summable__reindex,axiom,
    ! [F: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F,X2))
         => summable(real,comp(nat,real,nat,F,G)) ) ) ) ).

% summable_reindex
tff(fact_7883_suminf__reindex__mono,axiom,
    ! [F: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F,X2))
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),suminf(real,comp(nat,real,nat,F,G))),suminf(real,F)) ) ) ) ).

% suminf_reindex_mono
tff(fact_7884_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_7885_suminf__reindex,axiom,
    ! [F: fun(nat,real),G: fun(nat,nat)] :
      ( summable(real,F)
     => ( inj_on(nat,nat,G,top_top(set(nat)))
       => ( ! [X2: nat] : aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(nat,real,F,X2))
         => ( ! [X2: nat] :
                ( ~ member(nat,X2,aa(set(nat),set(nat),image(nat,nat,G),top_top(set(nat))))
               => ( aa(nat,real,F,X2) = zero_zero(real) ) )
           => ( suminf(real,comp(nat,real,nat,F,G)) = suminf(real,F) ) ) ) ) ) ).

% suminf_reindex
tff(fact_7886_inj__sgn__power,axiom,
    ! [Nb: nat] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => inj_on(real,real,aTP_Lamp_afk(nat,fun(real,real),Nb),top_top(set(real))) ) ).

% inj_sgn_power
tff(fact_7887_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_7888_inj__on__map__inv__f,axiom,
    ! [B: $tType,A: $tType,L: list(A),A3: set(A),F: fun(A,B)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),L)),A3)
     => ( inj_on(A,B,F,A3)
       => ( aa(list(B),list(A),map(B,A,inv_on(A,B,F,A3)),aa(list(A),list(B),map(A,B,F),L)) = L ) ) ) ).

% inj_on_map_inv_f
tff(fact_7889_Schroeder__Bernstein,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A3: set(A),B5: set(B),G: fun(B,A)] :
      ( inj_on(A,B,F,A3)
     => ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),aa(set(A),set(B),image(A,B,F),A3)),B5)
       => ( inj_on(B,A,G,B5)
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(set(B),set(A),image(B,A,G),B5)),A3)
           => ? [H2: fun(A,B)] : bij_betw(A,B,H2,A3,B5) ) ) ) ) ).

% Schroeder_Bernstein
tff(fact_7890_inv__on__f__f,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B),A3: set(A),Xb: A] :
      ( inj_on(A,B,F,A3)
     => ( member(A,Xb,A3)
       => ( aa(B,A,inv_on(A,B,F,A3),aa(A,B,F,Xb)) = Xb ) ) ) ).

% inv_on_f_f
tff(fact_7891_le__rel__bool__arg__iff,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [X5: fun($o,A),Y7: fun($o,A)] :
          ( aa(fun($o,A),$o,aa(fun($o,A),fun(fun($o,A),$o),ord_less_eq(fun($o,A)),X5),Y7)
        <=> ( aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,X5,$false)),aa($o,A,Y7,$false))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa($o,A,X5,$true)),aa($o,A,Y7,$true)) ) ) ) ).

% le_rel_bool_arg_iff
tff(fact_7892_inv__on__f__range,axiom,
    ! [A: $tType,B: $tType,Y: A,F: fun(B,A),A3: set(B)] :
      ( member(A,Y,aa(set(B),set(A),image(B,A,F),A3))
     => member(B,aa(A,B,inv_on(B,A,F,A3),Y),A3) ) ).

% inv_on_f_range
tff(fact_7893_f__inv__on__f,axiom,
    ! [B: $tType,A: $tType,Y: A,F: fun(B,A),A3: set(B)] :
      ( member(A,Y,aa(set(B),set(A),image(B,A,F),A3))
     => ( aa(B,A,F,aa(A,B,inv_on(B,A,F,A3),Y)) = Y ) ) ).

% f_inv_on_f
tff(fact_7894_inv__on__def,axiom,
    ! [A: $tType,B: $tType,F: fun(A,B),A3: set(A),Xb: B] : aa(B,A,inv_on(A,B,F,A3),Xb) = fChoice(A,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_anr(fun(A,B),fun(set(A),fun(B,fun(A,$o))),F),A3),Xb)) ).

% inv_on_def
tff(fact_7895_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_7896_all__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),S2: set(B),P: fun(set(A),$o)] :
      ( ! [T8: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T8),aa(set(B),set(A),image(B,A,F),S2))
         => aa(set(A),$o,P,T8) )
    <=> ! [T8: set(B)] :
          ( ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T8),S2)
            & inj_on(B,A,F,T8) )
         => aa(set(A),$o,P,aa(set(B),set(A),image(B,A,F),T8)) ) ) ).

% all_subset_image_inj
tff(fact_7897_ex__subset__image__inj,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),S2: set(B),P: fun(set(A),$o)] :
      ( ? [T8: set(A)] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),T8),aa(set(B),set(A),image(B,A,F),S2))
          & aa(set(A),$o,P,T8) )
    <=> ? [T8: set(B)] :
          ( aa(set(B),$o,aa(set(B),fun(set(B),$o),ord_less_eq(set(B)),T8),S2)
          & inj_on(B,A,F,T8)
          & aa(set(A),$o,P,aa(set(B),set(A),image(B,A,F),T8)) ) ) ).

% ex_subset_image_inj
tff(fact_7898_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))),F: fun(nat,$o),Nb: nat,W: B] :
          ( word_T1870391057261780392y_misc(A,B,Of_word,Word_of,Signed_drop_bit,Of_nat,Nat_of,Of_int,Int_of,Of_integer,Integer_of,Size,Set_bits_aux)
         => ( aa(B,B,aa(nat,fun(B,B),aa(fun(nat,$o),fun(nat,fun(B,B)),Set_bits_aux,F),Nb),W) = $ite(
                Nb = zero_zero(nat),
                W,
                $let(
                  n: nat,
                  n:= aa(nat,nat,minus_minus(nat,Nb),one_one(nat)),
                  aa(B,B,aa(nat,fun(B,B),aa(fun(nat,$o),fun(nat,fun(B,B)),Set_bits_aux,F),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,F,n),one_one(B),zero_zero(B)))) ) ) ) ) ) ).

% word_type_copy_misc.set_bits_aux_code
tff(fact_7899_uint32__test__bit__def,axiom,
    ! [Xb: uint32,Nb: code_integer] :
      ( uint32_test_bit(Xb,Nb)
    <=> $ite(
          ( aa(code_integer,$o,ord_less(code_integer,Nb),zero_zero(code_integer))
          | aa(code_integer,$o,ord_less(code_integer,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),Nb) ),
          aa(code_integer,$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,$o)))),bit_se5641148757651400278ts_bit(uint32)),Xb),Nb),
          aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),Xb),aa(code_integer,nat,code_nat_of_integer,Nb)) ) ) ).

% uint32_test_bit_def
tff(fact_7900_nat__of__integer__numeral,axiom,
    ! [Nb: num] : aa(code_integer,nat,code_nat_of_integer,aa(num,code_integer,numeral_numeral(code_integer),Nb)) = aa(num,nat,numeral_numeral(nat),Nb) ).

% nat_of_integer_numeral
tff(fact_7901_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_7902_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_7903_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$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_7904_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_7905_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_7906_nat__of__integer__less__iff,axiom,
    ! [Xb: code_integer,Y: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),zero_zero(code_integer)),Xb)
     => ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$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_7907_image__atLeastZeroLessThan__integer,axiom,
    ! [U: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$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_7908_integer__set__bit__code,axiom,
    ! [Xb: code_integer,Nb: code_integer,B2: $o] :
      bits_integer_set_bit(Xb,Nb,(B2)) = $ite(
        aa(code_integer,$o,ord_less(code_integer,Nb),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),Nb),(B2)),
        $ite((B2),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,Nb)),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,Nb)),one_one(code_integer))))) ) ).

% integer_set_bit_code
tff(fact_7909_uint32__shiftr__def,axiom,
    ! [Xb: uint32,Nb: code_integer] :
      uint32_shiftr(Xb,Nb) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,Nb),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Nb) ),
        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),Nb),
        aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4197421643247451524op_bit(uint32),aa(code_integer,nat,code_nat_of_integer,Nb)),Xb) ) ).

% uint32_shiftr_def
tff(fact_7910_uint32__shiftl__def,axiom,
    ! [Xb: uint32,Nb: code_integer] :
      uint32_shiftl(Xb,Nb) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,Nb),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Nb) ),
        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),Nb),
        aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4730199178511100633sh_bit(uint32),aa(code_integer,nat,code_nat_of_integer,Nb)),Xb) ) ).

% uint32_shiftl_def
tff(fact_7911_uint32__set__bit__def,axiom,
    ! [Xb: uint32,Nb: code_integer,B2: $o] :
      uint32_set_bit(Xb,Nb,(B2)) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,Nb),zero_zero(code_integer))
        | aa(code_integer,$o,ord_less(code_integer,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),Nb) ),
        aa($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),Nb),(B2)),
        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,Nb)),(B2)) ) ).

% uint32_set_bit_def
tff(fact_7912_nat__of__integer__code,axiom,
    ! [K: code_integer] :
      aa(code_integer,nat,code_nat_of_integer,K) = $ite(aa(code_integer,$o,aa(code_integer,fun(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_ans(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_7913_nth_H__def,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [A2: array(A)] : array_nth2(A,A2) = comp(nat,heap_Time_Heap(A),code_integer,array_nth(A,A2),code_nat_of_integer) ) ).

% nth'_def
tff(fact_7914_fst__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_fst(code_integer,code_integer),code_divmod_integer(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),K),L) ).

% fst_divmod_integer
tff(fact_7915_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_snd(code_integer,code_integer),code_divmod_integer(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),K),L) ).

% snd_divmod_integer
tff(fact_7916_divmod__integer__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_integer(K,L) = 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),divide_divide(code_integer),K),L)),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),K),L)) ).

% divmod_integer_def
tff(fact_7917_num__of__integer__code,axiom,
    ! [K: code_integer] :
      code_num_of_integer(K) = $ite(aa(code_integer,$o,aa(code_integer,fun(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_ant(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_7918_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_anu(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_7919_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_7920_one__integer_Orep__eq,axiom,
    code_int_of_integer(one_one(code_integer)) = one_one(int) ).

% one_integer.rep_eq
tff(fact_7921_plus__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),plus_plus(code_integer),Xb),Xaa)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% plus_integer.rep_eq
tff(fact_7922_minus__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] : code_int_of_integer(aa(code_integer,code_integer,minus_minus(code_integer,Xb),Xaa)) = aa(int,int,minus_minus(int,code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% minus_integer.rep_eq
tff(fact_7923_times__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),times_times(code_integer),Xb),Xaa)) = aa(int,int,aa(int,fun(int,int),times_times(int),code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ).

% times_integer.rep_eq
tff(fact_7924_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_7925_integer__less__eq__iff,axiom,
    ! [K: code_integer,L: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),K),L)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),code_int_of_integer(K)),code_int_of_integer(L)) ) ).

% integer_less_eq_iff
tff(fact_7926_less__eq__integer_Orep__eq,axiom,
    ! [Xb: code_integer,Xaa: code_integer] :
      ( aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),Xb),Xaa)
    <=> aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),code_int_of_integer(Xb)),code_int_of_integer(Xaa)) ) ).

% less_eq_integer.rep_eq
tff(fact_7927_int__of__integer__pow,axiom,
    ! [Xb: code_integer,Nb: nat] : code_int_of_integer(aa(nat,code_integer,aa(code_integer,fun(nat,code_integer),power_power(code_integer),Xb),Nb)) = aa(nat,int,aa(int,fun(nat,int),power_power(int),code_int_of_integer(Xb)),Nb) ).

% int_of_integer_pow
tff(fact_7928_dup_Orep__eq,axiom,
    ! [Xb: code_integer] : code_int_of_integer(code_dup(Xb)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),code_int_of_integer(Xb)),code_int_of_integer(Xb)) ).

% dup.rep_eq
tff(fact_7929_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_7930_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_7931_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_7932_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_7933_uint32__shiftr__code,axiom,
    ! [W: uint32,Nb: code_integer] :
      rep_uint322(uint32_shiftr(W,Nb)) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,Nb),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Nb) ),
        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),Nb)),
        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,Nb)),rep_uint322(W)) ) ).

% uint32_shiftr_code
tff(fact_7934_uint32__shiftl__code,axiom,
    ! [W: uint32,Nb: code_integer] :
      rep_uint322(uint32_shiftl(W,Nb)) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,Nb),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Nb) ),
        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),Nb)),
        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,Nb)),rep_uint322(W)) ) ).

% uint32_shiftl_code
tff(fact_7935_uint32_Oword__of__numeral,axiom,
    ! [Nb: num] : rep_uint322(aa(num,uint32,numeral_numeral(uint32),Nb)) = 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))))))),Nb) ).

% uint32.word_of_numeral
tff(fact_7936_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_7937_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_7938_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_7939_times__uint32_Orep__eq,axiom,
    ! [Xb: uint32,Xaa: uint32] : rep_uint322(aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),Xb),Xaa)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),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))))))),times_times(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint322(Xb)),rep_uint322(Xaa)) ).

% times_uint32.rep_eq
tff(fact_7940_uint32_Oword__of__mult,axiom,
    ! [P2: uint32,Q2: uint32] : rep_uint322(aa(uint32,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),P2),Q2)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),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))))))),times_times(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint322(P2)),rep_uint322(Q2)) ).

% uint32.word_of_mult
tff(fact_7941_divide__uint32_Orep__eq,axiom,
    ! [Xb: uint32,Xaa: uint32] : rep_uint322(aa(uint32,uint32,aa(uint32,fun(uint32,uint32),divide_divide(uint32),Xb),Xaa)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),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))))))),divide_divide(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint322(Xb)),rep_uint322(Xaa)) ).

% divide_uint32.rep_eq
tff(fact_7942_uint32_Oword__of__div,axiom,
    ! [P2: uint32,Q2: uint32] : rep_uint322(aa(uint32,uint32,aa(uint32,fun(uint32,uint32),divide_divide(uint32),P2),Q2)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),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))))))),divide_divide(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint322(P2)),rep_uint322(Q2)) ).

% uint32.word_of_div
tff(fact_7943_one__uint32_Orep__eq,axiom,
    rep_uint322(one_one(uint32)) = one_one(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))) ).

% one_uint32.rep_eq
tff(fact_7944_uint32_Oword__of__diff,axiom,
    ! [P2: uint32,Q2: uint32] : rep_uint322(aa(uint32,uint32,minus_minus(uint32,P2),Q2)) = 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)))))),minus_minus(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322(P2)),rep_uint322(Q2)) ).

% uint32.word_of_diff
tff(fact_7945_minus__uint32_Orep__eq,axiom,
    ! [Xb: uint32,Xaa: uint32] : rep_uint322(aa(uint32,uint32,minus_minus(uint32,Xb),Xaa)) = 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)))))),minus_minus(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),rep_uint322(Xb)),rep_uint322(Xaa)) ).

% minus_uint32.rep_eq
tff(fact_7946_plus__uint32_Orep__eq,axiom,
    ! [Xb: uint32,Xaa: uint32] : rep_uint322(aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),Xb),Xaa)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),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))))))),plus_plus(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint322(Xb)),rep_uint322(Xaa)) ).

% plus_uint32.rep_eq
tff(fact_7947_uint32_Oword__of__add,axiom,
    ! [P2: uint32,Q2: uint32] : rep_uint322(aa(uint32,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),P2),Q2)) = 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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),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))))))),plus_plus(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint322(P2)),rep_uint322(Q2)) ).

% uint32.word_of_add
tff(fact_7948_uint32_Oword__of__power,axiom,
    ! [P2: uint32,Nb: nat] : rep_uint322(aa(nat,uint32,aa(uint32,fun(nat,uint32),power_power(uint32),P2),Nb)) = aa(nat,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,word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),power_power(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint322(P2)),Nb) ).

% uint32.word_of_power
tff(fact_7949_less__eq__uint32_Orep__eq,axiom,
    ! [Xb: uint32,Xaa: uint32] :
      ( aa(uint32,$o,aa(uint32,fun(uint32,$o),ord_less_eq(uint32),Xb),Xaa)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o),ord_less_eq(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint322(Xb)),rep_uint322(Xaa)) ) ).

% less_eq_uint32.rep_eq
tff(fact_7950_uint32_Oless__eq__iff__word__of,axiom,
    ! [P2: uint32,Q2: uint32] :
      ( aa(uint32,$o,aa(uint32,fun(uint32,$o),ord_less_eq(uint32),P2),Q2)
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o),ord_less_eq(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),rep_uint322(P2)),rep_uint322(Q2)) ) ).

% uint32.less_eq_iff_word_of
tff(fact_7951_uint32__test__bit__code,axiom,
    ! [W: uint32,Nb: code_integer] :
      ( uint32_test_bit(W,Nb)
    <=> $ite(
          ( aa(code_integer,$o,ord_less(code_integer,Nb),zero_zero(code_integer))
          | aa(code_integer,$o,ord_less(code_integer,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),Nb) ),
          aa(code_integer,$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,$o)))),bit_se5641148757651400278ts_bit(uint32)),W),Nb),
          aa(nat,$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,Nb)) ) ) ).

% uint32_test_bit_code
tff(fact_7952_uint32__set__bit__code,axiom,
    ! [W: uint32,Nb: code_integer,B2: $o] :
      rep_uint322(uint32_set_bit(W,Nb,(B2))) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,Nb),zero_zero(code_integer))
        | aa(code_integer,$o,ord_less(code_integer,aa(num,code_integer,numeral_numeral(code_integer),aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,aa(num,num,bit1,one2)))))),Nb) ),
        rep_uint322(aa($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),Nb),(B2))),
        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,Nb)),(B2)) ) ).

% uint32_set_bit_code
tff(fact_7953_lift__def,axiom,
    ! [B: $tType,A: $tType,F: fun(A,B)] : heap_Time_lift(A,B,F) = comp(B,heap_Time_Heap(B),A,heap_Time_return(B),F) ).

% lift_def
tff(fact_7954_set__bit__uint32__code,axiom,
    ! [Xb: uint32,Nb: nat,B2: $o] :
      aa($o,uint32,aa(nat,fun($o,uint32),aa(uint32,fun(nat,fun($o,uint32)),generi7602027413899671122et_bit(uint32),Xb),Nb),(B2)) = $ite(aa(nat,$o,ord_less(nat,Nb),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),uint32_set_bit(Xb,code_integer_of_nat(Nb),(B2)),Xb) ).

% set_bit_uint32_code
tff(fact_7955_lift__collapse,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),Xb: B] : aa(B,heap_Time_Heap(A),heap_Time_lift(B,A,F),Xb) = aa(A,heap_Time_Heap(A),heap_Time_return(A),aa(B,A,F,Xb)) ).

% lift_collapse
tff(fact_7956_integer__of__nat__1,axiom,
    code_integer_of_nat(one_one(nat)) = one_one(code_integer) ).

% integer_of_nat_1
tff(fact_7957_one__uint32_Orsp,axiom,
    one_one(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))) = one_one(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))) ).

% one_uint32.rsp
tff(fact_7958_integer__of__nat__numeral,axiom,
    ! [Nb: num] : code_integer_of_nat(aa(num,nat,numeral_numeral(nat),Nb)) = aa(num,code_integer,numeral_numeral(code_integer),Nb) ).

% integer_of_nat_numeral
tff(fact_7959_bind__lift,axiom,
    ! [A: $tType,B: $tType,F: heap_Time_Heap(B),G: fun(B,A)] : heap_Time_bind(B,A,F,heap_Time_lift(B,A,G)) = heap_Time_bind(B,A,F,aTP_Lamp_anv(fun(B,A),fun(B,heap_Time_Heap(A)),G)) ).

% bind_lift
tff(fact_7960_test__bit__uint32__code,axiom,
    ! [Xb: uint32,Nb: nat] :
      ( aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),Xb),Nb)
    <=> ( aa(nat,$o,ord_less(nat,Nb),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2)))))))
        & uint32_test_bit(Xb,code_integer_of_nat(Nb)) ) ) ).

% test_bit_uint32_code
tff(fact_7961_shiftl__uint32__code,axiom,
    ! [Nb: nat,Xb: uint32] :
      aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4730199178511100633sh_bit(uint32),Nb),Xb) = $ite(aa(nat,$o,ord_less(nat,Nb),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),uint32_shiftl(Xb,code_integer_of_nat(Nb)),zero_zero(uint32)) ).

% shiftl_uint32_code
tff(fact_7962_shiftr__uint32__code,axiom,
    ! [Nb: nat,Xb: uint32] :
      aa(uint32,uint32,aa(nat,fun(uint32,uint32),bit_se4197421643247451524op_bit(uint32),Nb),Xb) = $ite(aa(nat,$o,ord_less(nat,Nb),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),uint32_shiftr(Xb,code_integer_of_nat(Nb)),zero_zero(uint32)) ).

% shiftr_uint32_code
tff(fact_7963_uint32__divmod__code,axiom,
    ! [Xb: uint32,Y: uint32] :
      uint32_divmod(Xb,Y) = $ite(
        aa(uint32,$o,aa(uint32,fun(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,aa(uint32,fun(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_7964_uint32_Oset__bits__aux__code,axiom,
    ! [F: fun(nat,$o),Nb: nat,W: uint32] :
      set_bits_aux_uint32(F,Nb,W) = $ite(
        Nb = zero_zero(nat),
        W,
        $let(
          n: nat,
          n:= aa(nat,nat,minus_minus(nat,Nb),one_one(nat)),
          set_bits_aux_uint32(F,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,F,n),one_one(uint32),zero_zero(uint32)))) ) ) ).

% uint32.set_bits_aux_code
tff(fact_7965_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_7966_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_7967_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_7968_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_7969_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_7970_sshiftr__uint32__code,axiom,
    ! [Nb: nat,Xb: uint32] :
      aa(uint32,uint32,aa(nat,fun(uint32,uint32),signed489701013188660438uint32,Nb),Xb) = $ite(
        aa(nat,$o,ord_less(nat,Nb),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(bit0(bit0(one2))))))),
        uint32_sshiftr(Xb,code_integer_of_nat(Nb)),
        $ite(aa(nat,$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_7971_uint32__div__def,axiom,
    ! [Xb: uint32,Y: uint32] : uint32_div(Xb,Y) = aa(product_prod(uint32,uint32),uint32,product_fst(uint32,uint32),uint32_divmod(Xb,Y)) ).

% uint32_div_def
tff(fact_7972_uint32__sshiftr__def,axiom,
    ! [Xb: uint32,Nb: code_integer] :
      uint32_sshiftr(Xb,Nb) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,Nb),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Nb) ),
        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),Nb),Xb),
        aa(uint32,uint32,aa(nat,fun(uint32,uint32),signed489701013188660438uint32,aa(code_integer,nat,code_nat_of_integer,Nb)),Xb) ) ).

% uint32_sshiftr_def
tff(fact_7973_uint32__sshiftr__code,axiom,
    ! [W: uint32,Nb: code_integer] :
      rep_uint322(uint32_sshiftr(W,Nb)) = $ite(
        ( aa(code_integer,$o,ord_less(code_integer,Nb),zero_zero(code_integer))
        | aa(code_integer,$o,aa(code_integer,fun(code_integer,$o),ord_less_eq(code_integer),aa(num,code_integer,numeral_numeral(code_integer),bit0(bit0(bit0(bit0(bit0(one2))))))),Nb) ),
        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),Nb),W)),
        signed_drop_bit(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))),aa(code_integer,nat,code_nat_of_integer,Nb),rep_uint322(W)) ) ).

% uint32_sshiftr_code
tff(fact_7974_integer__of__uint32__code,axiom,
    ! [Nb: uint32] :
      aa(uint32,code_integer,integer_of_uint32,Nb) = $ite(aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),Nb),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),Nb),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(Nb)) ).

% integer_of_uint32_code
tff(fact_7975_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_7976_less__eq__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,$o,aa(uint32,fun(uint32,$o),ord_less_eq(uint32),abs_uint32(Xaa)),abs_uint32(Xb))
    <=> aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o,aa(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),fun(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),$o),ord_less_eq(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),Xaa),Xb) ) ).

% less_eq_uint32.abs_eq
tff(fact_7977_plus__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,uint32,aa(uint32,fun(uint32,uint32),plus_plus(uint32),abs_uint32(Xaa)),abs_uint32(Xb)) = abs_uint32(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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),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))))))),plus_plus(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),Xaa),Xb)) ).

% plus_uint32.abs_eq
tff(fact_7978_minus__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,uint32,minus_minus(uint32,abs_uint32(Xaa)),abs_uint32(Xb)) = abs_uint32(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)))))),minus_minus(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),Xaa),Xb)) ).

% minus_uint32.abs_eq
tff(fact_7979_one__uint32__def,axiom,
    one_one(uint32) = abs_uint32(one_one(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))))) ).

% one_uint32_def
tff(fact_7980_divide__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,uint32,aa(uint32,fun(uint32,uint32),divide_divide(uint32),abs_uint32(Xaa)),abs_uint32(Xb)) = abs_uint32(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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),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))))))),divide_divide(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),Xaa),Xb)) ).

% divide_uint32.abs_eq
tff(fact_7981_times__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,uint32,aa(uint32,fun(uint32,uint32),times_times(uint32),abs_uint32(Xaa)),abs_uint32(Xb)) = abs_uint32(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(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1)))))),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))))))),times_times(word(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_bit0(numeral_num1))))))),Xaa),Xb)) ).

% times_uint32.abs_eq
tff(fact_7982_uint32_Oof__word__numeral,axiom,
    ! [Nb: num] : abs_uint32(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))))))),Nb)) = aa(num,uint32,numeral_numeral(uint32),Nb) ).

% uint32.of_word_numeral
tff(fact_7983_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_7984_integer__of__uint32__signed__def,axiom,
    ! [Nb: uint32] :
      intege5370686899274169573signed(Nb) = $ite(aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),Nb),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),Nb),aa(uint32,code_integer,integer_of_uint32,Nb)) ).

% integer_of_uint32_signed_def
tff(fact_7985_integer__of__uint32__signed__code,axiom,
    ! [Nb: uint32] :
      intege5370686899274169573signed(Nb) = $ite(aa(nat,$o,aa(uint32,fun(nat,$o),bit_se5641148757651400278ts_bit(uint32),Nb),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),Nb),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(Nb)))) ).

% integer_of_uint32_signed_code
tff(fact_7986_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_anw(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_7987_apsnd__conv,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: 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),F),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,F,Y)) ).

% apsnd_conv
tff(fact_7988_fst__apsnd,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: 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),F),Xb)) = aa(product_prod(A,C),A,product_fst(A,C),Xb) ).

% fst_apsnd
tff(fact_7989_apsnd__eq__conv,axiom,
    ! [B: $tType,C: $tType,A: $tType,F: 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),F),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,F,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_7990_snd__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: 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),F),Xb)) = aa(C,A,F,aa(product_prod(B,C),C,product_snd(B,C),Xb)) ).

% snd_apsnd
tff(fact_7991_fst__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: 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),F)) = product_fst(A,B) ).

% fst_comp_apsnd
tff(fact_7992_snd__comp__apsnd,axiom,
    ! [C: $tType,B: $tType,A: $tType,F: 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),F)) = comp(B,C,product_prod(A,B),F,product_snd(A,B)) ).

% snd_comp_apsnd
tff(fact_7993_inj__apsnd,axiom,
    ! [A: $tType,C: $tType,B: $tType,F: 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),F),top_top(set(product_prod(A,B))))
    <=> inj_on(B,C,F,top_top(set(B))) ) ).

% inj_apsnd
tff(fact_7994_fst__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_fst(code_integer,code_integer),code_divmod_abs(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),divide_divide(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L)) ).

% fst_divmod_abs
tff(fact_7995_snd__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] : aa(product_prod(code_integer,code_integer),code_integer,product_snd(code_integer,code_integer),code_divmod_abs(K,L)) = aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L)) ).

% snd_divmod_abs
tff(fact_7996_apsnd__compose,axiom,
    ! [C: $tType,B: $tType,D: $tType,A: $tType,F: fun(C,B),G: fun(D,C),Xb: product_prod(A,D)] : 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),F),aa(product_prod(A,D),product_prod(A,C),aa(fun(D,C),fun(product_prod(A,D),product_prod(A,C)),product_apsnd(D,C,A),G),Xb)) = aa(product_prod(A,D),product_prod(A,B),aa(fun(D,B),fun(product_prod(A,D),product_prod(A,B)),product_apsnd(D,B,A),comp(C,B,D,F,G)),Xb) ).

% apsnd_compose
tff(fact_7997_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_7998_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)),aa(code_integer,code_integer,abs_abs(code_integer),J)) ).

% divmod_abs_code(5)
tff(fact_7999_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_8000_divmod__abs__def,axiom,
    ! [K: code_integer,L: code_integer] : code_divmod_abs(K,L) = 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),divide_divide(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L))),aa(code_integer,code_integer,aa(code_integer,fun(code_integer,code_integer),modulo_modulo(code_integer),aa(code_integer,code_integer,abs_abs(code_integer),K)),aa(code_integer,code_integer,abs_abs(code_integer),L))) ).

% divmod_abs_def
tff(fact_8001_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_anx(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_any(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_8002_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,$o),aTP_Lamp_anz(code_integer,fun(code_integer,fun(code_integer,product_prod(code_integer,$o))),K)),code_divmod_abs(K,aa(num,code_integer,numeral_numeral(code_integer),bit0(one2))))) ).

% bit_cut_integer_code
tff(fact_8003_set__bits__int__def,axiom,
    ! [F: fun(nat,$o)] :
      bit_bi4170147762399595738t_bits(int,F) = $ite(
        ? [N6: nat] :
        ! [M7: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),M7)
         => ( aa(nat,$o,F,M7) = aa(nat,$o,F,N6) ) ),
        $let(
          n2: nat,
          n2:= ord_Least(nat,aTP_Lamp_aoa(fun(nat,$o),fun(nat,$o),F)),
          aa(int,int,bit_ri4674362597316999326ke_bit(int,n2),groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),aa(list(nat),list($o),map(nat,$o,F),upt(zero_zero(nat),aa(nat,nat,suc,n2))))) ),
        zero_zero(int) ) ).

% set_bits_int_def
tff(fact_8004_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,$o),aTP_Lamp_aob(fun(A,$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_aja(fun(B,fun(A,$o)),fun(A,fun(B,$o)),Q),Y3))) ) ) ) ).

% finite_Collect_bounded_ex
tff(fact_8005_eq__or__mem__image__simp,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A2: B,B5: set(B)] : collect(A,aa(set(B),fun(A,$o),aa(B,fun(set(B),fun(A,$o)),aTP_Lamp_aoc(fun(B,A),fun(B,fun(set(B),fun(A,$o))),F),A2),B5)) = aa(set(A),set(A),insert(A,aa(B,A,F,A2)),collect(A,aa(set(B),fun(A,$o),aTP_Lamp_aod(fun(B,A),fun(set(B),fun(A,$o)),F),B5))) ).

% eq_or_mem_image_simp
tff(fact_8006_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_8007_Least__le,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),K: A] :
          ( aa(A,$o,P,K)
         => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P)),K) ) ) ).

% Least_le
tff(fact_8008_LeastI2__wellorder__ex,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),Q: fun(A,$o)] :
          ( ? [X_12: A] : aa(A,$o,P,X_12)
         => ( ! [A4: A] :
                ( aa(A,$o,P,A4)
               => ( ! [B7: A] :
                      ( aa(A,$o,P,B7)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B7) )
                 => aa(A,$o,Q,A4) ) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2_wellorder_ex
tff(fact_8009_LeastI2__wellorder,axiom,
    ! [A: $tType] :
      ( wellorder(A)
     => ! [P: fun(A,$o),A2: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,A2)
         => ( ! [A4: A] :
                ( aa(A,$o,P,A4)
               => ( ! [B7: A] :
                      ( aa(A,$o,P,B7)
                     => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),A4),B7) )
                 => aa(A,$o,Q,A4) ) )
           => aa(A,$o,Q,ord_Least(A,P)) ) ) ) ).

% LeastI2_wellorder
tff(fact_8010_Least__equality,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y5: A] :
                ( aa(A,$o,P,Y5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y5) )
           => ( ord_Least(A,P) = Xb ) ) ) ) ).

% Least_equality
tff(fact_8011_LeastI2__order,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Xb: A,Q: fun(A,$o)] :
          ( aa(A,$o,P,Xb)
         => ( ! [Y5: A] :
                ( aa(A,$o,P,Y5)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Xb),Y5) )
           => ( ! [X2: A] :
                  ( aa(A,$o,P,X2)
                 => ( ! [Y4: A] :
                        ( aa(A,$o,P,Y4)
                       => aa(A,$o,aa(A,fun(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_8012_Least1__le,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o),Z: A] :
          ( ? [X3: A] :
              ( aa(A,$o,P,X3)
              & ! [Y5: A] :
                  ( aa(A,$o,P,Y5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y5) )
              & ! [Y5: A] :
                  ( ( aa(A,$o,P,Y5)
                    & ! [Ya2: A] :
                        ( aa(A,$o,P,Ya2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y5),Ya2) ) )
                 => ( Y5 = X3 ) ) )
         => ( aa(A,$o,P,Z)
           => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),ord_Least(A,P)),Z) ) ) ) ).

% Least1_le
tff(fact_8013_Least1I,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [P: fun(A,$o)] :
          ( ? [X3: A] :
              ( aa(A,$o,P,X3)
              & ! [Y5: A] :
                  ( aa(A,$o,P,Y5)
                 => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),X3),Y5) )
              & ! [Y5: A] :
                  ( ( aa(A,$o,P,Y5)
                    & ! [Ya2: A] :
                        ( aa(A,$o,P,Ya2)
                       => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y5),Ya2) ) )
                 => ( Y5 = X3 ) ) )
         => aa(A,$o,P,ord_Least(A,P)) ) ) ).

% Least1I
tff(fact_8014_finite__image__set,axiom,
    ! [B: $tType,A: $tType,P: fun(A,$o),F: fun(A,B)] :
      ( finite_finite2(A,collect(A,P))
     => finite_finite2(B,collect(B,aa(fun(A,B),fun(B,$o),aTP_Lamp_aoe(fun(A,$o),fun(fun(A,B),fun(B,$o)),P),F))) ) ).

% finite_image_set
tff(fact_8015_finite__image__set2,axiom,
    ! [C: $tType,B: $tType,A: $tType,P: fun(A,$o),Q: fun(B,$o),F: fun(A,fun(B,C))] :
      ( finite_finite2(A,collect(A,P))
     => ( finite_finite2(B,collect(B,Q))
       => finite_finite2(C,collect(C,aa(fun(A,fun(B,C)),fun(C,$o),aa(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o)),aTP_Lamp_aof(fun(A,$o),fun(fun(B,$o),fun(fun(A,fun(B,C)),fun(C,$o))),P),Q),F))) ) ) ).

% finite_image_set2
tff(fact_8016_set__Cons__def,axiom,
    ! [A: $tType,A3: set(A),XS: set(list(A))] : set_Cons(A,A3,XS) = collect(list(A),aa(set(list(A)),fun(list(A),$o),aTP_Lamp_aog(set(A),fun(set(list(A)),fun(list(A),$o)),A3),XS)) ).

% set_Cons_def
tff(fact_8017_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_8018_some__theI,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)
     => ( ! [B1: B,B22: B] :
            ( ? [A4: A] : aa(B,$o,aa(A,fun(B,$o),P,A4),B1)
           => ( ? [A4: A] : aa(B,$o,aa(A,fun(B,$o),P,A4),B22)
             => ( B1 = B22 ) ) )
       => aa(B,$o,aa(A,fun(B,$o),P,fChoice(A,aTP_Lamp_aoh(fun(A,fun(B,$o)),fun(A,$o),P))),the(B,aTP_Lamp_aoi(fun(A,fun(B,$o)),fun(B,$o),P))) ) ) ).

% some_theI
tff(fact_8019_ran__def,axiom,
    ! [A: $tType,B: $tType,M: fun(B,option(A))] : ran(B,A,M) = collect(A,aTP_Lamp_aoj(fun(B,option(A)),fun(A,$o),M)) ).

% ran_def
tff(fact_8020_full__SetCompr__eq,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A)] : collect(A,aTP_Lamp_aok(fun(B,A),fun(A,$o),F)) = aa(set(B),set(A),image(B,A,F),top_top(set(B))) ).

% full_SetCompr_eq
tff(fact_8021_Setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A3: set(B)] : collect(A,aa(set(B),fun(A,$o),aTP_Lamp_aod(fun(B,A),fun(set(B),fun(A,$o)),F),A3)) = aa(set(B),set(A),image(B,A,F),A3) ).

% Setcompr_eq_image
tff(fact_8022_setcompr__eq__image,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),P: fun(B,$o)] : collect(A,aa(fun(B,$o),fun(A,$o),aTP_Lamp_aol(fun(B,A),fun(fun(B,$o),fun(A,$o)),F),P)) = aa(set(B),set(A),image(B,A,F),collect(B,P)) ).

% setcompr_eq_image
tff(fact_8023_Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S2: set(A),P: fun(A,$o)] : bleast(A,S2,P) = ord_Least(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aom(set(A),fun(fun(A,$o),fun(A,$o)),S2),P)) ) ).

% Bleast_def
tff(fact_8024_abort__Bleast__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [S2: set(A),P: fun(A,$o)] : abort_Bleast(A,S2,P) = ord_Least(A,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aom(set(A),fun(fun(A,$o),fun(A,$o)),S2),P)) ) ).

% abort_Bleast_def
tff(fact_8025_Least__Suc,axiom,
    ! [P: fun(nat,$o),Nb: nat] :
      ( aa(nat,$o,P,Nb)
     => ( ~ aa(nat,$o,P,zero_zero(nat))
       => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,aTP_Lamp_amq(fun(nat,$o),fun(nat,$o),P))) ) ) ) ).

% Least_Suc
tff(fact_8026_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_8027_Least__Suc2,axiom,
    ! [P: fun(nat,$o),Nb: nat,Q: fun(nat,$o),M: nat] :
      ( aa(nat,$o,P,Nb)
     => ( aa(nat,$o,Q,M)
       => ( ~ aa(nat,$o,P,zero_zero(nat))
         => ( ! [K3: nat] :
                ( aa(nat,$o,P,aa(nat,nat,suc,K3))
              <=> aa(nat,$o,Q,K3) )
           => ( ord_Least(nat,P) = aa(nat,nat,suc,ord_Least(nat,Q)) ) ) ) ) ) ).

% Least_Suc2
tff(fact_8028_fs__contract,axiom,
    ! [A: $tType,B: $tType,C: $tType,F: fun(A,fun(B,C)),S2: set(C)] : aa(set(product_prod(A,B)),set(A),image(product_prod(A,B),A,product_fst(A,B)),collect(product_prod(A,B),aa(set(C),fun(product_prod(A,B),$o),aTP_Lamp_aon(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),$o)),F),S2))) = collect(A,aa(set(C),fun(A,$o),aTP_Lamp_aoo(fun(A,fun(B,C)),fun(set(C),fun(A,$o)),F),S2)) ).

% fs_contract
tff(fact_8029_set__map__filter,axiom,
    ! [A: $tType,B: $tType,G: fun(B,option(A)),Xs: list(B)] : aa(list(A),set(A),set2(A),map_filter(B,A,G,Xs)) = collect(A,aa(list(B),fun(A,$o),aTP_Lamp_aop(fun(B,option(A)),fun(list(B),fun(A,$o)),G),Xs)) ).

% set_map_filter
tff(fact_8030_set__conv__nth,axiom,
    ! [A: $tType,Xs: list(A)] : aa(list(A),set(A),set2(A),Xs) = collect(A,aTP_Lamp_aoq(list(A),fun(A,$o),Xs)) ).

% set_conv_nth
tff(fact_8031_Least__def,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [P: fun(A,$o)] : ord_Least(A,P) = the(A,aTP_Lamp_aor(fun(A,$o),fun(A,$o),P)) ) ).

% Least_def
tff(fact_8032_set__bits__int__unfold_H,axiom,
    ! [F: fun(nat,$o)] :
      bit_bi4170147762399595738t_bits(int,F) = $ite(
        ? [N6: nat] :
        ! [N11: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N11)
         => ~ aa(nat,$o,F,N11) ),
        groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),aa(list(nat),list($o),map(nat,$o,F),upt(zero_zero(nat),ord_Least(nat,aTP_Lamp_aos(fun(nat,$o),fun(nat,$o),F))))),
        $ite(
          ? [N6: nat] :
          ! [N11: nat] :
            ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),N6),N11)
           => aa(nat,$o,F,N11) ),
          $let(
            n2: nat,
            n2:= ord_Least(nat,aTP_Lamp_aot(fun(nat,$o),fun(nat,$o),F)),
            aa(int,int,bit_ri4674362597316999326ke_bit(int,n2),groups4207007520872428315er_sum($o,int,zero_neq_one_of_bool(int),aa(num,int,numeral_numeral(int),bit0(one2)),aa(list($o),list($o),append($o,aa(list(nat),list($o),map(nat,$o,F),upt(zero_zero(nat),n2))),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),$true),nil($o))))) ),
          zero_zero(int) ) ) ).

% set_bits_int_unfold'
tff(fact_8033_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_8034_pairself__image__eq,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),P: fun(B,fun(B,$o))] : aa(set(product_prod(B,B)),set(product_prod(A,A)),image(product_prod(B,B),product_prod(A,A),pairself(B,A,F)),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),$o),aTP_Lamp_aou(fun(B,A),fun(fun(B,fun(B,$o)),fun(product_prod(A,A),$o)),F),P)) ).

% pairself_image_eq
tff(fact_8035_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xb,Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa = one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg2,TreeList,Summary) )
             => ( ( Deg2 = Xaa )
                & $let(
                    n2: nat,
                    n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                    $let(
                      m: nat,
                      m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                      ( ! [X: vEBT_VEBT] :
                          ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                         => vEBT_VEBT_valid(X,n2) )
                      & vEBT_VEBT_valid(Summary,m)
                      & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                      & case_option($o,product_prod(nat,nat),
                          ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                          & ! [X: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X7) ) ),
                          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,$o))))),aTP_Lamp_aov(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList),Summary),n2),m)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
tff(fact_8036_valid__eq,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(Ta,D2)
    <=> vEBT_invar_vebt(Ta,D2) ) ).

% valid_eq
tff(fact_8037_valid__eq1,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( vEBT_invar_vebt(Ta,D2)
     => vEBT_VEBT_valid(Ta,D2) ) ).

% valid_eq1
tff(fact_8038_valid__eq2,axiom,
    ! [Ta: vEBT_VEBT,D2: nat] :
      ( vEBT_VEBT_valid(Ta,D2)
     => vEBT_invar_vebt(Ta,D2) ) ).

% valid_eq2
tff(fact_8039_Ball__comp__iff,axiom,
    ! [A: $tType,B: $tType,C: $tType,A3: fun(B,set(C)),F: fun(C,$o),G: fun(A,B),X3: A] :
      ( aa(A,$o,comp(B,$o,A,aa(fun(C,$o),fun(B,$o),aTP_Lamp_aow(fun(B,set(C)),fun(fun(C,$o),fun(B,$o)),A3),F),G),X3)
    <=> ! [Xa3: C] :
          ( member(C,Xa3,aa(A,set(C),comp(B,set(C),A,A3,G),X3))
         => aa(C,$o,F,Xa3) ) ) ).

% Ball_comp_iff
tff(fact_8040_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 )
     => ~ ! [A4: B,B3: B] :
            ( ( Xaa = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A4),B3) )
           => ( Y != aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Xb,A4)),aa(B,A,Xb,B3)) ) ) ) ).

% pairself.elims
tff(fact_8041_pairself_Osimps,axiom,
    ! [A: $tType,B: $tType,F: fun(B,A),A2: B,B2: B] : aa(product_prod(B,B),product_prod(A,A),pairself(B,A,F),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A2),B2)) = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,F,A2)),aa(B,A,F,B2)) ).

% pairself.simps
tff(fact_8042_Ball__fold,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] :
      ( finite_finite2(A,A3)
     => ( ! [X: A] :
            ( member(A,X,A3)
           => aa(A,$o,P,X) )
      <=> finite_fold(A,$o,aTP_Lamp_aox(fun(A,$o),fun(A,fun($o,$o)),P),$true,A3) ) ) ).

% Ball_fold
tff(fact_8043_in__set__simps_I4_J,axiom,
    ! [A: $tType,P: fun(A,$o),X3: A] :
      ( member(A,X3,aa(list(A),set(A),set2(A),nil(A)))
     => aa(A,$o,P,X3) ) ).

% in_set_simps(4)
tff(fact_8044_admissible__ball,axiom,
    ! [A: $tType,B: $tType,A3: set(A),Lub: fun(set(B),B),Ord: fun(B,fun(B,$o)),P: fun(B,fun(A,$o))] :
      ( ! [Y5: A] :
          ( member(A,Y5,A3)
         => comple1908693960933563346ssible(B,Lub,Ord,aa(A,fun(B,$o),aTP_Lamp_aja(fun(B,fun(A,$o)),fun(A,fun(B,$o)),P),Y5)) )
     => comple1908693960933563346ssible(B,Lub,Ord,aa(fun(B,fun(A,$o)),fun(B,$o),aTP_Lamp_aoy(set(A),fun(fun(B,fun(A,$o)),fun(B,$o)),A3),P)) ) ).

% admissible_ball
tff(fact_8045_in__set__simps_I6_J,axiom,
    ! [A: $tType,Xb: A,Y: A,Zs: list(A),P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs))))
         => aa(A,$o,P,X) )
    <=> ( aa(A,$o,P,Xb)
        & ! [X: A] :
            ( member(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Zs)))
           => aa(A,$o,P,X) ) ) ) ).

% in_set_simps(6)
tff(fact_8046_in__set__simps_I5_J,axiom,
    ! [A: $tType,Xb: A,P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,aa(list(A),set(A),set2(A),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Xb),nil(A))))
         => aa(A,$o,P,X) )
    <=> aa(A,$o,P,Xb) ) ).

% in_set_simps(5)
tff(fact_8047_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D2: nat] :
      ( vEBT_VEBT_valid(vEBT_Leaf((Uu),(Uv)),D2)
    <=> ( D2 = one_one(nat) ) ) ).

% VEBT_internal.valid'.simps(1)
tff(fact_8048_Ball__Collect,axiom,
    ! [A: $tType,A3: set(A),P: fun(A,$o)] :
      ( ! [X: A] :
          ( member(A,X,A3)
         => aa(A,$o,P,X) )
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),A3),collect(A,P)) ) ).

% Ball_Collect
tff(fact_8049_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option(product_prod(nat,nat)),Deg: nat,TreeLista: list(vEBT_VEBT),Summarya: vEBT_VEBT,Deg4: nat] :
      ( vEBT_VEBT_valid(vEBT_Node(Mima2,Deg,TreeLista,Summarya),Deg4)
    <=> ( ( Deg = Deg4 )
        & $let(
            n2: nat,
            n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg),aa(num,nat,numeral_numeral(nat),bit0(one2))),
            $let(
              m: nat,
              m:= aa(nat,nat,minus_minus(nat,Deg),n2),
              ( ! [X: vEBT_VEBT] :
                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                 => vEBT_VEBT_valid(X,n2) )
              & vEBT_VEBT_valid(Summarya,m)
              & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeLista) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
              & case_option($o,product_prod(nat,nat),
                  ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summarya),X7)
                  & ! [X: vEBT_VEBT] :
                      ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeLista))
                     => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X7) ) ),
                  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,$o))))),aTP_Lamp_aov(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg),TreeLista),Summarya),n2),m)),Mima2) ) ) ) ) ) ).

% VEBT_internal.valid'.simps(2)
tff(fact_8050_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_valid(Xb,Xaa)
      <=> (Y) )
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( (Y)
          <=> ( Xaa != one_one(nat) ) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg2,TreeList,Summary) )
             => ( (Y)
              <=> ~ ( ( Deg2 = Xaa )
                    & $let(
                        n2: nat,
                        n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        $let(
                          m: nat,
                          m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                          ( ! [X: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => vEBT_VEBT_valid(X,n2) )
                          & vEBT_VEBT_valid(Summary,m)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                              & ! [X: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X7) ) ),
                              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,$o))))),aTP_Lamp_aov(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList),Summary),n2),m)),Mima) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
tff(fact_8051_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xb,Xaa)
     => ( ( ? [Uu2: $o,Uv2: $o] : Xb = vEBT_Leaf((Uu2),(Uv2))
         => ( Xaa != one_one(nat) ) )
       => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
              ( ( Xb = vEBT_Node(Mima,Deg2,TreeList,Summary) )
             => ~ ( ( Deg2 = Xaa )
                  & $let(
                      n2: nat,
                      n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                      $let(
                        m: nat,
                        m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                        ( ! [X: vEBT_VEBT] :
                            ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                           => vEBT_VEBT_valid(X,n2) )
                        & vEBT_VEBT_valid(Summary,m)
                        & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                        & case_option($o,product_prod(nat,nat),
                            ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                            & ! [X: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X7) ) ),
                            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,$o))))),aTP_Lamp_aov(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList),Summary),n2),m)),Mima) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
tff(fact_8052_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat,Y: $o] :
      ( ( vEBT_VEBT_valid(Xb,Xaa)
      <=> (Y) )
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( ( (Y)
                <=> ( Xaa = one_one(nat) ) )
               => ~ aa(product_prod(vEBT_VEBT,nat),$o,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),(Uv2))),Xaa)) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg2,TreeList,Summary) )
               => ( ( (Y)
                  <=> ( ( Deg2 = Xaa )
                      & $let(
                          n2: nat,
                          n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $let(
                            m: nat,
                            m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                            ( ! [X: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => vEBT_VEBT_valid(X,n2) )
                            & vEBT_VEBT_valid(Summary,m)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                                & ! [X: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X7) ) ),
                                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,$o))))),aTP_Lamp_aov(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList),Summary),n2),m)),Mima) ) ) ) ) )
                 => ~ aa(product_prod(vEBT_VEBT,nat),$o,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,Deg2,TreeList,Summary)),Xaa)) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
tff(fact_8053_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( vEBT_VEBT_valid(Xb,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,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),(Uv2))),Xaa))
               => ( Xaa != one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg2,TreeList,Summary) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,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,Deg2,TreeList,Summary)),Xaa))
                 => ~ ( ( Deg2 = Xaa )
                      & $let(
                          n2: nat,
                          n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                          $let(
                            m: nat,
                            m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                            ( ! [X: vEBT_VEBT] :
                                ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                               => vEBT_VEBT_valid(X,n2) )
                            & vEBT_VEBT_valid(Summary,m)
                            & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                            & case_option($o,product_prod(nat,nat),
                                ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                                & ! [X: vEBT_VEBT] :
                                    ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                   => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X7) ) ),
                                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,$o))))),aTP_Lamp_aov(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList),Summary),n2),m)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
tff(fact_8054_fi__match__entails,axiom,
    ! [M: list(product_prod(assn,assn))] :
      ( ! [X2: product_prod(assn,assn)] :
          ( member(product_prod(assn,assn),X2,aa(list(product_prod(assn,assn)),set(product_prod(assn,assn)),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),aa(list(product_prod(assn,assn)),list(assn),map(product_prod(assn,assn),assn,product_fst(assn,assn)),M),one_one(assn))),foldr(assn,assn,times_times(assn),aa(list(product_prod(assn,assn)),list(assn),map(product_prod(assn,assn),assn,product_snd(assn,assn)),M),one_one(assn))) ) ).

% fi_match_entails
tff(fact_8055_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [Xb: vEBT_VEBT,Xaa: nat] :
      ( ~ vEBT_VEBT_valid(Xb,Xaa)
     => ( aa(product_prod(vEBT_VEBT,nat),$o,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,Uv2: $o] :
              ( ( Xb = vEBT_Leaf((Uu2),(Uv2)) )
             => ( aa(product_prod(vEBT_VEBT,nat),$o,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),(Uv2))),Xaa))
               => ( Xaa = one_one(nat) ) ) )
         => ~ ! [Mima: option(product_prod(nat,nat)),Deg2: nat,TreeList: list(vEBT_VEBT),Summary: vEBT_VEBT] :
                ( ( Xb = vEBT_Node(Mima,Deg2,TreeList,Summary) )
               => ( aa(product_prod(vEBT_VEBT,nat),$o,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,Deg2,TreeList,Summary)),Xaa))
                 => ( ( Deg2 = Xaa )
                    & $let(
                        n2: nat,
                        n2:= aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),Deg2),aa(num,nat,numeral_numeral(nat),bit0(one2))),
                        $let(
                          m: nat,
                          m:= aa(nat,nat,minus_minus(nat,Deg2),n2),
                          ( ! [X: vEBT_VEBT] :
                              ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                             => vEBT_VEBT_valid(X,n2) )
                          & vEBT_VEBT_valid(Summary,m)
                          & ( aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),TreeList) = aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),m) )
                          & case_option($o,product_prod(nat,nat),
                              ( ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(Summary),X7)
                              & ! [X: vEBT_VEBT] :
                                  ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),TreeList))
                                 => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X7) ) ),
                              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,$o))))),aTP_Lamp_aov(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,$o)))))),Deg2),TreeList),Summary),n2),m)),Mima) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
tff(fact_8056_pairself_Opelims,axiom,
    ! [A: $tType,B: $tType,Xb: fun(B,A),Xaa: product_prod(B,B),Y: product_prod(A,A)] :
      ( ( aa(product_prod(B,B),product_prod(A,A),pairself(B,A,Xb),Xaa) = Y )
     => ( aa(product_prod(fun(B,A),product_prod(B,B)),$o,accp(product_prod(fun(B,A),product_prod(B,B)),pairself_rel(B,A)),aa(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B)),aa(fun(B,A),fun(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B))),product_Pair(fun(B,A),product_prod(B,B)),Xb),Xaa))
       => ~ ! [A4: B,B3: B] :
              ( ( Xaa = aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A4),B3) )
             => ( ( Y = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Xb,A4)),aa(B,A,Xb,B3)) )
               => ~ aa(product_prod(fun(B,A),product_prod(B,B)),$o,accp(product_prod(fun(B,A),product_prod(B,B)),pairself_rel(B,A)),aa(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B)),aa(fun(B,A),fun(product_prod(B,B),product_prod(fun(B,A),product_prod(B,B))),product_Pair(fun(B,A),product_prod(B,B)),Xb),aa(B,product_prod(B,B),aa(B,fun(B,product_prod(B,B)),product_Pair(B,B),A4),B3))) ) ) ) ) ).

% pairself.pelims
tff(fact_8057_lexn__conv,axiom,
    ! [A: $tType,R3: set(product_prod(A,A)),Nb: nat] : lexn(A,R3,Nb) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),$o,aa(nat,fun(list(A),fun(list(A),$o)),aTP_Lamp_aoz(set(product_prod(A,A)),fun(nat,fun(list(A),fun(list(A),$o))),R3),Nb))) ).

% lexn_conv
tff(fact_8058_lexn__length,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R3: set(product_prod(A,A)),Nb: 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,R3,Nb))
     => ( ( aa(list(A),nat,size_size(list(A)),Xs) = Nb )
        & ( aa(list(A),nat,size_size(list(A)),Ys) = Nb ) ) ) ).

% lexn_length
tff(fact_8059_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,aa(list(product_prod(assn,assn)),set(product_prod(assn,assn)),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),aa(list(product_prod(assn,assn)),list(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),aa(list(product_prod(assn,assn)),list(assn),map(product_prod(assn,assn),assn,product_snd(assn,assn)),M8),one_one(assn))),UQ)),F5)) ) ) ).

% FI_RESULT_def
tff(fact_8060_FI__def,axiom,
    ! [M: list(product_prod(assn,assn)),P2: assn,Q2: assn,Up: assn,Uq: assn,F: assn] :
      ( fi(M,P2,Q2,Up,Uq,F)
    <=> ( ! [X: product_prod(assn,assn)] :
            ( member(product_prod(assn,assn),X,aa(list(product_prod(assn,assn)),set(product_prod(assn,assn)),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),aa(list(product_prod(assn,assn)),list(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),aa(list(product_prod(assn,assn)),list(assn),map(product_prod(assn,assn),assn,product_snd(assn,assn)),M),one_one(assn))),Q2)),Uq)),F)) ) ) ).

% FI_def
tff(fact_8061_entails__solve__finalize_I2_J,axiom,
    ! [M8: list(product_prod(assn,assn))] : fI_RESULT(M8,one_one(assn),one_one(assn),one_one(assn)) ).

% entails_solve_finalize(2)
tff(fact_8062_frame__inference__finalize,axiom,
    ! [M8: list(product_prod(assn,assn)),F5: assn] : fI_RESULT(M8,F5,one_one(assn),F5) ).

% frame_inference_finalize
tff(fact_8063_FI__p__nomatch,axiom,
    ! [M: list(product_prod(assn,assn)),Ps2: assn,Qs: assn,Q2: assn,P2: assn,Up: assn,Uq: assn,F: assn] :
      ( fi(M,Ps2,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qs),Q2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),Up),Uq,F)
     => fi(M,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Ps2),P2),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Qs),Q2),Up,Uq,F) ) ).

% FI_p_nomatch
tff(fact_8064_FI__finalize,axiom,
    ! [M: list(product_prod(assn,assn)),P2: assn,Up: assn,Q2: assn,Uq: assn,F: assn] :
      ( fI_RESULT(M,aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),P2),Up),aa(assn,assn,aa(assn,fun(assn,assn),times_times(assn),Q2),Uq),F)
     => fi(M,P2,Q2,Up,Uq,F) ) ).

% FI_finalize
tff(fact_8065_entails__solve__finalize_I1_J,axiom,
    ! [M8: list(product_prod(assn,assn)),P: assn] : fI_RESULT(M8,P,one_one(assn),top_top(assn)) ).

% entails_solve_finalize(1)
tff(fact_8066_lex__conv,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : lex(A,R3) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),$o,aTP_Lamp_apa(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R3))) ).

% lex_conv
tff(fact_8067_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,C72: $o] :
      ( aa(char,$o,aa(char,fun(char,$o),ord_less_eq(char),aa($o,char,char2((B0),(B12),(B23),(B32),(B42),(B52),(B62)),(B72))),aa($o,char,char2((C0),(C1),(C22),(C32),(C42),(C52),(C62)),(C72)))
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),foldr($o,nat,aTP_Lamp_apb($o,fun(nat,nat)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B0)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B12)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B23)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B32)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B42)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B52)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B62)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B72)),nil($o))))))))),zero_zero(nat))),foldr($o,nat,aTP_Lamp_apb($o,fun(nat,nat)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C0)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C1)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C22)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C32)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C42)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C52)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C62)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C72)),nil($o))))))))),zero_zero(nat))) ) ).

% less_eq_char_simp
tff(fact_8068_Cons__in__lex,axiom,
    ! [A: $tType,Xb: A,Xs: list(A),Y: A,Ys: list(A),R3: 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),aa(A,fun(list(A),list(A)),cons(A),Xb),Xs)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y),Ys)),lex(A,R3))
    <=> ( ( 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),R3)
          & ( 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,R3)) ) ) ) ).

% Cons_in_lex
tff(fact_8069_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,C72: $o] :
      ( aa(char,$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)),(C72)))
    <=> aa(nat,$o,ord_less(nat,foldr($o,nat,aTP_Lamp_apb($o,fun(nat,nat)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B0)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B12)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B23)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B32)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B42)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B52)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B62)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B72)),nil($o))))))))),zero_zero(nat))),foldr($o,nat,aTP_Lamp_apb($o,fun(nat,nat)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C0)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C1)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C22)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C32)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C42)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C52)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C62)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(C72)),nil($o))))))))),zero_zero(nat))) ) ).

% less_char_simp
tff(fact_8070_lex__append__leftI,axiom,
    ! [A: $tType,Ys: list(A),Zs: list(A),R3: set(product_prod(A,A)),Xs: 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)),Ys),Zs),lex(A,R3))
     => 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),append(A,Xs),Ys)),aa(list(A),list(A),append(A,Xs),Zs)),lex(A,R3)) ) ).

% lex_append_leftI
tff(fact_8071_Nil__notin__lex,axiom,
    ! [A: $tType,Ys: list(A),R3: 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)),nil(A)),Ys),lex(A,R3)) ).

% Nil_notin_lex
tff(fact_8072_Nil2__notin__lex,axiom,
    ! [A: $tType,Xs: list(A),R3: 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)),Xs),nil(A)),lex(A,R3)) ).

% Nil2_notin_lex
tff(fact_8073_lex__append__leftD,axiom,
    ! [A: $tType,R3: 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),R3)
     => ( 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),append(A,Xs),Ys)),aa(list(A),list(A),append(A,Xs),Zs)),lex(A,R3))
       => 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,R3)) ) ) ).

% lex_append_leftD
tff(fact_8074_lex__append__left__iff,axiom,
    ! [A: $tType,R3: 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),R3)
     => ( 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),append(A,Xs),Ys)),aa(list(A),list(A),append(A,Xs),Zs)),lex(A,R3))
      <=> 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,R3)) ) ) ).

% lex_append_left_iff
tff(fact_8075_lex__append__rightI,axiom,
    ! [A: $tType,Xs: list(A),Ys: list(A),R3: 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,R3))
     => ( ( 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)),aa(list(A),list(A),append(A,Xs),Us)),aa(list(A),list(A),append(A,Ys),Vs)),lex(A,R3)) ) ) ).

% lex_append_rightI
tff(fact_8076_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_8077_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($o),aa($o,fun(list($o),list($o)),cons($o),(B0)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B12)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B23)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B32)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B42)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B52)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B62)),aa(list($o),list($o),aa($o,fun(list($o),list($o)),cons($o),(B72)),nil($o)))))))))) ) ).

% of_char_Char
tff(fact_8078_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_8079_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_8080_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_8081_less__eq__char__def,axiom,
    ! [C1: char,C22: char] :
      ( aa(char,$o,aa(char,fun(char,$o),ord_less_eq(char),C1),C22)
    <=> aa(nat,$o,aa(nat,fun(nat,$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_8082_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_8083_char_Osize_I2_J,axiom,
    ! [X1: $o,X23: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X8: $o] : aa(char,nat,size_size(char),aa($o,char,char2((X1),(X23),(X33),(X42),(X52),(X62),(X72)),(X8))) = zero_zero(nat) ).

% char.size(2)
tff(fact_8084_char__of__def,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: A] : aa(A,char,unique5772411509450598832har_of(A),Nb) = aa($o,char,char2(~ aa(A,$o,dvd_dvd(A,aa(num,A,numeral_numeral(A),bit0(one2))),Nb),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Nb),one_one(nat)),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Nb),aa(num,nat,numeral_numeral(nat),bit0(one2))),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,one2))),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Nb),aa(num,nat,numeral_numeral(nat),bit0(bit0(one2)))),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,bit0(one2)))),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Nb),aa(num,nat,numeral_numeral(nat),bit0(aa(num,num,bit1,one2))))),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Nb),aa(num,nat,numeral_numeral(nat),aa(num,num,bit1,aa(num,num,bit1,one2))))) ) ).

% char_of_def
tff(fact_8085_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_apj(code_integer,fun($o,char))),code_bit_cut_integer(K)) ).

% char_of_integer_code
tff(fact_8086_char__of__quasi__inj,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [M: A,Nb: A] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),M) = aa(A,char,unique5772411509450598832har_of(A),Nb) )
        <=> ( 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),Nb),aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ) ) ) ).

% char_of_quasi_inj
tff(fact_8087_char__of__mod__256,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: A] : aa(A,char,unique5772411509450598832har_of(A),aa(A,A,aa(A,fun(A,A),modulo_modulo(A),Nb),aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2))))))))))) = aa(A,char,unique5772411509450598832har_of(A),Nb) ) ).

% char_of_mod_256
tff(fact_8088_of__char__of,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [A2: A] : aa(char,A,comm_s6883823935334413003f_char(A),aa(A,char,unique5772411509450598832har_of(A),A2)) = aa(A,A,aa(A,fun(A,A),modulo_modulo(A),A2),aa(num,A,numeral_numeral(A),bit0(bit0(bit0(bit0(bit0(bit0(bit0(bit0(one2)))))))))) ) ).

% of_char_of
tff(fact_8089_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_8090_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_8091_char__of__take__bit__eq,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: nat,M: A] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),Nb)
         => ( aa(A,char,unique5772411509450598832har_of(A),aa(A,A,bit_se2584673776208193580ke_bit(A,Nb),M)) = aa(A,char,unique5772411509450598832har_of(A),M) ) ) ) ).

% char_of_take_bit_eq
tff(fact_8092_char__of__eq__iff,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Nb: A,C2: char] :
          ( ( aa(A,char,unique5772411509450598832har_of(A),Nb) = C2 )
        <=> ( aa(A,A,bit_se2584673776208193580ke_bit(A,aa(num,nat,numeral_numeral(nat),bit0(bit0(bit0(one2))))),Nb) = aa(char,A,comm_s6883823935334413003f_char(A),C2) ) ) ) ).

% char_of_eq_iff
tff(fact_8093_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_8094_lenlex__conv,axiom,
    ! [A: $tType,R3: set(product_prod(A,A))] : lenlex(A,R3) = collect(product_prod(list(A),list(A)),product_case_prod(list(A),list(A),$o,aTP_Lamp_apk(set(product_prod(A,A)),fun(list(A),fun(list(A),$o)),R3))) ).

% lenlex_conv
tff(fact_8095_Nil__lenlex__iff1,axiom,
    ! [A: $tType,Ns: list(A),R3: 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)),nil(A)),Ns),lenlex(A,R3))
    <=> ( Ns != nil(A) ) ) ).

% Nil_lenlex_iff1
tff(fact_8096_Nil__lenlex__iff2,axiom,
    ! [A: $tType,Ns: list(A),R3: 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)),Ns),nil(A)),lenlex(A,R3)) ).

% Nil_lenlex_iff2
tff(fact_8097_lenlex__irreflexive,axiom,
    ! [A: $tType,R3: 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),R3)
     => ~ 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,R3)) ) ).

% lenlex_irreflexive
tff(fact_8098_lenlex__length,axiom,
    ! [A: $tType,Ms: list(A),Ns: list(A),R3: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),Ms),Ns),lenlex(A,R3))
     => aa(nat,$o,aa(nat,fun(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_8099_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)),aa(list(A),list(A),append(A,Us),Vs)),aa(list(A),list(A),append(A,Xs),Ys)),lenlex(A,R)) ) ) ).

% lenlex_append1
tff(fact_8100_Cons__lenlex__iff,axiom,
    ! [A: $tType,M: A,Ms: list(A),Nb: A,Ns: list(A),R3: set(product_prod(A,A))] :
      ( member(product_prod(list(A),list(A)),aa(list(A),product_prod(list(A),list(A)),aa(list(A),fun(list(A),product_prod(list(A),list(A))),product_Pair(list(A),list(A)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),M),Ms)),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Nb),Ns)),lenlex(A,R3))
    <=> ( aa(nat,$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),Nb),R3) )
        | ( ( M = Nb )
          & 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,R3)) ) ) ) ).

% Cons_lenlex_iff
tff(fact_8101_Eps__Opt__def,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      eps_Opt(A,P) = $ite(
        ? [X7: A] : aa(A,$o,P,X7),
        aa(A,option(A),some(A),fChoice(A,P)),
        none(A) ) ).

% Eps_Opt_def
tff(fact_8102_DERIV__even__real__root,axiom,
    ! [Nb: nat,Xb: real] :
      ( aa(nat,$o,ord_less(nat,zero_zero(nat)),Nb)
     => ( aa(nat,$o,dvd_dvd(nat,aa(num,nat,numeral_numeral(nat),bit0(one2))),Nb)
       => ( aa(real,$o,ord_less(real,Xb),zero_zero(real))
         => has_field_derivative(real,root(Nb),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),Nb))),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,root(Nb),Xb)),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ) ) ).

% DERIV_even_real_root
tff(fact_8103_some__opt__sym__eq__trivial,axiom,
    ! [A: $tType,Xb: A] : eps_Opt(A,aa(A,fun(A,$o),fequal(A),Xb)) = aa(A,option(A),some(A),Xb) ).

% some_opt_sym_eq_trivial
tff(fact_8104_some__opt__eq__trivial,axiom,
    ! [A: $tType,Xb: A] : eps_Opt(A,aTP_Lamp_dc(A,fun(A,$o),Xb)) = aa(A,option(A),some(A),Xb) ).

% some_opt_eq_trivial
tff(fact_8105_some__opt__false__trivial,axiom,
    ! [A: $tType] : eps_Opt(A,aTP_Lamp_bm(A,$o)) = none(A) ).

% some_opt_false_trivial
tff(fact_8106_Eps__Opt__eq__None,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( ( eps_Opt(A,P) = none(A) )
    <=> ~ ? [X_1: A] : aa(A,$o,P,X_1) ) ).

% Eps_Opt_eq_None
tff(fact_8107_DERIV__at__within__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),Y: A,Z: A,Xb: A,S2: set(A)] :
          ( has_field_derivative(A,F,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)),S2)))
        <=> has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_apl(fun(A,A),fun(A,fun(A,A)),F),Z),Y,topolo174197925503356063within(A,Xb,S2)) ) ) ).

% DERIV_at_within_shift
tff(fact_8108_DERIV__at__within__shift__lemma,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),Y: A,Z: A,Xb: A,S2: set(A)] :
          ( has_field_derivative(A,F,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)),S2)))
         => has_field_derivative(A,comp(A,A,A,F,aa(A,fun(A,A),plus_plus(A),Z)),Y,topolo174197925503356063within(A,Xb,S2)) ) ) ).

% DERIV_at_within_shift_lemma
tff(fact_8109_DERIV__image__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),Da: A,G: fun(A,A),Xb: A,S3: set(A),Db: A] :
          ( has_field_derivative(A,F,Da,topolo174197925503356063within(A,aa(A,A,G,Xb),aa(set(A),set(A),image(A,A,G),S3)))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S3))
           => has_field_derivative(A,comp(A,A,A,F,G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_image_chain
tff(fact_8110_at__within__Icc__at__left,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),B2)
         => ( topolo174197925503356063within(A,B2,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,B2,set_ord_lessThan(A,B2)) ) ) ) ).

% at_within_Icc_at_left
tff(fact_8111_has__real__derivative__neg__dec__left,axiom,
    ! [F: fun(real,real),L: real,Xb: real,S2: set(real)] :
      ( has_field_derivative(real,F,L,topolo174197925503356063within(real,Xb,S2))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D3)
            & ! [H6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H6)
               => ( member(real,aa(real,real,minus_minus(real,Xb),H6),S2)
                 => ( aa(real,$o,ord_less(real,H6),D3)
                   => aa(real,$o,ord_less(real,aa(real,real,F,Xb)),aa(real,real,F,aa(real,real,minus_minus(real,Xb),H6))) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
tff(fact_8112_has__real__derivative__pos__inc__left,axiom,
    ! [F: fun(real,real),L: real,Xb: real,S2: set(real)] :
      ( has_field_derivative(real,F,L,topolo174197925503356063within(real,Xb,S2))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D3)
            & ! [H6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H6)
               => ( member(real,aa(real,real,minus_minus(real,Xb),H6),S2)
                 => ( aa(real,$o,ord_less(real,H6),D3)
                   => aa(real,$o,ord_less(real,aa(real,real,F,aa(real,real,minus_minus(real,Xb),H6))),aa(real,real,F,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
tff(fact_8113_has__real__derivative__neg__dec__right,axiom,
    ! [F: fun(real,real),L: real,Xb: real,S2: set(real)] :
      ( has_field_derivative(real,F,L,topolo174197925503356063within(real,Xb,S2))
     => ( aa(real,$o,ord_less(real,L),zero_zero(real))
       => ? [D3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D3)
            & ! [H6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H6)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H6),S2)
                 => ( aa(real,$o,ord_less(real,H6),D3)
                   => aa(real,$o,ord_less(real,aa(real,real,F,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H6))),aa(real,real,F,Xb)) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
tff(fact_8114_has__real__derivative__pos__inc__right,axiom,
    ! [F: fun(real,real),L: real,Xb: real,S2: set(real)] :
      ( has_field_derivative(real,F,L,topolo174197925503356063within(real,Xb,S2))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),L)
       => ? [D3: real] :
            ( aa(real,$o,ord_less(real,zero_zero(real)),D3)
            & ! [H6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H6)
               => ( member(real,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H6),S2)
                 => ( aa(real,$o,ord_less(real,H6),D3)
                   => aa(real,$o,ord_less(real,aa(real,real,F,Xb)),aa(real,real,F,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H6))) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
tff(fact_8115_DERIV__pow,axiom,
    ! [Nb: nat,Xb: real,S3: set(real)] : has_field_derivative(real,aTP_Lamp_apm(nat,fun(real,real),Nb),aa(real,real,aa(real,fun(real,real),times_times(real),aa(nat,real,semiring_1_of_nat(real),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),Xb),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat))))),topolo174197925503356063within(real,Xb,S3)) ).

% DERIV_pow
tff(fact_8116_DERIV__power,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),D6: A,Xb: A,S3: set(A),Nb: nat] :
          ( has_field_derivative(A,F,D6,topolo174197925503356063within(A,Xb,S3))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_apn(fun(A,A),fun(nat,fun(A,A)),F),Nb),aa(A,A,aa(A,fun(A,A),times_times(A),aa(nat,A,semiring_1_of_nat(A),Nb)),aa(A,A,aa(A,fun(A,A),times_times(A),D6),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F,Xb)),aa(nat,nat,minus_minus(nat,Nb),aa(nat,nat,suc,zero_zero(nat)))))),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_power
tff(fact_8117_DERIV__quotient,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),D2: A,Xb: A,S3: set(A),G: fun(A,A),E3: A] :
          ( has_field_derivative(A,F,D2,topolo174197925503356063within(A,Xb,S3))
         => ( has_field_derivative(A,G,E3,topolo174197925503356063within(A,Xb,S3))
           => ( ( aa(A,A,G,Xb) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_apo(fun(A,A),fun(fun(A,A),fun(A,A)),F),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),D2),aa(A,A,G,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),E3),aa(A,A,F,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,S3)) ) ) ) ) ).

% DERIV_quotient
tff(fact_8118_DERIV__divide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),D6: A,Xb: A,S3: set(A),G: fun(A,A),E6: A] :
          ( has_field_derivative(A,F,D6,topolo174197925503356063within(A,Xb,S3))
         => ( has_field_derivative(A,G,E6,topolo174197925503356063within(A,Xb,S3))
           => ( ( aa(A,A,G,Xb) != zero_zero(A) )
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_apo(fun(A,A),fun(fun(A,A),fun(A,A)),F),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),D6),aa(A,A,G,Xb))),aa(A,A,aa(A,fun(A,A),times_times(A),aa(A,A,F,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,S3)) ) ) ) ) ).

% DERIV_divide
tff(fact_8119_DERIV__const,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [K: A,F5: filter(A)] : has_field_derivative(A,aTP_Lamp_app(A,fun(A,A),K),zero_zero(A),F5) ) ).

% DERIV_const
tff(fact_8120_Eps__Opt__eq__Some,axiom,
    ! [A: $tType,P: fun(A,$o),Xb: A] :
      ( ! [X6: A] :
          ( aa(A,$o,P,Xb)
         => ( aa(A,$o,P,X6)
           => ( X6 = Xb ) ) )
     => ( ( eps_Opt(A,P) = aa(A,option(A),some(A),Xb) )
      <=> aa(A,$o,P,Xb) ) ) ).

% Eps_Opt_eq_Some
tff(fact_8121_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_8122_DERIV__sum,axiom,
    ! [A: $tType,C: $tType,B: $tType] :
      ( real_V3459762299906320749_field(B)
     => ! [S2: set(A),F: fun(B,fun(A,B)),F4: fun(C,fun(A,B)),Xb: C,F5: filter(B)] :
          ( ! [N2: A] :
              ( member(A,N2,S2)
             => has_field_derivative(B,aa(A,fun(B,B),aTP_Lamp_apq(fun(B,fun(A,B)),fun(A,fun(B,B)),F),N2),aa(A,B,aa(C,fun(A,B),F4,Xb),N2),F5) )
         => has_field_derivative(B,aa(fun(B,fun(A,B)),fun(B,B),aTP_Lamp_apr(set(A),fun(fun(B,fun(A,B)),fun(B,B)),S2),F),aa(set(A),B,groups7311177749621191930dd_sum(A,B,aa(C,fun(A,B),F4,Xb)),S2),F5) ) ) ).

% DERIV_sum
tff(fact_8123_has__field__derivative__scaleR__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),D6: A,F5: filter(A),C2: real] :
          ( has_field_derivative(A,F,D6,F5)
         => has_field_derivative(A,aa(real,fun(A,A),aTP_Lamp_aps(fun(A,A),fun(real,fun(A,A)),F),C2),aa(A,A,real_V8093663219630862766scaleR(A,C2),D6),F5) ) ) ).

% has_field_derivative_scaleR_right
tff(fact_8124_DERIV__mult_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),D6: A,Xb: A,S3: set(A),G: fun(A,A),E6: A] :
          ( has_field_derivative(A,F,D6,topolo174197925503356063within(A,Xb,S3))
         => ( has_field_derivative(A,G,E6,topolo174197925503356063within(A,Xb,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_apt(fun(A,A),fun(fun(A,A),fun(A,A)),F),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,F,Xb)),E6)),aa(A,A,aa(A,fun(A,A),times_times(A),D6),aa(A,A,G,Xb))),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_mult'
tff(fact_8125_DERIV__mult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),Da: A,Xb: A,S3: set(A),G: fun(A,A),Db: A] :
          ( has_field_derivative(A,F,Da,topolo174197925503356063within(A,Xb,S3))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_apt(fun(A,A),fun(fun(A,A),fun(A,A)),F),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,F,Xb))),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_mult
tff(fact_8126_DERIV__cmult__right,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),D6: A,Xb: A,S3: set(A),C2: A] :
          ( has_field_derivative(A,F,D6,topolo174197925503356063within(A,Xb,S3))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_apu(fun(A,A),fun(A,fun(A,A)),F),C2),aa(A,A,aa(A,fun(A,A),times_times(A),D6),C2),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_cmult_right
tff(fact_8127_DERIV__cmult,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),D6: A,Xb: A,S3: set(A),C2: A] :
          ( has_field_derivative(A,F,D6,topolo174197925503356063within(A,Xb,S3))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_apv(fun(A,A),fun(A,fun(A,A)),F),C2),aa(A,A,aa(A,fun(A,A),times_times(A),C2),D6),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_cmult
tff(fact_8128_DERIV__cmult__Id,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [C2: A,Xb: A,S3: set(A)] : has_field_derivative(A,aa(A,fun(A,A),times_times(A),C2),C2,topolo174197925503356063within(A,Xb,S3)) ) ).

% DERIV_cmult_Id
tff(fact_8129_has__field__derivative__sinh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S3))
         => has_field_derivative(A,aTP_Lamp_apw(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,S3)) ) ) ).

% has_field_derivative_sinh
tff(fact_8130_has__field__derivative__cosh,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [G: fun(A,A),Db: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S3))
         => has_field_derivative(A,aTP_Lamp_apx(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,S3)) ) ) ).

% has_field_derivative_cosh
tff(fact_8131_DERIV__cdivide,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),D6: A,Xb: A,S3: set(A),C2: A] :
          ( has_field_derivative(A,F,D6,topolo174197925503356063within(A,Xb,S3))
         => has_field_derivative(A,aa(A,fun(A,A),aTP_Lamp_apy(fun(A,A),fun(A,fun(A,A)),F),C2),aa(A,A,aa(A,fun(A,A),divide_divide(A),D6),C2),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_cdivide
tff(fact_8132_DERIV__ident,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F5: filter(A)] : has_field_derivative(A,aTP_Lamp_apz(A,A),one_one(A),F5) ) ).

% DERIV_ident
tff(fact_8133_field__differentiable__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),F4: A,F5: filter(A),G: fun(A,A),G6: A] :
          ( has_field_derivative(A,F,F4,F5)
         => ( has_field_derivative(A,G,G6,F5)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aqa(fun(A,A),fun(fun(A,A),fun(A,A)),F),G),aa(A,A,minus_minus(A,F4),G6),F5) ) ) ) ).

% field_differentiable_diff
tff(fact_8134_DERIV__diff,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),D6: A,Xb: A,S3: set(A),G: fun(A,A),E6: A] :
          ( has_field_derivative(A,F,D6,topolo174197925503356063within(A,Xb,S3))
         => ( has_field_derivative(A,G,E6,topolo174197925503356063within(A,Xb,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aqa(fun(A,A),fun(fun(A,A),fun(A,A)),F),G),aa(A,A,minus_minus(A,D6),E6),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_diff
tff(fact_8135_field__differentiable__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),F4: A,F5: filter(A),G: fun(A,A),G6: A] :
          ( has_field_derivative(A,F,F4,F5)
         => ( has_field_derivative(A,G,G6,F5)
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aqb(fun(A,A),fun(fun(A,A),fun(A,A)),F),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),F4),G6),F5) ) ) ) ).

% field_differentiable_add
tff(fact_8136_DERIV__add,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),D6: A,Xb: A,S3: set(A),G: fun(A,A),E6: A] :
          ( has_field_derivative(A,F,D6,topolo174197925503356063within(A,Xb,S3))
         => ( has_field_derivative(A,G,E6,topolo174197925503356063within(A,Xb,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aqb(fun(A,A),fun(fun(A,A),fun(A,A)),F),G),aa(A,A,aa(A,fun(A,A),plus_plus(A),D6),E6),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_add
tff(fact_8137_DERIV__power__Suc,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),D6: A,Xb: A,S3: set(A),Nb: nat] :
          ( has_field_derivative(A,F,D6,topolo174197925503356063within(A,Xb,S3))
         => has_field_derivative(A,aa(nat,fun(A,A),aTP_Lamp_aqc(fun(A,A),fun(nat,fun(A,A)),F),Nb),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),Nb))),aa(A,A,aa(A,fun(A,A),times_times(A),D6),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F,Xb)),Nb))),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_power_Suc
tff(fact_8138_DERIV__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),D6: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F,D6,topolo174197925503356063within(A,Xb,S3))
         => has_field_derivative(A,aTP_Lamp_aqd(fun(A,A),fun(A,A),F),aa(A,A,uminus_uminus(A),D6),topolo174197925503356063within(A,Xb,S3)) ) ) ).

% DERIV_minus
tff(fact_8139_field__differentiable__minus,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),F4: A,F5: filter(A)] :
          ( has_field_derivative(A,F,F4,F5)
         => has_field_derivative(A,aTP_Lamp_aqd(fun(A,A),fun(A,A),F),aa(A,A,uminus_uminus(A),F4),F5) ) ) ).

% field_differentiable_minus
tff(fact_8140_DERIV__inverse__fun,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),D2: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F,D2,topolo174197925503356063within(A,Xb,S3))
         => ( ( aa(A,A,F,Xb) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_aqe(fun(A,A),fun(A,A),F),aa(A,A,uminus_uminus(A),aa(A,A,aa(A,fun(A,A),times_times(A),D2),aa(A,A,inverse_inverse(A),aa(nat,A,aa(A,fun(nat,A),power_power(A),aa(A,A,F,Xb)),aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))))),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_inverse_fun
tff(fact_8141_DERIV__inverse_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),D6: A,Xb: A,S3: set(A)] :
          ( has_field_derivative(A,F,D6,topolo174197925503356063within(A,Xb,S3))
         => ( ( aa(A,A,F,Xb) != zero_zero(A) )
           => has_field_derivative(A,aTP_Lamp_aqe(fun(A,A),fun(A,A),F),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,F,Xb))),D6)),aa(A,A,inverse_inverse(A),aa(A,A,F,Xb)))),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_inverse'
tff(fact_8142_DERIV__inverse,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Xb: A,S3: 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,S3)) ) ) ).

% DERIV_inverse
tff(fact_8143_has__field__derivative__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),Y: A,Xb: A,S3: set(A),Ta: set(A)] :
          ( has_field_derivative(A,F,Y,topolo174197925503356063within(A,Xb,S3))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),S3)
           => has_field_derivative(A,F,Y,topolo174197925503356063within(A,Xb,Ta)) ) ) ) ).

% has_field_derivative_subset
tff(fact_8144_DERIV__subset,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),F4: A,Xb: A,S3: set(A),Ta: set(A)] :
          ( has_field_derivative(A,F,F4,topolo174197925503356063within(A,Xb,S3))
         => ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Ta),S3)
           => has_field_derivative(A,F,F4,topolo174197925503356063within(A,Xb,Ta)) ) ) ) ).

% DERIV_subset
tff(fact_8145_at__le,axiom,
    ! [A: $tType] :
      ( topolo4958980785337419405_space(A)
     => ! [S3: set(A),Ta: set(A),Xb: A] :
          ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),S3),Ta)
         => aa(filter(A),$o,aa(filter(A),fun(filter(A),$o),ord_less_eq(filter(A)),topolo174197925503356063within(A,Xb,S3)),topolo174197925503356063within(A,Xb,Ta)) ) ) ).

% at_le
tff(fact_8146_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A2: real,B2: real,F: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2)
             => ? [Y4: real] :
                  ( has_field_derivative(real,F,Y4,topolo174197925503356063within(real,X2,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Y4),zero_zero(real)) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F,B2)),aa(real,real,F,A2)) ) ) ).

% DERIV_nonpos_imp_nonincreasing
tff(fact_8147_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A2: real,B2: real,F: fun(real,real)] :
      ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
     => ( ! [X2: real] :
            ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),X2)
           => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),X2),B2)
             => ? [Y4: real] :
                  ( has_field_derivative(real,F,Y4,topolo174197925503356063within(real,X2,top_top(set(real))))
                  & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Y4) ) ) )
       => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,F,A2)),aa(real,real,F,B2)) ) ) ).

% DERIV_nonneg_imp_nondecreasing
tff(fact_8148_deriv__nonneg__imp__mono,axiom,
    ! [A2: real,B2: real,G: fun(real,real),G6: fun(real,real)] :
      ( ! [X2: real] :
          ( member(real,X2,set_or1337092689740270186AtMost(real,A2,B2))
         => has_field_derivative(real,G,aa(real,real,G6,X2),topolo174197925503356063within(real,X2,top_top(set(real)))) )
     => ( ! [X2: real] :
            ( member(real,X2,set_or1337092689740270186AtMost(real,A2,B2))
           => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),aa(real,real,G6,X2)) )
       => ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),A2),B2)
         => aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),aa(real,real,G,A2)),aa(real,real,G,B2)) ) ) ) ).

% deriv_nonneg_imp_mono
tff(fact_8149_DERIV__mirror,axiom,
    ! [F: fun(real,real),Y: real,Xb: real] :
      ( has_field_derivative(real,F,Y,topolo174197925503356063within(real,aa(real,real,uminus_uminus(real),Xb),top_top(set(real))))
    <=> has_field_derivative(real,aTP_Lamp_aqf(fun(real,real),fun(real,real),F),aa(real,real,uminus_uminus(real),Y),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_mirror
tff(fact_8150_DERIV__cos__add,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K: A,Xaa: A] : has_field_derivative(A,aTP_Lamp_aqg(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_8151_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_aqh(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_8152_DERIV__fun__pow,axiom,
    ! [G: fun(real,real),M: real,Xb: real,Nb: 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_aqi(fun(real,real),fun(nat,fun(real,real)),G),Nb),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),Nb)),aa(nat,real,aa(real,fun(nat,real),power_power(real),aa(real,real,G,Xb)),aa(nat,nat,minus_minus(nat,Nb),one_one(nat))))),M),topolo174197925503356063within(real,Xb,top_top(set(real)))) ) ).

% DERIV_fun_pow
tff(fact_8153_DERIV__const__average,axiom,
    ! [A2: real,B2: real,V: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [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),A2),B2)),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,A2)),aa(real,real,V,B2))),aa(num,real,numeral_numeral(real),bit0(one2))) ) ) ) ).

% DERIV_const_average
tff(fact_8154_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_aqj(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_8155_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_aqk(fun(A,A),fun(A,A),G),aa(A,A,aa(A,fun(A,A),times_times(A),exp(A,aa(A,A,G,Xb))),M),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_fun_exp
tff(fact_8156_DERIV__chain__s,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [S3: set(A),G: fun(A,A),G6: fun(A,A),F: fun(A,A),F4: A,Xb: A] :
          ( ! [X2: A] :
              ( member(A,X2,S3)
             => has_field_derivative(A,G,aa(A,A,G6,X2),topolo174197925503356063within(A,X2,top_top(set(A)))) )
         => ( has_field_derivative(A,F,F4,topolo174197925503356063within(A,Xb,top_top(set(A))))
           => ( member(A,aa(A,A,F,Xb),S3)
             => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aql(fun(A,A),fun(fun(A,A),fun(A,A)),G),F),aa(A,A,aa(A,fun(A,A),times_times(A),F4),aa(A,A,G6,aa(A,A,F,Xb))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ) ).

% DERIV_chain_s
tff(fact_8157_DERIV__chain3,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [G: fun(A,A),G6: fun(A,A),F: fun(A,A),F4: A,Xb: A] :
          ( ! [X2: A] : has_field_derivative(A,G,aa(A,A,G6,X2),topolo174197925503356063within(A,X2,top_top(set(A))))
         => ( has_field_derivative(A,F,F4,topolo174197925503356063within(A,Xb,top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aql(fun(A,A),fun(fun(A,A),fun(A,A)),G),F),aa(A,A,aa(A,fun(A,A),times_times(A),F4),aa(A,A,G6,aa(A,A,F,Xb))),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ) ).

% DERIV_chain3
tff(fact_8158_DERIV__chain2,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),Da: A,G: fun(A,A),Xb: A,Db: A,S3: set(A)] :
          ( has_field_derivative(A,F,Da,topolo174197925503356063within(A,aa(A,A,G,Xb),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S3))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aql(fun(A,A),fun(fun(A,A),fun(A,A)),F),G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_chain2
tff(fact_8159_DERIV__chain_H,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),D6: A,Xb: A,S3: set(A),G: fun(A,A),E6: A] :
          ( has_field_derivative(A,F,D6,topolo174197925503356063within(A,Xb,S3))
         => ( has_field_derivative(A,G,E6,topolo174197925503356063within(A,aa(A,A,F,Xb),top_top(set(A))))
           => has_field_derivative(A,aa(fun(A,A),fun(A,A),aTP_Lamp_aqm(fun(A,A),fun(fun(A,A),fun(A,A)),F),G),aa(A,A,aa(A,fun(A,A),times_times(A),E6),D6),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_chain'
tff(fact_8160_DERIV__const__ratio__const,axiom,
    ! [A2: real,B2: real,F: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X2: real] : has_field_derivative(real,F,K,topolo174197925503356063within(real,X2,top_top(set(real))))
       => ( aa(real,real,minus_minus(real,aa(real,real,F,B2)),aa(real,real,F,A2)) = aa(real,real,aa(real,fun(real,real),times_times(real),aa(real,real,minus_minus(real,B2),A2)),K) ) ) ) ).

% DERIV_const_ratio_const
tff(fact_8161_DERIV__const__ratio__const2,axiom,
    ! [A2: real,B2: real,F: fun(real,real),K: real] :
      ( ( A2 != B2 )
     => ( ! [X2: real] : has_field_derivative(real,F,K,topolo174197925503356063within(real,X2,top_top(set(real))))
       => ( aa(real,real,aa(real,fun(real,real),divide_divide(real),aa(real,real,minus_minus(real,aa(real,real,F,B2)),aa(real,real,F,A2))),aa(real,real,minus_minus(real,B2),A2)) = K ) ) ) ).

% DERIV_const_ratio_const2
tff(fact_8162_DERIV__shift,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),Y: A,Xb: A,Z: A] :
          ( has_field_derivative(A,F,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_aqn(fun(A,A),fun(A,fun(A,A)),F),Z),Y,topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% DERIV_shift
tff(fact_8163_termdiffs__strong__converges__everywhere,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [C2: fun(nat,A),Xb: A] :
          ( ! [Y5: A] : summable(A,aa(A,fun(nat,A),aTP_Lamp_acv(fun(nat,A),fun(A,fun(nat,A)),C2),Y5))
         => has_field_derivative(A,aTP_Lamp_aqo(fun(nat,A),fun(A,A),C2),suminf(A,aa(A,fun(nat,A),aTP_Lamp_acw(fun(nat,A),fun(A,fun(nat,A)),C2),Xb)),topolo174197925503356063within(A,Xb,top_top(set(A)))) ) ) ).

% termdiffs_strong_converges_everywhere
tff(fact_8164_DERIV__chain,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [F: fun(A,A),Da: A,G: fun(A,A),Xb: A,Db: A,S3: set(A)] :
          ( has_field_derivative(A,F,Da,topolo174197925503356063within(A,aa(A,A,G,Xb),top_top(set(A))))
         => ( has_field_derivative(A,G,Db,topolo174197925503356063within(A,Xb,S3))
           => has_field_derivative(A,comp(A,A,A,F,G),aa(A,A,aa(A,fun(A,A),times_times(A),Da),Db),topolo174197925503356063within(A,Xb,S3)) ) ) ) ).

% DERIV_chain
tff(fact_8165_termdiffs__sums__strong,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [K6: real,C2: fun(nat,A),F: fun(A,A),F4: A,Z: A] :
          ( ! [Z3: A] :
              ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z3)),K6)
             => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_acv(fun(nat,A),fun(A,fun(nat,A)),C2),Z3)),aa(A,A,F,Z3)) )
         => ( has_field_derivative(A,F,F4,topolo174197925503356063within(A,Z,top_top(set(A))))
           => ( aa(real,$o,ord_less(real,real_V7770717601297561774m_norm(A,Z)),K6)
             => aa(A,$o,sums(A,aa(A,fun(nat,A),aTP_Lamp_acw(fun(nat,A),fun(A,fun(nat,A)),C2),Z)),F4) ) ) ) ) ).

% termdiffs_sums_strong
tff(fact_8166_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_8167_DERIV__neg__dec__left,axiom,
    ! [F: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F,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)
            & ! [H6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H6)
               => ( aa(real,$o,ord_less(real,H6),D3)
                 => aa(real,$o,ord_less(real,aa(real,real,F,Xb)),aa(real,real,F,aa(real,real,minus_minus(real,Xb),H6))) ) ) ) ) ) ).

% DERIV_neg_dec_left
tff(fact_8168_DERIV__pos__inc__left,axiom,
    ! [F: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F,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)
            & ! [H6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H6)
               => ( aa(real,$o,ord_less(real,H6),D3)
                 => aa(real,$o,ord_less(real,aa(real,real,F,aa(real,real,minus_minus(real,Xb),H6))),aa(real,real,F,Xb)) ) ) ) ) ) ).

% DERIV_pos_inc_left
tff(fact_8169_DERIV__neg__dec__right,axiom,
    ! [F: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F,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)
            & ! [H6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H6)
               => ( aa(real,$o,ord_less(real,H6),D3)
                 => aa(real,$o,ord_less(real,aa(real,real,F,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H6))),aa(real,real,F,Xb)) ) ) ) ) ) ).

% DERIV_neg_dec_right
tff(fact_8170_DERIV__pos__inc__right,axiom,
    ! [F: fun(real,real),L: real,Xb: real] :
      ( has_field_derivative(real,F,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)
            & ! [H6: real] :
                ( aa(real,$o,ord_less(real,zero_zero(real)),H6)
               => ( aa(real,$o,ord_less(real,H6),D3)
                 => aa(real,$o,ord_less(real,aa(real,real,F,Xb)),aa(real,real,F,aa(real,real,aa(real,fun(real,real),plus_plus(real),Xb),H6))) ) ) ) ) ) ).

% DERIV_pos_inc_right
tff(fact_8171_at__within__Icc__at,axiom,
    ! [A: $tType] :
      ( topolo2564578578187576103pology(A)
     => ! [A2: A,Xb: A,B2: A] :
          ( aa(A,$o,ord_less(A,A2),Xb)
         => ( aa(A,$o,ord_less(A,Xb),B2)
           => ( topolo174197925503356063within(A,Xb,set_or1337092689740270186AtMost(A,A2,B2)) = topolo174197925503356063within(A,Xb,top_top(set(A))) ) ) ) ) ).

% at_within_Icc_at
tff(fact_8172_DERIV__local__const,axiom,
    ! [F: fun(real,real),L: real,Xb: real,D2: real] :
      ( has_field_derivative(real,F,L,topolo174197925503356063within(real,Xb,top_top(set(real))))
     => ( aa(real,$o,ord_less(real,zero_zero(real)),D2)
       => ( ! [Y5: real] :
              ( aa(real,$o,ord_less(real,aa(real,real,abs_abs(real),aa(real,real,minus_minus(real,Xb),Y5))),D2)
             => ( aa(real,real,F,Xb) = aa(real,real,F,Y5) ) )
         => ( L = zero_zero(real) ) ) ) ) ).

% DERIV_local_const
tff(fact_8173_ATP_Olambda__1,axiom,
    ! [Uu: product_prod(nat,nat)] :
      aa(product_prod(nat,nat),heap_Time_Heap(option(nat)),aTP_Lamp_am(product_prod(nat,nat),heap_Time_Heap(option(nat))),Uu) = $ite(
        aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),x12),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),Uu)),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),Uu))),heap_Time_bind(nat,option(nat),vEBT_VEBT_lowi(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),x12),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aTP_Lamp_al(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),Uu))) ) ).

% ATP.lambda_1
tff(fact_8174_ATP_Olambda__2,axiom,
    ! [Uu: product_unit] : aa(product_unit,heap_Time_Heap(option(nat)),aTP_Lamp_an(product_unit,heap_Time_Heap(option(nat))),Uu) = 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)),aTP_Lamp_am(product_prod(nat,nat),heap_Time_Heap(option(nat))),x11) ).

% ATP.lambda_2
tff(fact_8175_ATP_Olambda__3,axiom,
    ! [Uu: option(nat)] : aa(option(nat),assn,aTP_Lamp_aw(option(nat),assn),Uu) = 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_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,summary)),tia)),pure_assn(Uu = 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,summary),xa))) ).

% ATP.lambda_3
tff(fact_8176_ATP_Olambda__4,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_abr(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_4
tff(fact_8177_ATP_Olambda__5,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_vg(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_5
tff(fact_8178_ATP_Olambda__6,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_aet(real,$o),Uu)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uu)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uu),aa(num,real,numeral_numeral(real),bit0(one2)))
        & ( cos(real,Uu) = zero_zero(real) ) ) ) ).

% ATP.lambda_6
tff(fact_8179_ATP_Olambda__7,axiom,
    ! [Uu: product_unit] :
      aa(product_unit,heap_Time_Heap(option(nat)),aTP_Lamp_ao(product_unit,heap_Time_Heap(option(nat))),Uu) = heap_Time_bind(product_unit,option(nat),
        refine_Imp_assert(( ( 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)) = x11 )
          & ( aa(nat,nat,suc,aa(nat,nat,suc,va)) = x12 ) )),
        aTP_Lamp_an(product_unit,heap_Time_Heap(option(nat)))) ).

% ATP.lambda_7
tff(fact_8180_ATP_Olambda__8,axiom,
    ! [Uu: nat] : aa(nat,real,aTP_Lamp_xf(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_8
tff(fact_8181_ATP_Olambda__9,axiom,
    ! [B: $tType,A: $tType] :
      ( ( type_len(A)
        & type_len(B) )
     => ! [Uu: word(B)] :
          ( aa(word(B),$o,aTP_Lamp_amk(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_9
tff(fact_8182_ATP_Olambda__10,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_ajm(nat,nat),Uu) = aa(nat,nat,minus_minus(nat,Uu),aa(nat,nat,suc,zero_zero(nat))) ).

% ATP.lambda_10
tff(fact_8183_ATP_Olambda__11,axiom,
    ! [B: $tType,Uu: B] : aa(B,product_prod(B,B),aTP_Lamp_amt(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_11
tff(fact_8184_ATP_Olambda__12,axiom,
    ! [A: $tType,Uu: A] : aa(A,product_prod(A,A),aTP_Lamp_amu(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_12
tff(fact_8185_ATP_Olambda__13,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_wj(A,A),Uu) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uu),one_one(A)) ) ).

% ATP.lambda_13
tff(fact_8186_ATP_Olambda__14,axiom,
    ! [A: $tType,Uu: A] : aa(A,list(A),aTP_Lamp_aik(A,list(A)),Uu) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).

% ATP.lambda_14
tff(fact_8187_ATP_Olambda__15,axiom,
    ! [A: $tType,Uu: A] : aa(A,set(A),aTP_Lamp_anl(A,set(A)),Uu) = aa(set(A),set(A),insert(A,Uu),bot_bot(set(A))) ).

% ATP.lambda_15
tff(fact_8188_ATP_Olambda__16,axiom,
    ! [Uu: int] :
      ( aa(int,$o,aTP_Lamp_ajf(int,$o),Uu)
    <=> ~ aa(int,$o,dvd_dvd(int,aa(num,int,numeral_numeral(int),bit0(one2))),Uu) ) ).

% ATP.lambda_16
tff(fact_8189_ATP_Olambda__17,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_aku(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_17
tff(fact_8190_ATP_Olambda__18,axiom,
    ! [B: $tType,Uu: list(B)] :
      ( aa(list(B),$o,aTP_Lamp_akv(list(B),$o),Uu)
    <=> ( Uu != nil(B) ) ) ).

% ATP.lambda_18
tff(fact_8191_ATP_Olambda__19,axiom,
    ! [A: $tType,Uu: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_alm(list(A),$o),Uu)
    <=> ( Uu != nil(A) ) ) ).

% ATP.lambda_19
tff(fact_8192_ATP_Olambda__20,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(product_prod(heap_ext(product_unit),nat),A),aTP_Lamp_akg(A,fun(product_prod(heap_ext(product_unit),nat),A)),Uu) = product_case_prod(heap_ext(product_unit),nat,A,aTP_Lamp_akf(A,fun(heap_ext(product_unit),fun(nat,A)),Uu)) ).

% ATP.lambda_20
tff(fact_8193_ATP_Olambda__21,axiom,
    ! [Uu: option(nat)] : aa(option(nat),assn,aTP_Lamp_aq(option(nat),assn),Uu) = ex_assn(list(vEBT_VEBTi),aTP_Lamp_ap(option(nat),fun(list(vEBT_VEBTi),assn),Uu)) ).

% ATP.lambda_21
tff(fact_8194_ATP_Olambda__22,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B)] : aa(fun(A,B),set(product_prod(A,B)),aTP_Lamp_anq(fun(A,B),set(product_prod(A,B))),Uu) = collect(product_prod(A,B),product_case_prod(A,B,$o,aTP_Lamp_anp(fun(A,B),fun(A,fun(B,$o)),Uu))) ).

% ATP.lambda_22
tff(fact_8195_ATP_Olambda__23,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_add(nat,A),Uu) = aa(A,A,inverse_inverse(A),semiring_char_0_fact(A,Uu)) ) ).

% ATP.lambda_23
tff(fact_8196_ATP_Olambda__24,axiom,
    ! [B: $tType,Uu: list(B)] : aa(list(B),fun(nat,nat),aTP_Lamp_akt(list(B),fun(nat,nat)),Uu) = ord_max(nat,aa(list(B),nat,size_size(list(B)),Uu)) ).

% ATP.lambda_24
tff(fact_8197_ATP_Olambda__25,axiom,
    ! [A: $tType,Uu: list(A)] : aa(list(A),fun(nat,nat),aTP_Lamp_all(list(A),fun(nat,nat)),Uu) = ord_max(nat,aa(list(A),nat,size_size(list(A)),Uu)) ).

% ATP.lambda_25
tff(fact_8198_ATP_Olambda__26,axiom,
    ! [A: $tType] :
      ( semiring_1(A)
     => ! [Uu: nat] : aa(nat,heap_Time_Heap(A),aTP_Lamp_agk(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_26
tff(fact_8199_ATP_Olambda__27,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_ael(num,option(num)),Uu) = aa(num,option(num),some(num),aa(num,num,bit1,Uu)) ).

% ATP.lambda_27
tff(fact_8200_ATP_Olambda__28,axiom,
    ! [Uu: num] : aa(num,option(num),aTP_Lamp_aei(num,option(num)),Uu) = aa(num,option(num),some(num),bit0(Uu)) ).

% ATP.lambda_28
tff(fact_8201_ATP_Olambda__29,axiom,
    ! [Uu: int] : aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ags(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_29
tff(fact_8202_ATP_Olambda__30,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),aa(int,int,abs_abs(int),Uu)) ).

% ATP.lambda_30
tff(fact_8203_ATP_Olambda__31,axiom,
    ! [Uu: nat] : aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_ez(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_31
tff(fact_8204_ATP_Olambda__32,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(product_prod(heap_ext(product_unit),nat),heap_ext(product_unit)),aTP_Lamp_aki(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_akh(heap_ext(product_unit),fun(nat,heap_ext(product_unit)))) ).

% ATP.lambda_32
tff(fact_8205_ATP_Olambda__33,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(product_prod(heap_ext(product_unit),nat),nat),aTP_Lamp_ajz(A,fun(product_prod(heap_ext(product_unit),nat),nat)),Uu) = product_case_prod(heap_ext(product_unit),nat,nat,aTP_Lamp_ajy(heap_ext(product_unit),fun(nat,nat))) ).

% ATP.lambda_33
tff(fact_8206_ATP_Olambda__34,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,heap_Time_Heap(B),aTP_Lamp_se(A,heap_Time_Heap(B)),Uu) = aa(fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(B),heap_Time_Heap2(B),aTP_Lamp_sd(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat))))) ).

% ATP.lambda_34
tff(fact_8207_ATP_Olambda__35,axiom,
    ! [Uu: nat] : aa(nat,option(num),aTP_Lamp_aes(nat,option(num)),Uu) = aa(num,option(num),some(num),one2) ).

% ATP.lambda_35
tff(fact_8208_ATP_Olambda__36,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_aey(num,fun(nat,option(num)),Uu),Uua) = case_num(option(num),aa(num,option(num),some(num),one2),aTP_Lamp_aew(nat,fun(num,option(num)),Uua),aTP_Lamp_aex(nat,fun(num,option(num)),Uua),Uu) ).

% ATP.lambda_36
tff(fact_8209_ATP_Olambda__37,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_adl(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_37
tff(fact_8210_ATP_Olambda__38,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_ym(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_38
tff(fact_8211_ATP_Olambda__39,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] :
          aa(nat,A,aTP_Lamp_adk(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_39
tff(fact_8212_ATP_Olambda__40,axiom,
    ! [Uu: fun(nat,real),Uua: nat] :
      aa(nat,real,aTP_Lamp_va(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_40
tff(fact_8213_ATP_Olambda__41,axiom,
    ! [A: $tType,Uu: list(A),Uua: nat] :
      aa(nat,option(A),aTP_Lamp_ago(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_41
tff(fact_8214_ATP_Olambda__42,axiom,
    ! [Uu: int,Uua: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_agp(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)),aa(int,int,abs_abs(int),Uu))) ).

% ATP.lambda_42
tff(fact_8215_ATP_Olambda__43,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] :
          aa(nat,A,aTP_Lamp_yl(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_43
tff(fact_8216_ATP_Olambda__44,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_br(option(A),fun(A,$o),Uu),Uua)
        <=> case_option($o,A,$true,aTP_Lamp_bq(A,fun(A,$o),Uua),Uu) ) ) ).

% ATP.lambda_44
tff(fact_8217_ATP_Olambda__45,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aa(nat,fun(num,option(num)),aTP_Lamp_aez(nat,fun(num,option(num))),Uu),Uua) = case_nat(option(num),none(num),aTP_Lamp_aey(num,fun(nat,option(num)),Uua),Uu) ).

% ATP.lambda_45
tff(fact_8218_ATP_Olambda__46,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: option(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_bt(option(A),fun(A,$o),Uu),Uua)
        <=> case_option($o,A,$false,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_46
tff(fact_8219_ATP_Olambda__47,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_aew(nat,fun(num,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_aei(num,option(num)),bit_take_bit_num(Uu,Uua)) ).

% ATP.lambda_47
tff(fact_8220_ATP_Olambda__48,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_aeu(num,fun(nat,option(num)),Uu),Uua) = case_option(option(num),num,none(num),aTP_Lamp_aei(num,option(num)),bit_take_bit_num(Uua,Uu)) ).

% ATP.lambda_48
tff(fact_8221_ATP_Olambda__49,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,$o)),Uua: product_prod(A,B)] :
      ( aa(product_prod(A,B),$o,aTP_Lamp_afg(fun(A,fun(B,$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_49
tff(fact_8222_ATP_Olambda__50,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat] : aa(nat,list(A),aTP_Lamp_alr(list(list(A)),fun(nat,list(A)),Uu),Uua) = aa(list(nat),list(A),map(nat,A,aa(nat,fun(nat,A),aTP_Lamp_alq(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_50
tff(fact_8223_ATP_Olambda__51,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_ze(fun(nat,fun(nat,A)),fun(nat,A),Uu),Uua) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_zd(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_51
tff(fact_8224_ATP_Olambda__52,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_zc(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_zb(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua)),set_ord_atMost(nat,Uua)) ) ).

% ATP.lambda_52
tff(fact_8225_ATP_Olambda__53,axiom,
    ! [Uu: option(nat),Uua: list(vEBT_VEBTi)] :
      aa(list(vEBT_VEBTi),assn,aTP_Lamp_ap(option(nat),fun(list(vEBT_VEBTi),assn),Uu),Uua) = 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,summary),x14)),aa(list(vEBT_VEBTi),assn,snga_assn(vEBT_VEBTi,x13),Uua))),aa(list(vEBT_VEBTi),assn,vEBT_List_list_assn(vEBT_VEBT,vEBT_VEBTi,vEBT_vebt_assn_raw,treeList),Uua))),
        pure_assn(( ( x11 = 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)) )
          & ( x12 = aa(nat,nat,suc,aa(nat,nat,suc,va)) )
          & ( Uu = $ite(
                aa(nat,$o,ord_less(nat,ma),xa),
                aa(nat,option(nat),some(nat),ma),
                $let(
                  l: nat,
                  l:= vEBT_VEBT_low(xa,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(xa,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(summary,h),
                            $ite(
                              pr = none(nat),
                              $ite(aa(nat,$o,ord_less(nat,mi),xa),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) ) ) ) ) ) ))) ).

% ATP.lambda_53
tff(fact_8226_ATP_Olambda__54,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_abl(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_54
tff(fact_8227_ATP_Olambda__55,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_aas(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_55
tff(fact_8228_ATP_Olambda__56,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_aat(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_56
tff(fact_8229_ATP_Olambda__57,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_aap(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_57
tff(fact_8230_ATP_Olambda__58,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_zx(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_58
tff(fact_8231_ATP_Olambda__59,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_aev(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$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,$o,aa(real,fun(real,$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_59
tff(fact_8232_ATP_Olambda__60,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_aen(real,fun(real,$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))))
        & ( tan(real,Uua) = Uu ) ) ) ).

% ATP.lambda_60
tff(fact_8233_ATP_Olambda__61,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,int,aa(code_integer,fun(code_integer,int),aTP_Lamp_anu(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_61
tff(fact_8234_ATP_Olambda__62,axiom,
    ! [Uu: nat,Uua: nat] :
      aa(nat,a,aa(nat,fun(nat,a),aTP_Lamp_ew(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_62
tff(fact_8235_ATP_Olambda__63,axiom,
    ! [Uu: complex,Uua: real] :
      ( aa(real,$o,aTP_Lamp_afm(complex,fun(real,$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,aa(real,fun(real,$o),ord_less_eq(real),Uua),pi) ) ) ).

% ATP.lambda_63
tff(fact_8236_ATP_Olambda__64,axiom,
    ! [Uu: real,Uua: int] :
      ( aa(int,$o,aTP_Lamp_afh(real,fun(int,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$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_64
tff(fact_8237_ATP_Olambda__65,axiom,
    ! [Uu: rat,Uua: int] :
      ( aa(int,$o,aTP_Lamp_afj(rat,fun(int,$o),Uu),Uua)
    <=> ( aa(rat,$o,aa(rat,fun(rat,$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_65
tff(fact_8238_ATP_Olambda__66,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_abt(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_66
tff(fact_8239_ATP_Olambda__67,axiom,
    ! [A: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_aaq(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_67
tff(fact_8240_ATP_Olambda__68,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_yz(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_68
tff(fact_8241_ATP_Olambda__69,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_yj(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_69
tff(fact_8242_ATP_Olambda__70,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aaj(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_70
tff(fact_8243_ATP_Olambda__71,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_zs(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_71
tff(fact_8244_ATP_Olambda__72,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_acy(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_72
tff(fact_8245_ATP_Olambda__73,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_to(set(A),fun(set(A),$o)),Uu),Uua)
    <=> ( aa(set(A),$o,ord_less(set(A),Uu),Uua)
        & finite_finite2(A,Uua) ) ) ).

% ATP.lambda_73
tff(fact_8246_ATP_Olambda__74,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_yx(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_74
tff(fact_8247_ATP_Olambda__75,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_yw(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_75
tff(fact_8248_ATP_Olambda__76,axiom,
    ! [Uu: nat,Uua: complex] :
      ( aa(complex,$o,aTP_Lamp_vo(nat,fun(complex,$o),Uu),Uua)
    <=> ( aa(nat,complex,aa(complex,fun(nat,complex),power_power(complex),Uua),Uu) = one_one(complex) ) ) ).

% ATP.lambda_76
tff(fact_8249_ATP_Olambda__77,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: nat,Uua: A] :
          ( aa(A,$o,aTP_Lamp_so(nat,fun(A,$o),Uu),Uua)
        <=> ( aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uu) = one_one(A) ) ) ) ).

% ATP.lambda_77
tff(fact_8250_ATP_Olambda__78,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_agm(A,fun(A,$o),Uu),Uua)
        <=> ( member(A,Uua,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),aa(A,A,abs_abs(A),Uua)),Uu) ) ) ) ).

% ATP.lambda_78
tff(fact_8251_ATP_Olambda__79,axiom,
    ! [Uu: product_prod(nat,nat),Uua: nat] : aa(nat,heap_Time_Heap(option(nat)),aTP_Lamp_al(product_prod(nat,nat),fun(nat,heap_Time_Heap(option(nat))),Uu),Uua) = heap_Time_bind(nat,option(nat),vEBT_VEBT_highi(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),x12),aa(num,nat,numeral_numeral(nat),bit0(one2)))),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aTP_Lamp_ak(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),Uu),Uua)) ).

% ATP.lambda_79
tff(fact_8252_ATP_Olambda__80,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_abs(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_80
tff(fact_8253_ATP_Olambda__81,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_aer(real,fun(real,$o),Uu),Uua)
    <=> ( aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),zero_zero(real)),Uua)
        & aa(real,$o,aa(real,fun(real,$o),ord_less_eq(real),Uua),pi)
        & ( cos(real,Uua) = Uu ) ) ) ).

% ATP.lambda_81
tff(fact_8254_ATP_Olambda__82,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(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_82
tff(fact_8255_ATP_Olambda__83,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_uz(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_83
tff(fact_8256_ATP_Olambda__84,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_uu(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_84
tff(fact_8257_ATP_Olambda__85,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_adv(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_85
tff(fact_8258_ATP_Olambda__86,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_adt(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_86
tff(fact_8259_ATP_Olambda__87,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_tv(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_87
tff(fact_8260_ATP_Olambda__88,axiom,
    ! [A: $tType] :
      ( real_V2822296259951069270ebra_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_abo(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_88
tff(fact_8261_ATP_Olambda__89,axiom,
    ! [A: $tType] :
      ( division_ring(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_tw(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_89
tff(fact_8262_ATP_Olambda__90,axiom,
    ! [A: $tType] :
      ( ( ring_1(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_acc(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_90
tff(fact_8263_ATP_Olambda__91,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_uy(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_91
tff(fact_8264_ATP_Olambda__92,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ux(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_92
tff(fact_8265_ATP_Olambda__93,axiom,
    ! [A: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_wp(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_93
tff(fact_8266_ATP_Olambda__94,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_aor(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ( aa(A,$o,Uu,Uua)
            & ! [Y3: A] :
                ( aa(A,$o,Uu,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Y3) ) ) ) ) ).

% ATP.lambda_94
tff(fact_8267_ATP_Olambda__95,axiom,
    ! [A: $tType] :
      ( order(A)
     => ! [Uu: fun(A,$o),Uua: A] :
          ( aa(A,$o,aTP_Lamp_afl(fun(A,$o),fun(A,$o),Uu),Uua)
        <=> ( aa(A,$o,Uu,Uua)
            & ! [Y3: A] :
                ( aa(A,$o,Uu,Y3)
               => aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Y3),Uua) ) ) ) ) ).

% ATP.lambda_95
tff(fact_8268_ATP_Olambda__96,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_yt(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_96
tff(fact_8269_ATP_Olambda__97,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,fun(nat,A)),Uua: nat] : aa(nat,A,aTP_Lamp_yq(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_97
tff(fact_8270_ATP_Olambda__98,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(B,A),aTP_Lamp_anm(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_98
tff(fact_8271_ATP_Olambda__99,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,complex,aTP_Lamp_ada(fun(A,real),fun(A,complex),Uu),Uua) = complex2(aa(A,real,Uu,Uua),zero_zero(real)) ).

% ATP.lambda_99
tff(fact_8272_ATP_Olambda__100,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: B] : aa(B,list(A),aTP_Lamp_alt(fun(B,A),fun(B,list(A)),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),aa(B,A,Uu,Uua)),nil(A)) ).

% ATP.lambda_100
tff(fact_8273_ATP_Olambda__101,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ur(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = zero_zero(B) ) ) ) ).

% ATP.lambda_101
tff(fact_8274_ATP_Olambda__102,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: fun(A,B),Uua: A] :
          ( aa(A,$o,aTP_Lamp_ya(fun(A,B),fun(A,$o),Uu),Uua)
        <=> ( aa(A,B,Uu,Uua) = one_one(B) ) ) ) ).

% ATP.lambda_102
tff(fact_8275_ATP_Olambda__103,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_ajw(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_ajv(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_103
tff(fact_8276_ATP_Olambda__104,axiom,
    ! [Uu: code_integer,Uua: $o] : aa($o,char,aa(code_integer,fun($o,char),aTP_Lamp_apj(code_integer,fun($o,char)),Uu),(Uua)) = aa(product_prod(code_integer,$o),char,product_case_prod(code_integer,$o,char,aTP_Lamp_api($o,fun(code_integer,fun($o,char)),(Uua))),code_bit_cut_integer(Uu)) ).

% ATP.lambda_104
tff(fact_8277_ATP_Olambda__105,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_rh(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_rg(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_105
tff(fact_8278_ATP_Olambda__106,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_qs(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_qr(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_106
tff(fact_8279_ATP_Olambda__107,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_qk(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_qj(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_107
tff(fact_8280_ATP_Olambda__108,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($o),aTP_Lamp_pk(product_prod(vEBT_VEBTi,nat),fun(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($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($o),aTP_Lamp_pj(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),Uua)),Uu) ).

% ATP.lambda_108
tff(fact_8281_ATP_Olambda__109,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),list(list(A)),aTP_Lamp_alv(list(A),fun(list(A),list(list(A))),Uu),Uua) = aa(list(A),list(list(A)),map(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_alu(list(A),fun(A,list(A))),Uua)),Uu) ).

% ATP.lambda_109
tff(fact_8282_ATP_Olambda__110,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($o),aTP_Lamp_qc(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),fun(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($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_qb(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_110
tff(fact_8283_ATP_Olambda__111,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_oj(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_oi(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_111
tff(fact_8284_ATP_Olambda__112,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_nq(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_np(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_112
tff(fact_8285_ATP_Olambda__113,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_mz(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_my(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_113
tff(fact_8286_ATP_Olambda__114,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_adc(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_114
tff(fact_8287_ATP_Olambda__115,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_adi(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_115
tff(fact_8288_ATP_Olambda__116,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_adf(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_116
tff(fact_8289_ATP_Olambda__117,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ade(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_117
tff(fact_8290_ATP_Olambda__118,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_tp(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_118
tff(fact_8291_ATP_Olambda__119,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: num,Uua: num] : aa(num,word(A),aTP_Lamp_aeh(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_119
tff(fact_8292_ATP_Olambda__120,axiom,
    ! [Uu: num,Uua: num] : aa(num,int,aTP_Lamp_aej(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_120
tff(fact_8293_ATP_Olambda__121,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_abi(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_121
tff(fact_8294_ATP_Olambda__122,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_afk(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),aa(real,real,abs_abs(real),Uua)),Uu)) ).

% ATP.lambda_122
tff(fact_8295_ATP_Olambda__123,axiom,
    ! [Uu: $o,Uua: nat] : aa(nat,nat,aa($o,fun(nat,nat),aTP_Lamp_apb($o,fun(nat,nat)),(Uu)),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),aa($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_123
tff(fact_8296_ATP_Olambda__124,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_abf(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_124
tff(fact_8297_ATP_Olambda__125,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_abg(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_125
tff(fact_8298_ATP_Olambda__126,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_abk(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_126
tff(fact_8299_ATP_Olambda__127,axiom,
    ! [Uu: real,Uua: nat] : aa(nat,real,aTP_Lamp_abj(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_127
tff(fact_8300_ATP_Olambda__128,axiom,
    ! [Uu: option(nat),Uua: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap(option(nat)),aTP_Lamp_ab(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),Uu),Uua) = heap_Time_bind(option(nat),option(nat),vEBT_vebt_maxti(Uua),aTP_Lamp_aa(option(nat),fun(option(nat),heap_Time_Heap(option(nat))),Uu)) ).

% ATP.lambda_128
tff(fact_8301_ATP_Olambda__129,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: A] : aa(A,list(list(A)),aTP_Lamp_alw(list(list(A)),fun(A,list(list(A))),Uu),Uua) = aa(list(list(A)),list(list(A)),map(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua)),product_lists(A,Uu)) ).

% ATP.lambda_129
tff(fact_8302_ATP_Olambda__130,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,num,aa(code_integer,fun(code_integer,num),aTP_Lamp_ant(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_130
tff(fact_8303_ATP_Olambda__131,axiom,
    ! [Uu: code_integer,Uua: code_integer] :
      aa(code_integer,nat,aa(code_integer,fun(code_integer,nat),aTP_Lamp_ans(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_131
tff(fact_8304_ATP_Olambda__132,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat] : aa(nat,A,aTP_Lamp_yk(nat,fun(nat,A),Uu),Uua) = aa(nat,A,gbinomial(A,aa(nat,A,semiring_1_of_nat(A),Uua)),Uu) ) ).

% ATP.lambda_132
tff(fact_8305_ATP_Olambda__133,axiom,
    ! [A: $tType,B: $tType,Uu: list(B),Uua: A] : aa(A,list(product_prod(A,B)),aTP_Lamp_alz(list(B),fun(A,list(product_prod(A,B))),Uu),Uua) = aa(list(B),list(product_prod(A,B)),map(B,product_prod(A,B),aa(A,fun(B,product_prod(A,B)),product_Pair(A,B),Uua)),Uu) ).

% ATP.lambda_133
tff(fact_8306_ATP_Olambda__134,axiom,
    ! [Uu: real,Uua: real] :
      ( aa(real,$o,aTP_Lamp_aep(real,fun(real,$o),Uu),Uua)
    <=> ( exp(real,Uua) = Uu ) ) ).

% ATP.lambda_134
tff(fact_8307_ATP_Olambda__135,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_lx(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_135
tff(fact_8308_ATP_Olambda__136,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_ly(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_136
tff(fact_8309_ATP_Olambda__137,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_xe(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_137
tff(fact_8310_ATP_Olambda__138,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_akw(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_138
tff(fact_8311_ATP_Olambda__139,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: list(A),Uua: A] :
          ( aa(A,$o,aTP_Lamp_akx(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_139
tff(fact_8312_ATP_Olambda__140,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aac(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_140
tff(fact_8313_ATP_Olambda__141,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_agi(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_141
tff(fact_8314_ATP_Olambda__142,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_agh(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_142
tff(fact_8315_ATP_Olambda__143,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ajc(set(A),fun(A,$o),Uu),Uua)
    <=> ( Uu = aa(set(A),set(A),insert(A,Uua),bot_bot(set(A))) ) ) ).

% ATP.lambda_143
tff(fact_8316_ATP_Olambda__144,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_ajo(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_144
tff(fact_8317_ATP_Olambda__145,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,product_prod(A,B),aTP_Lamp_ann(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_145
tff(fact_8318_ATP_Olambda__146,axiom,
    ! [A: $tType] :
      ( bit_un5681908812861735899ations(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aiz(A,fun(nat,A),Uu),Uua) = aa(A,A,aa(nat,fun(A,A),bit_se4730199178511100633sh_bit(A),Uua),aa($o,A,zero_neq_one_of_bool(A),aa(nat,$o,aa(A,fun(nat,$o),bit_se5641148757651400278ts_bit(A),Uu),Uua))) ) ).

% ATP.lambda_146
tff(fact_8319_ATP_Olambda__147,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aa(int,fun(int,int),aTP_Lamp_aef(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_147
tff(fact_8320_ATP_Olambda__148,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] :
      ( aa(list(A),$o,aTP_Lamp_alp(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_148
tff(fact_8321_ATP_Olambda__149,axiom,
    ! [A: $tType] :
      ( ( semiring_char_0(A)
        & semidom_divide(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aam(A,fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_149
tff(fact_8322_ATP_Olambda__150,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_aec(A,fun(nat,A),Uu),Uua) = aa(A,A,minus_minus(A,Uu),aa(nat,A,semiring_1_of_nat(A),Uua)) ) ).

% ATP.lambda_150
tff(fact_8323_ATP_Olambda__151,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_yg(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_151
tff(fact_8324_ATP_Olambda__152,axiom,
    ! [Uu: nat,Uua: vEBT_VEBTi] : aa(vEBT_VEBTi,heap_Time_Heap($o),aTP_Lamp_ed(nat,fun(vEBT_VEBTi,heap_Time_Heap($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_152
tff(fact_8325_ATP_Olambda__153,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A)] :
      ( aa(set(A),$o,aTP_Lamp_sr(set(A),fun(set(A),$o),Uu),Uua)
    <=> aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),Uua),Uu) ) ).

% ATP.lambda_153
tff(fact_8326_ATP_Olambda__154,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_sp(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),Uu) ) ).

% ATP.lambda_154
tff(fact_8327_ATP_Olambda__155,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_yn(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uua),Uu) ) ) ).

% ATP.lambda_155
tff(fact_8328_ATP_Olambda__156,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aif(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),modulo_modulo(nat),Uua),Uu) ).

% ATP.lambda_156
tff(fact_8329_ATP_Olambda__157,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ahk(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_157
tff(fact_8330_ATP_Olambda__158,axiom,
    ! [A: $tType] :
      ( field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_and(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),divide_divide(A),Uua),Uu) ) ).

% ATP.lambda_158
tff(fact_8331_ATP_Olambda__159,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_sq(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,ord_less(nat,Uua),Uu) ) ).

% ATP.lambda_159
tff(fact_8332_ATP_Olambda__160,axiom,
    ! [A: $tType] :
      ( preorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_bq(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,ord_less(A,Uua),Uu) ) ) ).

% ATP.lambda_160
tff(fact_8333_ATP_Olambda__161,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_wk(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,ord_less(A,Uua),Uu) ) ) ).

% ATP.lambda_161
tff(fact_8334_ATP_Olambda__162,axiom,
    ! [A: $tType] :
      ( ab_semigroup_mult(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_bs(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_162
tff(fact_8335_ATP_Olambda__163,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ahz(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),times_times(A),Uua),Uu) ) ).

% ATP.lambda_163
tff(fact_8336_ATP_Olambda__164,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aij(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,minus_minus(nat,Uua),Uu) ).

% ATP.lambda_164
tff(fact_8337_ATP_Olambda__165,axiom,
    ! [A: $tType] :
      ( linordered_idom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ahi(A,fun(A,A),Uu),Uua) = aa(A,A,minus_minus(A,Uua),Uu) ) ).

% ATP.lambda_165
tff(fact_8338_ATP_Olambda__166,axiom,
    ! [A: $tType] :
      ( ab_group_add(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_166
tff(fact_8339_ATP_Olambda__167,axiom,
    ! [Uu: nat,Uua: real] : aa(real,real,aTP_Lamp_apm(nat,fun(real,real),Uu),Uua) = aa(nat,real,aa(real,fun(nat,real),power_power(real),Uua),Uu) ).

% ATP.lambda_167
tff(fact_8340_ATP_Olambda__168,axiom,
    ! [Uu: code_integer,Uua: code_integer] : aa(code_integer,code_integer,aTP_Lamp_aii(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_168
tff(fact_8341_ATP_Olambda__169,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_ajl(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uu) ).

% ATP.lambda_169
tff(fact_8342_ATP_Olambda__170,axiom,
    ! [Uu: int,Uua: int] : aa(int,int,aTP_Lamp_aih(int,fun(int,int),Uu),Uua) = aa(int,int,aa(int,fun(int,int),plus_plus(int),Uua),Uu) ).

% ATP.lambda_170
tff(fact_8343_ATP_Olambda__171,axiom,
    ! [A: $tType] :
      ( cancel_semigroup_add(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_anf(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_171
tff(fact_8344_ATP_Olambda__172,axiom,
    ! [A: $tType] :
      ( linordered_semidom(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_ahh(A,fun(A,A),Uu),Uua) = aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu) ) ).

% ATP.lambda_172
tff(fact_8345_ATP_Olambda__173,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: list(A),Uua: array(A)] : aa(array(A),assn,aTP_Lamp_pl(list(A),fun(array(A),assn),Uu),Uua) = aa(list(A),assn,snga_assn(A,Uua),Uu) ) ).

% ATP.lambda_173
tff(fact_8346_ATP_Olambda__174,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_sn(nat,fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,dvd_dvd(nat,Uua),Uu) ) ).

% ATP.lambda_174
tff(fact_8347_ATP_Olambda__175,axiom,
    ! [Uu: int,Uua: int] :
      ( aa(int,$o,aTP_Lamp_sz(int,fun(int,$o),Uu),Uua)
    <=> aa(int,$o,dvd_dvd(int,Uua),Uu) ) ).

% ATP.lambda_175
tff(fact_8348_ATP_Olambda__176,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ns(A,fun(A,$o),Uu),Uua)
        <=> aa(A,$o,dvd_dvd(A,Uua),Uu) ) ) ).

% ATP.lambda_176
tff(fact_8349_ATP_Olambda__177,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_ek(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_177
tff(fact_8350_ATP_Olambda__178,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,product_prod(nat,A),aTP_Lamp_ajq(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_178
tff(fact_8351_ATP_Olambda__179,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_ti(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_179
tff(fact_8352_ATP_Olambda__180,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_yo(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,binomial(Uua),Uu) ).

% ATP.lambda_180
tff(fact_8353_ATP_Olambda__181,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] : aa(A,list(A),aa(list(A),fun(A,list(A)),aTP_Lamp_alu(list(A),fun(A,list(A))),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uu) ).

% ATP.lambda_181
tff(fact_8354_ATP_Olambda__182,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_av(set(A),fun(A,$o),Uu),Uua)
    <=> member(A,Uua,Uu) ) ).

% ATP.lambda_182
tff(fact_8355_ATP_Olambda__183,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A)] : aa(list(A),A,aTP_Lamp_alo(nat,fun(list(A),A),Uu),Uua) = aa(nat,A,nth(A,Uua),Uu) ).

% ATP.lambda_183
tff(fact_8356_ATP_Olambda__184,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_dc(A,fun(A,$o),Uu),Uua)
    <=> ( Uua = Uu ) ) ).

% ATP.lambda_184
tff(fact_8357_ATP_Olambda__185,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_alx(A,fun(list(A),list(A))),Uu),Uua) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),nil(A)) ).

% ATP.lambda_185
tff(fact_8358_ATP_Olambda__186,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(list(A)),aa(A,fun(list(A),list(list(A))),aTP_Lamp_aly(A,fun(list(A),list(list(A)))),Uu),Uua) = aa(list(list(A)),list(list(A)),aa(list(A),fun(list(list(A)),list(list(A))),cons(list(A)),Uua),nil(list(A))) ).

% ATP.lambda_186
tff(fact_8359_ATP_Olambda__187,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] :
      ( aa(A,$o,aTP_Lamp_tc(fun(A,nat),fun(A,$o),Uu),Uua)
    <=> aa(nat,$o,ord_less(nat,zero_zero(nat)),aa(A,nat,Uu,Uua)) ) ).

% ATP.lambda_187
tff(fact_8360_ATP_Olambda__188,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_xb(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_188
tff(fact_8361_ATP_Olambda__189,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_xa(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_189
tff(fact_8362_ATP_Olambda__190,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_ox(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_190
tff(fact_8363_ATP_Olambda__191,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_rl(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_191
tff(fact_8364_ATP_Olambda__192,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)),$o,aTP_Lamp_rq(fun(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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_192
tff(fact_8365_ATP_Olambda__193,axiom,
    ! [Uu: fun(real,real),Uua: real] : aa(real,real,aTP_Lamp_aqf(fun(real,real),fun(real,real),Uu),Uua) = aa(real,real,Uu,aa(real,real,uminus_uminus(real),Uua)) ).

% ATP.lambda_193
tff(fact_8366_ATP_Olambda__194,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(B,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_er(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_194
tff(fact_8367_ATP_Olambda__195,axiom,
    ! [C: $tType,B: $tType,A: $tType,Uu: fun(A,C),Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_eo(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_195
tff(fact_8368_ATP_Olambda__196,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_ri(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_196
tff(fact_8369_ATP_Olambda__197,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_ru(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_197
tff(fact_8370_ATP_Olambda__198,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)),$o,aTP_Lamp_ry(fun(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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_198
tff(fact_8371_ATP_Olambda__199,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_amq(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> aa(nat,$o,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_199
tff(fact_8372_ATP_Olambda__200,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_wb(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_200
tff(fact_8373_ATP_Olambda__201,axiom,
    ! [A: $tType] :
      ( ( topolo5987344860129210374id_add(A)
        & topological_t2_space(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_wc(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_201
tff(fact_8374_ATP_Olambda__202,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_xx(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_202
tff(fact_8375_ATP_Olambda__203,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_up(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ) ).

% ATP.lambda_203
tff(fact_8376_ATP_Olambda__204,axiom,
    ! [A: $tType,Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_ajp(fun(nat,A),fun(nat,A),Uu),Uua) = aa(nat,A,Uu,aa(nat,nat,suc,Uua)) ).

% ATP.lambda_204
tff(fact_8377_ATP_Olambda__205,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: fun(A,B)] : aa(fun(A,B),B,aTP_Lamp_afw(A,fun(fun(A,B),B),Uu),Uua) = aa(A,B,Uua,Uu) ).

% ATP.lambda_205
tff(fact_8378_ATP_Olambda__206,axiom,
    ! [B: $tType,A: $tType,Uu: fun(set(A),A),Uua: B] : aa(B,A,aTP_Lamp_agb(fun(set(A),A),fun(B,A),Uu),Uua) = aa(set(A),A,Uu,bot_bot(set(A))) ).

% ATP.lambda_206
tff(fact_8379_ATP_Olambda__207,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: option(A)] : aa(option(A),option(A),aa(A,fun(option(A),option(A)),aTP_Lamp_amx(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_207
tff(fact_8380_ATP_Olambda__208,axiom,
    ! [Uu: nat,Uua: num] : aa(num,option(num),aTP_Lamp_aex(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_208
tff(fact_8381_ATP_Olambda__209,axiom,
    ! [Uu: num,Uua: nat] : aa(nat,option(num),aTP_Lamp_aem(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_209
tff(fact_8382_ATP_Olambda__210,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option($o)),aTP_Lamp_amg(list(A),fun(list(A),fun(product_prod(A,list(A)),option($o))),Uu),Uua) = product_case_prod(A,list(A),option($o),aa(list(A),fun(A,fun(list(A),option($o))),aTP_Lamp_amf(list(A),fun(list(A),fun(A,fun(list(A),option($o)))),Uu),Uua)) ).

% ATP.lambda_210
tff(fact_8383_ATP_Olambda__211,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A))))),aTP_Lamp_amd(A,fun(list(A),fun(product_prod(A,list(A)),option(product_prod(list(A),product_prod(A,list(A)))))),Uu),Uua) = product_case_prod(A,list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_amc(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua)) ).

% ATP.lambda_211
tff(fact_8384_ATP_Olambda__212,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_akz(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_aky(fun(A,B),fun(list(A),fun(list(A),fun(list(A),list(A)))),Uu),Uua)) ) ).

% ATP.lambda_212
tff(fact_8385_ATP_Olambda__213,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_akb(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_aka(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_213
tff(fact_8386_ATP_Olambda__214,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,assn)),Uua: A] : aa(A,assn,aTP_Lamp_cx(fun(B,fun(A,assn)),fun(A,assn),Uu),Uua) = ex_assn(B,aa(A,fun(B,assn),aTP_Lamp_cw(fun(B,fun(A,assn)),fun(A,fun(B,assn)),Uu),Uua)) ).

% ATP.lambda_214
tff(fact_8387_ATP_Olambda__215,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(nat,A),Uua: A] : aa(A,A,aTP_Lamp_aqo(fun(nat,A),fun(A,A),Uu),Uua) = suminf(A,aa(A,fun(nat,A),aTP_Lamp_acv(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua)) ) ).

% ATP.lambda_215
tff(fact_8388_ATP_Olambda__216,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,complex,aTP_Lamp_adm(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_216
tff(fact_8389_ATP_Olambda__217,axiom,
    ! [Uu: nat,Uua: option(nat)] : aa(option(nat),heap_Time_Heap(option(nat)),aTP_Lamp_a(nat,fun(option(nat),heap_Time_Heap(option(nat))),Uu),Uua) = 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),x12),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),aa(nat,option(nat),some(nat),Uu))),Uua)) ).

% ATP.lambda_217
tff(fact_8390_ATP_Olambda__218,axiom,
    ! [Uu: option(nat),Uua: option(nat)] : aa(option(nat),heap_Time_Heap(option(nat)),aTP_Lamp_aa(option(nat),fun(option(nat),heap_Time_Heap(option(nat))),Uu),Uua) = 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),x12),aa(num,nat,numeral_numeral(nat),bit0(one2)))))),Uu)),Uua)) ).

% ATP.lambda_218
tff(fact_8391_ATP_Olambda__219,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_sf(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_219
tff(fact_8392_ATP_Olambda__220,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_amv(fun(B,option(A)),fun(B,$o),Uu),Uua)
    <=> ( aa(B,option(A),Uu,Uua) != none(A) ) ) ).

% ATP.lambda_220
tff(fact_8393_ATP_Olambda__221,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_adg(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_221
tff(fact_8394_ATP_Olambda__222,axiom,
    ! [Uu: nat,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aeg(nat,fun(nat,$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_222
tff(fact_8395_ATP_Olambda__223,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_abh(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_223
tff(fact_8396_ATP_Olambda__224,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_aci(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_224
tff(fact_8397_ATP_Olambda__225,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: A,Uua: nat] : aa(nat,real,aTP_Lamp_acj(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_225
tff(fact_8398_ATP_Olambda__226,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_aag(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_226
tff(fact_8399_ATP_Olambda__227,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_xc(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_227
tff(fact_8400_ATP_Olambda__228,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),heap_Time_Heap(list(A)),aTP_Lamp_aft(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),aa(A,fun(list(A),list(A)),cons(A),Uu),Uua)) ).

% ATP.lambda_228
tff(fact_8401_ATP_Olambda__229,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_aqg(A,fun(A,A),Uu),Uua) = cos(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Uua),Uu)) ) ).

% ATP.lambda_229
tff(fact_8402_ATP_Olambda__230,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_amo(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,ord_min(nat,Uu,Uua)) ).

% ATP.lambda_230
tff(fact_8403_ATP_Olambda__231,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_amn(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,ord_min(nat,Uua,Uu)) ).

% ATP.lambda_231
tff(fact_8404_ATP_Olambda__232,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_aff(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,Uu),Uua)) ).

% ATP.lambda_232
tff(fact_8405_ATP_Olambda__233,axiom,
    ! [Uu: nat,Uua: nat] : aa(nat,nat,aTP_Lamp_afe(nat,fun(nat,nat),Uu),Uua) = aa(nat,nat,suc,aa(nat,nat,ord_max(nat,Uua),Uu)) ).

% ATP.lambda_233
tff(fact_8406_ATP_Olambda__234,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: A] :
          ( aa(A,$o,aTP_Lamp_ami(A,fun(A,$o),Uu),Uua)
        <=> ~ aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu),Uua) ) ) ).

% ATP.lambda_234
tff(fact_8407_ATP_Olambda__235,axiom,
    ! [A: $tType,Uu: set(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aar(set(A),fun(A,$o),Uu),Uua)
    <=> ~ member(A,Uua,Uu) ) ).

% ATP.lambda_235
tff(fact_8408_ATP_Olambda__236,axiom,
    ! [A: $tType,Uu: A,Uua: A] :
      ( aa(A,$o,aTP_Lamp_akn(A,fun(A,$o),Uu),Uua)
    <=> ( Uua != Uu ) ) ).

% ATP.lambda_236
tff(fact_8409_ATP_Olambda__237,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_ach(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_237
tff(fact_8410_ATP_Olambda__238,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V4412858255891104859lgebra(A) )
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_abz(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_238
tff(fact_8411_ATP_Olambda__239,axiom,
    ! [A: $tType] :
      ( real_Vector_banach(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,real,aTP_Lamp_abw(fun(nat,A),fun(nat,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_239
tff(fact_8412_ATP_Olambda__240,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_ub(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_240
tff(fact_8413_ATP_Olambda__241,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2822296259951069270ebra_1(A) )
     => ! [Uu: fun(B,A),Uua: B] : aa(B,real,aTP_Lamp_xl(fun(B,A),fun(B,real),Uu),Uua) = real_V7770717601297561774m_norm(A,aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_241
tff(fact_8414_ATP_Olambda__242,axiom,
    ! [B: $tType,A: $tType] :
      ( ( real_V8999393235501362500lgebra(B)
        & comm_semiring_1(B) )
     => ! [Uu: fun(A,B),Uua: A] : aa(A,real,aTP_Lamp_xm(fun(A,B),fun(A,real),Uu),Uua) = real_V7770717601297561774m_norm(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_242
tff(fact_8415_ATP_Olambda__243,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_aqe(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,inverse_inverse(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_243
tff(fact_8416_ATP_Olambda__244,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_xh(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_244
tff(fact_8417_ATP_Olambda__245,axiom,
    ! [A: $tType,B: $tType] :
      ( semiring_1(A)
     => ! [Uu: fun(B,nat),Uua: B] : aa(B,A,aTP_Lamp_tr(fun(B,nat),fun(B,A),Uu),Uua) = aa(nat,A,semiring_1_of_nat(A),aa(B,nat,Uu,Uua)) ) ).

% ATP.lambda_245
tff(fact_8418_ATP_Olambda__246,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,real,aTP_Lamp_ec(fun(A,nat),fun(A,real),Uu),Uua) = aa(nat,real,semiring_1_of_nat(real),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_246
tff(fact_8419_ATP_Olambda__247,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A] : aa(A,int,aTP_Lamp_vl(fun(A,nat),fun(A,int),Uu),Uua) = aa(nat,int,semiring_1_of_nat(int),aa(A,nat,Uu,Uua)) ).

% ATP.lambda_247
tff(fact_8420_ATP_Olambda__248,axiom,
    ! [A: $tType,Uu: fun(A,real),Uua: A] : aa(A,real,aTP_Lamp_aau(fun(A,real),fun(A,real),Uu),Uua) = aa(real,real,ln_ln(real),aa(A,real,Uu,Uua)) ).

% ATP.lambda_248
tff(fact_8421_ATP_Olambda__249,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,complex,aTP_Lamp_aco(fun(nat,real),fun(nat,complex),Uu),Uua) = real_Vector_of_real(complex,aa(nat,real,Uu,Uua)) ).

% ATP.lambda_249
tff(fact_8422_ATP_Olambda__250,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_acu(fun(nat,real),fun(nat,A),Uu),Uua) = real_Vector_of_real(A,aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_250
tff(fact_8423_ATP_Olambda__251,axiom,
    ! [A: $tType] :
      ( ( real_V2191834092415804123ebra_1(A)
        & real_V822414075346904944vector(A) )
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_act(fun(nat,real),fun(nat,A),Uu),Uua) = real_Vector_of_real(A,aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_251
tff(fact_8424_ATP_Olambda__252,axiom,
    ! [A: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu: fun(nat,real),Uua: nat] : aa(nat,A,aTP_Lamp_acr(fun(nat,real),fun(nat,A),Uu),Uua) = real_Vector_of_real(A,aa(nat,real,Uu,Uua)) ) ).

% ATP.lambda_252
tff(fact_8425_ATP_Olambda__253,axiom,
    ! [A: $tType,B: $tType] :
      ( real_V2191834092415804123ebra_1(A)
     => ! [Uu: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_acp(fun(B,real),fun(B,A),Uu),Uua) = real_Vector_of_real(A,aa(B,real,Uu,Uua)) ) ).

% ATP.lambda_253
tff(fact_8426_ATP_Olambda__254,axiom,
    ! [A: $tType,B: $tType] :
      ( ( comm_monoid_mult(A)
        & real_V2191834092415804123ebra_1(A) )
     => ! [Uu: fun(B,real),Uua: B] : aa(B,A,aTP_Lamp_acq(fun(B,real),fun(B,A),Uu),Uua) = real_Vector_of_real(A,aa(B,real,Uu,Uua)) ) ).

% ATP.lambda_254
tff(fact_8427_ATP_Olambda__255,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_aai(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_255
tff(fact_8428_ATP_Olambda__256,axiom,
    ! [A: $tType] :
      ( ordered_ab_group_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_adb(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_256
tff(fact_8429_ATP_Olambda__257,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,A,aTP_Lamp_acs(fun(nat,A),fun(nat,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(nat,A,Uu,Uua)) ) ).

% ATP.lambda_257
tff(fact_8430_ATP_Olambda__258,axiom,
    ! [A: $tType,B: $tType] :
      ( ab_group_add(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_aah(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_258
tff(fact_8431_ATP_Olambda__259,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_aqd(fun(A,A),fun(A,A),Uu),Uua) = aa(A,A,uminus_uminus(A),aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_259
tff(fact_8432_ATP_Olambda__260,axiom,
    ! [A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_yd(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_260
tff(fact_8433_ATP_Olambda__261,axiom,
    ! [A: $tType,B: $tType] :
      ( comm_ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_xi(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_261
tff(fact_8434_ATP_Olambda__262,axiom,
    ! [A: $tType,B: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(B,int),Uua: B] : aa(B,A,aTP_Lamp_ts(fun(B,int),fun(B,A),Uu),Uua) = aa(int,A,ring_1_of_int(A),aa(B,int,Uu,Uua)) ) ).

% ATP.lambda_262
tff(fact_8435_ATP_Olambda__263,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: B] : aa(B,heap_Time_Heap(A),aTP_Lamp_anv(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_263
tff(fact_8436_ATP_Olambda__264,axiom,
    ! [A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: fun(nat,A),Uua: nat] : aa(nat,fun(A,A),aTP_Lamp_uv(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_264
tff(fact_8437_ATP_Olambda__265,axiom,
    ! [Uu: fun(nat,real),Uua: nat] : aa(nat,real,aTP_Lamp_abu(fun(nat,real),fun(nat,real),Uu),Uua) = aa(real,real,abs_abs(real),aa(nat,real,Uu,Uua)) ).

% ATP.lambda_265
tff(fact_8438_ATP_Olambda__266,axiom,
    ! [A: $tType,B: $tType] :
      ( ordere166539214618696060dd_abs(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_abm(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_266
tff(fact_8439_ATP_Olambda__267,axiom,
    ! [A: $tType,B: $tType] :
      ( linordered_field(A)
     => ! [Uu: fun(B,A),Uua: B] : aa(B,A,aTP_Lamp_abp(fun(B,A),fun(B,A),Uu),Uua) = aa(A,A,abs_abs(A),aa(B,A,Uu,Uua)) ) ).

% ATP.lambda_267
tff(fact_8440_ATP_Olambda__268,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_apw(fun(A,A),fun(A,A),Uu),Uua) = sinh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_268
tff(fact_8441_ATP_Olambda__269,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_apx(fun(A,A),fun(A,A),Uu),Uua) = cosh(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_269
tff(fact_8442_ATP_Olambda__270,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_aqj(fun(A,A),fun(A,A),Uu),Uua) = sin(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_270
tff(fact_8443_ATP_Olambda__271,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_aqk(fun(A,A),fun(A,A),Uu),Uua) = exp(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_271
tff(fact_8444_ATP_Olambda__272,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_acz(fun(A,B),fun(A,B),Uu),Uua) = exp(B,aa(A,B,Uu,Uua)) ) ).

% ATP.lambda_272
tff(fact_8445_ATP_Olambda__273,axiom,
    ! [A: $tType] :
      ( ( real_Vector_banach(A)
        & real_V3459762299906320749_field(A) )
     => ! [Uu: fun(A,A),Uua: A] : aa(A,A,aTP_Lamp_aqh(fun(A,A),fun(A,A),Uu),Uua) = cos(A,aa(A,A,Uu,Uua)) ) ).

% ATP.lambda_273
tff(fact_8446_ATP_Olambda__274,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,fun(B,assn)),Uua: A] : aa(A,assn,aTP_Lamp_bh(fun(A,fun(B,assn)),fun(A,assn),Uu),Uua) = ex_assn(B,aa(A,fun(B,assn),Uu,Uua)) ).

% ATP.lambda_274
tff(fact_8447_ATP_Olambda__275,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_aby(fun(A,fun(nat,B)),fun(A,B),Uu),Uua) = suminf(B,aa(A,fun(nat,B),Uu,Uua)) ) ).

% ATP.lambda_275
tff(fact_8448_ATP_Olambda__276,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A] : aa(A,fun(set(B),set(B)),aTP_Lamp_amw(fun(A,B),fun(A,fun(set(B),set(B))),Uu),Uua) = insert(B,aa(A,B,Uu,Uua)) ).

% ATP.lambda_276
tff(fact_8449_ATP_Olambda__277,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A] :
      ( aa(A,$o,aTP_Lamp_ss(fun(A,$o),fun(A,$o),Uu),Uua)
    <=> ~ aa(A,$o,Uu,Uua) ) ).

% ATP.lambda_277
tff(fact_8450_ATP_Olambda__278,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,fun(A,$o)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_ajb(fun(B,fun(A,$o)),fun(B,$o),Uu),Uua)
    <=> ! [X_1: A] : aa(A,$o,aa(B,fun(A,$o),Uu,Uua),X_1) ) ).

% ATP.lambda_278
tff(fact_8451_ATP_Olambda__279,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aoh(fun(A,fun(B,$o)),fun(A,$o),Uu),Uua)
    <=> ? [X_1: B] : aa(B,$o,aa(A,fun(B,$o),Uu,Uua),X_1) ) ).

% ATP.lambda_279
tff(fact_8452_ATP_Olambda__280,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,fun(A,$o)),Uua: fun(B,A)] :
      ( aa(fun(B,A),$o,aTP_Lamp_agd(fun(B,fun(A,$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_280
tff(fact_8453_ATP_Olambda__281,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: B] :
      ( aa(B,$o,aTP_Lamp_aoi(fun(A,fun(B,$o)),fun(B,$o),Uu),Uua)
    <=> ? [A5: A] : aa(B,$o,aa(A,fun(B,$o),Uu,A5),Uua) ) ).

% ATP.lambda_281
tff(fact_8454_ATP_Olambda__282,axiom,
    ! [A: $tType,Uu: list(A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aoq(list(A),fun(A,$o),Uu),Uua)
    <=> ? [I2: nat] :
          ( ( Uua = aa(nat,A,nth(A,Uu),I2) )
          & aa(nat,$o,ord_less(nat,I2),aa(list(A),nat,size_size(list(A)),Uu)) ) ) ).

% ATP.lambda_282
tff(fact_8455_ATP_Olambda__283,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aoa(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> ! [M7: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),M7)
         => ( aa(nat,$o,Uu,M7)
          <=> aa(nat,$o,Uu,Uua) ) ) ) ).

% ATP.lambda_283
tff(fact_8456_ATP_Olambda__284,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aot(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> ! [N11: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),N11)
         => aa(nat,$o,Uu,N11) ) ) ).

% ATP.lambda_284
tff(fact_8457_ATP_Olambda__285,axiom,
    ! [Uu: fun(nat,$o),Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_aos(fun(nat,$o),fun(nat,$o),Uu),Uua)
    <=> ! [N11: nat] :
          ( aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),Uua),N11)
         => ~ aa(nat,$o,Uu,N11) ) ) ).

% ATP.lambda_285
tff(fact_8458_ATP_Olambda__286,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_wy(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_286
tff(fact_8459_ATP_Olambda__287,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_wx(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_287
tff(fact_8460_ATP_Olambda__288,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aoj(fun(B,option(A)),fun(A,$o),Uu),Uua)
    <=> ? [A5: B] : aa(B,option(A),Uu,A5) = aa(A,option(A),some(A),Uua) ) ).

% ATP.lambda_288
tff(fact_8461_ATP_Olambda__289,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: A] :
      ( aa(A,$o,aTP_Lamp_aok(fun(B,A),fun(A,$o),Uu),Uua)
    <=> ? [X: B] : Uua = aa(B,A,Uu,X) ) ).

% ATP.lambda_289
tff(fact_8462_ATP_Olambda__290,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_xd(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_290
tff(fact_8463_ATP_Olambda__291,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,fun(B,$o)),Uua: fun(A,option(B))] :
      ( aa(fun(A,option(B)),$o,aTP_Lamp_aiv(fun(A,fun(B,$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_291
tff(fact_8464_ATP_Olambda__292,axiom,
    ! [D: $tType,B: $tType,A: $tType,C: $tType,Uu: fun(A,fun(B,fun(C,nat))),Uua: fun(product_prod(product_prod(A,B),C),heap_Time_Heap(D))] :
      ( aa(fun(product_prod(product_prod(A,B),C),heap_Time_Heap(D)),$o,aTP_Lamp_wi(fun(A,fun(B,fun(C,nat))),fun(fun(product_prod(product_prod(A,B),C),heap_Time_Heap(D)),$o),Uu),Uua)
    <=> ! [X: A,Xa3: B,Xb4: C] : time_TBOUND(D,aa(C,heap_Time_Heap(D),aa(B,fun(C,heap_Time_Heap(D)),aa(A,fun(B,fun(C,heap_Time_Heap(D))),product_curry(A,B,fun(C,heap_Time_Heap(D)),product_curry(product_prod(A,B),C,heap_Time_Heap(D),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_292
tff(fact_8465_ATP_Olambda__293,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)),$o,aTP_Lamp_wm(fun(A,fun(heap_ext(product_unit),fun(heap_ext(product_unit),fun(B,fun(nat,$o))))),fun(fun(A,heap_Time_Heap(B)),$o),Uu),Uua)
    <=> ! [X: A,H4: heap_ext(product_unit),H9: heap_ext(product_unit),R6: B,N6: nat] :
          ( heap_Time_effect(B,aa(A,heap_Time_Heap(B),Uua,X),H4,H9,R6,N6)
         => aa(nat,$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),H4),H9),R6),N6) ) ) ).

% ATP.lambda_293
tff(fact_8466_ATP_Olambda__294,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: array(A)] : aa(array(A),heap_Time_Heap(array(A)),aTP_Lamp_agn(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_294
tff(fact_8467_ATP_Olambda__295,axiom,
    ! [Uu: vEBT_VEBTi,Uua: nat] : aa(nat,heap_Time_Heap($o),aa(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)),aTP_Lamp_rz(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),Uu),Uua) = aa(fun(heap_ext(product_unit),option(product_prod($o,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap($o),heap_Time_Heap2($o),aTP_Lamp_rr(heap_ext(product_unit),option(product_prod($o,product_prod(heap_ext(product_unit),nat))))) ).

% ATP.lambda_295
tff(fact_8468_ATP_Olambda__296,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_rv(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),Uu),Uua) = aa(fun(heap_ext(product_unit),option(product_prod(option(nat),product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(option(nat)),heap_Time_Heap2(option(nat)),aTP_Lamp_rm(heap_ext(product_unit),option(product_prod(option(nat),product_prod(heap_ext(product_unit),nat))))) ).

% ATP.lambda_296
tff(fact_8469_ATP_Olambda__297,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_rj(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),Uu),Uua) = aa(fun(heap_ext(product_unit),option(product_prod(vEBT_VEBTi,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(vEBT_VEBTi),heap_Time_Heap2(vEBT_VEBTi),aTP_Lamp_oy(heap_ext(product_unit),option(product_prod(vEBT_VEBTi,product_prod(heap_ext(product_unit),nat))))) ).

% ATP.lambda_297
tff(fact_8470_ATP_Olambda__298,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_rg(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_rf(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_dy(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),Uub),Uua) ).

% ATP.lambda_298
tff(fact_8471_ATP_Olambda__299,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_qr(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_qq(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_cq(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uub),Uua) ).

% ATP.lambda_299
tff(fact_8472_ATP_Olambda__300,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_qj(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_qi(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_ch(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uub),Uua) ).

% ATP.lambda_300
tff(fact_8473_ATP_Olambda__301,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($o)),aTP_Lamp_pj(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($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($o))))),aTP_Lamp_pi(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))))),Uu),Uub),aTP_Lamp_ej(nat,fun($o,fun($o,heap_Time_Heap($o))),Uub),Uua) ).

% ATP.lambda_301
tff(fact_8474_ATP_Olambda__302,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($o)),aTP_Lamp_sa(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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($o))))),aTP_Lamp_gi(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o)))))),Uu),Uub),aTP_Lamp_ej(nat,fun($o,fun($o,heap_Time_Heap($o))),Uub),Uua) ).

% ATP.lambda_302
tff(fact_8475_ATP_Olambda__303,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_rx(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_gc(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_cq(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uub),Uua) ).

% ATP.lambda_303
tff(fact_8476_ATP_Olambda__304,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_rw(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_fw(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_ch(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uub),Uua) ).

% ATP.lambda_304
tff(fact_8477_ATP_Olambda__305,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_rk(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_fq(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_dy(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),Uub),Uua) ).

% ATP.lambda_305
tff(fact_8478_ATP_Olambda__306,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_ch(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uu),(Uua)),(Uub)) = $ite(
        aa(nat,$o,aa(nat,fun(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_306
tff(fact_8479_ATP_Olambda__307,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_vb(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_307
tff(fact_8480_ATP_Olambda__308,axiom,
    ! [Uu: fun(nat,real),Uua: fun(nat,real),Uub: nat] :
      aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_wz(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_308
tff(fact_8481_ATP_Olambda__309,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_gj(num,fun(code_integer,fun(code_integer,product_prod(code_integer,code_integer))),Uu),Uua),Uub) = $ite(aa(code_integer,$o,aa(code_integer,fun(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_309
tff(fact_8482_ATP_Olambda__310,axiom,
    ! [Uu: num,Uua: nat,Uub: nat] :
      aa(nat,product_prod(nat,nat),aa(nat,fun(nat,product_prod(nat,nat)),aTP_Lamp_fc(num,fun(nat,fun(nat,product_prod(nat,nat))),Uu),Uua),Uub) = $ite(aa(nat,$o,aa(nat,fun(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_310
tff(fact_8483_ATP_Olambda__311,axiom,
    ! [Uu: num,Uua: int,Uub: int] :
      aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_fb(num,fun(int,fun(int,product_prod(int,int))),Uu),Uua),Uub) = $ite(aa(int,$o,aa(int,fun(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_311
tff(fact_8484_ATP_Olambda__312,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_fd(num,fun(A,fun(A,product_prod(A,A))),Uu),Uua),Uub) = $ite(aa(A,$o,aa(A,fun(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_312
tff(fact_8485_ATP_Olambda__313,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_wf(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_313
tff(fact_8486_ATP_Olambda__314,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_tt(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_314
tff(fact_8487_ATP_Olambda__315,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_xj(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uu = Uub,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_315
tff(fact_8488_ATP_Olambda__316,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_vp(nat,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(nat,A,Uua,Uub),zero_zero(A)) ) ).

% ATP.lambda_316
tff(fact_8489_ATP_Olambda__317,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_tu(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),zero_zero(B)) ) ).

% ATP.lambda_317
tff(fact_8490_ATP_Olambda__318,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_xk(A,fun(fun(A,B),fun(A,B)),Uu),Uua),Uub) = $ite(Uub = Uu,aa(A,B,Uua,Uub),one_one(B)) ) ).

% ATP.lambda_318
tff(fact_8491_ATP_Olambda__319,axiom,
    ! [Uu: product_prod(nat,nat),Uua: option(nat),Uub: product_unit] :
      aa(product_unit,heap_Time_Heap(option(nat)),aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aTP_Lamp_ac(product_prod(nat,nat),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),Uu),Uua),Uub) = $ite(
        Uua = none(nat),
        $ite(aa(nat,$o,ord_less(nat,aa(product_prod(nat,nat),nat,product_fst(nat,nat),Uu)),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),Uu))),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,x13),aa(option(nat),nat,the2(nat),Uua)),aTP_Lamp_ab(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),Uua)) ) ).

% ATP.lambda_319
tff(fact_8492_ATP_Olambda__320,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_cq(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_320
tff(fact_8493_ATP_Olambda__321,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_any(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_321
tff(fact_8494_ATP_Olambda__322,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_anw(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,abs_abs(code_integer),Uu)),Uub))) ).

% ATP.lambda_322
tff(fact_8495_ATP_Olambda__323,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_anx(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_323
tff(fact_8496_ATP_Olambda__324,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_dy(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_324
tff(fact_8497_ATP_Olambda__325,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_cj(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_ci(nat,fun(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),Uu),Uub))) ).

% ATP.lambda_325
tff(fact_8498_ATP_Olambda__326,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_akm(fun(A,$o),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = $ite(aa(A,$o,Uu,Uua),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub),Uub) ).

% ATP.lambda_326
tff(fact_8499_ATP_Olambda__327,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_anb(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_327
tff(fact_8500_ATP_Olambda__328,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_we(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_328
tff(fact_8501_ATP_Olambda__329,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,$o),fun(A,B),aTP_Lamp_uo(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),zero_zero(B)) ) ).

% ATP.lambda_329
tff(fact_8502_ATP_Olambda__330,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,$o),fun(A,B),aTP_Lamp_xw(fun(A,B),fun(fun(A,$o),fun(A,B)),Uu),Uua),Uub) = $ite(aa(A,$o,Uua,Uub),aa(A,B,Uu,Uub),one_one(B)) ) ).

% ATP.lambda_330
tff(fact_8503_ATP_Olambda__331,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_agj(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_331
tff(fact_8504_ATP_Olambda__332,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: B] :
      aa(B,option(A),aa(fun(B,$o),fun(B,option(A)),aTP_Lamp_ame(fun(B,A),fun(fun(B,$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_332
tff(fact_8505_ATP_Olambda__333,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: list(A),Uub: list(A)] : aa(list(A),list(A),aa(list(A),fun(list(A),list(A)),aTP_Lamp_agg(fun(A,fun(A,$o)),fun(list(A),fun(list(A),list(A))),Uu),Uua),Uub) = merges9089515139780605204_merge(A,Uu,aa(list(A),list(A),mergesort_by_rel(A,Uu),Uua),aa(list(A),list(A),mergesort_by_rel(A,Uu),Uub)) ).

% ATP.lambda_333
tff(fact_8506_ATP_Olambda__334,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($o)),aTP_Lamp_pd(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($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_334
tff(fact_8507_ATP_Olambda__335,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_ake(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_akb(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))),aa(heap_Time_Heap(B),fun(heap_ext(product_unit),option(product_prod(B,product_prod(heap_ext(product_unit),nat)))),heap_Time_execute(B),Uu),Uub)) ).

% ATP.lambda_335
tff(fact_8508_ATP_Olambda__336,axiom,
    ! [D: $tType,C: $tType,B: $tType,A: $tType,Uu: fun(D,fun(B,C)),Uua: fun(A,D),Uub: product_prod(A,B)] : aa(product_prod(A,B),C,aa(fun(A,D),fun(product_prod(A,B),C),aTP_Lamp_eu(fun(D,fun(B,C)),fun(fun(A,D),fun(product_prod(A,B),C)),Uu),Uua),Uub) = aa(B,C,aa(D,fun(B,C),Uu,aa(A,D,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_336
tff(fact_8509_ATP_Olambda__337,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_zd(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_337
tff(fact_8510_ATP_Olambda__338,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_zb(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_338
tff(fact_8511_ATP_Olambda__339,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($o)),aTP_Lamp_gd(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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_339
tff(fact_8512_ATP_Olambda__340,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_yu(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_340
tff(fact_8513_ATP_Olambda__341,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_yr(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_341
tff(fact_8514_ATP_Olambda__342,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_agc(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_342
tff(fact_8515_ATP_Olambda__343,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_sg(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_343
tff(fact_8516_ATP_Olambda__344,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_ait(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_344
tff(fact_8517_ATP_Olambda__345,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_aiy(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_345
tff(fact_8518_ATP_Olambda__346,axiom,
    ! [C: $tType,D: $tType,B: $tType,A: $tType,Uu: fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),Uua: fun(A,heap_Time_Heap(B)),Uub: C] : aa(C,heap_Time_Heap(D),aa(fun(A,heap_Time_Heap(B)),fun(C,heap_Time_Heap(D)),aTP_Lamp_ov(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),fun(fun(A,heap_Time_Heap(B)),fun(C,heap_Time_Heap(D))),Uu),Uua),Uub) = aa(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D),aa(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D)),Uu,Uub),Uua) ).

% ATP.lambda_346
tff(fact_8519_ATP_Olambda__347,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_apq(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_347
tff(fact_8520_ATP_Olambda__348,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_xo(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_348
tff(fact_8521_ATP_Olambda__349,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_tz(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_349
tff(fact_8522_ATP_Olambda__350,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_cw(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_350
tff(fact_8523_ATP_Olambda__351,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_aja(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_351
tff(fact_8524_ATP_Olambda__352,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_afx(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_352
tff(fact_8525_ATP_Olambda__353,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_vv(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_353
tff(fact_8526_ATP_Olambda__354,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_bg(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_354
tff(fact_8527_ATP_Olambda__355,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_xu(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_355
tff(fact_8528_ATP_Olambda__356,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_ul(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_356
tff(fact_8529_ATP_Olambda__357,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_acb(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_aca(fun(nat,A),fun(fun(nat,A),fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_357
tff(fact_8530_ATP_Olambda__358,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_aba(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_aaz(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_358
tff(fact_8531_ATP_Olambda__359,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_aay(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_aax(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_359
tff(fact_8532_ATP_Olambda__360,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_aaw(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_aav(A,fun(A,fun(nat,fun(nat,A))),Uu),Uua),Uub)),set_ord_atMost(nat,Uub)) ) ).

% ATP.lambda_360
tff(fact_8533_ATP_Olambda__361,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_vk(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_361
tff(fact_8534_ATP_Olambda__362,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_anz(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_362
tff(fact_8535_ATP_Olambda__363,axiom,
    ! [A: $tType,Uu: fun(A,fun(A,$o)),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_aje(fun(A,fun(A,$o)),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,$o,aa(A,fun(A,$o),Uu,Uua),Uub)
        | ( Uua = Uub ) ) ) ).

% ATP.lambda_363
tff(fact_8536_ATP_Olambda__364,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_yv(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uua),Uub) = groups7121269368397514597t_prod(nat,A,aa(nat,fun(nat,A),aTP_Lamp_yu(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_364
tff(fact_8537_ATP_Olambda__365,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_ys(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_yr(fun(nat,fun(nat,A)),fun(nat,fun(nat,A)),Uu),Uub)),set_or1337092689740270186AtMost(nat,aa(nat,nat,suc,Uub),Uua)) ) ).

% ATP.lambda_365
tff(fact_8538_ATP_Olambda__366,axiom,
    ! [Uu: $o,Uua: code_integer,Uub: $o] : aa($o,char,aa(code_integer,fun($o,char),aTP_Lamp_api($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($o,char)),aTP_Lamp_aph($o,fun($o,fun(code_integer,fun($o,char))),(Uu)),(Uub))),code_bit_cut_integer(Uua)) ).

% ATP.lambda_366
tff(fact_8539_ATP_Olambda__367,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ahf(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_ahe(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_367
tff(fact_8540_ATP_Olambda__368,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ahd(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_ahc(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_368
tff(fact_8541_ATP_Olambda__369,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_ahb(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_aha(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_369
tff(fact_8542_ATP_Olambda__370,axiom,
    ! [Uu: rat,Uua: int,Uub: int] : aa(int,product_prod(int,int),aa(int,fun(int,product_prod(int,int)),aTP_Lamp_agz(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_agy(int,fun(int,fun(int,fun(int,product_prod(int,int)))),Uua),Uub)),quotient_of(Uu)) ).

% ATP.lambda_370
tff(fact_8543_ATP_Olambda__371,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_agw(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_agv(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_371
tff(fact_8544_ATP_Olambda__372,axiom,
    ! [Uu: rat,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_agu(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_agt(int,fun(int,fun(int,fun(int,$o))),Uua),Uub)),quotient_of(Uu)) ) ).

% ATP.lambda_372
tff(fact_8545_ATP_Olambda__373,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_xp(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_xo(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub),Uua) ) ).

% ATP.lambda_373
tff(fact_8546_ATP_Olambda__374,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_vw(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_vv(fun(A,fun(nat,B)),fun(nat,fun(A,B)),Uua),Uub)),Uu) ) ).

% ATP.lambda_374
tff(fact_8547_ATP_Olambda__375,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_ua(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_tz(fun(B,fun(C,A)),fun(C,fun(B,A)),Uu),Uub)),Uua) ) ).

% ATP.lambda_375
tff(fact_8548_ATP_Olambda__376,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_aak(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_376
tff(fact_8549_ATP_Olambda__377,axiom,
    ! [A: $tType] :
      ( ( real_V8999393235501362500lgebra(A)
        & idom(A) )
     => ! [Uu: fun(nat,A),Uua: nat,Uub: A] :
          ( aa(A,$o,aa(nat,fun(A,$o),aTP_Lamp_zh(fun(nat,A),fun(nat,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(set(nat),A,groups7311177749621191930dd_sum(nat,A,aa(A,fun(nat,A),aTP_Lamp_yp(fun(nat,A),fun(A,fun(nat,A)),Uu),Uub)),set_ord_atMost(nat,Uua)) = zero_zero(A) ) ) ) ).

% ATP.lambda_377
tff(fact_8550_ATP_Olambda__378,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_tx(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_378
tff(fact_8551_ATP_Olambda__379,axiom,
    ! [Uu: real,Uua: fun(nat,real),Uub: nat] : aa(nat,real,aa(fun(nat,real),fun(nat,real),aTP_Lamp_aal(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_379
tff(fact_8552_ATP_Olambda__380,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_acn(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_380
tff(fact_8553_ATP_Olambda__381,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),$o),aTP_Lamp_apk(set(product_prod(A,A)),fun(list(A),fun(list(A),$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_381
tff(fact_8554_ATP_Olambda__382,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),$o),aTP_Lamp_apa(set(product_prod(A,A)),fun(list(A),fun(list(A),$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),Ys7: list(A)] :
            ( ( Uua = aa(list(A),list(A),append(A,Xys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs6)) )
            & ( Uub = aa(list(A),list(A),append(A,Xys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys7)) )
            & 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_382
tff(fact_8555_ATP_Olambda__383,axiom,
    ! [A: $tType,Uu: nat,Uua: list(A),Uub: list(A)] :
      ( aa(list(A),$o,aa(list(A),fun(list(A),$o),aTP_Lamp_als(nat,fun(list(A),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( ( aa(list(A),nat,size_size(list(A)),Uub) = Uu )
        & aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),aa(list(A),set(A),set2(A),Uua)) ) ) ).

% ATP.lambda_383
tff(fact_8556_ATP_Olambda__384,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_sm(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(list(A),nat,size_size(list(A)),Uub)),Uua) ) ) ).

% ATP.lambda_384
tff(fact_8557_ATP_Olambda__385,axiom,
    ! [A: $tType,Uu: set(A),Uua: nat,Uub: list(A)] :
      ( aa(list(A),$o,aa(nat,fun(list(A),$o),aTP_Lamp_sl(set(A),fun(nat,fun(list(A),$o)),Uu),Uua),Uub)
    <=> ( aa(set(A),$o,aa(set(A),fun(set(A),$o),ord_less_eq(set(A)),aa(list(A),set(A),set2(A),Uub)),Uu)
        & ( aa(list(A),nat,size_size(list(A)),Uub) = Uua ) ) ) ).

% ATP.lambda_385
tff(fact_8558_ATP_Olambda__386,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: A] :
      ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_akl(fun(A,$o),fun(list(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,aa(list(A),set(A),set2(A),Uua))
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_386
tff(fact_8559_ATP_Olambda__387,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aeb(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_387
tff(fact_8560_ATP_Olambda__388,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_ba(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_388
tff(fact_8561_ATP_Olambda__389,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_acm(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),diffs(A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_389
tff(fact_8562_ATP_Olambda__390,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_acw(fun(nat,A),fun(A,fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),diffs(A,Uu,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uua),Uub)) ) ).

% ATP.lambda_390
tff(fact_8563_ATP_Olambda__391,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_acx(A,fun(fun(nat,A),fun(nat,A)),Uu),Uua),Uub) = aa(A,A,aa(A,fun(A,A),times_times(A),diffs(A,Uua,Uub)),aa(nat,A,aa(A,fun(nat,A),power_power(A),Uu),Uub)) ) ).

% ATP.lambda_391
tff(fact_8564_ATP_Olambda__392,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_zf(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_392
tff(fact_8565_ATP_Olambda__393,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_df(A,fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        | member(A,Uub,Uua) ) ) ).

% ATP.lambda_393
tff(fact_8566_ATP_Olambda__394,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_adr(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uua)
        & aa(int,$o,ord_less(int,Uua),Uub) ) ) ).

% ATP.lambda_394
tff(fact_8567_ATP_Olambda__395,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: word(A),Uua: word(A),Uub: word(A)] :
          ( aa(word(A),$o,aa(word(A),fun(word(A),$o),aTP_Lamp_db(word(A),fun(word(A),fun(word(A),$o)),Uu),Uua),Uub)
        <=> ( aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Uu),Uub)
            & aa(word(A),$o,aa(word(A),fun(word(A),$o),ord_less_eq(word(A)),Uub),Uua) ) ) ) ).

% ATP.lambda_395
tff(fact_8568_ATP_Olambda__396,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_sv(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_396
tff(fact_8569_ATP_Olambda__397,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_adq(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,ord_less(int,Uua),Uub) ) ) ).

% ATP.lambda_397
tff(fact_8570_ATP_Olambda__398,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_sx(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uu),Uub)
        & aa(int,$o,ord_less(int,Uub),Uua) ) ) ).

% ATP.lambda_398
tff(fact_8571_ATP_Olambda__399,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_sy(int,fun(int,fun(int,$o)),Uu),Uua),Uub)
    <=> ( aa(int,$o,ord_less(int,Uu),Uub)
        & aa(int,$o,aa(int,fun(int,$o),ord_less_eq(int),Uub),Uua) ) ) ).

% ATP.lambda_399
tff(fact_8572_ATP_Olambda__400,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_sw(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_400
tff(fact_8573_ATP_Olambda__401,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ct(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_401
tff(fact_8574_ATP_Olambda__402,axiom,
    ! [Uu: vEBT_VEBT,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_cu(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_402
tff(fact_8575_ATP_Olambda__403,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_aig(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_403
tff(fact_8576_ATP_Olambda__404,axiom,
    ! [Uu: int,Uua: int,Uub: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aTP_Lamp_aie(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_404
tff(fact_8577_ATP_Olambda__405,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: set(A)] :
      ( aa(set(A),$o,aa(set(A),fun(set(A),$o),aTP_Lamp_si(set(A),fun(set(A),fun(set(A),$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_405
tff(fact_8578_ATP_Olambda__406,axiom,
    ! [A: $tType] :
      ( ord(A)
     => ! [Uu: set(A),Uua: fun(A,$o),Uub: A] :
          ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_aom(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & aa(A,$o,Uua,Uub) ) ) ) ).

% ATP.lambda_406
tff(fact_8579_ATP_Olambda__407,axiom,
    ! [A: $tType,Uu: set(A),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_cv(set(A),fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_407
tff(fact_8580_ATP_Olambda__408,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_anc(fun(A,$o),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uua)
        & aa(A,$o,Uu,Uub) ) ) ).

% ATP.lambda_408
tff(fact_8581_ATP_Olambda__409,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dh(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uu = Uub )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_409
tff(fact_8582_ATP_Olambda__410,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_dg(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub = Uu )
        & aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_410
tff(fact_8583_ATP_Olambda__411,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_afu(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_aft(A,fun(list(A),heap_Time_Heap(list(A))),Uub)) ).

% ATP.lambda_411
tff(fact_8584_ATP_Olambda__412,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_afv(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_aft(A,fun(list(A),heap_Time_Heap(list(A))),Uub)) ).

% ATP.lambda_412
tff(fact_8585_ATP_Olambda__413,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,$o),aTP_Lamp_tm(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_413
tff(fact_8586_ATP_Olambda__414,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,$o),aTP_Lamp_ajg(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_414
tff(fact_8587_ATP_Olambda__415,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,$o),aTP_Lamp_tk(set(A),fun(fun(A,B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,Uu)
            & ( aa(A,B,Uua,Uub) != one_one(B) ) ) ) ) ).

% ATP.lambda_415
tff(fact_8588_ATP_Olambda__416,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,$o),aTP_Lamp_ajh(fun(B,A),fun(set(B),fun(B,$o)),Uu),Uua),Uub)
        <=> ( member(B,Uub,Uua)
            & ( aa(B,A,Uu,Uub) != zero_zero(A) ) ) ) ) ).

% ATP.lambda_416
tff(fact_8589_ATP_Olambda__417,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: $o] : aa($o,assn,aa(vEBT_VEBTi,fun($o,assn),aTP_Lamp_bv(vEBT_VEBT,fun(vEBT_VEBTi,fun($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_417
tff(fact_8590_ATP_Olambda__418,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option(nat)] : aa(option(nat),assn,aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_ar(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_418
tff(fact_8591_ATP_Olambda__419,axiom,
    ! [Uu: vEBT_VEBT,Uua: vEBT_VEBTi,Uub: option(nat)] : aa(option(nat),assn,aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_as(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_419
tff(fact_8592_ATP_Olambda__420,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: array(A),Uua: list(A),Uub: nat] : aa(nat,assn,aa(list(A),fun(nat,assn),aTP_Lamp_eb(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_420
tff(fact_8593_ATP_Olambda__421,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_tj(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_421
tff(fact_8594_ATP_Olambda__422,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_cz(set(A),fun(set(A),fun(A,$o)),Uu),Uua),Uub)
    <=> ( member(A,Uub,Uu)
        & ~ member(A,Uub,Uua) ) ) ).

% ATP.lambda_422
tff(fact_8595_ATP_Olambda__423,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_ahy(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Uub),Uua)),Uu) ) ).

% ATP.lambda_423
tff(fact_8596_ATP_Olambda__424,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_za(nat,fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),Uua),Uub)),Uu) ) ).

% ATP.lambda_424
tff(fact_8597_ATP_Olambda__425,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_zj(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_425
tff(fact_8598_ATP_Olambda__426,axiom,
    ! [A: $tType] :
      ( linordered_field(A)
     => ! [Uu: A,Uua: A,Uub: A] : aa(A,A,aa(A,fun(A,A),aTP_Lamp_aid(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_426
tff(fact_8599_ATP_Olambda__427,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,minus_minus(A,aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_427
tff(fact_8600_ATP_Olambda__428,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,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),divide_divide(A),Uub),Uu)),Uua) ) ).

% ATP.lambda_428
tff(fact_8601_ATP_Olambda__429,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,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),times_times(A),Uu),Uub)),Uua) ) ).

% ATP.lambda_429
tff(fact_8602_ATP_Olambda__430,axiom,
    ! [A: $tType,B: $tType,Uu: set(product_prod(B,A)),Uua: B,Uub: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_afo(set(product_prod(B,A)),fun(B,fun(A,$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_430
tff(fact_8603_ATP_Olambda__431,axiom,
    ! [B: $tType,A: $tType,Uu: set(product_prod(A,B)),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_sh(set(product_prod(A,B)),fun(A,fun(B,$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_431
tff(fact_8604_ATP_Olambda__432,axiom,
    ! [A: $tType,Uu: list(list(A)),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_alq(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_432
tff(fact_8605_ATP_Olambda__433,axiom,
    ! [Uu: nat,Uua: complex,Uub: complex] :
      ( aa(complex,$o,aa(complex,fun(complex,$o),aTP_Lamp_tb(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_433
tff(fact_8606_ATP_Olambda__434,axiom,
    ! [Uu: complex,Uua: nat,Uub: complex] :
      ( aa(complex,$o,aa(nat,fun(complex,$o),aTP_Lamp_adp(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_434
tff(fact_8607_ATP_Olambda__435,axiom,
    ! [A: $tType] :
      ( archim2362893244070406136eiling(A)
     => ! [Uu: A,Uua: A,Uub: A] :
          ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_agl(A,fun(A,fun(A,$o)),Uu),Uua),Uub)
        <=> ( member(A,Uub,ring_1_Ints(A))
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uu),Uub)
            & aa(A,$o,aa(A,fun(A,$o),ord_less_eq(A),Uub),Uua) ) ) ) ).

% ATP.lambda_435
tff(fact_8608_ATP_Olambda__436,axiom,
    ! [Uu: nat,Uua: vEBT_VEBTi,Uub: option(nat)] : aa(option(nat),assn,aa(vEBT_VEBTi,fun(option(nat),assn),aTP_Lamp_at(nat,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,summary),Uua)),pure_assn(Uub = vEBT_vebt_pred(summary,Uu))) ).

% ATP.lambda_436
tff(fact_8609_ATP_Olambda__437,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(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_437
tff(fact_8610_ATP_Olambda__438,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_acf(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_438
tff(fact_8611_ATP_Olambda__439,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_ace(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_439
tff(fact_8612_ATP_Olambda__440,axiom,
    ! [C: $tType,D: $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(D))),Uub: fun(A,heap_Time_Heap(B))] : aa(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D),aa(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D)),aTP_Lamp_ow(fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(C)),fun(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),Uu),Uua),Uub) = heap_Time_bind(C,D,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(D)),aTP_Lamp_ov(fun(C,fun(fun(A,heap_Time_Heap(B)),heap_Time_Heap(D))),fun(fun(A,heap_Time_Heap(B)),fun(C,heap_Time_Heap(D))),Uua),Uub)) ).

% ATP.lambda_440
tff(fact_8613_ATP_Olambda__441,axiom,
    ! [Uu: fun(nat,real),Uua: real,Uub: nat] : aa(nat,real,aa(real,fun(nat,real),aTP_Lamp_ack(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_441
tff(fact_8614_ATP_Olambda__442,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_zo(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_442
tff(fact_8615_ATP_Olambda__443,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_yp(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_443
tff(fact_8616_ATP_Olambda__444,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_aad(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_444
tff(fact_8617_ATP_Olambda__445,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_acd(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_445
tff(fact_8618_ATP_Olambda__446,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_acv(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_446
tff(fact_8619_ATP_Olambda__447,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_yy(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_447
tff(fact_8620_ATP_Olambda__448,axiom,
    ! [A: $tType] :
      ( idom(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_zi(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_448
tff(fact_8621_ATP_Olambda__449,axiom,
    ! [Uu: fun(nat,$o),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_sj(fun(nat,$o),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> ( aa(nat,$o,Uu,Uub)
        & aa(nat,$o,ord_less(nat,Uub),Uua) ) ) ).

% ATP.lambda_449
tff(fact_8622_ATP_Olambda__450,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: list(A),Uub: A] :
          ( aa(A,$o,aa(list(A),fun(A,$o),aTP_Lamp_ako(fun(A,B),fun(list(A),fun(A,$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_450
tff(fact_8623_ATP_Olambda__451,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(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_451
tff(fact_8624_ATP_Olambda__452,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_xr(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_452
tff(fact_8625_ATP_Olambda__453,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_apo(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_453
tff(fact_8626_ATP_Olambda__454,axiom,
    ! [Uu: fun(nat,nat),Uua: fun(nat,nat),Uub: nat] : aa(nat,nat,aa(fun(nat,nat),fun(nat,nat),aTP_Lamp_aee(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_454
tff(fact_8627_ATP_Olambda__455,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_aed(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_455
tff(fact_8628_ATP_Olambda__456,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_xq(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_456
tff(fact_8629_ATP_Olambda__457,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,aa(A,fun(A,A),times_times(A),aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_457
tff(fact_8630_ATP_Olambda__458,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_abq(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_458
tff(fact_8631_ATP_Olambda__459,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_vt(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_459
tff(fact_8632_ATP_Olambda__460,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_uh(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_460
tff(fact_8633_ATP_Olambda__461,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,minus_minus(A,aa(A,A,Uu,Uub)),aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_461
tff(fact_8634_ATP_Olambda__462,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_vi(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_462
tff(fact_8635_ATP_Olambda__463,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_wo(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_463
tff(fact_8636_ATP_Olambda__464,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_vm(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_464
tff(fact_8637_ATP_Olambda__465,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_vq(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_465
tff(fact_8638_ATP_Olambda__466,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_uc(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_466
tff(fact_8639_ATP_Olambda__467,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_aqb(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_467
tff(fact_8640_ATP_Olambda__468,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_aji(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_468
tff(fact_8641_ATP_Olambda__469,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_su(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_469
tff(fact_8642_ATP_Olambda__470,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_ana(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_470
tff(fact_8643_ATP_Olambda__471,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_st(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_471
tff(fact_8644_ATP_Olambda__472,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_akk(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_472
tff(fact_8645_ATP_Olambda__473,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_ane(fun(A,B),fun(A,fun(A,$o)),Uu),Uua),Uub)
    <=> ( aa(A,B,Uu,Uua) = aa(A,B,Uu,Uub) ) ) ).

% ATP.lambda_473
tff(fact_8646_ATP_Olambda__474,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_aqc(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_474
tff(fact_8647_ATP_Olambda__475,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,real),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_abe(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_475
tff(fact_8648_ATP_Olambda__476,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_abd(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_476
tff(fact_8649_ATP_Olambda__477,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_xn(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_477
tff(fact_8650_ATP_Olambda__478,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_ty(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_478
tff(fact_8651_ATP_Olambda__479,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_apr(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_479
tff(fact_8652_ATP_Olambda__480,axiom,
    ! [Uu: fun(nat,nat),Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_sk(fun(nat,nat),fun(nat,fun(nat,$o)),Uu),Uua),Uub)
    <=> aa(nat,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),aa(nat,nat,Uu,Uub)),Uua) ) ).

% ATP.lambda_480
tff(fact_8653_ATP_Olambda__481,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_un(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_481
tff(fact_8654_ATP_Olambda__482,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_vu(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_482
tff(fact_8655_ATP_Olambda__483,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_ui(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_483
tff(fact_8656_ATP_Olambda__484,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_apy(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_484
tff(fact_8657_ATP_Olambda__485,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_alg(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,aa(A,B,Uu,Uub)),Uua) ) ) ).

% ATP.lambda_485
tff(fact_8658_ATP_Olambda__486,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_vr(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_486
tff(fact_8659_ATP_Olambda__487,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_ue(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_487
tff(fact_8660_ATP_Olambda__488,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_apu(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_488
tff(fact_8661_ATP_Olambda__489,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_bf(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_489
tff(fact_8662_ATP_Olambda__490,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_vy(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_490
tff(fact_8663_ATP_Olambda__491,axiom,
    ! [A: $tType] :
      ( ring_1(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_wq(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_491
tff(fact_8664_ATP_Olambda__492,axiom,
    ! [Uu: fun(real,real),Uua: nat,Uub: real] : aa(real,real,aa(nat,fun(real,real),aTP_Lamp_aqi(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_492
tff(fact_8665_ATP_Olambda__493,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_xs(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_493
tff(fact_8666_ATP_Olambda__494,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_apn(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_494
tff(fact_8667_ATP_Olambda__495,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_ahl(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_495
tff(fact_8668_ATP_Olambda__496,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_pb(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_496
tff(fact_8669_ATP_Olambda__497,axiom,
    ! [B: $tType,A: $tType,Uu: set(A),Uua: fun(B,A),Uub: B] :
      ( aa(B,$o,aa(fun(B,A),fun(B,$o),aTP_Lamp_anj(set(A),fun(fun(B,A),fun(B,$o)),Uu),Uua),Uub)
    <=> member(A,aa(B,A,Uua,Uub),Uu) ) ).

% ATP.lambda_497
tff(fact_8670_ATP_Olambda__498,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,$o),aTP_Lamp_ali(fun(A,word(B)),fun(word(B),fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,word(B),Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_498
tff(fact_8671_ATP_Olambda__499,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_akr(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> ( aa(A,B,Uu,Uub) = Uua ) ) ) ).

% ATP.lambda_499
tff(fact_8672_ATP_Olambda__500,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,aa(list(nat),list(A),map(nat,A,Uua),upt(zero_zero(nat),Uu)),Uub)) ) ).

% ATP.lambda_500
tff(fact_8673_ATP_Olambda__501,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_ajx(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_501
tff(fact_8674_ATP_Olambda__502,axiom,
    ! [Uu: product_prod(nat,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_ad(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uu),Uua),Uub) = heap_Time_bind(product_unit,option(nat),
        refine_Imp_assert(( ( Uub = none(nat) )
          = ( vEBT_vebt_pred(summary,Uua) = none(nat) ) )),
        aa(option(nat),fun(product_unit,heap_Time_Heap(option(nat))),aTP_Lamp_ac(product_prod(nat,nat),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat)))),Uu),Uub)) ).

% ATP.lambda_502
tff(fact_8675_ATP_Olambda__503,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_oq(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_op(nat,fun(nat,fun(list(vEBT_VEBTi),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),Uu),Uua),Uub)) ).

% ATP.lambda_503
tff(fact_8676_ATP_Olambda__504,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_on(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_om(nat,fun(nat,fun(list(vEBT_VEBTi),fun(product_unit,heap_Time_Heap(vEBT_VEBTi)))),Uu),Uua),Uub)) ).

% ATP.lambda_504
tff(fact_8677_ATP_Olambda__505,axiom,
    ! [Uu: product_prod(nat,nat),Uua: nat,Uub: nat] : aa(nat,heap_Time_Heap(option(nat)),aa(nat,fun(nat,heap_Time_Heap(option(nat))),aTP_Lamp_ak(product_prod(nat,nat),fun(nat,fun(nat,heap_Time_Heap(option(nat)))),Uu),Uua),Uub) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uua = vEBT_VEBT_low(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),x12),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)))),aTP_Lamp_aj(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),Uu),Uua),Uub)) ).

% ATP.lambda_505
tff(fact_8678_ATP_Olambda__506,axiom,
    ! [A: $tType,Uu: A,Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_de(A,fun(fun(A,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ( ( Uub != Uu )
       => aa(A,$o,Uua,Uub) ) ) ).

% ATP.lambda_506
tff(fact_8679_ATP_Olambda__507,axiom,
    ! [A: $tType] :
      ( field_char_0(A)
     => ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_aea(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_507
tff(fact_8680_ATP_Olambda__508,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_acl(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_508
tff(fact_8681_ATP_Olambda__509,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_adj(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_509
tff(fact_8682_ATP_Olambda__510,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_oo(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_ok(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),Uu),Uub)) ).

% ATP.lambda_510
tff(fact_8683_ATP_Olambda__511,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_ot(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_ok(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),Uu),Uub)) ).

% ATP.lambda_511
tff(fact_8684_ATP_Olambda__512,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_ef(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_ee(array(vEBT_VEBTi),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))),Uu),Uua),Uub)) ).

% ATP.lambda_512
tff(fact_8685_ATP_Olambda__513,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_ol(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_ok(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),Uu),Uub)) ).

% ATP.lambda_513
tff(fact_8686_ATP_Olambda__514,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_or(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_ok(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))),Uu),Uub)) ).

% ATP.lambda_514
tff(fact_8687_ATP_Olambda__515,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_ci(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_by(nat,fun(option(nat),fun(option(nat),heap_Time_Heap(option(nat)))),Uu),Uua)) ).

% ATP.lambda_515
tff(fact_8688_ATP_Olambda__516,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_bz(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_by(nat,fun(option(nat),fun(option(nat),heap_Time_Heap(option(nat)))),Uu),Uua)) ).

% ATP.lambda_516
tff(fact_8689_ATP_Olambda__517,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_ou(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_ot(nat,fun(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Uu),Uua)) ).

% ATP.lambda_517
tff(fact_8690_ATP_Olambda__518,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_os(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_or(nat,fun(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Uu),Uua)) ).

% ATP.lambda_518
tff(fact_8691_ATP_Olambda__519,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_alc(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_alb(fun(A,B),fun(list(A),fun(A,fun(A,fun(list(A),list(A))))),Uu),Uua),Uub)) ) ).

% ATP.lambda_519
tff(fact_8692_ATP_Olambda__520,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: nat,Uua: nat,Uub: word(A)] :
          ( aa(word(A),$o,aa(nat,fun(word(A),$o),aTP_Lamp_aml(nat,fun(nat,fun(word(A),$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_520
tff(fact_8693_ATP_Olambda__521,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] :
      ( aa(nat,$o,aa(nat,fun(nat,$o),aTP_Lamp_amm(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_521
tff(fact_8694_ATP_Olambda__522,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_adx(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_522
tff(fact_8695_ATP_Olambda__523,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_ady(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_523
tff(fact_8696_ATP_Olambda__524,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_adw(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_524
tff(fact_8697_ATP_Olambda__525,axiom,
    ! [A: $tType] :
      ( euclid5411537665997757685th_nat(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_vf(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_525
tff(fact_8698_ATP_Olambda__526,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_vd(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_526
tff(fact_8699_ATP_Olambda__527,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_527
tff(fact_8700_ATP_Olambda__528,axiom,
    ! [A: $tType,Uu: A,Uua: set(A),Uub: A] :
      ( aa(A,$o,aa(set(A),fun(A,$o),aTP_Lamp_ail(A,fun(set(A),fun(A,$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_528
tff(fact_8701_ATP_Olambda__529,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_ajv(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_529
tff(fact_8702_ATP_Olambda__530,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_ajr(nat,fun(fun(nat,A),fun(array(A),assn)),Uu),Uua),Uub) = aa(list(A),assn,snga_assn(A,Uub),aa(list(nat),list(A),map(nat,A,Uua),upt(zero_zero(nat),Uu))) ) ).

% ATP.lambda_530
tff(fact_8703_ATP_Olambda__531,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_ww(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_531
tff(fact_8704_ATP_Olambda__532,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_us(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_532
tff(fact_8705_ATP_Olambda__533,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,nat,aa(nat,fun(nat,nat),aTP_Lamp_ve(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_533
tff(fact_8706_ATP_Olambda__534,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: nat,Uua: A,Uub: array(A)] : aa(array(A),assn,aa(A,fun(array(A),assn),aTP_Lamp_afq(nat,fun(A,fun(array(A),assn)),Uu),Uua),Uub) = aa(list(A),assn,snga_assn(A,Uub),replicate(A,Uu,Uua)) ) ).

% ATP.lambda_534
tff(fact_8707_ATP_Olambda__535,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_amh(fun(B,option(A)),fun(list(B),fun(A,list(A))),Uu),Uua),Uub) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),map_filter(B,A,Uu,Uua)) ).

% ATP.lambda_535
tff(fact_8708_ATP_Olambda__536,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_abb(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_536
tff(fact_8709_ATP_Olambda__537,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: real,Uub: nat] : aa(nat,A,aa(real,fun(nat,A),aTP_Lamp_abc(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_537
tff(fact_8710_ATP_Olambda__538,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_aps(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_538
tff(fact_8711_ATP_Olambda__539,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_ams(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_539
tff(fact_8712_ATP_Olambda__540,axiom,
    ! [B: $tType,A: $tType] :
      ( linorder(B)
     => ! [Uu: fun(A,B),Uua: B,Uub: A] :
          ( aa(A,$o,aa(B,fun(A,$o),aTP_Lamp_alh(fun(A,B),fun(B,fun(A,$o)),Uu),Uua),Uub)
        <=> aa(B,$o,ord_less(B,Uua),aa(A,B,Uu,Uub)) ) ) ).

% ATP.lambda_540
tff(fact_8713_ATP_Olambda__541,axiom,
    ! [B: $tType,Uu: assn,Uua: fun(B,assn),Uub: B] : aa(B,assn,aa(fun(B,assn),fun(B,assn),aTP_Lamp_bi(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_541
tff(fact_8714_ATP_Olambda__542,axiom,
    ! [A: $tType,Uu: assn,Uua: fun(A,assn),Uub: A] : aa(A,assn,aa(fun(A,assn),fun(A,assn),aTP_Lamp_bl(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_542
tff(fact_8715_ATP_Olambda__543,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_wa(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_543
tff(fact_8716_ATP_Olambda__544,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_ud(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_544
tff(fact_8717_ATP_Olambda__545,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_vz(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_545
tff(fact_8718_ATP_Olambda__546,axiom,
    ! [A: $tType] :
      ( real_V4412858255891104859lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,A,aa(A,fun(nat,A),aTP_Lamp_vs(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_546
tff(fact_8719_ATP_Olambda__547,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,aa(A,fun(A,A),times_times(A),Uua),aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_547
tff(fact_8720_ATP_Olambda__548,axiom,
    ! [A: $tType,Uu: fun(A,assn),Uua: assn,Uub: A] : aa(A,assn,aa(assn,fun(A,assn),aTP_Lamp_bp(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_548
tff(fact_8721_ATP_Olambda__549,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_xy(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_549
tff(fact_8722_ATP_Olambda__550,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: A,Uub: $o] :
      ( aa($o,$o,aa(A,fun($o,$o),aTP_Lamp_aox(fun(A,$o),fun(A,fun($o,$o)),Uu),Uua),(Uub))
    <=> ( (Uub)
        & aa(A,$o,Uu,Uua) ) ) ).

% ATP.lambda_550
tff(fact_8723_ATP_Olambda__551,axiom,
    ! [B: $tType,A: $tType,Uu: fun(A,B),Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_anp(fun(A,B),fun(A,fun(B,$o)),Uu),Uua),Uub)
    <=> ( Uub = aa(A,B,Uu,Uua) ) ) ).

% ATP.lambda_551
tff(fact_8724_ATP_Olambda__552,axiom,
    ! [A: $tType,Uu: assn,Uua: A,Uub: A] : aa(A,assn,aa(A,fun(A,assn),aTP_Lamp_bj(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_552
tff(fact_8725_ATP_Olambda__553,axiom,
    ! [A: $tType] :
      ( comm_semiring_1(A)
     => ! [Uu: A,Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_yh(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_553
tff(fact_8726_ATP_Olambda__554,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_ama(list(A),fun(A,fun(list(A),list(A))),Uu),Uua),Uub) = aa(list(A),list(A),append(A,Uub),Uu) ).

% ATP.lambda_554
tff(fact_8727_ATP_Olambda__555,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: A,Uub: A] :
      ( aa(A,$o,aa(A,fun(A,$o),aTP_Lamp_te(fun(A,nat),fun(A,fun(A,$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_555
tff(fact_8728_ATP_Olambda__556,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,$o),Uub: A] :
      ( aa(A,$o,aa(fun(A,$o),fun(A,$o),aTP_Lamp_tf(fun(A,nat),fun(fun(A,$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_556
tff(fact_8729_ATP_Olambda__557,axiom,
    ! [A: $tType,Uu: fun(A,nat),Uua: fun(A,nat),Uub: A] :
      ( aa(A,$o,aa(fun(A,nat),fun(A,$o),aTP_Lamp_td(fun(A,nat),fun(fun(A,nat),fun(A,$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_557
tff(fact_8730_ATP_Olambda__558,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_yb(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_558
tff(fact_8731_ATP_Olambda__559,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_wn(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_559
tff(fact_8732_ATP_Olambda__560,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_apl(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_560
tff(fact_8733_ATP_Olambda__561,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: fun(nat,A),Uua: nat,Uub: nat] : aa(nat,A,aa(nat,fun(nat,A),aTP_Lamp_wr(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_561
tff(fact_8734_ATP_Olambda__562,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_xz(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_562
tff(fact_8735_ATP_Olambda__563,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_uq(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_563
tff(fact_8736_ATP_Olambda__564,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_aqn(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_564
tff(fact_8737_ATP_Olambda__565,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_afp(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_565
tff(fact_8738_ATP_Olambda__566,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_en(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_566
tff(fact_8739_ATP_Olambda__567,axiom,
    ! [A: $tType,Uu: fun(A,$o),Uua: list(A),Uub: nat] :
      ( aa(nat,$o,aa(list(A),fun(nat,$o),aTP_Lamp_akj(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_567
tff(fact_8740_ATP_Olambda__568,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_wd(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_568
tff(fact_8741_ATP_Olambda__569,axiom,
    ! [B: $tType,C: $tType,D: $tType,A: $tType,Uu: fun(D,fun(B,C)),Uua: fun(A,D),Uub: A] : aa(A,fun(B,C),aa(fun(A,D),fun(A,fun(B,C)),aTP_Lamp_ev(fun(D,fun(B,C)),fun(fun(A,D),fun(A,fun(B,C))),Uu),Uua),Uub) = aa(D,fun(B,C),Uu,aa(A,D,Uua,Uub)) ).

% ATP.lambda_569
tff(fact_8742_ATP_Olambda__570,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_aho(fun(C,A),fun(fun(B,C),fun(B,A)),Uu),Uua),Uub) = aa(C,A,Uu,aa(B,C,Uua,Uub)) ).

% ATP.lambda_570
tff(fact_8743_ATP_Olambda__571,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_afd(fun(B,A),fun(fun(num,B),fun(num,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(num,B,Uua,Uub)) ).

% ATP.lambda_571
tff(fact_8744_ATP_Olambda__572,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_aeo(fun(B,A),fun(fun(nat,B),fun(nat,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(nat,B,Uua,Uub)) ).

% ATP.lambda_572
tff(fact_8745_ATP_Olambda__573,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_bk(fun(B,A),fun(fun(C,B),fun(C,A)),Uu),Uua),Uub) = aa(B,A,Uu,aa(C,B,Uua,Uub)) ).

% ATP.lambda_573
tff(fact_8746_ATP_Olambda__574,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_aql(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uu,aa(A,A,Uua,Uub)) ) ).

% ATP.lambda_574
tff(fact_8747_ATP_Olambda__575,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_aht(fun(B,A),fun(fun(A,C),fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uua,aa(B,A,Uu,Uub)) ).

% ATP.lambda_575
tff(fact_8748_ATP_Olambda__576,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_aqm(fun(A,A),fun(fun(A,A),fun(A,A)),Uu),Uua),Uub) = aa(A,A,Uua,aa(A,A,Uu,Uub)) ) ).

% ATP.lambda_576
tff(fact_8749_ATP_Olambda__577,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_ado(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_577
tff(fact_8750_ATP_Olambda__578,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_adn(fun(A,B),fun(fun(B,C),fun(A,C)),Uu),Uua),Uub) = aa(B,C,Uua,aa(A,B,Uu,Uub)) ) ).

% ATP.lambda_578
tff(fact_8751_ATP_Olambda__579,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_ep(fun(A,C),fun(A,fun(B,C)),Uu),Uua),Uub) = aa(A,C,Uu,Uua) ).

% ATP.lambda_579
tff(fact_8752_ATP_Olambda__580,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_ang(fun(A,B),fun(A,fun(C,B)),Uu),Uua),Uub) = aa(A,B,Uu,Uua) ).

% ATP.lambda_580
tff(fact_8753_ATP_Olambda__581,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_ain(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_581
tff(fact_8754_ATP_Olambda__582,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_ok(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_582
tff(fact_8755_ATP_Olambda__583,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),$o)),aTP_Lamp_akd(heap_Time_Heap(A),fun(heap_ext(product_unit),fun(A,fun(product_prod(heap_ext(product_unit),nat),$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,$o))),aTP_Lamp_akc(heap_Time_Heap(A),fun(heap_ext(product_unit),fun(A,fun(heap_ext(product_unit),fun(nat,$o)))),Uu),Uua),Uub)) ).

% ATP.lambda_583
tff(fact_8756_ATP_Olambda__584,axiom,
    ! [Uu: nat,Uua: $o,Uub: $o] :
      aa($o,heap_Time_Heap($o),aa($o,fun($o,heap_Time_Heap($o)),aTP_Lamp_ej(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_584
tff(fact_8757_ATP_Olambda__585,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_bx(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_585
tff(fact_8758_ATP_Olambda__586,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_by(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_586
tff(fact_8759_ATP_Olambda__587,axiom,
    ! [A: $tType] :
      ( real_V8999393235501362500lgebra(A)
     => ! [Uu: fun(nat,A),Uua: A,Uub: nat] : aa(nat,real,aa(A,fun(nat,real),aTP_Lamp_abx(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_587
tff(fact_8760_ATP_Olambda__588,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_yf(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_588
tff(fact_8761_ATP_Olambda__589,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,assn,aa(nat,fun(nat,assn),aTP_Lamp_ay(nat,fun(nat,fun(nat,assn)),Uu),Uua),Uub) = pure_assn(Uub = vEBT_VEBT_high(Uu,Uua)) ).

% ATP.lambda_589
tff(fact_8762_ATP_Olambda__590,axiom,
    ! [Uu: nat,Uua: nat,Uub: nat] : aa(nat,assn,aa(nat,fun(nat,assn),aTP_Lamp_ax(nat,fun(nat,fun(nat,assn)),Uu),Uua),Uub) = pure_assn(Uub = vEBT_VEBT_low(Uu,Uua)) ).

% ATP.lambda_590
tff(fact_8763_ATP_Olambda__591,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,$o),aTP_Lamp_aoo(fun(A,fun(B,C)),fun(set(C),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [B6: B] : member(C,aa(B,C,aa(A,fun(B,C),Uu,Uub),B6),Uua) ) ).

% ATP.lambda_591
tff(fact_8764_ATP_Olambda__592,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: set(B),Uub: A] :
      ( aa(A,$o,aa(set(B),fun(A,$o),aTP_Lamp_aod(fun(B,A),fun(set(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [L4: B] :
          ( ( Uub = aa(B,A,Uu,L4) )
          & member(B,L4,Uua) ) ) ).

% ATP.lambda_592
tff(fact_8765_ATP_Olambda__593,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,$o),aTP_Lamp_aow(fun(B,set(C)),fun(fun(C,$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_593
tff(fact_8766_ATP_Olambda__594,axiom,
    ! [A: $tType,B: $tType,Uu: fun(B,A),Uua: fun(B,$o),Uub: A] :
      ( aa(A,$o,aa(fun(B,$o),fun(A,$o),aTP_Lamp_aol(fun(B,A),fun(fun(B,$o),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X: B] :
          ( ( Uub = aa(B,A,Uu,X) )
          & aa(B,$o,Uua,X) ) ) ).

% ATP.lambda_594
tff(fact_8767_ATP_Olambda__595,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_aoe(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_595
tff(fact_8768_ATP_Olambda__596,axiom,
    ! [B: $tType,A: $tType,Uu: fun(B,option(A)),Uua: list(B),Uub: A] :
      ( aa(A,$o,aa(list(B),fun(A,$o),aTP_Lamp_aop(fun(B,option(A)),fun(list(B),fun(A,$o)),Uu),Uua),Uub)
    <=> ? [X: B] :
          ( member(B,X,aa(list(B),set(B),set2(B),Uua))
          & ( aa(B,option(A),Uu,X) = aa(A,option(A),some(A),Uub) ) ) ) ).

% ATP.lambda_596
tff(fact_8769_ATP_Olambda__597,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,$o),aTP_Lamp_aoy(set(A),fun(fun(B,fun(A,$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_597
tff(fact_8770_ATP_Olambda__598,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),$o),aTP_Lamp_aon(fun(A,fun(B,C)),fun(set(C),fun(product_prod(A,B),$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_598
tff(fact_8771_ATP_Olambda__599,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_aob(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_599
tff(fact_8772_ATP_Olambda__600,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),$o),aTP_Lamp_aix(B,fun(fun(A,fun(B,$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_600
tff(fact_8773_ATP_Olambda__601,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),$o),aTP_Lamp_aou(fun(B,A),fun(fun(B,fun(B,$o)),fun(product_prod(A,A),$o)),Uu),Uua),Uub)
    <=> ? [A5: B,B6: B] :
          ( ( Uub = aa(A,product_prod(A,A),aa(A,fun(A,product_prod(A,A)),product_Pair(A,A),aa(B,A,Uu,A5)),aa(B,A,Uu,B6)) )
          & aa(B,$o,aa(B,fun(B,$o),Uua,A5),B6) ) ) ).

% ATP.lambda_601
tff(fact_8774_ATP_Olambda__602,axiom,
    ! [A: $tType,Uu: set(A),Uua: set(list(A)),Uub: list(A)] :
      ( aa(list(A),$o,aa(set(list(A)),fun(list(A),$o),aTP_Lamp_aog(set(A),fun(set(list(A)),fun(list(A),$o)),Uu),Uua),Uub)
    <=> ? [X: A,Xs3: list(A)] :
          ( ( Uub = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs3) )
          & member(A,X,Uu)
          & member(list(A),Xs3,Uua) ) ) ).

% ATP.lambda_602
tff(fact_8775_ATP_Olambda__603,axiom,
    ! [B: $tType,A: $tType,E: $tType,C: $tType,D: $tType,Uu: fun(E,fun(A,fun(C,fun(B,fun(D,$o))))),Uua: E,Uub: fun(A,option(product_prod(B,product_prod(C,D))))] :
      ( aa(fun(A,option(product_prod(B,product_prod(C,D)))),$o,aa(E,fun(fun(A,option(product_prod(B,product_prod(C,D)))),$o),aTP_Lamp_aiw(fun(E,fun(A,fun(C,fun(B,fun(D,$o))))),fun(E,fun(fun(A,option(product_prod(B,product_prod(C,D)))),$o)),Uu),Uua),Uub)
    <=> ! [H4: A,H9: C,R6: B,N6: D] :
          ( ( aa(A,option(product_prod(B,product_prod(C,D))),Uub,H4) = aa(product_prod(B,product_prod(C,D)),option(product_prod(B,product_prod(C,D))),some(product_prod(B,product_prod(C,D))),aa(product_prod(C,D),product_prod(B,product_prod(C,D)),aa(B,fun(product_prod(C,D),product_prod(B,product_prod(C,D))),product_Pair(B,product_prod(C,D)),R6),aa(D,product_prod(C,D),aa(C,fun(D,product_prod(C,D)),product_Pair(C,D),H9),N6))) )
         => aa(D,$o,aa(B,fun(D,$o),aa(C,fun(B,fun(D,$o)),aa(A,fun(C,fun(B,fun(D,$o))),aa(E,fun(A,fun(C,fun(B,fun(D,$o)))),Uu,Uua),H4),H9),R6),N6) ) ) ).

% ATP.lambda_603
tff(fact_8776_ATP_Olambda__604,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($o))),aTP_Lamp_rs(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),Uu),Uua),Uub) = aa(fun(heap_ext(product_unit),option(product_prod($o,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap($o),heap_Time_Heap2($o),aTP_Lamp_rr(heap_ext(product_unit),option(product_prod($o,product_prod(heap_ext(product_unit),nat))))) ).

% ATP.lambda_604
tff(fact_8777_ATP_Olambda__605,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_rn(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat))))),Uu),Uua),Uub) = aa(fun(heap_ext(product_unit),option(product_prod(option(nat),product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(option(nat)),heap_Time_Heap2(option(nat)),aTP_Lamp_rm(heap_ext(product_unit),option(product_prod(option(nat),product_prod(heap_ext(product_unit),nat))))) ).

% ATP.lambda_605
tff(fact_8778_ATP_Olambda__606,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_oz(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),Uu),Uua),Uub) = aa(fun(heap_ext(product_unit),option(product_prod(vEBT_VEBTi,product_prod(heap_ext(product_unit),nat)))),heap_Time_Heap(vEBT_VEBTi),heap_Time_Heap2(vEBT_VEBTi),aTP_Lamp_oy(heap_ext(product_unit),option(product_prod(vEBT_VEBTi,product_prod(heap_ext(product_unit),nat))))) ).

% ATP.lambda_606
tff(fact_8779_ATP_Olambda__607,axiom,
    ! [C: $tType,A: $tType,B: $tType,D: $tType,Uu: fun(B,fun(C,fun(D,A))),Uua: D,Uub: B,Uuc: C] : aa(C,A,aa(B,fun(C,A),aa(D,fun(B,fun(C,A)),aTP_Lamp_el(fun(B,fun(C,fun(D,A))),fun(D,fun(B,fun(C,A))),Uu),Uua),Uub),Uuc) = aa(D,A,aa(C,fun(D,A),aa(B,fun(C,fun(D,A)),Uu,Uub),Uuc),Uua) ).

% ATP.lambda_607
tff(fact_8780_ATP_Olambda__608,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_qb(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($o)))))),aTP_Lamp_qa(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($o)),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))))),Uu),Uua),Uuc),aTP_Lamp_ej(nat,fun($o,fun($o,heap_Time_Heap($o))),Uuc),Uub) ).

% ATP.lambda_608
tff(fact_8781_ATP_Olambda__609,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_oi(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_oh(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_ch(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uuc),Uub) ).

% ATP.lambda_609
tff(fact_8782_ATP_Olambda__610,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_np(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_no(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_cq(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uuc),Uub) ).

% ATP.lambda_610
tff(fact_8783_ATP_Olambda__611,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_my(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_mx(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_dy(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),Uuc),Uub) ).

% ATP.lambda_611
tff(fact_8784_ATP_Olambda__612,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_ro(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_kk(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_cq(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uuc),Uub) ).

% ATP.lambda_612
tff(fact_8785_ATP_Olambda__613,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_rp(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_jv(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_ch(nat,fun($o,fun($o,heap_Time_Heap(option(nat)))),Uuc),Uub) ).

% ATP.lambda_613
tff(fact_8786_ATP_Olambda__614,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_pa(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_je(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_dy(nat,fun($o,fun($o,heap_Time_Heap(vEBT_VEBTi))),Uuc),Uub) ).

% ATP.lambda_614
tff(fact_8787_ATP_Olambda__615,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_rt(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($o)))))),aTP_Lamp_ht(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o)))),fun(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($o))))))),Uu),Uua),Uuc),aTP_Lamp_ej(nat,fun($o,fun($o,heap_Time_Heap($o))),Uuc),Uub) ).

% ATP.lambda_615
tff(fact_8788_ATP_Olambda__616,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_aaz(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_616
tff(fact_8789_ATP_Olambda__617,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_aav(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_617
tff(fact_8790_ATP_Olambda__618,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_aax(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_618
tff(fact_8791_ATP_Olambda__619,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($o))),aTP_Lamp_ee(array(vEBT_VEBTi),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o)))),Uu),Uua),Uub),Uuc) = $ite(aa(nat,$o,ord_less(nat,Uua),Uuc),heap_Time_bind(vEBT_VEBTi,$o,aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,Uu),Uua),aTP_Lamp_ed(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)),Uub)),aa($o,heap_Time_Heap($o),heap_Time_return($o),$false)) ).

% ATP.lambda_619
tff(fact_8792_ATP_Olambda__620,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_zt(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_620
tff(fact_8793_ATP_Olambda__621,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_zv(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_621
tff(fact_8794_ATP_Olambda__622,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_zw(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_622
tff(fact_8795_ATP_Olambda__623,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_zu(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_623
tff(fact_8796_ATP_Olambda__624,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_vj(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_624
tff(fact_8797_ATP_Olambda__625,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_ye(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_625
tff(fact_8798_ATP_Olambda__626,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_uw(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_626
tff(fact_8799_ATP_Olambda__627,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_agq(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_627
tff(fact_8800_ATP_Olambda__628,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($o))),aTP_Lamp_hu(list(vEBT_VEBT),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($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_628
tff(fact_8801_ATP_Olambda__629,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_gk(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_ci(nat,fun(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),Uu),Uub))) ).

% ATP.lambda_629
tff(fact_8802_ATP_Olambda__630,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($o,fun(A,B)),aTP_Lamp_afy(fun(A,B),fun(fun(A,B),fun($o,fun(A,B))),Uu),Uua),(Uub)),Uuc) = $ite((Uub),aa(A,B,Uu,Uuc),aa(A,B,Uua,Uuc)) ).

% ATP.lambda_630
tff(fact_8803_ATP_Olambda__631,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_wh(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_wg(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_631
tff(fact_8804_ATP_Olambda__632,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($o,char)),aTP_Lamp_aph($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($o,char))),aTP_Lamp_apg($o,fun($o,fun($o,fun(code_integer,fun($o,char)))),(Uu)),(Uua)),(Uuc))),code_bit_cut_integer(Uub)) ).

% ATP.lambda_632
tff(fact_8805_ATP_Olambda__633,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_ug(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_uf(fun(B,A),fun(fun(C,A),fun(B,fun(C,A))),Uu),Uua),Uuc)),Uub) ) ).

% ATP.lambda_633
tff(fact_8806_ATP_Olambda__634,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_alb(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,aa(B,fun(B,$o),ord_less_eq(B),aa(A,B,Uu,Uub)),aa(A,B,Uu,Uuc)),Uua,aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uuc),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uub),nil(A)))),
            aa(list(A),fun(A,fun(list(A),list(A))),aTP_Lamp_ala(fun(A,B),fun(list(A),fun(A,fun(list(A),list(A)))),Uu),Uua)) ) ).

% ATP.lambda_634
tff(fact_8807_ATP_Olambda__635,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_aao(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_635
tff(fact_8808_ATP_Olambda__636,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_xv(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,C,aa(B,fun(A,C),aTP_Lamp_xu(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc),collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_ta(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_636
tff(fact_8809_ATP_Olambda__637,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_um(set(A),fun(fun(A,fun(B,C)),fun(fun(A,fun(B,$o)),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,aa(B,fun(A,C),aTP_Lamp_ul(fun(A,fun(B,C)),fun(B,fun(A,C)),Uua),Uuc)),collect(A,aa(B,fun(A,$o),aa(fun(A,fun(B,$o)),fun(B,fun(A,$o)),aTP_Lamp_ta(set(A),fun(fun(A,fun(B,$o)),fun(B,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_637
tff(fact_8810_ATP_Olambda__638,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_zq(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_zp(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_638
tff(fact_8811_ATP_Olambda__639,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_zm(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_zl(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_639
tff(fact_8812_ATP_Olambda__640,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_ws(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_640
tff(fact_8813_ATP_Olambda__641,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_wg(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_641
tff(fact_8814_ATP_Olambda__642,axiom,
    ! [Uu: nat,Uua: nat,Uub: vEBT_VEBTi,Uuc: option(nat)] : aa(option(nat),assn,aa(vEBT_VEBTi,fun(option(nat),assn),aa(nat,fun(vEBT_VEBTi,fun(option(nat),assn)),aTP_Lamp_au(nat,fun(nat,fun(vEBT_VEBTi,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,aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),Uua)),Uub)),pure_assn(Uuc = vEBT_vebt_pred(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),Uua),Uu))) ).

% ATP.lambda_642
tff(fact_8815_ATP_Olambda__643,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_fa(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_643
tff(fact_8816_ATP_Olambda__644,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_zr(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_644
tff(fact_8817_ATP_Olambda__645,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_zk(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_645
tff(fact_8818_ATP_Olambda__646,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_zn(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_646
tff(fact_8819_ATP_Olambda__647,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_adh(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_647
tff(fact_8820_ATP_Olambda__648,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_aoz(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),Ys7: list(A)] :
            ( ( Uub = aa(list(A),list(A),append(A,Xys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),X),Xs6)) )
            & ( Uuc = aa(list(A),list(A),append(A,Xys),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Y3),Ys7)) )
            & 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_648
tff(fact_8821_ATP_Olambda__649,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($o))),aTP_Lamp_eg(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($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_ef(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))),Uub),Uuc)) ).

% ATP.lambda_649
tff(fact_8822_ATP_Olambda__650,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_wu(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_650
tff(fact_8823_ATP_Olambda__651,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_afi($o,fun(A,fun(A,fun(A,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( ( (Uu)
         => ( Uuc = Uua ) )
        & ( ~ (Uu)
         => ( Uuc = Uub ) ) ) ) ).

% ATP.lambda_651
tff(fact_8824_ATP_Olambda__652,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,$o)),aTP_Lamp_ahp(fun(B,A),fun(set(B),fun(fun(A,$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_652
tff(fact_8825_ATP_Olambda__653,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_uj(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_653
tff(fact_8826_ATP_Olambda__654,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_ta(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_654
tff(fact_8827_ATP_Olambda__655,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_wt(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_655
tff(fact_8828_ATP_Olambda__656,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_ahr(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_656
tff(fact_8829_ATP_Olambda__657,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_afn(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_657
tff(fact_8830_ATP_Olambda__658,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_ahm(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_658
tff(fact_8831_ATP_Olambda__659,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_ahv(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_659
tff(fact_8832_ATP_Olambda__660,axiom,
    ! [A: $tType,B: $tType,Uu: fun(A,B),Uua: set(A),Uub: B,Uuc: A] :
      ( aa(A,$o,aa(B,fun(A,$o),aa(set(A),fun(B,fun(A,$o)),aTP_Lamp_anr(fun(A,B),fun(set(A),fun(B,fun(A,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(A,Uuc,Uua)
        & ( aa(A,B,Uu,Uuc) = Uub ) ) ) ).

% ATP.lambda_660
tff(fact_8833_ATP_Olambda__661,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_ank(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_661
tff(fact_8834_ATP_Olambda__662,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_wv(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_662
tff(fact_8835_ATP_Olambda__663,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_zg(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_663
tff(fact_8836_ATP_Olambda__664,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_aky(fun(A,B),fun(list(A),fun(list(A),fun(list(A),list(A)))),Uu),Uua),Uub),Uuc) = aa(list(A),list(A),append(A,aa(list(A),list(A),linorder_sort_key(A,B,Uu),Uua)),aa(list(A),list(A),append(A,Uub),aa(list(A),list(A),linorder_sort_key(A,B,Uu),Uuc))) ) ).

% ATP.lambda_664
tff(fact_8837_ATP_Olambda__665,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_agv(int,fun(int,fun(int,fun(int,$o))),Uu),Uua),Uub),Uuc)
    <=> aa(int,$o,aa(int,fun(int,$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_665
tff(fact_8838_ATP_Olambda__666,axiom,
    ! [Uu: int,Uua: int,Uub: int,Uuc: int] :
      ( aa(int,$o,aa(int,fun(int,$o),aa(int,fun(int,fun(int,$o)),aTP_Lamp_agt(int,fun(int,fun(int,fun(int,$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_666
tff(fact_8839_ATP_Olambda__667,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_afa(A,fun(B,fun(A,fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( ( Uu = Uub )
        & ( Uua = Uuc ) ) ) ).

% ATP.lambda_667
tff(fact_8840_ATP_Olambda__668,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,$o)),aTP_Lamp_ahq(fun(B,A),fun(set(B),fun(fun(A,$o),fun(B,$o))),Uu),Uua),Uub),Uuc)
    <=> ( member(B,Uuc,Uua)
        & aa(A,$o,Uub,aa(B,A,Uu,Uuc)) ) ) ).

% ATP.lambda_668
tff(fact_8841_ATP_Olambda__669,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_tl(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_669
tff(fact_8842_ATP_Olambda__670,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_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),plus_plus(B),aa(A,B,Uua,Uuc)),aa(A,B,Uub,Uuc)) != zero_zero(B) ) ) ) ) ).

% ATP.lambda_670
tff(fact_8843_ATP_Olambda__671,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_be(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_671
tff(fact_8844_ATP_Olambda__672,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_az(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_672
tff(fact_8845_ATP_Olambda__673,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_bb(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_673
tff(fact_8846_ATP_Olambda__674,axiom,
    ! [Uu: product_prod(nat,nat),Uua: nat,Uub: nat,Uuc: 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)))),aTP_Lamp_ah(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),Uu),Uua),Uub),Uuc) = heap_Time_bind(vEBT_VEBTi,option(nat),aa(nat,heap_Time_Heap(vEBT_VEBTi),array_nth(vEBT_VEBTi,x13),Uub),aa(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),aTP_Lamp_ag(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),Uu),Uua),Uub)) ).

% ATP.lambda_674
tff(fact_8847_ATP_Olambda__675,axiom,
    ! [Uu: product_prod(nat,nat),Uua: nat,Uub: nat,Uuc: 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)))),aTP_Lamp_ag(product_prod(nat,nat),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap(option(nat))))),Uu),Uua),Uub),Uuc) = $let(
        aktnode: vEBT_VEBT,
        aktnode:= aa(nat,vEBT_VEBT,nth(vEBT_VEBT,treeList),Uub),
        heap_Time_bind(option(nat),option(nat),vEBT_vebt_minti(Uuc),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)))))),aTP_Lamp_af(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),aktnode)) ) ).

% ATP.lambda_675
tff(fact_8848_ATP_Olambda__676,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_acg(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_676
tff(fact_8849_ATP_Olambda__677,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_ajj(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_677
tff(fact_8850_ATP_Olambda__678,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_aae(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_678
tff(fact_8851_ATP_Olambda__679,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_ajk(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_679
tff(fact_8852_ATP_Olambda__680,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_zp(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_680
tff(fact_8853_ATP_Olambda__681,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_aca(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_681
tff(fact_8854_ATP_Olambda__682,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_zl(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_682
tff(fact_8855_ATP_Olambda__683,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_uf(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_683
tff(fact_8856_ATP_Olambda__684,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_xt(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_uj(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_684
tff(fact_8857_ATP_Olambda__685,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_uk(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_uj(set(B),fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_685
tff(fact_8858_ATP_Olambda__686,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_ala(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_akz(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_686
tff(fact_8859_ATP_Olambda__687,axiom,
    ! [Uu: product_prod(nat,nat),Uua: nat,Uub: nat,Uuc: 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)))),aTP_Lamp_aj(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),Uu),Uua),Uub),Uuc) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uub = vEBT_VEBT_high(xa,aa(nat,nat,aa(nat,fun(nat,nat),divide_divide(nat),x12),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)))),aTP_Lamp_ai(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),Uu),Uua),Uub)) ).

% ATP.lambda_687
tff(fact_8860_ATP_Olambda__688,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_bc(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_688
tff(fact_8861_ATP_Olambda__689,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_bd(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_689
tff(fact_8862_ATP_Olambda__690,axiom,
    ! [Uu: product_prod(nat,nat),Uua: nat,Uub: nat,Uuc: 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)))),aTP_Lamp_ai(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),Uu),Uua),Uub),Uuc) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(aa(nat,$o,ord_less(nat,Uub),aa(list(vEBT_VEBT),nat,size_size(list(vEBT_VEBT)),treeList))),aa(nat,fun(product_unit,heap_Time_Heap(option(nat))),aa(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat)))),aTP_Lamp_ah(product_prod(nat,nat),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap(option(nat))))),Uu),Uua),Uub)) ).

% ATP.lambda_690
tff(fact_8863_ATP_Olambda__691,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_pf(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_pe(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_691
tff(fact_8864_ATP_Olambda__692,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_gf(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_ge(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_692
tff(fact_8865_ATP_Olambda__693,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_op(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_oo(nat,fun(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Uu),Uua)) ).

% ATP.lambda_693
tff(fact_8866_ATP_Olambda__694,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_om(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_ol(nat,fun(nat,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))),Uu),Uua)) ).

% ATP.lambda_694
tff(fact_8867_ATP_Olambda__695,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_afs(A,fun(list(A),fun(A,fun(nat,list(A)))),Uu),Uua),Uub),Uuc) = aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),list_update(A,Uua,Uuc,Uub)) ).

% ATP.lambda_695
tff(fact_8868_ATP_Olambda__696,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_tg(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_696
tff(fact_8869_ATP_Olambda__697,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_hd($o,fun(fun(A,fun(B,$o)),fun(A,fun(B,$o))),(Uu)),Uua),Uub),Uuc)
    <=> ( (Uu)
        & aa(B,$o,aa(A,fun(B,$o),Uua,Uub),Uuc) ) ) ).

% ATP.lambda_697
tff(fact_8870_ATP_Olambda__698,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_aka(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))),aa(heap_Time_Heap(A),fun(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_698
tff(fact_8871_ATP_Olambda__699,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_gz(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_699
tff(fact_8872_ATP_Olambda__700,axiom,
    ! [A: $tType,Uu: list(A),Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option($o),aa(A,fun(list(A),option($o)),aa(list(A),fun(A,fun(list(A),option($o))),aTP_Lamp_amf(list(A),fun(list(A),fun(A,fun(list(A),option($o)))),Uu),Uua),Uub),Uuc) = subset_eq_mset_impl(A,Uu,aa(list(A),list(A),append(A,Uua),Uuc)) ).

% ATP.lambda_700
tff(fact_8873_ATP_Olambda__701,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ahn(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,B,Uua,collect(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_ahm(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_701
tff(fact_8874_ATP_Olambda__702,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_mult(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_ahx(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = groups7121269368397514597t_prod(A,C,Uub,collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_ahv(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_702
tff(fact_8875_ATP_Olambda__703,axiom,
    ! [B: $tType,A: $tType,C: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: C] : aa(C,B,aa(fun(A,C),fun(C,B),aa(fun(A,B),fun(fun(A,C),fun(C,B)),aTP_Lamp_ahu(set(A),fun(fun(A,B),fun(fun(A,C),fun(C,B))),Uu),Uua),Uub),Uuc) = aa(set(A),B,groups7311177749621191930dd_sum(A,B,Uua),collect(A,aa(C,fun(A,$o),aa(fun(A,C),fun(C,fun(A,$o)),aTP_Lamp_ahm(set(A),fun(fun(A,C),fun(C,fun(A,$o))),Uu),Uub),Uuc))) ) ).

% ATP.lambda_703
tff(fact_8876_ATP_Olambda__704,axiom,
    ! [C: $tType,A: $tType,B: $tType] :
      ( comm_monoid_add(C)
     => ! [Uu: set(A),Uua: fun(A,B),Uub: fun(A,C),Uuc: B] : aa(B,C,aa(fun(A,C),fun(B,C),aa(fun(A,B),fun(fun(A,C),fun(B,C)),aTP_Lamp_ahw(set(A),fun(fun(A,B),fun(fun(A,C),fun(B,C))),Uu),Uua),Uub),Uuc) = aa(set(A),C,groups7311177749621191930dd_sum(A,C,Uub),collect(A,aa(B,fun(A,$o),aa(fun(A,B),fun(B,fun(A,$o)),aTP_Lamp_ahv(set(A),fun(fun(A,B),fun(B,fun(A,$o))),Uu),Uua),Uuc))) ) ).

% ATP.lambda_704
tff(fact_8877_ATP_Olambda__705,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_amp(A,fun(list(A),fun(A,fun(list(A),A))),Uu),Uua),Uub),Uuc) = ord_min(A,Uu,min_list(A,Uua)) ) ).

% ATP.lambda_705
tff(fact_8878_ATP_Olambda__706,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_cs(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_706
tff(fact_8879_ATP_Olambda__707,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_ald(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_707
tff(fact_8880_ATP_Olambda__708,axiom,
    ! [A: $tType,C: $tType,B: $tType,D: $tType,Uu: fun(B,A),Uua: fun(C,fun(D,B)),Uub: C,Uuc: D] : aa(D,A,aa(C,fun(D,A),aa(fun(C,fun(D,B)),fun(C,fun(D,A)),aTP_Lamp_em(fun(B,A),fun(fun(C,fun(D,B)),fun(C,fun(D,A))),Uu),Uua),Uub),Uuc) = aa(B,A,Uu,aa(D,B,aa(C,fun(D,B),Uua,Uub),Uuc)) ).

% ATP.lambda_708
tff(fact_8881_ATP_Olambda__709,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_adu(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_709
tff(fact_8882_ATP_Olambda__710,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_ads(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_710
tff(fact_8883_ATP_Olambda__711,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_yc(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_711
tff(fact_8884_ATP_Olambda__712,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_ut(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_712
tff(fact_8885_ATP_Olambda__713,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_aga(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_713
tff(fact_8886_ATP_Olambda__714,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_sc(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_714
tff(fact_8887_ATP_Olambda__715,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_ais(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_715
tff(fact_8888_ATP_Olambda__716,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_aip(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_716
tff(fact_8889_ATP_Olambda__717,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_alf(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_ale(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_717
tff(fact_8890_ATP_Olambda__718,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_agy(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_718
tff(fact_8891_ATP_Olambda__719,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_ahe(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_719
tff(fact_8892_ATP_Olambda__720,axiom,
    ! [A: $tType,Uu: A,Uua: list(A),Uub: A,Uuc: list(A)] : aa(list(A),option(product_prod(list(A),product_prod(A,list(A)))),aa(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))),aa(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A)))))),aTP_Lamp_amc(A,fun(list(A),fun(A,fun(list(A),option(product_prod(list(A),product_prod(A,list(A))))))),Uu),Uua),Uub),Uuc) = aa(product_prod(list(A),product_prod(A,list(A))),option(product_prod(list(A),product_prod(A,list(A)))),some(product_prod(list(A),product_prod(A,list(A)))),aa(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A))),aa(list(A),fun(product_prod(A,list(A)),product_prod(list(A),product_prod(A,list(A)))),product_Pair(list(A),product_prod(A,list(A))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uu),Uua)),aa(list(A),product_prod(A,list(A)),aa(A,fun(list(A),product_prod(A,list(A))),product_Pair(A,list(A)),Uub),Uuc))) ).

% ATP.lambda_720
tff(fact_8893_ATP_Olambda__721,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_aha(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_721
tff(fact_8894_ATP_Olambda__722,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_ahc(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_722
tff(fact_8895_ATP_Olambda__723,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_wl(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_723
tff(fact_8896_ATP_Olambda__724,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_aoc(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_724
tff(fact_8897_ATP_Olambda__725,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,$o)),aTP_Lamp_aof(fun(A,$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_725
tff(fact_8898_ATP_Olambda__726,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($o)))),aTP_Lamp_pn(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($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_726
tff(fact_8899_ATP_Olambda__727,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($o)))),aTP_Lamp_hg(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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_727
tff(fact_8900_ATP_Olambda__728,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_dx(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,aa(nat,fun(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_dw(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_728
tff(fact_8901_ATP_Olambda__729,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($o)))),aTP_Lamp_pe(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($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_pd(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_729
tff(fact_8902_ATP_Olambda__730,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($o)))),aTP_Lamp_ge(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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_gd(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_730
tff(fact_8903_ATP_Olambda__731,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_ca(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_bz(nat,fun(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),Uua),Uud)) ) ).

% ATP.lambda_731
tff(fact_8904_ATP_Olambda__732,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_cf(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,aa(nat,fun(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_ce(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_732
tff(fact_8905_ATP_Olambda__733,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_co(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,aa(nat,fun(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,aa(nat,fun(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_cn(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))))),Uu),Uua),Uub),Uuc))) ) ) ).

% ATP.lambda_733
tff(fact_8906_ATP_Olambda__734,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_eh(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc),Uud) = $ite(
        aa(nat,$o,aa(nat,fun(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($o))),aTP_Lamp_eg(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,heap_Time_Heap($o)))),Uu),Uua),Uub))) ) ) ) ) ).

% ATP.lambda_734
tff(fact_8907_ATP_Olambda__735,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_cp(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_co(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_735
tff(fact_8908_ATP_Olambda__736,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_cg(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_cf(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_736
tff(fact_8909_ATP_Olambda__737,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_ei(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($o)))),aTP_Lamp_eh(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o))))),Uu),Uub),Uuc)),Uua) ).

% ATP.lambda_737
tff(fact_8910_ATP_Olambda__738,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_afz(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_738
tff(fact_8911_ATP_Olambda__739,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_sb(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_739
tff(fact_8912_ATP_Olambda__740,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_air(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_740
tff(fact_8913_ATP_Olambda__741,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_aio(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_741
tff(fact_8914_ATP_Olambda__742,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_aab(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_aaa(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_742
tff(fact_8915_ATP_Olambda__743,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_apg($o,fun($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($o,char)))),aTP_Lamp_apf($o,fun($o,fun($o,fun($o,fun(code_integer,fun($o,char))))),(Uu)),(Uua)),(Uub)),(Uud))),code_bit_cut_integer(Uuc)) ).

% ATP.lambda_743
tff(fact_8916_ATP_Olambda__744,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_age(fun(code_integer,product_prod(A,A)),fun(A,fun(code_integer,fun(A,fun(A,list(A))))),Uu),Uua),Uub),Uuc),Uud) = aa(list(A),list(A),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),aa(A,fun(list(A),list(A)),cons(A),Uuc),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uud),nil(A)))) ) ).

% ATP.lambda_744
tff(fact_8917_ATP_Olambda__745,axiom,
    ! [A: $tType] :
      ( ( code_term_of(A)
        & quickc6926020345158392990erm_of(A) )
     => ! [Uu: fun(code_integer,product_prod(A,A)),Uua: code_integer,Uub: A,Uuc: A,Uud: A] : aa(A,list(A),aa(A,fun(A,list(A)),aa(A,fun(A,fun(A,list(A))),aa(code_integer,fun(A,fun(A,fun(A,list(A)))),aTP_Lamp_agf(fun(code_integer,product_prod(A,A)),fun(code_integer,fun(A,fun(A,fun(A,list(A))))),Uu),Uua),Uub),Uuc),Uud) = aa(list(A),list(A),append(A,code_T4080844693773952564amples(A,Uu,Uub,aa(code_integer,code_integer,minus_minus(code_integer,Uua),one_one(code_integer)))),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uuc),aa(list(A),list(A),aa(A,fun(list(A),list(A)),cons(A),Uud),nil(A)))) ) ).

% ATP.lambda_745
tff(fact_8918_ATP_Olambda__746,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_adz(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_746
tff(fact_8919_ATP_Olambda__747,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_aaf(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_aae(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_747
tff(fact_8920_ATP_Olambda__748,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_zz(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_748
tff(fact_8921_ATP_Olambda__749,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_zy(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_749
tff(fact_8922_ATP_Olambda__750,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_aan(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_750
tff(fact_8923_ATP_Olambda__751,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_aaa(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_751
tff(fact_8924_ATP_Olambda__752,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_ea(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_752
tff(fact_8925_ATP_Olambda__753,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_ia(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_hz(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_753
tff(fact_8926_ATP_Olambda__754,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_ib(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($o)))),aTP_Lamp_ia(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,heap_Time_Heap($o))))),Uu),Uua),Uub),Uuc)) ).

% ATP.lambda_754
tff(fact_8927_ATP_Olambda__755,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_cn(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_cm(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,heap_Time_Heap(option(nat)))))),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_755
tff(fact_8928_ATP_Olambda__756,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($o)))),aTP_Lamp_pg(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($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_pf(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_756
tff(fact_8929_ATP_Olambda__757,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($o)))),aTP_Lamp_gg(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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_gf(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_757
tff(fact_8930_ATP_Olambda__758,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_cm(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_cl(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_758
tff(fact_8931_ATP_Olambda__759,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_hv(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_hu(list(vEBT_VEBT),fun(nat,fun(nat,fun(vEBT_VEBTi,heap_Time_Heap($o)))),Uua),Uub),Uuc)) ).

% ATP.lambda_759
tff(fact_8932_ATP_Olambda__760,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,$o))),aTP_Lamp_akc(heap_Time_Heap(A),fun(heap_ext(product_unit),fun(A,fun(heap_ext(product_unit),fun(nat,$o)))),Uu),Uua),Uub),Uuc),Uud)
    <=> ( aa(heap_ext(product_unit),option(product_prod(A,product_prod(heap_ext(product_unit),nat))),aa(heap_Time_Heap(A),fun(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_760
tff(fact_8933_ATP_Olambda__761,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_gl(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_gk(nat,fun(array(vEBT_VEBTi),fun(option(nat),fun(product_unit,heap_Time_Heap(option(nat))))),Uu),Uua),Uud)) ).

% ATP.lambda_761
tff(fact_8934_ATP_Olambda__762,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_th(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_762
tff(fact_8935_ATP_Olambda__763,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_aek(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_763
tff(fact_8936_ATP_Olambda__764,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_ie(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_id(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_764
tff(fact_8937_ATP_Olambda__765,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_dj(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_765
tff(fact_8938_ATP_Olambda__766,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_rf(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,aa(nat,fun(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_re(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_766
tff(fact_8939_ATP_Olambda__767,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_lh(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,aa(nat,fun(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_lg(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_767
tff(fact_8940_ATP_Olambda__768,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_fq(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,aa(nat,fun(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_fp(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_768
tff(fact_8941_ATP_Olambda__769,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_jf(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_bz(nat,fun(option(nat),fun(vEBT_VEBTi,heap_Time_Heap(option(nat)))),Uua),Uud)) ) ).

% ATP.lambda_769
tff(fact_8942_ATP_Olambda__770,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_dw(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,aa(nat,fun(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_dv(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_770
tff(fact_8943_ATP_Olambda__771,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_qh(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,aa(nat,fun(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_qg(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_771
tff(fact_8944_ATP_Olambda__772,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_fv(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,aa(nat,fun(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_fu(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_772
tff(fact_8945_ATP_Olambda__773,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_qp(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,aa(nat,fun(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,aa(nat,fun(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_qo(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_773
tff(fact_8946_ATP_Olambda__774,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_gb(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,aa(nat,fun(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,aa(nat,fun(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_ga(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_774
tff(fact_8947_ATP_Olambda__775,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_ph(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,aa(nat,fun(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_pg(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_775
tff(fact_8948_ATP_Olambda__776,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_gh(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,aa(nat,fun(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_gg(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_776
tff(fact_8949_ATP_Olambda__777,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_qq(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_qp(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_777
tff(fact_8950_ATP_Olambda__778,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_qi(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_qh(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_778
tff(fact_8951_ATP_Olambda__779,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_gc(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_gb(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_779
tff(fact_8952_ATP_Olambda__780,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_fw(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_fv(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_780
tff(fact_8953_ATP_Olambda__781,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($o))))),aTP_Lamp_ig(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($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_if(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_781
tff(fact_8954_ATP_Olambda__782,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_pi(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($o))))),aTP_Lamp_ph(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap($o)),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))))),Uu),Uua),Uuc),Uud)),Uub) ).

% ATP.lambda_782
tff(fact_8955_ATP_Olambda__783,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_gi(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($o))))),aTP_Lamp_gh(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($o))),fun(nat,fun(nat,fun(array(vEBT_VEBTi),fun(nat,fun(nat,heap_Time_Heap($o)))))),Uu),Uua),Uuc),Uud)),Uub) ).

% ATP.lambda_783
tff(fact_8956_ATP_Olambda__784,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_apf($o,fun($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($o,char))))),aTP_Lamp_ape($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_784
tff(fact_8957_ATP_Olambda__785,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_du(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_dt(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_785
tff(fact_8958_ATP_Olambda__786,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_dk(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_dj(nat,fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))))),Uu),Uuc),Uud),Uue)) ).

% ATP.lambda_786
tff(fact_8959_ATP_Olambda__787,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_ce(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_cd(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_787
tff(fact_8960_ATP_Olambda__788,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_pt(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_ps(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_788
tff(fact_8961_ATP_Olambda__789,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_hm(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_hl(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_789
tff(fact_8962_ATP_Olambda__790,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_qo(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_qn(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_790
tff(fact_8963_ATP_Olambda__791,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_pu(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($o))))),aTP_Lamp_pt(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($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_791
tff(fact_8964_ATP_Olambda__792,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_hn(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($o))))),aTP_Lamp_hm(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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_792
tff(fact_8965_ATP_Olambda__793,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_ga(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_fz(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_793
tff(fact_8966_ATP_Olambda__794,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_qn(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_qm(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_794
tff(fact_8967_ATP_Olambda__795,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_fz(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_fy(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_795
tff(fact_8968_ATP_Olambda__796,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_po(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_pn(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_796
tff(fact_8969_ATP_Olambda__797,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_hh(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_hg(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_797
tff(fact_8970_ATP_Olambda__798,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_ale(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),aa(a,fun(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),aa(a,fun(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),aa(a,fun(list(a),list(a)),cons(a),Uub),Uud)),Uue))) ) ) ).

% ATP.lambda_798
tff(fact_8971_ATP_Olambda__799,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($o))))),aTP_Lamp_hz(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,heap_Time_Heap($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_hy(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_799
tff(fact_8972_ATP_Olambda__800,axiom,
    ! [Uu: product_prod(nat,nat),Uua: nat,Uub: nat,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: 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)))))),aTP_Lamp_af(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) = heap_Time_bind(product_unit,option(nat),refine_Imp_assert(Uue = vEBT_vebt_mint(Uud)),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))))))),aTP_Lamp_ae(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)) ).

% ATP.lambda_800
tff(fact_8973_ATP_Olambda__801,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_lw(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_lv(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_801
tff(fact_8974_ATP_Olambda__802,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_hc(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_hb(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_802
tff(fact_8975_ATP_Olambda__803,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($o))))),aTP_Lamp_ih(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,heap_Time_Heap($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($o))))),aTP_Lamp_ig(vEBT_VEBT,fun(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(product_unit,heap_Time_Heap($o)))))),Uu),Uua),Uub),Uuc),Uud)) ).

% ATP.lambda_803
tff(fact_8976_ATP_Olambda__804,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_dv(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_du(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_804
tff(fact_8977_ATP_Olambda__805,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_cl(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_ck(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_805
tff(fact_8978_ATP_Olambda__806,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_cr(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_806
tff(fact_8979_ATP_Olambda__807,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_hf(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_807
tff(fact_8980_ATP_Olambda__808,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_px(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_pw(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_808
tff(fact_8981_ATP_Olambda__809,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($o)))))),aTP_Lamp_id(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(product_prod(list(vEBT_VEBT),vEBT_VEBT),heap_Time_Heap($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_ic(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_809
tff(fact_8982_ATP_Olambda__810,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_hq(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_hp(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_810
tff(fact_8983_ATP_Olambda__811,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_mx(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,aa(nat,fun(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_mw(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_811
tff(fact_8984_ATP_Olambda__812,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_je(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,aa(nat,fun(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_jd(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_812
tff(fact_8985_ATP_Olambda__813,axiom,
    ! [Uu: product_prod(nat,nat),Uua: nat,Uub: nat,Uuc: vEBT_VEBTi,Uud: vEBT_VEBT,Uue: option(nat),Uuf: 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))))))),aTP_Lamp_ae(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) = $ite(
        ( ( Uue != none(nat) )
        & vEBT_VEBT_greater(aa(nat,option(nat),some(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))),aa(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(option(nat)))),vEBT_VEBT_vebt_predi,Uud),Uuc),Uua),aTP_Lamp_a(nat,fun(option(nat),heap_Time_Heap(option(nat))),Uub)),
        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,summary),x14),Uub),aa(nat,fun(option(nat),heap_Time_Heap(option(nat))),aTP_Lamp_ad(product_prod(nat,nat),fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))),Uu),Uub)) ) ).

% ATP.lambda_813
tff(fact_8986_ATP_Olambda__814,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_ck(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_bx(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_cj(nat,fun(array(vEBT_VEBTi),fun(option(nat),heap_Time_Heap(option(nat)))),Uu),Uua)) ) ).

% ATP.lambda_814
tff(fact_8987_ATP_Olambda__815,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_lg(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,aa(nat,fun(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_lf(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_815
tff(fact_8988_ATP_Olambda__816,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_re(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,aa(nat,fun(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_rd(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_816
tff(fact_8989_ATP_Olambda__817,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_fp(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,aa(nat,fun(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_fo(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_817
tff(fact_8990_ATP_Olambda__818,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_lq(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,aa(nat,fun(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_lp(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_818
tff(fact_8991_ATP_Olambda__819,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_gu(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,aa(nat,fun(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,aa(nat,fun(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_gt(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_819
tff(fact_8992_ATP_Olambda__820,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_if(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,aa(nat,fun(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($o))))),aTP_Lamp_ie(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT)),heap_Time_Heap($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_gz(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_ha($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uu))) ) ) ) ) ).

% ATP.lambda_820
tff(fact_8993_ATP_Olambda__821,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($o)))))),aTP_Lamp_pz(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($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_py(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_821
tff(fact_8994_ATP_Olambda__822,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($o)))))),aTP_Lamp_hs(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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_hr(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_822
tff(fact_8995_ATP_Olambda__823,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_lv(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_lu(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_gz(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_ha($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uu)) ).

% ATP.lambda_823
tff(fact_8996_ATP_Olambda__824,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_hb(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_gy(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_gz(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_ha($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uu)) ).

% ATP.lambda_824
tff(fact_8997_ATP_Olambda__825,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($o,char))))),aTP_Lamp_ape($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($o,char)))))),aTP_Lamp_apd($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_825
tff(fact_8998_ATP_Olambda__826,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_dt(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_ds(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_826
tff(fact_8999_ATP_Olambda__827,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_rc(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_rb(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_827
tff(fact_9000_ATP_Olambda__828,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_fn(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_fm(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_828
tff(fact_9001_ATP_Olambda__829,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_gt(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_gs(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_829
tff(fact_9002_ATP_Olambda__830,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_qg(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_qf(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_830
tff(fact_9003_ATP_Olambda__831,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_fu(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_ft(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_831
tff(fact_9004_ATP_Olambda__832,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_cd(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_cc(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_832
tff(fact_9005_ATP_Olambda__833,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_aov(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,aa(nat,fun(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))
        & ! [I2: nat] :
            ( aa(nat,$o,ord_less(nat,I2),aa(nat,nat,aa(nat,fun(nat,nat),power_power(nat),aa(num,nat,numeral_numeral(nat),bit0(one2))),Uud))
           => ( ? [X_1: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(aa(nat,vEBT_VEBT,nth(vEBT_VEBT,Uua),I2)),X_1)
            <=> aa(nat,$o,vEBT_V8194947554948674370ptions(Uub),I2) ) )
        & $ite(
            Uue = Uuf,
            ! [X: vEBT_VEBT] :
              ( member(vEBT_VEBT,X,aa(list(vEBT_VEBT),set(vEBT_VEBT),set2(vEBT_VEBT),Uua))
             => ~ ? [X7: nat] : aa(nat,$o,vEBT_V8194947554948674370ptions(X),X7) ),
            ( 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,$o,aa(nat,fun(nat,$o),ord_less_eq(nat),X),Uuf) ) ) ) ) ) ) ) ).

% ATP.lambda_833
tff(fact_9006_ATP_Olambda__834,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($o)))))),aTP_Lamp_ps(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($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_pr(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_834
tff(fact_9007_ATP_Olambda__835,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($o)))))),aTP_Lamp_hl(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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_hk(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_835
tff(fact_9008_ATP_Olambda__836,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_jg(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_jf(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_836
tff(fact_9009_ATP_Olambda__837,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_hx(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_hw(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_837
tff(fact_9010_ATP_Olambda__838,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_oh(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_og(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_838
tff(fact_9011_ATP_Olambda__839,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_no(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_nn(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_839
tff(fact_9012_ATP_Olambda__840,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_kk(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_kj(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_840
tff(fact_9013_ATP_Olambda__841,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_jv(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_ju(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_841
tff(fact_9014_ATP_Olambda__842,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($o)))))),aTP_Lamp_qa(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($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($o)))))),aTP_Lamp_pz(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($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_842
tff(fact_9015_ATP_Olambda__843,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($o)))))),aTP_Lamp_ht(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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($o)))))),aTP_Lamp_hs(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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_843
tff(fact_9016_ATP_Olambda__844,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_rd(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_rc(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_844
tff(fact_9017_ATP_Olambda__845,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_fo(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_fn(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_845
tff(fact_9018_ATP_Olambda__846,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_qm(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_ql(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_846
tff(fact_9019_ATP_Olambda__847,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_fy(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_fx(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_847
tff(fact_9020_ATP_Olambda__848,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_hy(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($o)))))),aTP_Lamp_hx(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,heap_Time_Heap($o))))))),Uu),Uua),Uub),Uuc),Uud),Uue)) ).

% ATP.lambda_848
tff(fact_9021_ATP_Olambda__849,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_afr(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),aa(A,fun(list(A),list(A)),cons(A),Uua),Uub)),Uue)),aa(nat,B,nth(B,aa(list(B),list(B),aa(B,fun(list(B),list(B)),cons(B),Uuc),Uud)),Uue))) ).

% ATP.lambda_849
tff(fact_9022_ATP_Olambda__850,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($o))))))),aTP_Lamp_pw(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($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_pv(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_850
tff(fact_9023_ATP_Olambda__851,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_lu(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_lt(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_851
tff(fact_9024_ATP_Olambda__852,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_le(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_ld(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_852
tff(fact_9025_ATP_Olambda__853,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($o))))))),aTP_Lamp_hp(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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_ho(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_853
tff(fact_9026_ATP_Olambda__854,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_gy(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_gx(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_854
tff(fact_9027_ATP_Olambda__855,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($o,char))))))),aTP_Lamp_apc($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_855
tff(fact_9028_ATP_Olambda__856,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_ql(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_bx(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_cj(nat,fun(array(vEBT_VEBTi),fun(option(nat),heap_Time_Heap(option(nat)))),Uua),Uub)) ) ).

% ATP.lambda_856
tff(fact_9029_ATP_Olambda__857,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_fx(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_bx(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_cj(nat,fun(array(vEBT_VEBTi),fun(option(nat),heap_Time_Heap(option(nat)))),Uua),Uub)) ) ).

% ATP.lambda_857
tff(fact_9030_ATP_Olambda__858,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($o))))))),aTP_Lamp_hw(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($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($o)))),aTP_Lamp_hv(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(nat,fun(nat,fun(product_unit,heap_Time_Heap($o))))),Uub),Uuc),Uud),Uue)),
        aa($o,heap_Time_Heap($o),heap_Time_return($o),$false) ) ).

% ATP.lambda_858
tff(fact_9031_ATP_Olambda__859,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_mw(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,aa(nat,fun(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_mv(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_859
tff(fact_9032_ATP_Olambda__860,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_jd(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,aa(nat,fun(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_jc(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_860
tff(fact_9033_ATP_Olambda__861,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_ob(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,aa(nat,fun(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_oa(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_861
tff(fact_9034_ATP_Olambda__862,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_jp(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,aa(nat,fun(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_jo(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_862
tff(fact_9035_ATP_Olambda__863,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_ni(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,aa(nat,fun(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,aa(nat,fun(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_nh(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_863
tff(fact_9036_ATP_Olambda__864,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_ke(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,aa(nat,fun(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,aa(nat,fun(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_kd(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_864
tff(fact_9037_ATP_Olambda__865,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_py(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,aa(nat,fun(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_px(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_gz(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_ha($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uua))) ) ) ) ) ).

% ATP.lambda_865
tff(fact_9038_ATP_Olambda__866,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_hr(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,aa(nat,fun(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_hq(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_gz(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_ha($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uua))) ) ) ) ) ).

% ATP.lambda_866
tff(fact_9039_ATP_Olambda__867,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_lr(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_lq(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_867
tff(fact_9040_ATP_Olambda__868,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_gv(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_gu(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_868
tff(fact_9041_ATP_Olambda__869,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_og(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_of(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_gz(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_ha($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uua)) ).

% ATP.lambda_869
tff(fact_9042_ATP_Olambda__870,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_nn(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_nm(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_gz(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_ha($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uua)) ).

% ATP.lambda_870
tff(fact_9043_ATP_Olambda__871,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_kj(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_ki(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_gz(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_ha($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uua)) ).

% ATP.lambda_871
tff(fact_9044_ATP_Olambda__872,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_ju(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_jt(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_gz(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_ha($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uua)) ).

% ATP.lambda_872
tff(fact_9045_ATP_Olambda__873,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($o,char)))))),aTP_Lamp_apd($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($o,char))))))),aTP_Lamp_apc($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_873
tff(fact_9046_ATP_Olambda__874,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_lf(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_le(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_gz(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_ha($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uu)) ).

% ATP.lambda_874
tff(fact_9047_ATP_Olambda__875,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_rb(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_ra(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_875
tff(fact_9048_ATP_Olambda__876,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_fm(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_fl(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_876
tff(fact_9049_ATP_Olambda__877,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($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_dl(nat,fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($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_dk(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_877
tff(fact_9050_ATP_Olambda__878,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_lp(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_lo(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_878
tff(fact_9051_ATP_Olambda__879,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_nh(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_ng(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_879
tff(fact_9052_ATP_Olambda__880,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_kd(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_kc(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_880
tff(fact_9053_ATP_Olambda__881,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_ds(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_dr(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_881
tff(fact_9054_ATP_Olambda__882,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_qf(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_qe(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_882
tff(fact_9055_ATP_Olambda__883,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_ft(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_fs(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_883
tff(fact_9056_ATP_Olambda__884,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_gp(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_go(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_884
tff(fact_9057_ATP_Olambda__885,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_go(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_gn(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_885
tff(fact_9058_ATP_Olambda__886,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($o))))))),aTP_Lamp_ic(nat,fun(option(product_prod(nat,nat)),fun(nat,fun(array(vEBT_VEBTi),fun(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,heap_Time_Heap($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_ib(nat,fun(nat,fun(array(vEBT_VEBTi),fun(list(vEBT_VEBT),fun(product_unit,heap_Time_Heap($o))))),Uu),Uub),Uuc),Uuf)) ).

% ATP.lambda_886
tff(fact_9059_ATP_Olambda__887,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_gs(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_gr(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_887
tff(fact_9060_ATP_Olambda__888,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_pq(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_pp(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_888
tff(fact_9061_ATP_Olambda__889,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_hj(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_hi(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_889
tff(fact_9062_ATP_Olambda__890,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_gq(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_gp(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_890
tff(fact_9063_ATP_Olambda__891,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_cc(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_cb(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_891
tff(fact_9064_ATP_Olambda__892,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_pr(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($o))))))),aTP_Lamp_pq(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($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_892
tff(fact_9065_ATP_Olambda__893,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_hk(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($o))))))),aTP_Lamp_hj(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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_893
tff(fact_9066_ATP_Olambda__894,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_of(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_oe(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_894
tff(fact_9067_ATP_Olambda__895,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_nm(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_nl(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_895
tff(fact_9068_ATP_Olambda__896,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_mu(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_mt(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_896
tff(fact_9069_ATP_Olambda__897,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_lt(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_ls(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_897
tff(fact_9070_ATP_Olambda__898,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_ld(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_lc(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_898
tff(fact_9071_ATP_Olambda__899,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_ki(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_kh(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_899
tff(fact_9072_ATP_Olambda__900,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_jt(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_js(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_900
tff(fact_9073_ATP_Olambda__901,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_jb(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_ja(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_901
tff(fact_9074_ATP_Olambda__902,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_gx(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_gw(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_902
tff(fact_9075_ATP_Olambda__903,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_cb(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_bx(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_ca(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_903
tff(fact_9076_ATP_Olambda__904,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($o)))))))),aTP_Lamp_pp(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($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_po(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_904
tff(fact_9077_ATP_Olambda__905,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($o)))))))),aTP_Lamp_hi(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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_hh(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_905
tff(fact_9078_ATP_Olambda__906,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_oc(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_ob(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_906
tff(fact_9079_ATP_Olambda__907,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_nj(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_ni(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_907
tff(fact_9080_ATP_Olambda__908,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_kf(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_ke(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_908
tff(fact_9081_ATP_Olambda__909,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_jq(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_jp(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_909
tff(fact_9082_ATP_Olambda__910,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_mv(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_mu(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_gz(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_ha($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uua)) ).

% ATP.lambda_910
tff(fact_9083_ATP_Olambda__911,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_jc(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_jb(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_gz(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_ha($o,fun($o,product_prod(option(product_prod(nat,nat)),product_prod(nat,product_prod(list(vEBT_VEBT),vEBT_VEBT))))),Uua)) ).

% ATP.lambda_911
tff(fact_9084_ATP_Olambda__912,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_kl(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_dk(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_912
tff(fact_9085_ATP_Olambda__913,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_qt(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_dk(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_913
tff(fact_9086_ATP_Olambda__914,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_fe(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_dk(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_914
tff(fact_9087_ATP_Olambda__915,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_dm(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($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))),aTP_Lamp_dl(nat,fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))),Uu),Uub),Uuc),Uud),Uue),Uuf),(Uug))) ).

% ATP.lambda_915
tff(fact_9088_ATP_Olambda__916,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_oa(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_nz(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_916
tff(fact_9089_ATP_Olambda__917,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_jo(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_jn(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_917
tff(fact_9090_ATP_Olambda__918,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_dr(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_dq(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_918
tff(fact_9091_ATP_Olambda__919,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_ra(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_qz(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_919
tff(fact_9092_ATP_Olambda__920,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_fl(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_fk(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_920
tff(fact_9093_ATP_Olambda__921,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_nd(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_nc(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_921
tff(fact_9094_ATP_Olambda__922,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_jz(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_jy(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_922
tff(fact_9095_ATP_Olambda__923,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_nc(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_nb(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_923
tff(fact_9096_ATP_Olambda__924,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_jy(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_jx(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_924
tff(fact_9097_ATP_Olambda__925,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($o)))))))),aTP_Lamp_pv(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap($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_pu(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_925
tff(fact_9098_ATP_Olambda__926,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($o)))))))),aTP_Lamp_ho(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap($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_hn(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_926
tff(fact_9099_ATP_Olambda__927,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_lo(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_ln(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_927
tff(fact_9100_ATP_Olambda__928,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_ng(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_nf(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_928
tff(fact_9101_ATP_Olambda__929,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_kc(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_kb(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_929
tff(fact_9102_ATP_Olambda__930,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_gr(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_gq(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_930
tff(fact_9103_ATP_Olambda__931,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_gn(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_gm(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_931
tff(fact_9104_ATP_Olambda__932,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_ne(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_nd(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_932
tff(fact_9105_ATP_Olambda__933,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_ka(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_jz(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_933
tff(fact_9106_ATP_Olambda__934,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_do(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($o,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_dn(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh)) ).

% ATP.lambda_934
tff(fact_9107_ATP_Olambda__935,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_qe(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_qd(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_935
tff(fact_9108_ATP_Olambda__936,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_fs(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_fr(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_936
tff(fact_9109_ATP_Olambda__937,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_oe(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_od(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_937
tff(fact_9110_ATP_Olambda__938,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_nl(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_nk(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_938
tff(fact_9111_ATP_Olambda__939,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_mt(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_ms(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_939
tff(fact_9112_ATP_Olambda__940,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_kh(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_kg(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_940
tff(fact_9113_ATP_Olambda__941,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_js(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_jr(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_941
tff(fact_9114_ATP_Olambda__942,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_ja(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_iz(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_942
tff(fact_9115_ATP_Olambda__943,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_gm(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_bx(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_gl(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))),Uu),Uua),Uuc),Uue)) ) ).

% ATP.lambda_943
tff(fact_9116_ATP_Olambda__944,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_qd(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_bx(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_ca(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_944
tff(fact_9117_ATP_Olambda__945,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_fr(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_bx(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_ca(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_945
tff(fact_9118_ATP_Olambda__946,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_lb(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_la(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_946
tff(fact_9119_ATP_Olambda__947,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_ky(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_kx(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_947
tff(fact_9120_ATP_Olambda__948,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($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_mb(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($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_dk(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_948
tff(fact_9121_ATP_Olambda__949,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($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_ii(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($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_dk(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_949
tff(fact_9122_ATP_Olambda__950,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($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_km(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($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($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_kl(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uub),Uuc),Uud),Uue),Uuf),Uug),(Uuh))) ).

% ATP.lambda_950
tff(fact_9123_ATP_Olambda__951,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($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_qu(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($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($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_qt(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),(Uuh))) ).

% ATP.lambda_951
tff(fact_9124_ATP_Olambda__952,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($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_ff(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($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($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))),aTP_Lamp_fe(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),(Uuh))) ).

% ATP.lambda_952
tff(fact_9125_ATP_Olambda__953,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_qz(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_qy(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_953
tff(fact_9126_ATP_Olambda__954,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_fk(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_fj(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_954
tff(fact_9127_ATP_Olambda__955,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_ll(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_lk(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_955
tff(fact_9128_ATP_Olambda__956,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_dn(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_dm(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_956
tff(fact_9129_ATP_Olambda__957,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_lk(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_lj(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_957
tff(fact_9130_ATP_Olambda__958,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_ls(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_lr(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_958
tff(fact_9131_ATP_Olambda__959,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_gw(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_gv(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_959
tff(fact_9132_ATP_Olambda__960,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_lc(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_lb(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_960
tff(fact_9133_ATP_Olambda__961,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_nz(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_ny(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_961
tff(fact_9134_ATP_Olambda__962,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_jn(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_jm(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_962
tff(fact_9135_ATP_Olambda__963,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_ln(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_lm(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_963
tff(fact_9136_ATP_Olambda__964,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_nf(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_ne(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_964
tff(fact_9137_ATP_Olambda__965,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_kb(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_ka(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_965
tff(fact_9138_ATP_Olambda__966,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_nb(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_na(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_966
tff(fact_9139_ATP_Olambda__967,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_jx(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_jw(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_967
tff(fact_9140_ATP_Olambda__968,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_lm(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_ll(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_968
tff(fact_9141_ATP_Olambda__969,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_kz(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_ky(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_969
tff(fact_9142_ATP_Olambda__970,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_la(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_kz(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_970
tff(fact_9143_ATP_Olambda__971,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_qw(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_qv(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_971
tff(fact_9144_ATP_Olambda__972,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_fh(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_fg(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_972
tff(fact_9145_ATP_Olambda__973,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_dq(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_dp(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_973
tff(fact_9146_ATP_Olambda__974,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_dp(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_do(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_974
tff(fact_9147_ATP_Olambda__975,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_na(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_bx(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_gl(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))),Uua),Uub),Uud),Uuf)) ) ).

% ATP.lambda_975
tff(fact_9148_ATP_Olambda__976,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_jw(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_bx(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_gl(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBT,fun(nat,fun(option(nat),heap_Time_Heap(option(nat)))))),Uua),Uub),Uud),Uuf)) ) ).

% ATP.lambda_976
tff(fact_9149_ATP_Olambda__977,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_mr(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_mq(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_977
tff(fact_9150_ATP_Olambda__978,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_iy(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_ix(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_978
tff(fact_9151_ATP_Olambda__979,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_mo(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_mn(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_979
tff(fact_9152_ATP_Olambda__980,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_iv(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_iu(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_980
tff(fact_9153_ATP_Olambda__981,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_kx(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_kw(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_981
tff(fact_9154_ATP_Olambda__982,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($o,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_qv(fun(product_prod(vEBT_VEBTi,nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($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_qu(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_982
tff(fact_9155_ATP_Olambda__983,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_mc(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($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_mb(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui))) ).

% ATP.lambda_983
tff(fact_9156_ATP_Olambda__984,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_ij(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($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_ii(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(array(vEBT_VEBTi),heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uuc),Uud),Uue),Uuf),Uug),Uuh),(Uui))) ).

% ATP.lambda_984
tff(fact_9157_ATP_Olambda__985,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($o,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_fg(fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($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_ff(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_985
tff(fact_9158_ATP_Olambda__986,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_kq(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_kp(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_986
tff(fact_9159_ATP_Olambda__987,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_nw(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_nv(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_987
tff(fact_9160_ATP_Olambda__988,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_jk(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_jj(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_988
tff(fact_9161_ATP_Olambda__989,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_nv(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_nu(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_989
tff(fact_9162_ATP_Olambda__990,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_jj(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_ji(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_990
tff(fact_9163_ATP_Olambda__991,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_od(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_oc(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_991
tff(fact_9164_ATP_Olambda__992,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_nk(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_nj(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_992
tff(fact_9165_ATP_Olambda__993,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_kg(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_kf(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_993
tff(fact_9166_ATP_Olambda__994,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_jr(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_jq(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_994
tff(fact_9167_ATP_Olambda__995,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_ms(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_mr(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_995
tff(fact_9168_ATP_Olambda__996,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_iz(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_iy(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_996
tff(fact_9169_ATP_Olambda__997,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_ny(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_nx(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_997
tff(fact_9170_ATP_Olambda__998,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_jm(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_jl(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_998
tff(fact_9171_ATP_Olambda__999,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_lj(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_li(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_999
tff(fact_9172_ATP_Olambda__1000,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_kr(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_kq(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_1000
tff(fact_9173_ATP_Olambda__1001,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_nx(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_nw(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_1001
tff(fact_9174_ATP_Olambda__1002,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_jl(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_jk(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_1002
tff(fact_9175_ATP_Olambda__1003,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_mp(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_mo(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_1003
tff(fact_9176_ATP_Olambda__1004,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_iw(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_iv(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_1004
tff(fact_9177_ATP_Olambda__1005,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_mq(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_mp(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_1005
tff(fact_9178_ATP_Olambda__1006,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_ix(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_iw(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_1006
tff(fact_9179_ATP_Olambda__1007,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_kp(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($o,heap_Time_Heap(vEBT_VEBTi))))))))))),aTP_Lamp_ko(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($o,heap_Time_Heap(vEBT_VEBTi)))))))))))),Uu),Uua),Uub),Uuc),Uud),Uue),Uuf),Uug),Uuh),Uui),Uuj)) ).

% ATP.lambda_1007
tff(fact_9180_ATP_Olambda__1008,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_qy(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_qx(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_1008
tff(fact_9181_ATP_Olambda__1009,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_fj(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_fi(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_1009
tff(fact_9182_ATP_Olambda__1010,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_qx(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_qw(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_1010
tff(fact_9183_ATP_Olambda__1011,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_fi(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_fh(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_1011
tff(fact_9184_ATP_Olambda__1012,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_li(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_bx(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_jg(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_1012
tff(fact_9185_ATP_Olambda__1013,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_mn(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_mm(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_1013
tff(fact_9186_ATP_Olambda__1014,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_iu(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_it(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_1014
tff(fact_9187_ATP_Olambda__1015,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_kw(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_kv(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_1015
tff(fact_9188_ATP_Olambda__1016,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_mg(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_mf(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_1016
tff(fact_9189_ATP_Olambda__1017,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_in(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_im(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_1017
tff(fact_9190_ATP_Olambda__1018,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_ko(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_kn(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_1018
tff(fact_9191_ATP_Olambda__1019,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_nu(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_nt(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_1019
tff(fact_9192_ATP_Olambda__1020,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_ji(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_jh(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_1020
tff(fact_9193_ATP_Olambda__1021,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_mh(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_mg(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_1021
tff(fact_9194_ATP_Olambda__1022,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_io(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_in(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_1022
tff(fact_9195_ATP_Olambda__1023,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_mf(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_me(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_1023
tff(fact_9196_ATP_Olambda__1024,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_im(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_il(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_1024
tff(fact_9197_ATP_Olambda__1025,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_nt(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_bx(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_jg(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_1025
tff(fact_9198_ATP_Olambda__1026,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_jh(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_bx(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_jg(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_1026
tff(fact_9199_ATP_Olambda__1027,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_kn(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($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))),aTP_Lamp_km(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),Uu),Uua),Uub),Uud),Uue),Uuf),Uug),Uui),(Uuk))) ).

% ATP.lambda_1027
tff(fact_9200_ATP_Olambda__1028,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_mm(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_ml(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_1028
tff(fact_9201_ATP_Olambda__1029,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_it(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_is(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_1029
tff(fact_9202_ATP_Olambda__1030,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_kv(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_ku(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_1030
tff(fact_9203_ATP_Olambda__1031,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($o,heap_Time_Heap(vEBT_VEBTi)))))))))))),aTP_Lamp_me(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($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_md(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_1031
tff(fact_9204_ATP_Olambda__1032,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($o,heap_Time_Heap(vEBT_VEBTi)))))))))))),aTP_Lamp_il(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($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_ik(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_1032
tff(fact_9205_ATP_Olambda__1033,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($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))),aTP_Lamp_md(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($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($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_mc(fun(product_prod(product_prod(vEBT_VEBT,vEBT_VEBTi),nat),heap_Time_Heap(vEBT_VEBTi)),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uuj),(Uul))) ).

% ATP.lambda_1033
tff(fact_9206_ATP_Olambda__1034,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($o,fun(product_unit,heap_Time_Heap(vEBT_VEBTi))))))))))))),aTP_Lamp_ik(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(list(vEBT_VEBT),fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun(nat,fun(vEBT_VEBTi,fun($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($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi)))))))))),aTP_Lamp_ij(fun(vEBT_VEBT,fun(vEBT_VEBTi,fun(nat,heap_Time_Heap(vEBT_VEBTi)))),fun(nat,fun(array(vEBT_VEBTi),fun(vEBT_VEBTi,fun(vEBT_VEBT,fun(nat,fun(nat,fun(nat,fun(nat,fun($o,fun(vEBT_VEBTi,heap_Time_Heap(vEBT_VEBTi))))))))))),Uu),Uua),Uub),Uuc),Uue),Uuf),Uug),Uuh),Uuj),(Uul))) ).

% ATP.lambda_1034
tff(fact_9207_ATP_Olambda__1035,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_ku(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_kt(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_1035
tff(fact_9208_ATP_Olambda__1036,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_ml(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_mk(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_1036
tff(fact_9209_ATP_Olambda__1037,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_is(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_ir(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_1037
tff(fact_9210_ATP_Olambda__1038,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_kt(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_ks(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_1038
tff(fact_9211_ATP_Olambda__1039,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_ks(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_kr(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_1039
tff(fact_9212_ATP_Olambda__1040,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_mk(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_mj(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_1040
tff(fact_9213_ATP_Olambda__1041,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_ir(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_iq(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_1041
tff(fact_9214_ATP_Olambda__1042,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_mj(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_mi(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_1042
tff(fact_9215_ATP_Olambda__1043,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_iq(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_ip(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_1043
tff(fact_9216_ATP_Olambda__1044,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_mi(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_mh(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_1044
tff(fact_9217_ATP_Olambda__1045,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_ip(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_io(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_1045
tff(fact_9218_ATP_Olambda__1046,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_lz(option(product_prod(nat,nat)),fun(nat,fun(list(vEBT_VEBT),fun(vEBT_VEBT,$o)))),Uu),Uua),Uub),Uuc)
    <=> $true ) ).

% ATP.lambda_1046
tff(fact_9219_ATP_Olambda__1047,axiom,
    ! [B: $tType,A: $tType,Uu: $o,Uua: A,Uub: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_ey($o,fun(A,fun(B,$o)),(Uu)),Uua),Uub)
    <=> (Uu) ) ).

% ATP.lambda_1047
tff(fact_9220_ATP_Olambda__1048,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_akf(A,fun(heap_ext(product_unit),fun(nat,A)),Uu),Uua),Uub) = Uu ).

% ATP.lambda_1048
tff(fact_9221_ATP_Olambda__1049,axiom,
    ! [A: $tType,B: $tType,Uu: heap_Time_Heap(B),Uua: A] : aa(A,heap_Time_Heap(B),aTP_Lamp_pm(heap_Time_Heap(B),fun(A,heap_Time_Heap(B)),Uu),Uua) = Uu ).

% ATP.lambda_1049
tff(fact_9222_ATP_Olambda__1050,axiom,
    ! [A: $tType,Uu: heap_Time_Heap(A),Uua: product_unit] : aa(product_unit,heap_Time_Heap(A),aTP_Lamp_cy(heap_Time_Heap(A),fun(product_unit,heap_Time_Heap(A)),Uu),Uua) = Uu ).

% ATP.lambda_1050
tff(fact_9223_ATP_Olambda__1051,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_akh(heap_ext(product_unit),fun(nat,heap_ext(product_unit))),Uu),Uua) = Uu ).

% ATP.lambda_1051
tff(fact_9224_ATP_Olambda__1052,axiom,
    ! [Uu: assn,Uua: product_unit] : aa(product_unit,assn,aTP_Lamp_da(assn,fun(product_unit,assn),Uu),Uua) = Uu ).

% ATP.lambda_1052
tff(fact_9225_ATP_Olambda__1053,axiom,
    ! [A: $tType,Uu: assn,Uua: A] : aa(A,assn,aTP_Lamp_bu(assn,fun(A,assn),Uu),Uua) = Uu ).

% ATP.lambda_1053
tff(fact_9226_ATP_Olambda__1054,axiom,
    ! [A: $tType,B: $tType,Uu: option(B),Uua: A] : aa(A,option(B),aTP_Lamp_aiu(option(B),fun(A,option(B)),Uu),Uua) = Uu ).

% ATP.lambda_1054
tff(fact_9227_ATP_Olambda__1055,axiom,
    ! [Uu: $o,Uua: nat] :
      ( aa(nat,$o,aTP_Lamp_amr($o,fun(nat,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_1055
tff(fact_9228_ATP_Olambda__1056,axiom,
    ! [A: $tType,Uu: $o,Uua: A] :
      ( aa(A,$o,aTP_Lamp_ex($o,fun(A,$o),(Uu)),Uua)
    <=> (Uu) ) ).

% ATP.lambda_1056
tff(fact_9229_ATP_Olambda__1057,axiom,
    ! [A: $tType,C: $tType,B: $tType,Uu: fun(B,C),Uua: A] : aa(A,fun(B,C),aTP_Lamp_es(fun(B,C),fun(A,fun(B,C)),Uu),Uua) = Uu ).

% ATP.lambda_1057
tff(fact_9230_ATP_Olambda__1058,axiom,
    ! [A: $tType,B: $tType,C: $tType,Uu: C,Uua: product_prod(A,B)] : aa(product_prod(A,B),C,aTP_Lamp_nr(C,fun(product_prod(A,B),C),Uu),Uua) = Uu ).

% ATP.lambda_1058
tff(fact_9231_ATP_Olambda__1059,axiom,
    ! [A: $tType,B: $tType] :
      ( linorder(B)
     => ! [Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ahj(B,fun(A,B),Uu),Uua) = Uu ) ).

% ATP.lambda_1059
tff(fact_9232_ATP_Olambda__1060,axiom,
    ! [A: $tType,B: $tType,Uu: B,Uua: A] : aa(A,B,aTP_Lamp_ahs(B,fun(A,B),Uu),Uua) = Uu ).

% ATP.lambda_1060
tff(fact_9233_ATP_Olambda__1061,axiom,
    ! [A: $tType] :
      ( real_V822414075346904944vector(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_abv(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1061
tff(fact_9234_ATP_Olambda__1062,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A,Uua: A] : aa(A,A,aTP_Lamp_app(A,fun(A,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1062
tff(fact_9235_ATP_Olambda__1063,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_aln(A,fun(list(A),A)),Uu),Uua) = Uu ) ).

% ATP.lambda_1063
tff(fact_9236_ATP_Olambda__1064,axiom,
    ! [A: $tType] :
      ( heap(A)
     => ! [Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ajs(A,fun(nat,A),Uu),Uua) = Uu ) ).

% ATP.lambda_1064
tff(fact_9237_ATP_Olambda__1065,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),A,aa(A,fun(list(A),A),aTP_Lamp_amj(A,fun(list(A),A)),Uu),Uua) = Uu ).

% ATP.lambda_1065
tff(fact_9238_ATP_Olambda__1066,axiom,
    ! [A: $tType,Uu: A,Uua: nat] : aa(nat,A,aTP_Lamp_ajn(A,fun(nat,A),Uu),Uua) = Uu ).

% ATP.lambda_1066
tff(fact_9239_ATP_Olambda__1067,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] : aa(B,A,aa(A,fun(B,A),aTP_Lamp_eq(A,fun(B,A)),Uu),Uua) = Uu ).

% ATP.lambda_1067
tff(fact_9240_ATP_Olambda__1068,axiom,
    ! [Uu: heap_ext(product_unit),Uua: nat] : aa(nat,nat,aa(heap_ext(product_unit),fun(nat,nat),aTP_Lamp_ajy(heap_ext(product_unit),fun(nat,nat)),Uu),Uua) = Uua ).

% ATP.lambda_1068
tff(fact_9241_ATP_Olambda__1069,axiom,
    ! [B: $tType,A: $tType,Uu: B,Uua: A] : aa(A,A,aa(B,fun(A,A),aTP_Lamp_et(B,fun(A,A)),Uu),Uua) = Uua ).

% ATP.lambda_1069
tff(fact_9242_ATP_Olambda__1070,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] : aa(list(A),list(A),aa(A,fun(list(A),list(A)),aTP_Lamp_amb(A,fun(list(A),list(A))),Uu),Uua) = Uua ).

% ATP.lambda_1070
tff(fact_9243_ATP_Olambda__1071,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_ha($o,fun($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_1071
tff(fact_9244_ATP_Olambda__1072,axiom,
    ! [Uu: $o,Uua: $o] :
      ( aa($o,$o,aa($o,fun($o,$o),aTP_Lamp_ma($o,fun($o,$o)),(Uu)),(Uua))
    <=> $false ) ).

% ATP.lambda_1072
tff(fact_9245_ATP_Olambda__1073,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_alk(A,fun(list(A),$o)),Uu),Uua)
    <=> $false ) ).

% ATP.lambda_1073
tff(fact_9246_ATP_Olambda__1074,axiom,
    ! [A: $tType,Uu: A,Uua: list(A)] :
      ( aa(list(A),$o,aa(A,fun(list(A),$o),aTP_Lamp_alj(A,fun(list(A),$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1074
tff(fact_9247_ATP_Olambda__1075,axiom,
    ! [B: $tType,A: $tType,Uu: A,Uua: B] :
      ( aa(B,$o,aa(A,fun(B,$o),aTP_Lamp_he(A,fun(B,$o)),Uu),Uua)
    <=> $true ) ).

% ATP.lambda_1075
tff(fact_9248_ATP_Olambda__1076,axiom,
    ! [Uu: complex] : aa(complex,complex,aTP_Lamp_vn(complex,complex),Uu) = Uu ).

% ATP.lambda_1076
tff(fact_9249_ATP_Olambda__1077,axiom,
    ! [A: $tType] :
      ( type_len(A)
     => ! [Uu: word(A)] : aa(word(A),word(A),aTP_Lamp_ano(word(A),word(A)),Uu) = Uu ) ).

% ATP.lambda_1077
tff(fact_9250_ATP_Olambda__1078,axiom,
    ! [Uu: nat] : aa(nat,nat,aTP_Lamp_vc(nat,nat),Uu) = Uu ).

% ATP.lambda_1078
tff(fact_9251_ATP_Olambda__1079,axiom,
    ! [Uu: int] : aa(int,int,aTP_Lamp_vx(int,int),Uu) = Uu ).

% ATP.lambda_1079
tff(fact_9252_ATP_Olambda__1080,axiom,
    ! [B: $tType] :
      ( comm_monoid_mult(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_ani(B,B),Uu) = Uu ) ).

% ATP.lambda_1080
tff(fact_9253_ATP_Olambda__1081,axiom,
    ! [B: $tType] :
      ( comm_monoid_add(B)
     => ! [Uu: B] : aa(B,B,aTP_Lamp_anh(B,B),Uu) = Uu ) ).

% ATP.lambda_1081
tff(fact_9254_ATP_Olambda__1082,axiom,
    ! [A: $tType] :
      ( real_V3459762299906320749_field(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_apz(A,A),Uu) = Uu ) ).

% ATP.lambda_1082
tff(fact_9255_ATP_Olambda__1083,axiom,
    ! [A: $tType] :
      ( linorder(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_aks(A,A),Uu) = Uu ) ).

% ATP.lambda_1083
tff(fact_9256_ATP_Olambda__1084,axiom,
    ! [A: $tType] :
      ( monoid_mult(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_bo(A,A),Uu) = Uu ) ).

% ATP.lambda_1084
tff(fact_9257_ATP_Olambda__1085,axiom,
    ! [A: $tType,Uu: A] : aa(A,A,aTP_Lamp_dz(A,A),Uu) = Uu ).

% ATP.lambda_1085
tff(fact_9258_ATP_Olambda__1086,axiom,
    ! [C: $tType,B: $tType,Uu: C] : aa(C,set(B),aTP_Lamp_amy(C,set(B)),Uu) = bot_bot(set(B)) ).

% ATP.lambda_1086
tff(fact_9259_ATP_Olambda__1087,axiom,
    ! [A: $tType,D: $tType,Uu: A] : aa(A,set(D),aTP_Lamp_amz(A,set(D)),Uu) = bot_bot(set(D)) ).

% ATP.lambda_1087
tff(fact_9260_ATP_Olambda__1088,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topolo4958980785337419405_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_vh(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1088
tff(fact_9261_ATP_Olambda__1089,axiom,
    ! [A: $tType] :
      ( ( comm_monoid_add(A)
        & topological_t2_space(A) )
     => ! [Uu: nat] : aa(nat,A,aTP_Lamp_abn(nat,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1089
tff(fact_9262_ATP_Olambda__1090,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_add(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_tq(B,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1090
tff(fact_9263_ATP_Olambda__1091,axiom,
    ! [A: $tType] :
      ( mult_zero(A)
     => ! [Uu: A] : aa(A,A,aTP_Lamp_bw(A,A),Uu) = zero_zero(A) ) ).

% ATP.lambda_1091
tff(fact_9264_ATP_Olambda__1092,axiom,
    ! [A: $tType,B: $tType] :
      ( zero(B)
     => ! [Uu: A] : aa(A,B,aTP_Lamp_ajd(A,B),Uu) = zero_zero(B) ) ).

% ATP.lambda_1092
tff(fact_9265_ATP_Olambda__1093,axiom,
    ! [Uu: product_unit] : aa(product_unit,assn,aTP_Lamp_aim(product_unit,assn),Uu) = one_one(assn) ).

% ATP.lambda_1093
tff(fact_9266_ATP_Olambda__1094,axiom,
    ! [B: $tType,A: $tType] :
      ( comm_monoid_mult(A)
     => ! [Uu: B] : aa(B,A,aTP_Lamp_xg(B,A),Uu) = one_one(A) ) ).

% ATP.lambda_1094
tff(fact_9267_ATP_Olambda__1095,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_oy(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_1095
tff(fact_9268_ATP_Olambda__1096,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_rm(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_1096
tff(fact_9269_ATP_Olambda__1097,axiom,
    ! [Uu: heap_ext(product_unit)] : aa(heap_ext(product_unit),option(product_prod($o,product_prod(heap_ext(product_unit),nat))),aTP_Lamp_rr(heap_ext(product_unit),option(product_prod($o,product_prod(heap_ext(product_unit),nat)))),Uu) = none(product_prod($o,product_prod(heap_ext(product_unit),nat))) ).

% ATP.lambda_1097
tff(fact_9270_ATP_Olambda__1098,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_sd(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_1098
tff(fact_9271_ATP_Olambda__1099,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_pc(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_1099
tff(fact_9272_ATP_Olambda__1100,axiom,
    ! [B: $tType,A: $tType,Uu: B] : aa(B,option(A),aTP_Lamp_agx(B,option(A)),Uu) = none(A) ).

% ATP.lambda_1100
tff(fact_9273_ATP_Olambda__1101,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,option(B),aTP_Lamp_di(A,option(B)),Uu) = none(B) ).

% ATP.lambda_1101
tff(fact_9274_ATP_Olambda__1102,axiom,
    ! [A: $tType,B: $tType,Uu: A] : aa(A,B,aTP_Lamp_aiq(A,B),Uu) = undefined(B) ).

% ATP.lambda_1102
tff(fact_9275_ATP_Olambda__1103,axiom,
    ! [Uu: real] :
      ( aa(real,$o,aTP_Lamp_aeq(real,$o),Uu)
    <=> $false ) ).

% ATP.lambda_1103
tff(fact_9276_ATP_Olambda__1104,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_afc(nat,$o),Uu)
    <=> $false ) ).

% ATP.lambda_1104
tff(fact_9277_ATP_Olambda__1105,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_bm(A,$o),Uu)
    <=> $false ) ).

% ATP.lambda_1105
tff(fact_9278_ATP_Olambda__1106,axiom,
    ! [Uu: nat] :
      ( aa(nat,$o,aTP_Lamp_afb(nat,$o),Uu)
    <=> $true ) ).

% ATP.lambda_1106
tff(fact_9279_ATP_Olambda__1107,axiom,
    ! [A: $tType,Uu: A] :
      ( aa(A,$o,aTP_Lamp_bn(A,$o),Uu)
    <=> $true ) ).

% ATP.lambda_1107
tff(fact_9280_ATP_Olambda__1108,axiom,
    ! [A: $tType,Uu: A] : aa(A,fun(nat,nat),aTP_Lamp_dd(A,fun(nat,nat)),Uu) = suc ).

% ATP.lambda_1108

% Type constructors (1316)
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,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(heap_Time_Heap(A11)) ) ).

tff(tcon_Code__Numeral_Ointeger___Typerep_Otyperep_3,axiom,
    typerep(code_integer) ).

tff(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_4,axiom,
    typerep(code_term) ).

tff(tcon_Heap_Oheap_Oheap__ext___Typerep_Otyperep_5,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(heap_ext(A11)) ) ).

tff(tcon_Product__Type_Ounit___Typerep_Otyperep_6,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_Ounit___Finite__Set_Ofinite,axiom,
    finite_finite(product_unit) ).

tff(tcon_Product__Type_Oprod___Typerep_Otyperep_7,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( typerep(A11)
        & typerep(A12) )
     => typerep(product_prod(A11,A12)) ) ).

tff(tcon_Product__Type_Oprod___Enum_Oenum_8,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( enum(A11)
        & enum(A12) )
     => enum(product_prod(A11,A12)) ) ).

tff(tcon_Product__Type_Oprod___Finite__Set_Ofinite_9,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( finite_finite(A11)
        & finite_finite(A12) )
     => finite_finite(product_prod(A11,A12)) ) ).

tff(tcon_Numeral__Type_Onum1___Typerep_Otyperep_10,axiom,
    typerep(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Enum_Oenum_11,axiom,
    enum(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Finite__Set_Ofinite_12,axiom,
    finite_finite(numeral_num1) ).

tff(tcon_Numeral__Type_Obit1___Typerep_Otyperep_13,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Enum_Oenum_14,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => enum(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Finite__Set_Ofinite_15,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => finite_finite(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Typerep_Otyperep_16,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Enum_Oenum_17,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => enum(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Finite__Set_Ofinite_18,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => finite_finite(numeral_bit0(A11)) ) ).

tff(tcon_Multiset_Omultiset___Typerep_Otyperep_19,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(multiset(A11)) ) ).

tff(tcon_Extended__Nat_Oenat___Typerep_Otyperep_20,axiom,
    typerep(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder_21,axiom,
    comple5582772986160207858norder(extended_enat) ).

tff(tcon_Complex_Ocomplex___Typerep_Otyperep_22,axiom,
    typerep(complex) ).

tff(tcon_Assertions_Oassn___Typerep_Otyperep_23,axiom,
    typerep(assn) ).

tff(tcon_Uint32_Ouint32___Typerep_Otyperep_24,axiom,
    typerep(uint32) ).

tff(tcon_Option_Ooption___Typerep_Otyperep_25,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(option(A11)) ) ).

tff(tcon_Option_Ooption___Enum_Oenum_26,axiom,
    ! [A11: $tType] :
      ( enum(A11)
     => enum(option(A11)) ) ).

tff(tcon_Option_Ooption___Complete__Lattices_Ocomplete__linorder_27,axiom,
    ! [A11: $tType] :
      ( comple5582772986160207858norder(A11)
     => comple5582772986160207858norder(option(A11)) ) ).

tff(tcon_Option_Ooption___Finite__Set_Ofinite_28,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => finite_finite(option(A11)) ) ).

tff(tcon_Filter_Ofilter___Typerep_Otyperep_29,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(filter(A11)) ) ).

tff(tcon_Enum_Ofinite__3___Typerep_Otyperep_30,axiom,
    typerep(finite_3) ).

tff(tcon_Enum_Ofinite__3___Enum_Oenum_31,axiom,
    enum(finite_3) ).

tff(tcon_Enum_Ofinite__3___Complete__Lattices_Ocomplete__linorder_32,axiom,
    comple5582772986160207858norder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Finite__Set_Ofinite_33,axiom,
    finite_finite(finite_3) ).

tff(tcon_Enum_Ofinite__2___Typerep_Otyperep_34,axiom,
    typerep(finite_2) ).

tff(tcon_Enum_Ofinite__2___Enum_Oenum_35,axiom,
    enum(finite_2) ).

tff(tcon_Enum_Ofinite__2___Complete__Lattices_Ocomplete__linorder_36,axiom,
    comple5582772986160207858norder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Finite__Set_Ofinite_37,axiom,
    finite_finite(finite_2) ).

tff(tcon_Enum_Ofinite__1___Typerep_Otyperep_38,axiom,
    typerep(finite_1) ).

tff(tcon_Enum_Ofinite__1___Enum_Oenum_39,axiom,
    enum(finite_1) ).

tff(tcon_Enum_Ofinite__1___Complete__Lattices_Ocomplete__linorder_40,axiom,
    comple5582772986160207858norder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Finite__Set_Ofinite_41,axiom,
    finite_finite(finite_1) ).

tff(tcon_String_Ochar___Typerep_Otyperep_42,axiom,
    typerep(char) ).

tff(tcon_String_Ochar___Enum_Oenum_43,axiom,
    enum(char) ).

tff(tcon_String_Ochar___Finite__Set_Ofinite_44,axiom,
    finite_finite(char) ).

tff(tcon_Heap_Oarray___Typerep_Otyperep_45,axiom,
    ! [A11: $tType] : typerep(array(A11)) ).

tff(tcon_Word_Oword___Typerep_Otyperep_46,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(word(A11)) ) ).

tff(tcon_Word_Oword___Enum_Oenum_47,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => enum(word(A11)) ) ).

tff(tcon_Word_Oword___Finite__Set_Ofinite_48,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => finite_finite(word(A11)) ) ).

tff(tcon_Real_Oreal___Typerep_Otyperep_49,axiom,
    typerep(real) ).

tff(tcon_List_Olist___Typerep_Otyperep_50,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(list(A11)) ) ).

tff(tcon_Heap_Oref___Typerep_Otyperep_51,axiom,
    ! [A11: $tType] : typerep(ref(A11)) ).

tff(tcon_HOL_Obool___Typerep_Otyperep_52,axiom,
    typerep($o) ).

tff(tcon_HOL_Obool___Enum_Oenum_53,axiom,
    enum($o) ).

tff(tcon_HOL_Obool___Finite__Set_Ofinite_54,axiom,
    finite_finite($o) ).

tff(tcon_Set_Oset___Typerep_Otyperep_55,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => typerep(set(A11)) ) ).

tff(tcon_Set_Oset___Enum_Oenum_56,axiom,
    ! [A11: $tType] :
      ( enum(A11)
     => enum(set(A11)) ) ).

tff(tcon_Set_Oset___Finite__Set_Ofinite_57,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => finite_finite(set(A11)) ) ).

tff(tcon_Rat_Orat___Typerep_Otyperep_58,axiom,
    typerep(rat) ).

tff(tcon_Num_Onum___Typerep_Otyperep_59,axiom,
    typerep(num) ).

tff(tcon_Nat_Onat___Typerep_Otyperep_60,axiom,
    typerep(nat) ).

tff(tcon_Int_Oint___Typerep_Otyperep_61,axiom,
    typerep(int) ).

tff(tcon_fun___Typerep_Otyperep_62,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( typerep(A11)
        & typerep(A12) )
     => typerep(fun(A11,A12)) ) ).

tff(tcon_fun___Enum_Oenum_63,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( enum(A11)
        & enum(A12) )
     => enum(fun(A11,A12)) ) ).

tff(tcon_fun___Finite__Set_Ofinite_64,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( finite_finite(A11)
        & finite_finite(A12) )
     => finite_finite(fun(A11,A12)) ) ).

tff(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
    ! [A11: $tType,A12: $tType] :
      ( boolea8198339166811842893lgebra(A12)
     => boolea8198339166811842893lgebra(fun(A11,A12)) ) ).

tff(tcon_fun___Code__Evaluation_Oterm__of,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( typerep(A11)
        & typerep(A12) )
     => code_term_of(fun(A11,A12)) ) ).

tff(tcon_fun___Orderings_Oorder__top,axiom,
    ! [A11: $tType,A12: $tType] :
      ( order_top(A12)
     => order_top(fun(A11,A12)) ) ).

tff(tcon_fun___Orderings_Oorder__bot,axiom,
    ! [A11: $tType,A12: $tType] :
      ( order_bot(A12)
     => order_bot(fun(A11,A12)) ) ).

tff(tcon_fun___Orderings_Opreorder,axiom,
    ! [A11: $tType,A12: $tType] :
      ( preorder(A12)
     => preorder(fun(A11,A12)) ) ).

tff(tcon_fun___Orderings_Oorder,axiom,
    ! [A11: $tType,A12: $tType] :
      ( order(A12)
     => order(fun(A11,A12)) ) ).

tff(tcon_fun___Orderings_Oord,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ord(A12)
     => ord(fun(A11,A12)) ) ).

tff(tcon_fun___Groups_Ouminus,axiom,
    ! [A11: $tType,A12: $tType] :
      ( uminus(A12)
     => uminus(fun(A11,A12)) ) ).

tff(tcon_fun___Groups_Ominus,axiom,
    ! [A11: $tType,A12: $tType] :
      ( minus(A12)
     => minus(fun(A11,A12)) ) ).

tff(tcon_fun___HOL_Oequal,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( enum(A11)
        & cl_HOL_Oequal(A12) )
     => cl_HOL_Oequal(fun(A11,A12)) ) ).

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___Quickcheck__Narrowing_Opartial__term__of,axiom,
    quickc6926020345158392990erm_of(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___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___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___Groups_Ocancel__ab__semigroup__add,axiom,
    cancel2418104881723323429up_add(int) ).

tff(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
    ring_15535105094025558882visors(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___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___Groups_Oab__semigroup__add,axiom,
    ab_semigroup_add(int) ).

tff(tcon_Int_Oint___Generic__set__bit_Oset__bit,axiom,
    generic_set_set_bit(int) ).

tff(tcon_Int_Oint___Code__Evaluation_Oterm__of_65,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___Groups_Osemigroup__mult,axiom,
    semigroup_mult(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___Groups_Osemigroup__add,axiom,
    semigroup_add(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_66,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_67,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___Orderings_Oord_68,axiom,
    ord(int) ).

tff(tcon_Int_Oint___Groups_Ouminus_69,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___Groups_Ominus_70,axiom,
    minus(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_71,axiom,
    cl_HOL_Oequal(int) ).

tff(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_72,axiom,
    condit6923001295902523014norder(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_73,axiom,
    bit_un5681908812861735899ations(nat) ).

tff(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_74,axiom,
    semiri1453513574482234551roduct(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_75,axiom,
    euclid5411537665997757685th_nat(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_76,axiom,
    ordere1937475149494474687imp_le(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_77,axiom,
    euclid3128863361964157862miring(nat) ).

tff(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_78,axiom,
    euclid4440199948858584721cancel(nat) ).

tff(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_79,axiom,
    unique1627219031080169319umeral(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_80,axiom,
    semiri6575147826004484403cancel(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_81,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_82,axiom,
    ordere580206878836729694up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_83,axiom,
    ordere2412721322843649153imp_le(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_84,axiom,
    bit_se359711467146920520ations(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_85,axiom,
    linord2810124833399127020strict(nat) ).

tff(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_86,axiom,
    strict7427464778891057005id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_87,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_88,axiom,
    euclid3725896446679973847miring(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Otopological__space_89,axiom,
    topolo4958980785337419405_space(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_90,axiom,
    topolo1944317154257567458pology(nat) ).

tff(tcon_Nat_Onat___Quickcheck__Narrowing_Opartial__term__of_91,axiom,
    quickc6926020345158392990erm_of(nat) ).

tff(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_92,axiom,
    topolo5987344860129210374id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_93,axiom,
    linord4140545234300271783up_add(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_94,axiom,
    topolo2564578578187576103pology(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_95,axiom,
    semiri2026040879449505780visors(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_96,axiom,
    linord181362715937106298miring(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_97,axiom,
    linord8928482502909563296strict(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_98,axiom,
    semiri3467727345109120633visors(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_99,axiom,
    ordere6658533253407199908up_add(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_100,axiom,
    semiri6843258321239162965malize(nat) ).

tff(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_101,axiom,
    ordere6911136660526730532id_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_102,axiom,
    cancel2418104881723323429up_add(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_103,axiom,
    cancel1802427076303600483id_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_104,axiom,
    comm_s4317794764714335236cancel(nat) ).

tff(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_105,axiom,
    bit_semiring_bits(nat) ).

tff(tcon_Nat_Onat___Topological__Spaces_Ot2__space_106,axiom,
    topological_t2_space(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__comm__semiring_107,axiom,
    ordere2520102378445227354miring(nat) ).

tff(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_108,axiom,
    cancel_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semiring_109,axiom,
    linordered_semiring(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring__0_110,axiom,
    ordered_semiring_0(nat) ).

tff(tcon_Nat_Onat___Rings_Olinordered__semidom_111,axiom,
    linordered_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__mult_112,axiom,
    ab_semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1__cancel_113,axiom,
    semiring_1_cancel(nat) ).

tff(tcon_Nat_Onat___Rings_Oalgebraic__semidom_114,axiom,
    algebraic_semidom(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_115,axiom,
    comm_monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
    comm_monoid_diff(nat) ).

tff(tcon_Nat_Onat___Groups_Oab__semigroup__add_116,axiom,
    ab_semigroup_add(nat) ).

tff(tcon_Nat_Onat___Code__Evaluation_Oterm__of_117,axiom,
    code_term_of(nat) ).

tff(tcon_Nat_Onat___Rings_Oordered__semiring_118,axiom,
    ordered_semiring(nat) ).

tff(tcon_Nat_Onat___Parity_Osemiring__parity_119,axiom,
    semiring_parity(nat) ).

tff(tcon_Nat_Onat___Groups_Ocomm__monoid__add_120,axiom,
    comm_monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__modulo_121,axiom,
    semiring_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__1_122,axiom,
    comm_semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring__0_123,axiom,
    comm_semiring_0(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__mult_124,axiom,
    semigroup_mult(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__modulo_125,axiom,
    semidom_modulo(nat) ).

tff(tcon_Nat_Onat___Rings_Osemidom__divide_126,axiom,
    semidom_divide(nat) ).

tff(tcon_Nat_Onat___Num_Osemiring__numeral_127,axiom,
    semiring_numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Osemigroup__add_128,axiom,
    semigroup_add(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__less__one_129,axiom,
    zero_less_one(nat) ).

tff(tcon_Nat_Onat___Rings_Ocomm__semiring_130,axiom,
    comm_semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Owellorder,axiom,
    wellorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder__bot_131,axiom,
    order_bot(nat) ).

tff(tcon_Nat_Onat___Nat_Osemiring__char__0_132,axiom,
    semiring_char_0(nat) ).

tff(tcon_Nat_Onat___Rings_Ozero__neq__one_133,axiom,
    zero_neq_one(nat) ).

tff(tcon_Nat_Onat___Orderings_Opreorder_134,axiom,
    preorder(nat) ).

tff(tcon_Nat_Onat___Orderings_Olinorder_135,axiom,
    linorder(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__mult_136,axiom,
    monoid_mult(nat) ).

tff(tcon_Nat_Onat___Groups_Omonoid__add_137,axiom,
    monoid_add(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__1_138,axiom,
    semiring_1(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring__0_139,axiom,
    semiring_0(nat) ).

tff(tcon_Nat_Onat___Orderings_Ono__top_140,axiom,
    no_top(nat) ).

tff(tcon_Nat_Onat___GCD_Osemiring__gcd_141,axiom,
    semiring_gcd(nat) ).

tff(tcon_Nat_Onat___Rings_Omult__zero_142,axiom,
    mult_zero(nat) ).

tff(tcon_Nat_Onat___Orderings_Oorder_143,axiom,
    order(nat) ).

tff(tcon_Nat_Onat___Rings_Osemiring_144,axiom,
    semiring(nat) ).

tff(tcon_Nat_Onat___Orderings_Oord_145,axiom,
    ord(nat) ).

tff(tcon_Nat_Onat___Groups_Ominus_146,axiom,
    minus(nat) ).

tff(tcon_Nat_Onat___Power_Opower_147,axiom,
    power(nat) ).

tff(tcon_Nat_Onat___Num_Onumeral_148,axiom,
    numeral(nat) ).

tff(tcon_Nat_Onat___Groups_Ozero_149,axiom,
    zero(nat) ).

tff(tcon_Nat_Onat___Groups_Oplus_150,axiom,
    plus(nat) ).

tff(tcon_Nat_Onat___Groups_Oone_151,axiom,
    one(nat) ).

tff(tcon_Nat_Onat___Rings_Odvd_152,axiom,
    dvd(nat) ).

tff(tcon_Nat_Onat___Heap_Oheap_153,axiom,
    heap(nat) ).

tff(tcon_Nat_Onat___HOL_Oequal_154,axiom,
    cl_HOL_Oequal(nat) ).

tff(tcon_Nat_Onat___Nat_Osize,axiom,
    size(nat) ).

tff(tcon_Num_Onum___Quickcheck__Narrowing_Opartial__term__of_155,axiom,
    quickc6926020345158392990erm_of(num) ).

tff(tcon_Num_Onum___Code__Evaluation_Oterm__of_156,axiom,
    code_term_of(num) ).

tff(tcon_Num_Onum___Orderings_Opreorder_157,axiom,
    preorder(num) ).

tff(tcon_Num_Onum___Orderings_Olinorder_158,axiom,
    linorder(num) ).

tff(tcon_Num_Onum___Orderings_Oorder_159,axiom,
    order(num) ).

tff(tcon_Num_Onum___Orderings_Oord_160,axiom,
    ord(num) ).

tff(tcon_Num_Onum___Groups_Oplus_161,axiom,
    plus(num) ).

tff(tcon_Num_Onum___HOL_Oequal_162,axiom,
    cl_HOL_Oequal(num) ).

tff(tcon_Num_Onum___Nat_Osize_163,axiom,
    size(num) ).

tff(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_164,axiom,
    semiri1453513574482234551roduct(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_165,axiom,
    ordere1937475149494474687imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_166,axiom,
    semiri6575147826004484403cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_167,axiom,
    strict9044650504122735259up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_168,axiom,
    ordere580206878836729694up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_169,axiom,
    ordere2412721322843649153imp_le(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_170,axiom,
    linord2810124833399127020strict(rat) ).

tff(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_171,axiom,
    strict7427464778891057005id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_172,axiom,
    ordere8940638589300402666id_add(rat) ).

tff(tcon_Rat_Orat___Quickcheck__Narrowing_Opartial__term__of_173,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_174,axiom,
    linord715952674999750819strict(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_175,axiom,
    linord4140545234300271783up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_176,axiom,
    semiri2026040879449505780visors(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_177,axiom,
    linord181362715937106298miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_178,axiom,
    linord8928482502909563296strict(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_179,axiom,
    semiri3467727345109120633visors(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_180,axiom,
    ordere6658533253407199908up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_181,axiom,
    ordere166539214618696060dd_abs(rat) ).

tff(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
    archim2362893244070406136eiling(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_182,axiom,
    ordere6911136660526730532id_add(rat) ).

tff(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_183,axiom,
    linord5086331880401160121up_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_184,axiom,
    cancel2418104881723323429up_add(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_185,axiom,
    ring_15535105094025558882visors(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_186,axiom,
    cancel1802427076303600483id_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring__strict_187,axiom,
    linord4710134922213307826strict(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_188,axiom,
    comm_s4317794764714335236cancel(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__comm__semiring_189,axiom,
    ordere2520102378445227354miring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring__1_190,axiom,
    linord6961819062388156250ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Oordered__ab__group__add_191,axiom,
    ordered_ab_group_add(rat) ).

tff(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_192,axiom,
    cancel_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semiring_193,axiom,
    linordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring__0_194,axiom,
    ordered_semiring_0(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__semidom_195,axiom,
    linordered_semidom(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
    dense_linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__mult_196,axiom,
    ab_semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1__cancel_197,axiom,
    semiring_1_cancel(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_198,axiom,
    comm_monoid_mult(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__semigroup__add_199,axiom,
    ab_semigroup_add(rat) ).

tff(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
    linordered_field(rat) ).

tff(tcon_Rat_Orat___Code__Evaluation_Oterm__of_200,axiom,
    code_term_of(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__semiring_201,axiom,
    ordered_semiring(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring__abs_202,axiom,
    ordered_ring_abs(rat) ).

tff(tcon_Rat_Orat___Groups_Ocomm__monoid__add_203,axiom,
    comm_monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__ring_204,axiom,
    linordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Olinordered__idom_205,axiom,
    linordered_idom(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__1_206,axiom,
    comm_semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring__0_207,axiom,
    comm_semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Odense__order,axiom,
    dense_order(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__mult_208,axiom,
    semigroup_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Osemidom__divide_209,axiom,
    semidom_divide(rat) ).

tff(tcon_Rat_Orat___Num_Osemiring__numeral_210,axiom,
    semiring_numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Osemigroup__add_211,axiom,
    semigroup_add(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_212,axiom,
    zero_less_one(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__semiring_213,axiom,
    comm_semiring(rat) ).

tff(tcon_Rat_Orat___Nat_Osemiring__char__0_214,axiom,
    semiring_char_0(rat) ).

tff(tcon_Rat_Orat___Groups_Oab__group__add_215,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_216,axiom,
    zero_neq_one(rat) ).

tff(tcon_Rat_Orat___Rings_Oordered__ring_217,axiom,
    ordered_ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__abs__sgn_218,axiom,
    idom_abs_sgn(rat) ).

tff(tcon_Rat_Orat___Orderings_Opreorder_219,axiom,
    preorder(rat) ).

tff(tcon_Rat_Orat___Orderings_Olinorder_220,axiom,
    linorder(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__mult_221,axiom,
    monoid_mult(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom__divide_222,axiom,
    idom_divide(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring__1_223,axiom,
    comm_ring_1(rat) ).

tff(tcon_Rat_Orat___Groups_Omonoid__add_224,axiom,
    monoid_add(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__1_225,axiom,
    semiring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring__0_226,axiom,
    semiring_0(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__top_227,axiom,
    no_top(rat) ).

tff(tcon_Rat_Orat___Orderings_Ono__bot_228,axiom,
    no_bot(rat) ).

tff(tcon_Rat_Orat___Groups_Ogroup__add_229,axiom,
    group_add(rat) ).

tff(tcon_Rat_Orat___Rings_Omult__zero_230,axiom,
    mult_zero(rat) ).

tff(tcon_Rat_Orat___Rings_Ocomm__ring_231,axiom,
    comm_ring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oorder_232,axiom,
    order(rat) ).

tff(tcon_Rat_Orat___Num_Oneg__numeral_233,axiom,
    neg_numeral(rat) ).

tff(tcon_Rat_Orat___Nat_Oring__char__0_234,axiom,
    ring_char_0(rat) ).

tff(tcon_Rat_Orat___Rings_Osemiring_235,axiom,
    semiring(rat) ).

tff(tcon_Rat_Orat___Orderings_Oord_236,axiom,
    ord(rat) ).

tff(tcon_Rat_Orat___Groups_Ouminus_237,axiom,
    uminus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring__1_238,axiom,
    ring_1(rat) ).

tff(tcon_Rat_Orat___Rings_Oabs__if_239,axiom,
    abs_if(rat) ).

tff(tcon_Rat_Orat___Groups_Ominus_240,axiom,
    minus(rat) ).

tff(tcon_Rat_Orat___Fields_Ofield,axiom,
    field(rat) ).

tff(tcon_Rat_Orat___Power_Opower_241,axiom,
    power(rat) ).

tff(tcon_Rat_Orat___Num_Onumeral_242,axiom,
    numeral(rat) ).

tff(tcon_Rat_Orat___Groups_Ozero_243,axiom,
    zero(rat) ).

tff(tcon_Rat_Orat___Groups_Oplus_244,axiom,
    plus(rat) ).

tff(tcon_Rat_Orat___Rings_Oring_245,axiom,
    ring(rat) ).

tff(tcon_Rat_Orat___Rings_Oidom_246,axiom,
    idom(rat) ).

tff(tcon_Rat_Orat___Groups_Oone_247,axiom,
    one(rat) ).

tff(tcon_Rat_Orat___Rings_Odvd_248,axiom,
    dvd(rat) ).

tff(tcon_Rat_Orat___HOL_Oequal_249,axiom,
    cl_HOL_Oequal(rat) ).

tff(tcon_Set_Oset___Quickcheck__Narrowing_Opartial__term__of_250,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(set(A11)) ) ).

tff(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_251,axiom,
    ! [A11: $tType] : boolea8198339166811842893lgebra(set(A11)) ).

tff(tcon_Set_Oset___Code__Evaluation_Oterm__of_252,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(set(A11)) ) ).

tff(tcon_Set_Oset___Orderings_Oorder__top_253,axiom,
    ! [A11: $tType] : order_top(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oorder__bot_254,axiom,
    ! [A11: $tType] : order_bot(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Opreorder_255,axiom,
    ! [A11: $tType] : preorder(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oorder_256,axiom,
    ! [A11: $tType] : order(set(A11)) ).

tff(tcon_Set_Oset___Orderings_Oord_257,axiom,
    ! [A11: $tType] : ord(set(A11)) ).

tff(tcon_Set_Oset___Groups_Ouminus_258,axiom,
    ! [A11: $tType] : uminus(set(A11)) ).

tff(tcon_Set_Oset___Groups_Ominus_259,axiom,
    ! [A11: $tType] : minus(set(A11)) ).

tff(tcon_Set_Oset___HOL_Oequal_260,axiom,
    ! [A11: $tType] :
      ( cl_HOL_Oequal(A11)
     => cl_HOL_Oequal(set(A11)) ) ).

tff(tcon_HOL_Obool___Topological__Spaces_Otopological__space_261,axiom,
    topolo4958980785337419405_space($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_262,axiom,
    topolo1944317154257567458pology($o) ).

tff(tcon_HOL_Obool___Quickcheck__Narrowing_Opartial__term__of_263,axiom,
    quickc6926020345158392990erm_of($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_264,axiom,
    topolo2564578578187576103pology($o) ).

tff(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_265,axiom,
    boolea8198339166811842893lgebra($o) ).

tff(tcon_HOL_Obool___Topological__Spaces_Ot2__space_266,axiom,
    topological_t2_space($o) ).

tff(tcon_HOL_Obool___Code__Evaluation_Oterm__of_267,axiom,
    code_term_of($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__top_268,axiom,
    order_top($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder__bot_269,axiom,
    order_bot($o) ).

tff(tcon_HOL_Obool___Orderings_Opreorder_270,axiom,
    preorder($o) ).

tff(tcon_HOL_Obool___Orderings_Olinorder_271,axiom,
    linorder($o) ).

tff(tcon_HOL_Obool___Orderings_Oorder_272,axiom,
    order($o) ).

tff(tcon_HOL_Obool___Orderings_Oord_273,axiom,
    ord($o) ).

tff(tcon_HOL_Obool___Groups_Ouminus_274,axiom,
    uminus($o) ).

tff(tcon_HOL_Obool___Groups_Ominus_275,axiom,
    minus($o) ).

tff(tcon_HOL_Obool___Heap_Oheap_276,axiom,
    heap($o) ).

tff(tcon_HOL_Obool___HOL_Oequal_277,axiom,
    cl_HOL_Oequal($o) ).

tff(tcon_Heap_Oref___Quickcheck__Narrowing_Opartial__term__of_278,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(ref(A11)) ) ).

tff(tcon_Heap_Oref___Code__Evaluation_Oterm__of_279,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(ref(A11)) ) ).

tff(tcon_Heap_Oref___Heap_Oheap_280,axiom,
    ! [A11: $tType] : heap(ref(A11)) ).

tff(tcon_Heap_Oref___HOL_Oequal_281,axiom,
    ! [A11: $tType] : cl_HOL_Oequal(ref(A11)) ).

tff(tcon_Heap_Oref___Nat_Osize_282,axiom,
    ! [A11: $tType] : size(ref(A11)) ).

tff(tcon_List_Olist___Quickcheck__Narrowing_Opartial__term__of_283,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(list(A11)) ) ).

tff(tcon_List_Olist___Code__Evaluation_Oterm__of_284,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(list(A11)) ) ).

tff(tcon_List_Olist___Heap_Oheap_285,axiom,
    ! [A11: $tType] :
      ( heap(A11)
     => heap(list(A11)) ) ).

tff(tcon_List_Olist___HOL_Oequal_286,axiom,
    ! [A11: $tType] : cl_HOL_Oequal(list(A11)) ).

tff(tcon_List_Olist___Nat_Osize_287,axiom,
    ! [A11: $tType] : size(list(A11)) ).

tff(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_288,axiom,
    condit6923001295902523014norder(real) ).

tff(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_289,axiom,
    semiri1453513574482234551roduct(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_290,axiom,
    ordere1937475149494474687imp_le(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_291,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_292,axiom,
    strict9044650504122735259up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_293,axiom,
    ordere580206878836729694up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_294,axiom,
    ordere2412721322843649153imp_le(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_295,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_296,axiom,
    strict7427464778891057005id_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_297,axiom,
    ordere8940638589300402666id_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Otopological__space_298,axiom,
    topolo4958980785337419405_space(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_299,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_300,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_301,axiom,
    archim462609752435547400_field(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_302,axiom,
    linord715952674999750819strict(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_303,axiom,
    topolo5987344860129210374id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_304,axiom,
    linord4140545234300271783up_add(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_305,axiom,
    topolo2564578578187576103pology(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_306,axiom,
    semiri2026040879449505780visors(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_307,axiom,
    linord181362715937106298miring(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra__1,axiom,
    real_V2191834092415804123ebra_1(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_308,axiom,
    linord8928482502909563296strict(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_309,axiom,
    semiri3467727345109120633visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra,axiom,
    real_V6157519004096292374lgebra(real) ).

tff(tcon_Real_Oreal___Limits_Otopological__ab__group__add,axiom,
    topolo1287966508704411220up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_310,axiom,
    ordere6658533253407199908up_add(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_311,axiom,
    ordere166539214618696060dd_abs(real) ).

tff(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_312,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_313,axiom,
    ordere6911136660526730532id_add(real) ).

tff(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_314,axiom,
    linord5086331880401160121up_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_315,axiom,
    cancel2418104881723323429up_add(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_316,axiom,
    ring_15535105094025558882visors(real) ).

tff(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__field,axiom,
    real_V7773925162809079976_field(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_317,axiom,
    cancel1802427076303600483id_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring__strict_318,axiom,
    linord4710134922213307826strict(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_319,axiom,
    comm_s4317794764714335236cancel(real) ).

tff(tcon_Real_Oreal___Topological__Spaces_Ot2__space_320,axiom,
    topological_t2_space(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__comm__semiring_321,axiom,
    ordere2520102378445227354miring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring__1_322,axiom,
    linord6961819062388156250ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Oordered__ab__group__add_323,axiom,
    ordered_ab_group_add(real) ).

tff(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_324,axiom,
    cancel_semigroup_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semiring_325,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_326,axiom,
    ordered_semiring_0(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__semidom_327,axiom,
    linordered_semidom(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__linorder_328,axiom,
    dense_linorder(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__mult_329,axiom,
    ab_semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1__cancel_330,axiom,
    semiring_1_cancel(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_331,axiom,
    comm_monoid_mult(real) ).

tff(tcon_Real_Oreal___Groups_Oab__semigroup__add_332,axiom,
    ab_semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Olinordered__field_333,axiom,
    linordered_field(real) ).

tff(tcon_Real_Oreal___Code__Evaluation_Oterm__of_334,axiom,
    code_term_of(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__semiring_335,axiom,
    ordered_semiring(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring__abs_336,axiom,
    ordered_ring_abs(real) ).

tff(tcon_Real_Oreal___Groups_Ocomm__monoid__add_337,axiom,
    comm_monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__ring_338,axiom,
    linordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Olinordered__idom_339,axiom,
    linordered_idom(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__1_340,axiom,
    comm_semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring__0_341,axiom,
    comm_semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Odense__order_342,axiom,
    dense_order(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__mult_343,axiom,
    semigroup_mult(real) ).

tff(tcon_Real_Oreal___Rings_Osemidom__divide_344,axiom,
    semidom_divide(real) ).

tff(tcon_Real_Oreal___Num_Osemiring__numeral_345,axiom,
    semiring_numeral(real) ).

tff(tcon_Real_Oreal___Groups_Osemigroup__add_346,axiom,
    semigroup_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__abs__sgn_347,axiom,
    field_abs_sgn(real) ).

tff(tcon_Real_Oreal___Fields_Odivision__ring_348,axiom,
    division_ring(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__less__one_349,axiom,
    zero_less_one(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__semiring_350,axiom,
    comm_semiring(real) ).

tff(tcon_Real_Oreal___Nat_Osemiring__char__0_351,axiom,
    semiring_char_0(real) ).

tff(tcon_Real_Oreal___Groups_Oab__group__add_352,axiom,
    ab_group_add(real) ).

tff(tcon_Real_Oreal___Fields_Ofield__char__0_353,axiom,
    field_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Ozero__neq__one_354,axiom,
    zero_neq_one(real) ).

tff(tcon_Real_Oreal___Rings_Oordered__ring_355,axiom,
    ordered_ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__abs__sgn_356,axiom,
    idom_abs_sgn(real) ).

tff(tcon_Real_Oreal___Orderings_Opreorder_357,axiom,
    preorder(real) ).

tff(tcon_Real_Oreal___Orderings_Olinorder_358,axiom,
    linorder(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__mult_359,axiom,
    monoid_mult(real) ).

tff(tcon_Real_Oreal___Transcendental_Oln,axiom,
    ln(real) ).

tff(tcon_Real_Oreal___Rings_Oidom__divide_360,axiom,
    idom_divide(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring__1_361,axiom,
    comm_ring_1(real) ).

tff(tcon_Real_Oreal___Groups_Omonoid__add_362,axiom,
    monoid_add(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__1_363,axiom,
    semiring_1(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring__0_364,axiom,
    semiring_0(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__top_365,axiom,
    no_top(real) ).

tff(tcon_Real_Oreal___Orderings_Ono__bot_366,axiom,
    no_bot(real) ).

tff(tcon_Real_Oreal___Groups_Ogroup__add_367,axiom,
    group_add(real) ).

tff(tcon_Real_Oreal___Rings_Omult__zero_368,axiom,
    mult_zero(real) ).

tff(tcon_Real_Oreal___Rings_Ocomm__ring_369,axiom,
    comm_ring(real) ).

tff(tcon_Real_Oreal___Orderings_Oorder_370,axiom,
    order(real) ).

tff(tcon_Real_Oreal___Num_Oneg__numeral_371,axiom,
    neg_numeral(real) ).

tff(tcon_Real_Oreal___Nat_Oring__char__0_372,axiom,
    ring_char_0(real) ).

tff(tcon_Real_Oreal___Rings_Osemiring_373,axiom,
    semiring(real) ).

tff(tcon_Real_Oreal___Orderings_Oord_374,axiom,
    ord(real) ).

tff(tcon_Real_Oreal___Groups_Ouminus_375,axiom,
    uminus(real) ).

tff(tcon_Real_Oreal___Rings_Oring__1_376,axiom,
    ring_1(real) ).

tff(tcon_Real_Oreal___Rings_Oabs__if_377,axiom,
    abs_if(real) ).

tff(tcon_Real_Oreal___Groups_Ominus_378,axiom,
    minus(real) ).

tff(tcon_Real_Oreal___Fields_Ofield_379,axiom,
    field(real) ).

tff(tcon_Real_Oreal___Power_Opower_380,axiom,
    power(real) ).

tff(tcon_Real_Oreal___Num_Onumeral_381,axiom,
    numeral(real) ).

tff(tcon_Real_Oreal___Groups_Ozero_382,axiom,
    zero(real) ).

tff(tcon_Real_Oreal___Groups_Oplus_383,axiom,
    plus(real) ).

tff(tcon_Real_Oreal___Rings_Oring_384,axiom,
    ring(real) ).

tff(tcon_Real_Oreal___Rings_Oidom_385,axiom,
    idom(real) ).

tff(tcon_Real_Oreal___Groups_Oone_386,axiom,
    one(real) ).

tff(tcon_Real_Oreal___Rings_Odvd_387,axiom,
    dvd(real) ).

tff(tcon_Real_Oreal___HOL_Oequal_388,axiom,
    cl_HOL_Oequal(real) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bit__operations_389,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => bit_se359711467146920520ations(word(A11)) ) ).

tff(tcon_Word_Oword___Quickcheck__Narrowing_Opartial__term__of_390,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(word(A11)) ) ).

tff(tcon_Word_Oword___Bit__Comprehension_Obit__comprehension_391,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => bit_bi6583157726757044596ension(word(A11)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Oring__bit__operations_392,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => bit_ri3973907225187159222ations(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__ab__semigroup__add_393,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => cancel2418104881723323429up_add(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__comm__monoid__add_394,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => cancel1802427076303600483id_add(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1__cancel_395,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_s4317794764714335236cancel(word(A11)) ) ).

tff(tcon_Word_Oword___Bit__Operations_Osemiring__bits_396,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => bit_semiring_bits(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ocancel__semigroup__add_397,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => cancel_semigroup_add(word(A11)) ) ).

tff(tcon_Word_Oword___Least__significant__bit_Olsb_398,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => least_6119777620449941438nt_lsb(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__mult_399,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ab_semigroup_mult(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1__cancel_400,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_1_cancel(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__mult_401,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_monoid_mult(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Oab__semigroup__add_402,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ab_semigroup_add(word(A11)) ) ).

tff(tcon_Word_Oword___Generic__set__bit_Oset__bit_403,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => generic_set_set_bit(word(A11)) ) ).

tff(tcon_Word_Oword___Code__Evaluation_Oterm__of_404,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(word(A11)) ) ).

tff(tcon_Word_Oword___Parity_Osemiring__parity_405,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_parity(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ocomm__monoid__add_406,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_monoid_add(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__modulo_407,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_modulo(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__1_408,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_semiring_1(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring__0_409,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_semiring_0(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Osemigroup__mult_410,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semigroup_mult(word(A11)) ) ).

tff(tcon_Word_Oword___Num_Osemiring__numeral_411,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_numeral(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Osemigroup__add_412,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semigroup_add(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__semiring_413,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_semiring(word(A11)) ) ).

tff(tcon_Word_Oword___Orderings_Owellorder_414,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => wellorder(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Oab__group__add_415,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ab_group_add(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ozero__neq__one_416,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => zero_neq_one(word(A11)) ) ).

tff(tcon_Word_Oword___Parity_Oring__parity_417,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ring_parity(word(A11)) ) ).

tff(tcon_Word_Oword___Orderings_Opreorder_418,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => preorder(word(A11)) ) ).

tff(tcon_Word_Oword___Orderings_Olinorder_419,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => linorder(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__mult_420,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => monoid_mult(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring__1_421,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_ring_1(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Omonoid__add_422,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => monoid_add(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__1_423,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_1(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring__0_424,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring_0(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ogroup__add_425,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => group_add(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Omult__zero_426,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => mult_zero(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Ocomm__ring_427,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => comm_ring(word(A11)) ) ).

tff(tcon_Word_Oword___Orderings_Oorder_428,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => order(word(A11)) ) ).

tff(tcon_Word_Oword___Num_Oneg__numeral_429,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => neg_numeral(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Osemiring_430,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => semiring(word(A11)) ) ).

tff(tcon_Word_Oword___Orderings_Oord_431,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ord(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ouminus_432,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => uminus(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Oring__1_433,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ring_1(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ominus_434,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => minus(word(A11)) ) ).

tff(tcon_Word_Oword___Power_Opower_435,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => power(word(A11)) ) ).

tff(tcon_Word_Oword___Num_Onumeral_436,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => numeral(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Ozero_437,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => zero(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Oplus_438,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => plus(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Oring_439,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => ring(word(A11)) ) ).

tff(tcon_Word_Oword___Groups_Oone_440,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => one(word(A11)) ) ).

tff(tcon_Word_Oword___Rings_Odvd_441,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => dvd(word(A11)) ) ).

tff(tcon_Word_Oword___HOL_Oequal_442,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => cl_HOL_Oequal(word(A11)) ) ).

tff(tcon_Word_Oword___Nat_Osize_443,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => size(word(A11)) ) ).

tff(tcon_Heap_Oarray___Quickcheck__Narrowing_Opartial__term__of_444,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(array(A11)) ) ).

tff(tcon_Heap_Oarray___Code__Evaluation_Oterm__of_445,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(array(A11)) ) ).

tff(tcon_Heap_Oarray___Heap_Oheap_446,axiom,
    ! [A11: $tType] : heap(array(A11)) ).

tff(tcon_Heap_Oarray___HOL_Oequal_447,axiom,
    ! [A11: $tType] : cl_HOL_Oequal(array(A11)) ).

tff(tcon_Heap_Oarray___Nat_Osize_448,axiom,
    ! [A11: $tType] : size(array(A11)) ).

tff(tcon_String_Ochar___Quickcheck__Narrowing_Opartial__term__of_449,axiom,
    quickc6926020345158392990erm_of(char) ).

tff(tcon_String_Ochar___Code__Evaluation_Oterm__of_450,axiom,
    code_term_of(char) ).

tff(tcon_String_Ochar___Orderings_Opreorder_451,axiom,
    preorder(char) ).

tff(tcon_String_Ochar___Orderings_Olinorder_452,axiom,
    linorder(char) ).

tff(tcon_String_Ochar___Orderings_Oorder_453,axiom,
    order(char) ).

tff(tcon_String_Ochar___Orderings_Oord_454,axiom,
    ord(char) ).

tff(tcon_String_Ochar___Heap_Oheap_455,axiom,
    heap(char) ).

tff(tcon_String_Ochar___HOL_Oequal_456,axiom,
    cl_HOL_Oequal(char) ).

tff(tcon_String_Ochar___Nat_Osize_457,axiom,
    size(char) ).

tff(tcon_Enum_Ofinite__1___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_458,axiom,
    condit6923001295902523014norder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__monoid__add__imp__le_459,axiom,
    ordere1937475149494474687imp_le(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors__cancel_460,axiom,
    semiri6575147826004484403cancel(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__ab__semigroup__add_461,axiom,
    strict9044650504122735259up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__diff_462,axiom,
    ordere1170586879665033532d_diff(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__ab__semigroup__add_463,axiom,
    ordere580206878836729694up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add__imp__le_464,axiom,
    ordere2412721322843649153imp_le(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__comm__semiring__strict_465,axiom,
    linord2810124833399127020strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ostrict__ordered__comm__monoid__add_466,axiom,
    strict7427464778891057005id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__cancel__comm__monoid__add_467,axiom,
    ordere8940638589300402666id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocanonically__ordered__monoid__add_468,axiom,
    canoni5634975068530333245id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Quickcheck__Narrowing_Opartial__term__of_469,axiom,
    quickc6926020345158392990erm_of(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__semigroup__add_470,axiom,
    linord4140545234300271783up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring__strict_471,axiom,
    linord8928482502909563296strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Boolean__Algebras_Oboolean__algebra_472,axiom,
    boolea8198339166811842893lgebra(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__no__zero__divisors_473,axiom,
    semiri3467727345109120633visors(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__semigroup__add_474,axiom,
    ordere6658533253407199908up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add__abs_475,axiom,
    ordere166539214618696060dd_abs(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__comm__monoid__add_476,axiom,
    ordere6911136660526730532id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Olinordered__ab__group__add_477,axiom,
    linord5086331880401160121up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__ab__semigroup__add_478,axiom,
    cancel2418104881723323429up_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__comm__monoid__add_479,axiom,
    cancel1802427076303600483id_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__ring__strict_480,axiom,
    linord4710134922213307826strict(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__comm__semiring_481,axiom,
    ordere2520102378445227354miring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oordered__ab__group__add_482,axiom,
    ordered_ab_group_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocancel__semigroup__add_483,axiom,
    cancel_semigroup_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__semiring_484,axiom,
    linordered_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__semiring__0_485,axiom,
    ordered_semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Odense__linorder_486,axiom,
    dense_linorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__mult_487,axiom,
    ab_semigroup_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__mult_488,axiom,
    comm_monoid_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__diff_489,axiom,
    comm_monoid_diff(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__semigroup__add_490,axiom,
    ab_semigroup_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Code__Evaluation_Oterm__of_491,axiom,
    code_term_of(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__semiring_492,axiom,
    ordered_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__ring__abs_493,axiom,
    ordered_ring_abs(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ocomm__monoid__add_494,axiom,
    comm_monoid_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Olinordered__ring_495,axiom,
    linordered_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring__0_496,axiom,
    comm_semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Odense__order_497,axiom,
    dense_order(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Osemigroup__mult_498,axiom,
    semigroup_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Osemigroup__add_499,axiom,
    semigroup_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__semiring_500,axiom,
    comm_semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Owellorder_501,axiom,
    wellorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder__top_502,axiom,
    order_top(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder__bot_503,axiom,
    order_bot(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oab__group__add_504,axiom,
    ab_group_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oordered__ring_505,axiom,
    ordered_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Opreorder_506,axiom,
    preorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Olinorder_507,axiom,
    linorder(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Omonoid__mult_508,axiom,
    monoid_mult(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Omonoid__add_509,axiom,
    monoid_add(finite_1) ).

tff(tcon_Enum_Ofinite__1___Type__Length_Olen0,axiom,
    type_len0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring__0_510,axiom,
    semiring_0(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ogroup__add_511,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_512,axiom,
    mult_zero(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Ocomm__ring_513,axiom,
    comm_ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oorder_514,axiom,
    order(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Osemiring_515,axiom,
    semiring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Orderings_Oord_516,axiom,
    ord(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ouminus_517,axiom,
    uminus(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oabs__if_518,axiom,
    abs_if(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ominus_519,axiom,
    minus(finite_1) ).

tff(tcon_Enum_Ofinite__1___Power_Opower_520,axiom,
    power(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Ozero_521,axiom,
    zero(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oplus_522,axiom,
    plus(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Oring_523,axiom,
    ring(finite_1) ).

tff(tcon_Enum_Ofinite__1___Groups_Oone_524,axiom,
    one(finite_1) ).

tff(tcon_Enum_Ofinite__1___Rings_Odvd_525,axiom,
    dvd(finite_1) ).

tff(tcon_Enum_Ofinite__1___HOL_Oequal_526,axiom,
    cl_HOL_Oequal(finite_1) ).

tff(tcon_Enum_Ofinite__2___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_527,axiom,
    condit6923001295902523014norder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_528,axiom,
    semiri1453513574482234551roduct(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Ounique__euclidean__semiring_529,axiom,
    euclid3128863361964157862miring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring__cancel_530,axiom,
    euclid4440199948858584721cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors__cancel_531,axiom,
    semiri6575147826004484403cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Euclidean__Division_Oeuclidean__semiring_532,axiom,
    euclid3725896446679973847miring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Quickcheck__Narrowing_Opartial__term__of_533,axiom,
    quickc6926020345158392990erm_of(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1__no__zero__divisors_534,axiom,
    semiri2026040879449505780visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__no__zero__divisors_535,axiom,
    semiri3467727345109120633visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__ab__semigroup__add_536,axiom,
    cancel2418104881723323429up_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring__1__no__zero__divisors_537,axiom,
    ring_15535105094025558882visors(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__comm__monoid__add_538,axiom,
    cancel1802427076303600483id_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1__cancel_539,axiom,
    comm_s4317794764714335236cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocancel__semigroup__add_540,axiom,
    cancel_semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__mult_541,axiom,
    ab_semigroup_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1__cancel_542,axiom,
    semiring_1_cancel(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oalgebraic__semidom_543,axiom,
    algebraic_semidom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__mult_544,axiom,
    comm_monoid_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__semigroup__add_545,axiom,
    ab_semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Code__Evaluation_Oterm__of_546,axiom,
    code_term_of(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ocomm__monoid__add_547,axiom,
    comm_monoid_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__modulo_548,axiom,
    semiring_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__1_549,axiom,
    comm_semiring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring__0_550,axiom,
    comm_semiring_0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Osemigroup__mult_551,axiom,
    semigroup_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom__modulo_552,axiom,
    semidom_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemidom__divide_553,axiom,
    semidom_divide(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Osemiring__numeral_554,axiom,
    semiring_numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Osemigroup__add_555,axiom,
    semigroup_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Odivision__ring_556,axiom,
    division_ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__semiring_557,axiom,
    comm_semiring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Owellorder_558,axiom,
    wellorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder__top_559,axiom,
    order_top(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder__bot_560,axiom,
    order_bot(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oab__group__add_561,axiom,
    ab_group_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ozero__neq__one_562,axiom,
    zero_neq_one(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__abs__sgn_563,axiom,
    idom_abs_sgn(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Opreorder_564,axiom,
    preorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Olinorder_565,axiom,
    linorder(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Omonoid__mult_566,axiom,
    monoid_mult(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__modulo_567,axiom,
    idom_modulo(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom__divide_568,axiom,
    idom_divide(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__ring__1_569,axiom,
    comm_ring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Omonoid__add_570,axiom,
    monoid_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Type__Length_Olen0_571,axiom,
    type_len0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__1_572,axiom,
    semiring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring__0_573,axiom,
    semiring_0(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ogroup__add_574,axiom,
    group_add(finite_2) ).

tff(tcon_Enum_Ofinite__2___Type__Length_Olen_575,axiom,
    type_len(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Omult__zero_576,axiom,
    mult_zero(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Ocomm__ring_577,axiom,
    comm_ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oorder_578,axiom,
    order(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Oneg__numeral_579,axiom,
    neg_numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Osemiring_580,axiom,
    semiring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Orderings_Oord_581,axiom,
    ord(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ouminus_582,axiom,
    uminus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring__1_583,axiom,
    ring_1(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ominus_584,axiom,
    minus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Fields_Ofield_585,axiom,
    field(finite_2) ).

tff(tcon_Enum_Ofinite__2___Power_Opower_586,axiom,
    power(finite_2) ).

tff(tcon_Enum_Ofinite__2___Num_Onumeral_587,axiom,
    numeral(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Ozero_588,axiom,
    zero(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oplus_589,axiom,
    plus(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oring_590,axiom,
    ring(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Oidom_591,axiom,
    idom(finite_2) ).

tff(tcon_Enum_Ofinite__2___Groups_Oone_592,axiom,
    one(finite_2) ).

tff(tcon_Enum_Ofinite__2___Rings_Odvd_593,axiom,
    dvd(finite_2) ).

tff(tcon_Enum_Ofinite__2___HOL_Oequal_594,axiom,
    cl_HOL_Oequal(finite_2) ).

tff(tcon_Enum_Ofinite__3___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_595,axiom,
    condit6923001295902523014norder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_596,axiom,
    semiri1453513574482234551roduct(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Ounique__euclidean__semiring_597,axiom,
    euclid3128863361964157862miring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring__cancel_598,axiom,
    euclid4440199948858584721cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors__cancel_599,axiom,
    semiri6575147826004484403cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Euclidean__Division_Oeuclidean__semiring_600,axiom,
    euclid3725896446679973847miring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Quickcheck__Narrowing_Opartial__term__of_601,axiom,
    quickc6926020345158392990erm_of(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1__no__zero__divisors_602,axiom,
    semiri2026040879449505780visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__no__zero__divisors_603,axiom,
    semiri3467727345109120633visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__ab__semigroup__add_604,axiom,
    cancel2418104881723323429up_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring__1__no__zero__divisors_605,axiom,
    ring_15535105094025558882visors(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__comm__monoid__add_606,axiom,
    cancel1802427076303600483id_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1__cancel_607,axiom,
    comm_s4317794764714335236cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocancel__semigroup__add_608,axiom,
    cancel_semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__mult_609,axiom,
    ab_semigroup_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1__cancel_610,axiom,
    semiring_1_cancel(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oalgebraic__semidom_611,axiom,
    algebraic_semidom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__mult_612,axiom,
    comm_monoid_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__semigroup__add_613,axiom,
    ab_semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Code__Evaluation_Oterm__of_614,axiom,
    code_term_of(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ocomm__monoid__add_615,axiom,
    comm_monoid_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__modulo_616,axiom,
    semiring_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__1_617,axiom,
    comm_semiring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring__0_618,axiom,
    comm_semiring_0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Osemigroup__mult_619,axiom,
    semigroup_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom__modulo_620,axiom,
    semidom_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemidom__divide_621,axiom,
    semidom_divide(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Osemiring__numeral_622,axiom,
    semiring_numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Osemigroup__add_623,axiom,
    semigroup_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Odivision__ring_624,axiom,
    division_ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__semiring_625,axiom,
    comm_semiring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Owellorder_626,axiom,
    wellorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder__top_627,axiom,
    order_top(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder__bot_628,axiom,
    order_bot(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oab__group__add_629,axiom,
    ab_group_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ozero__neq__one_630,axiom,
    zero_neq_one(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__abs__sgn_631,axiom,
    idom_abs_sgn(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Opreorder_632,axiom,
    preorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Olinorder_633,axiom,
    linorder(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Omonoid__mult_634,axiom,
    monoid_mult(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__modulo_635,axiom,
    idom_modulo(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom__divide_636,axiom,
    idom_divide(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__ring__1_637,axiom,
    comm_ring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Omonoid__add_638,axiom,
    monoid_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Type__Length_Olen0_639,axiom,
    type_len0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__1_640,axiom,
    semiring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring__0_641,axiom,
    semiring_0(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ogroup__add_642,axiom,
    group_add(finite_3) ).

tff(tcon_Enum_Ofinite__3___Type__Length_Olen_643,axiom,
    type_len(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Omult__zero_644,axiom,
    mult_zero(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Ocomm__ring_645,axiom,
    comm_ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oorder_646,axiom,
    order(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Oneg__numeral_647,axiom,
    neg_numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Osemiring_648,axiom,
    semiring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Orderings_Oord_649,axiom,
    ord(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ouminus_650,axiom,
    uminus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring__1_651,axiom,
    ring_1(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ominus_652,axiom,
    minus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Fields_Ofield_653,axiom,
    field(finite_3) ).

tff(tcon_Enum_Ofinite__3___Power_Opower_654,axiom,
    power(finite_3) ).

tff(tcon_Enum_Ofinite__3___Num_Onumeral_655,axiom,
    numeral(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Ozero_656,axiom,
    zero(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oplus_657,axiom,
    plus(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oring_658,axiom,
    ring(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Oidom_659,axiom,
    idom(finite_3) ).

tff(tcon_Enum_Ofinite__3___Groups_Oone_660,axiom,
    one(finite_3) ).

tff(tcon_Enum_Ofinite__3___Rings_Odvd_661,axiom,
    dvd(finite_3) ).

tff(tcon_Enum_Ofinite__3___HOL_Oequal_662,axiom,
    cl_HOL_Oequal(finite_3) ).

tff(tcon_Filter_Ofilter___Quickcheck__Narrowing_Opartial__term__of_663,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(filter(A11)) ) ).

tff(tcon_Filter_Ofilter___Code__Evaluation_Oterm__of_664,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(filter(A11)) ) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__top_665,axiom,
    ! [A11: $tType] : order_top(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder__bot_666,axiom,
    ! [A11: $tType] : order_bot(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Opreorder_667,axiom,
    ! [A11: $tType] : preorder(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oorder_668,axiom,
    ! [A11: $tType] : order(filter(A11)) ).

tff(tcon_Filter_Ofilter___Orderings_Oord_669,axiom,
    ! [A11: $tType] : ord(filter(A11)) ).

tff(tcon_Filter_Ofilter___HOL_Oequal_670,axiom,
    ! [A11: $tType] :
      ( cl_HOL_Oequal(A11)
     => cl_HOL_Oequal(filter(A11)) ) ).

tff(tcon_Option_Ooption___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_671,axiom,
    ! [A11: $tType] :
      ( comple5582772986160207858norder(A11)
     => condit6923001295902523014norder(option(A11)) ) ).

tff(tcon_Option_Ooption___Quickcheck__Narrowing_Opartial__term__of_672,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(option(A11)) ) ).

tff(tcon_Option_Ooption___Code__Evaluation_Oterm__of_673,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Owellorder_674,axiom,
    ! [A11: $tType] :
      ( wellorder(A11)
     => wellorder(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__top_675,axiom,
    ! [A11: $tType] :
      ( order_top(A11)
     => order_top(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder__bot_676,axiom,
    ! [A11: $tType] :
      ( order(A11)
     => order_bot(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Opreorder_677,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => preorder(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Olinorder_678,axiom,
    ! [A11: $tType] :
      ( linorder(A11)
     => linorder(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Oorder_679,axiom,
    ! [A11: $tType] :
      ( order(A11)
     => order(option(A11)) ) ).

tff(tcon_Option_Ooption___Orderings_Oord_680,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => ord(option(A11)) ) ).

tff(tcon_Option_Ooption___Heap_Oheap_681,axiom,
    ! [A11: $tType] :
      ( heap(A11)
     => heap(option(A11)) ) ).

tff(tcon_Option_Ooption___HOL_Oequal_682,axiom,
    ! [A11: $tType] : cl_HOL_Oequal(option(A11)) ).

tff(tcon_Option_Ooption___Nat_Osize_683,axiom,
    ! [A11: $tType] : size(option(A11)) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bit__operations_684,axiom,
    bit_se359711467146920520ations(uint32) ).

tff(tcon_Uint32_Ouint32___Quickcheck__Narrowing_Opartial__term__of_685,axiom,
    quickc6926020345158392990erm_of(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Comprehension_Obit__comprehension_686,axiom,
    bit_bi6583157726757044596ension(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Oring__bit__operations_687,axiom,
    bit_ri3973907225187159222ations(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__ab__semigroup__add_688,axiom,
    cancel2418104881723323429up_add(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__comm__monoid__add_689,axiom,
    cancel1802427076303600483id_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1__cancel_690,axiom,
    comm_s4317794764714335236cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Bit__Operations_Osemiring__bits_691,axiom,
    bit_semiring_bits(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocancel__semigroup__add_692,axiom,
    cancel_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Least__significant__bit_Olsb_693,axiom,
    least_6119777620449941438nt_lsb(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__mult_694,axiom,
    ab_semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1__cancel_695,axiom,
    semiring_1_cancel(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__mult_696,axiom,
    comm_monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__semigroup__add_697,axiom,
    ab_semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Generic__set__bit_Oset__bit_698,axiom,
    generic_set_set_bit(uint32) ).

tff(tcon_Uint32_Ouint32___Code__Evaluation_Oterm__of_699,axiom,
    code_term_of(uint32) ).

tff(tcon_Uint32_Ouint32___Parity_Osemiring__parity_700,axiom,
    semiring_parity(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ocomm__monoid__add_701,axiom,
    comm_monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__modulo_702,axiom,
    semiring_modulo(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__1_703,axiom,
    comm_semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring__0_704,axiom,
    comm_semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Osemigroup__mult_705,axiom,
    semigroup_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Osemiring__numeral_706,axiom,
    semiring_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Osemigroup__add_707,axiom,
    semigroup_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__semiring_708,axiom,
    comm_semiring(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oab__group__add_709,axiom,
    ab_group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ozero__neq__one_710,axiom,
    zero_neq_one(uint32) ).

tff(tcon_Uint32_Ouint32___Parity_Oring__parity_711,axiom,
    ring_parity(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Opreorder_712,axiom,
    preorder(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Olinorder_713,axiom,
    linorder(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__mult_714,axiom,
    monoid_mult(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring__1_715,axiom,
    comm_ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Omonoid__add_716,axiom,
    monoid_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__1_717,axiom,
    semiring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring__0_718,axiom,
    semiring_0(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ogroup__add_719,axiom,
    group_add(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Omult__zero_720,axiom,
    mult_zero(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Ocomm__ring_721,axiom,
    comm_ring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oorder_722,axiom,
    order(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Oneg__numeral_723,axiom,
    neg_numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Osemiring_724,axiom,
    semiring(uint32) ).

tff(tcon_Uint32_Ouint32___Orderings_Oord_725,axiom,
    ord(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ouminus_726,axiom,
    uminus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring__1_727,axiom,
    ring_1(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ominus_728,axiom,
    minus(uint32) ).

tff(tcon_Uint32_Ouint32___Power_Opower_729,axiom,
    power(uint32) ).

tff(tcon_Uint32_Ouint32___Num_Onumeral_730,axiom,
    numeral(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Ozero_731,axiom,
    zero(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oplus_732,axiom,
    plus(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Oring_733,axiom,
    ring(uint32) ).

tff(tcon_Uint32_Ouint32___Groups_Oone_734,axiom,
    one(uint32) ).

tff(tcon_Uint32_Ouint32___Rings_Odvd_735,axiom,
    dvd(uint32) ).

tff(tcon_Uint32_Ouint32___HOL_Oequal_736,axiom,
    cl_HOL_Oequal(uint32) ).

tff(tcon_Uint32_Ouint32___Nat_Osize_737,axiom,
    size(uint32) ).

tff(tcon_Assertions_Oassn___Boolean__Algebras_Oboolean__algebra_738,axiom,
    boolea8198339166811842893lgebra(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oab__semigroup__mult_739,axiom,
    ab_semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ocomm__monoid__mult_740,axiom,
    comm_monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Groups_Osemigroup__mult_741,axiom,
    semigroup_mult(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__top_742,axiom,
    order_top(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder__bot_743,axiom,
    order_bot(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Opreorder_744,axiom,
    preorder(assn) ).

tff(tcon_Assertions_Oassn___Groups_Omonoid__mult_745,axiom,
    monoid_mult(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oorder_746,axiom,
    order(assn) ).

tff(tcon_Assertions_Oassn___Orderings_Oord_747,axiom,
    ord(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ouminus_748,axiom,
    uminus(assn) ).

tff(tcon_Assertions_Oassn___Groups_Ominus_749,axiom,
    minus(assn) ).

tff(tcon_Assertions_Oassn___Power_Opower_750,axiom,
    power(assn) ).

tff(tcon_Assertions_Oassn___Groups_Oone_751,axiom,
    one(assn) ).

tff(tcon_Assertions_Oassn___Rings_Odvd_752,axiom,
    dvd(assn) ).

tff(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_753,axiom,
    semiri1453513574482234551roduct(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_754,axiom,
    real_V8999393235501362500lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_755,axiom,
    real_V2822296259951069270ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_756,axiom,
    semiri6575147826004484403cancel(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_757,axiom,
    real_V4412858255891104859lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_758,axiom,
    real_V822414075346904944vector(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_759,axiom,
    topolo4958980785337419405_space(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_760,axiom,
    real_V3459762299906320749_field(complex) ).

tff(tcon_Complex_Ocomplex___Quickcheck__Narrowing_Opartial__term__of_761,axiom,
    quickc6926020345158392990erm_of(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_762,axiom,
    real_V5047593784448816457lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_763,axiom,
    topolo5987344860129210374id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_764,axiom,
    semiri2026040879449505780visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_765,axiom,
    real_V2191834092415804123ebra_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_766,axiom,
    semiri3467727345109120633visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_767,axiom,
    real_V6157519004096292374lgebra(complex) ).

tff(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_768,axiom,
    topolo1287966508704411220up_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_769,axiom,
    real_V4867850818363320053vector(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_770,axiom,
    cancel2418104881723323429up_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_771,axiom,
    ring_15535105094025558882visors(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_772,axiom,
    real_V7773925162809079976_field(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_773,axiom,
    cancel1802427076303600483id_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_774,axiom,
    comm_s4317794764714335236cancel(complex) ).

tff(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_775,axiom,
    topological_t2_space(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_776,axiom,
    cancel_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_777,axiom,
    real_Vector_banach(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_778,axiom,
    ab_semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_779,axiom,
    semiring_1_cancel(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_780,axiom,
    comm_monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_781,axiom,
    ab_semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Code__Evaluation_Oterm__of_782,axiom,
    code_term_of(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_783,axiom,
    comm_monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_784,axiom,
    comm_semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_785,axiom,
    comm_semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_786,axiom,
    semigroup_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemidom__divide_787,axiom,
    semidom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Num_Osemiring__numeral_788,axiom,
    semiring_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Osemigroup__add_789,axiom,
    semigroup_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_790,axiom,
    field_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Odivision__ring_791,axiom,
    division_ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_792,axiom,
    comm_semiring(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_793,axiom,
    semiring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oab__group__add_794,axiom,
    ab_group_add(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield__char__0_795,axiom,
    field_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_796,axiom,
    zero_neq_one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_797,axiom,
    idom_abs_sgn(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__mult_798,axiom,
    monoid_mult(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom__divide_799,axiom,
    idom_divide(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_800,axiom,
    comm_ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Omonoid__add_801,axiom,
    monoid_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__1_802,axiom,
    semiring_1(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring__0_803,axiom,
    semiring_0(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ogroup__add_804,axiom,
    group_add(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Omult__zero_805,axiom,
    mult_zero(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Ocomm__ring_806,axiom,
    comm_ring(complex) ).

tff(tcon_Complex_Ocomplex___Num_Oneg__numeral_807,axiom,
    neg_numeral(complex) ).

tff(tcon_Complex_Ocomplex___Nat_Oring__char__0_808,axiom,
    ring_char_0(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Osemiring_809,axiom,
    semiring(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ouminus_810,axiom,
    uminus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring__1_811,axiom,
    ring_1(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ominus_812,axiom,
    minus(complex) ).

tff(tcon_Complex_Ocomplex___Fields_Ofield_813,axiom,
    field(complex) ).

tff(tcon_Complex_Ocomplex___Power_Opower_814,axiom,
    power(complex) ).

tff(tcon_Complex_Ocomplex___Num_Onumeral_815,axiom,
    numeral(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Ozero_816,axiom,
    zero(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oplus_817,axiom,
    plus(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oring_818,axiom,
    ring(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Oidom_819,axiom,
    idom(complex) ).

tff(tcon_Complex_Ocomplex___Groups_Oone_820,axiom,
    one(complex) ).

tff(tcon_Complex_Ocomplex___Rings_Odvd_821,axiom,
    dvd(complex) ).

tff(tcon_Complex_Ocomplex___HOL_Oequal_822,axiom,
    cl_HOL_Oequal(complex) ).

tff(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_823,axiom,
    condit6923001295902523014norder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_824,axiom,
    strict9044650504122735259up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_825,axiom,
    strict7427464778891057005id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_826,axiom,
    canoni5634975068530333245id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Quickcheck__Narrowing_Opartial__term__of_827,axiom,
    quickc6926020345158392990erm_of(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_828,axiom,
    linord4140545234300271783up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_829,axiom,
    linord181362715937106298miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_830,axiom,
    semiri3467727345109120633visors(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_831,axiom,
    ordere6658533253407199908up_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_832,axiom,
    ordere6911136660526730532id_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_833,axiom,
    ordere2520102378445227354miring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_834,axiom,
    ab_semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_835,axiom,
    comm_monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_836,axiom,
    ab_semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Code__Evaluation_Oterm__of_837,axiom,
    code_term_of(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_838,axiom,
    ordered_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_839,axiom,
    comm_monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_840,axiom,
    comm_semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_841,axiom,
    comm_semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_842,axiom,
    semigroup_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_843,axiom,
    semiring_numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_844,axiom,
    semigroup_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_845,axiom,
    zero_less_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_846,axiom,
    comm_semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Owellorder_847,axiom,
    wellorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_848,axiom,
    order_top(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_849,axiom,
    order_bot(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_850,axiom,
    semiring_char_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_851,axiom,
    zero_neq_one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Opreorder_852,axiom,
    preorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Olinorder_853,axiom,
    linorder(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_854,axiom,
    monoid_mult(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_855,axiom,
    monoid_add(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_856,axiom,
    semiring_1(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_857,axiom,
    semiring_0(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Omult__zero_858,axiom,
    mult_zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oorder_859,axiom,
    order(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Osemiring_860,axiom,
    semiring(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Orderings_Oord_861,axiom,
    ord(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ominus_862,axiom,
    minus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Power_Opower_863,axiom,
    power(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Num_Onumeral_864,axiom,
    numeral(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Ozero_865,axiom,
    zero(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oplus_866,axiom,
    plus(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Groups_Oone_867,axiom,
    one(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___Rings_Odvd_868,axiom,
    dvd(extended_enat) ).

tff(tcon_Extended__Nat_Oenat___HOL_Oequal_869,axiom,
    cl_HOL_Oequal(extended_enat) ).

tff(tcon_Multiset_Omultiset___Quickcheck__Narrowing_Opartial__term__of_870,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Oordered__ab__semigroup__add_871,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => ordere6658533253407199908up_add(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__ab__semigroup__add_872,axiom,
    ! [A11: $tType] : cancel2418104881723323429up_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__comm__monoid__add_873,axiom,
    ! [A11: $tType] : cancel1802427076303600483id_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocancel__semigroup__add_874,axiom,
    ! [A11: $tType] : cancel_semigroup_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__diff_875,axiom,
    ! [A11: $tType] : comm_monoid_diff(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Oab__semigroup__add_876,axiom,
    ! [A11: $tType] : ab_semigroup_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Code__Evaluation_Oterm__of_877,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ocomm__monoid__add_878,axiom,
    ! [A11: $tType] : comm_monoid_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Osemigroup__add_879,axiom,
    ! [A11: $tType] : semigroup_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Orderings_Opreorder_880,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => preorder(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Omonoid__add_881,axiom,
    ! [A11: $tType] : monoid_add(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Orderings_Oorder_882,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => order(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Orderings_Oord_883,axiom,
    ! [A11: $tType] :
      ( preorder(A11)
     => ord(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Groups_Ominus_884,axiom,
    ! [A11: $tType] : minus(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Ozero_885,axiom,
    ! [A11: $tType] : zero(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___Groups_Oplus_886,axiom,
    ! [A11: $tType] : plus(multiset(A11)) ).

tff(tcon_Multiset_Omultiset___HOL_Oequal_887,axiom,
    ! [A11: $tType] :
      ( cl_HOL_Oequal(A11)
     => cl_HOL_Oequal(multiset(A11)) ) ).

tff(tcon_Multiset_Omultiset___Nat_Osize_888,axiom,
    ! [A11: $tType] : size(multiset(A11)) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__ab__semigroup__add_889,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel2418104881723323429up_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__comm__monoid__add_890,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel1802427076303600483id_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1__cancel_891,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_s4317794764714335236cancel(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocancel__semigroup__add_892,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel_semigroup_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__mult_893,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_semigroup_mult(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1__cancel_894,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_1_cancel(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__mult_895,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_monoid_mult(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__semigroup__add_896,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_semigroup_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Code__Evaluation_Oterm__of_897,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ocomm__monoid__add_898,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_monoid_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__1_899,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_semiring_1(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring__0_900,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_semiring_0(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Osemigroup__mult_901,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semigroup_mult(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Osemiring__numeral_902,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_numeral(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Osemigroup__add_903,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semigroup_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__semiring_904,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_semiring(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Owellorder_905,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => wellorder(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oab__group__add_906,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_group_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ozero__neq__one_907,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => zero_neq_one(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Opreorder_908,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => preorder(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Olinorder_909,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => linorder(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__mult_910,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => monoid_mult(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring__1_911,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_ring_1(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Omonoid__add_912,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => monoid_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen0_913,axiom,
    ! [A11: $tType] :
      ( type_len0(A11)
     => type_len0(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__1_914,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_1(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring__0_915,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_0(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ogroup__add_916,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => group_add(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Type__Length_Olen_917,axiom,
    ! [A11: $tType] :
      ( type_len(A11)
     => type_len(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Omult__zero_918,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => mult_zero(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Ocomm__ring_919,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_ring(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oorder_920,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => order(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Oneg__numeral_921,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => neg_numeral(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Osemiring_922,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Orderings_Oord_923,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ord(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ouminus_924,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => uminus(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring__1_925,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ring_1(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ominus_926,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => minus(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Power_Opower_927,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => power(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Num_Onumeral_928,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => numeral(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Ozero_929,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => zero(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oplus_930,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => plus(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Oring_931,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ring(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Groups_Oone_932,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => one(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___Rings_Odvd_933,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => dvd(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit0___HOL_Oequal_934,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cl_HOL_Oequal(numeral_bit0(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__ab__semigroup__add_935,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel2418104881723323429up_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__comm__monoid__add_936,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel1802427076303600483id_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1__cancel_937,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_s4317794764714335236cancel(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocancel__semigroup__add_938,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cancel_semigroup_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__mult_939,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_semigroup_mult(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1__cancel_940,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_1_cancel(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__mult_941,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_monoid_mult(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__semigroup__add_942,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_semigroup_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Code__Evaluation_Oterm__of_943,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ocomm__monoid__add_944,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_monoid_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__1_945,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_semiring_1(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring__0_946,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_semiring_0(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Osemigroup__mult_947,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semigroup_mult(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Osemiring__numeral_948,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_numeral(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Osemigroup__add_949,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semigroup_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__semiring_950,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_semiring(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Owellorder_951,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => wellorder(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oab__group__add_952,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ab_group_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ozero__neq__one_953,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => zero_neq_one(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Opreorder_954,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => preorder(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Olinorder_955,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => linorder(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__mult_956,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => monoid_mult(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring__1_957,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_ring_1(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Omonoid__add_958,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => monoid_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen0_959,axiom,
    ! [A11: $tType] :
      ( type_len0(A11)
     => type_len0(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__1_960,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_1(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring__0_961,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring_0(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ogroup__add_962,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => group_add(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Type__Length_Olen_963,axiom,
    ! [A11: $tType] :
      ( type_len0(A11)
     => type_len(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Omult__zero_964,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => mult_zero(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Ocomm__ring_965,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => comm_ring(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oorder_966,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => order(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Oneg__numeral_967,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => neg_numeral(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Osemiring_968,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => semiring(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Orderings_Oord_969,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ord(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ouminus_970,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => uminus(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring__1_971,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ring_1(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ominus_972,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => minus(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Power_Opower_973,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => power(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Num_Onumeral_974,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => numeral(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Ozero_975,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => zero(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oplus_976,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => plus(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Oring_977,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => ring(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Groups_Oone_978,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => one(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___Rings_Odvd_979,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => dvd(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Obit1___HOL_Oequal_980,axiom,
    ! [A11: $tType] :
      ( finite_finite(A11)
     => cl_HOL_Oequal(numeral_bit1(A11)) ) ).

tff(tcon_Numeral__Type_Onum1___Quickcheck__Narrowing_Opartial__term__of_981,axiom,
    quickc6926020345158392990erm_of(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__ab__semigroup__add_982,axiom,
    cancel2418104881723323429up_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__comm__monoid__add_983,axiom,
    cancel1802427076303600483id_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocancel__semigroup__add_984,axiom,
    cancel_semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__mult_985,axiom,
    ab_semigroup_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__mult_986,axiom,
    comm_monoid_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__semigroup__add_987,axiom,
    ab_semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Code__Evaluation_Oterm__of_988,axiom,
    code_term_of(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ocomm__monoid__add_989,axiom,
    comm_monoid_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring__0_990,axiom,
    comm_semiring_0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Osemigroup__mult_991,axiom,
    semigroup_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Osemigroup__add_992,axiom,
    semigroup_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__semiring_993,axiom,
    comm_semiring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Owellorder_994,axiom,
    wellorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oab__group__add_995,axiom,
    ab_group_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Opreorder_996,axiom,
    preorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Olinorder_997,axiom,
    linorder(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Omonoid__mult_998,axiom,
    monoid_mult(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Omonoid__add_999,axiom,
    monoid_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Type__Length_Olen0_1000,axiom,
    type_len0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Osemiring__0_1001,axiom,
    semiring_0(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ogroup__add_1002,axiom,
    group_add(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Type__Length_Olen_1003,axiom,
    type_len(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Omult__zero_1004,axiom,
    mult_zero(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Ocomm__ring_1005,axiom,
    comm_ring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Oorder_1006,axiom,
    order(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Osemiring_1007,axiom,
    semiring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Orderings_Oord_1008,axiom,
    ord(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ouminus_1009,axiom,
    uminus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ominus_1010,axiom,
    minus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Power_Opower_1011,axiom,
    power(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Num_Onumeral_1012,axiom,
    numeral(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Ozero_1013,axiom,
    zero(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oplus_1014,axiom,
    plus(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Oring_1015,axiom,
    ring(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Groups_Oone_1016,axiom,
    one(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___Rings_Odvd_1017,axiom,
    dvd(numeral_num1) ).

tff(tcon_Numeral__Type_Onum1___HOL_Oequal_1018,axiom,
    cl_HOL_Oequal(numeral_num1) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_1019,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( topolo4958980785337419405_space(A11)
        & topolo4958980785337419405_space(A12) )
     => topolo4958980785337419405_space(product_prod(A11,A12)) ) ).

tff(tcon_Product__Type_Oprod___Quickcheck__Narrowing_Opartial__term__of_1020,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( typerep(A11)
        & typerep(A12) )
     => quickc6926020345158392990erm_of(product_prod(A11,A12)) ) ).

tff(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_1021,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( topological_t2_space(A11)
        & topological_t2_space(A12) )
     => topological_t2_space(product_prod(A11,A12)) ) ).

tff(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_1022,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( typerep(A11)
        & typerep(A12) )
     => code_term_of(product_prod(A11,A12)) ) ).

tff(tcon_Product__Type_Oprod___Heap_Oheap_1023,axiom,
    ! [A11: $tType,A12: $tType] :
      ( ( heap(A11)
        & heap(A12) )
     => heap(product_prod(A11,A12)) ) ).

tff(tcon_Product__Type_Oprod___HOL_Oequal_1024,axiom,
    ! [A11: $tType,A12: $tType] : cl_HOL_Oequal(product_prod(A11,A12)) ).

tff(tcon_Product__Type_Oprod___Nat_Osize_1025,axiom,
    ! [A11: $tType,A12: $tType] : size(product_prod(A11,A12)) ).

tff(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_1026,axiom,
    condit6923001295902523014norder(product_unit) ).

tff(tcon_Product__Type_Ounit___Quickcheck__Narrowing_Opartial__term__of_1027,axiom,
    quickc6926020345158392990erm_of(product_unit) ).

tff(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_1028,axiom,
    boolea8198339166811842893lgebra(product_unit) ).

tff(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_1029,axiom,
    code_term_of(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Owellorder_1030,axiom,
    wellorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__top_1031,axiom,
    order_top(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder__bot_1032,axiom,
    order_bot(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Opreorder_1033,axiom,
    preorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Olinorder_1034,axiom,
    linorder(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oorder_1035,axiom,
    order(product_unit) ).

tff(tcon_Product__Type_Ounit___Orderings_Oord_1036,axiom,
    ord(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ouminus_1037,axiom,
    uminus(product_unit) ).

tff(tcon_Product__Type_Ounit___Groups_Ominus_1038,axiom,
    minus(product_unit) ).

tff(tcon_Product__Type_Ounit___Heap_Oheap_1039,axiom,
    heap(product_unit) ).

tff(tcon_Product__Type_Ounit___HOL_Oequal_1040,axiom,
    cl_HOL_Oequal(product_unit) ).

tff(tcon_Heap_Oheap_Oheap__ext___Quickcheck__Narrowing_Opartial__term__of_1041,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(heap_ext(A11)) ) ).

tff(tcon_Heap_Oheap_Oheap__ext___Code__Evaluation_Oterm__of_1042,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(heap_ext(A11)) ) ).

tff(tcon_Heap_Oheap_Oheap__ext___HOL_Oequal_1043,axiom,
    ! [A11: $tType] : cl_HOL_Oequal(heap_ext(A11)) ).

tff(tcon_Code__Evaluation_Oterm___Quickcheck__Narrowing_Opartial__term__of_1044,axiom,
    quickc6926020345158392990erm_of(code_term) ).

tff(tcon_Code__Evaluation_Oterm___Code__Evaluation_Oterm__of_1045,axiom,
    code_term_of(code_term) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_1046,axiom,
    bit_un5681908812861735899ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_1047,axiom,
    semiri1453513574482234551roduct(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_1048,axiom,
    euclid5411537665997757685th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_1049,axiom,
    euclid8789492081693882211th_nat(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_1050,axiom,
    ordere1937475149494474687imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_1051,axiom,
    euclid3128863361964157862miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_1052,axiom,
    euclid4440199948858584721cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_1053,axiom,
    unique1627219031080169319umeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_1054,axiom,
    euclid8851590272496341667cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_1055,axiom,
    semiri6575147826004484403cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_1056,axiom,
    strict9044650504122735259up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_1057,axiom,
    ordere580206878836729694up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_1058,axiom,
    ordere2412721322843649153imp_le(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_1059,axiom,
    bit_se359711467146920520ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_1060,axiom,
    linord2810124833399127020strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_1061,axiom,
    strict7427464778891057005id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_1062,axiom,
    ordere8940638589300402666id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_1063,axiom,
    euclid3725896446679973847miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Quickcheck__Narrowing_Opartial__term__of_1064,axiom,
    quickc6926020345158392990erm_of(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_1065,axiom,
    linord715952674999750819strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__semigroup__add_1066,axiom,
    linord4140545234300271783up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_1067,axiom,
    bit_ri3973907225187159222ations(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_1068,axiom,
    semiri2026040879449505780visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_1069,axiom,
    linord181362715937106298miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_1070,axiom,
    euclid5891614535332579305n_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_1071,axiom,
    linord8928482502909563296strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_1072,axiom,
    semiri3467727345109120633visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_1073,axiom,
    ordere6658533253407199908up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_1074,axiom,
    ordere166539214618696060dd_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_1075,axiom,
    ordere6911136660526730532id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_1076,axiom,
    linord5086331880401160121up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_1077,axiom,
    cancel2418104881723323429up_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_1078,axiom,
    ring_15535105094025558882visors(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_1079,axiom,
    cancel1802427076303600483id_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_1080,axiom,
    linord4710134922213307826strict(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_1081,axiom,
    comm_s4317794764714335236cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_1082,axiom,
    bit_semiring_bits(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_1083,axiom,
    ordere2520102378445227354miring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_1084,axiom,
    linord6961819062388156250ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_1085,axiom,
    ordered_ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_1086,axiom,
    cancel_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_1087,axiom,
    linordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Least__significant__bit_Olsb_1088,axiom,
    least_6119777620449941438nt_lsb(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_1089,axiom,
    ordered_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_1090,axiom,
    linordered_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_1091,axiom,
    ab_semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_1092,axiom,
    semiring_1_cancel(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_1093,axiom,
    algebraic_semidom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_1094,axiom,
    comm_monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_1095,axiom,
    ab_semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Generic__set__bit_Oset__bit_1096,axiom,
    generic_set_set_bit(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Code__Evaluation_Oterm__of_1097,axiom,
    code_term_of(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_1098,axiom,
    ordered_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_1099,axiom,
    ordered_ring_abs(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_1100,axiom,
    semiring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_1101,axiom,
    comm_monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_1102,axiom,
    semiring_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_1103,axiom,
    linordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_1104,axiom,
    linordered_idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_1105,axiom,
    comm_semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_1106,axiom,
    comm_semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_1107,axiom,
    semigroup_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_1108,axiom,
    semidom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_1109,axiom,
    semidom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_1110,axiom,
    semiring_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_1111,axiom,
    semigroup_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_1112,axiom,
    zero_less_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_1113,axiom,
    comm_semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_1114,axiom,
    semiring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_1115,axiom,
    ab_group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_1116,axiom,
    zero_neq_one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_1117,axiom,
    ordered_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_1118,axiom,
    idom_abs_sgn(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_1119,axiom,
    ring_parity(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_1120,axiom,
    preorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_1121,axiom,
    linorder(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_1122,axiom,
    monoid_mult(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_1123,axiom,
    idom_modulo(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_1124,axiom,
    idom_divide(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_1125,axiom,
    comm_ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_1126,axiom,
    monoid_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_1127,axiom,
    semiring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_1128,axiom,
    semiring_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_1129,axiom,
    group_add(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_1130,axiom,
    mult_zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_1131,axiom,
    comm_ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oorder_1132,axiom,
    order(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_1133,axiom,
    neg_numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_1134,axiom,
    ring_char_0(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Osemiring_1135,axiom,
    semiring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Orderings_Oord_1136,axiom,
    ord(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ouminus_1137,axiom,
    uminus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring__1_1138,axiom,
    ring_1(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_1139,axiom,
    abs_if(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ominus_1140,axiom,
    minus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Power_Opower_1141,axiom,
    power(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Num_Onumeral_1142,axiom,
    numeral(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Ozero_1143,axiom,
    zero(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oplus_1144,axiom,
    plus(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oring_1145,axiom,
    ring(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Oidom_1146,axiom,
    idom(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Groups_Oone_1147,axiom,
    one(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___Rings_Odvd_1148,axiom,
    dvd(code_integer) ).

tff(tcon_Code__Numeral_Ointeger___HOL_Oequal_1149,axiom,
    cl_HOL_Oequal(code_integer) ).

tff(tcon_Heap__Time__Monad_OHeap___Quickcheck__Narrowing_Opartial__term__of_1150,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => quickc6926020345158392990erm_of(heap_Time_Heap(A11)) ) ).

tff(tcon_Heap__Time__Monad_OHeap___Code__Evaluation_Oterm__of_1151,axiom,
    ! [A11: $tType] :
      ( typerep(A11)
     => code_term_of(heap_Time_Heap(A11)) ) ).

tff(tcon_Heap__Time__Monad_OHeap___HOL_Oequal_1152,axiom,
    ! [A11: $tType] : cl_HOL_Oequal(heap_Time_Heap(A11)) ).

tff(tcon_Heap__Time__Monad_OHeap___Nat_Osize_1153,axiom,
    ! [A11: $tType] : size(heap_Time_Heap(A11)) ).

tff(tcon_VEBT__Definitions_OVEBT___Quickcheck__Narrowing_Opartial__term__of_1154,axiom,
    quickc6926020345158392990erm_of(vEBT_VEBT) ).

tff(tcon_VEBT__Definitions_OVEBT___Code__Evaluation_Oterm__of_1155,axiom,
    code_term_of(vEBT_VEBT) ).

tff(tcon_VEBT__Definitions_OVEBT___HOL_Oequal_1156,axiom,
    cl_HOL_Oequal(vEBT_VEBT) ).

tff(tcon_VEBT__Definitions_OVEBT___Nat_Osize_1157,axiom,
    size(vEBT_VEBT) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Quickcheck__Narrowing_Opartial__term__of_1158,axiom,
    quickc6926020345158392990erm_of(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Code__Evaluation_Oterm__of_1159,axiom,
    code_term_of(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Heap_Oheap_1160,axiom,
    heap(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___HOL_Oequal_1161,axiom,
    cl_HOL_Oequal(vEBT_VEBTi) ).

tff(tcon_VEBT__BuildupMemImp_OVEBTi___Nat_Osize_1162,axiom,
    size(vEBT_VEBTi) ).

% Helper facts (3)
tff(help_fequal_2_1_T,axiom,
    ! [A: $tType,X9: A,Y: A] :
      ( ( X9 != Y )
      | aa(A,$o,aa(A,fun(A,$o),fequal(A),X9),Y) ) ).

tff(help_fequal_1_1_T,axiom,
    ! [A: $tType,X9: A,Y: A] :
      ( ~ aa(A,$o,aa(A,fun(A,$o),fequal(A),X9),Y)
      | ( X9 = Y ) ) ).

tff(help_fChoice_1_1_T,axiom,
    ! [A: $tType,P: fun(A,$o)] :
      ( aa(A,$o,P,fChoice(A,P))
      = ( ? [X7: A] : aa(A,$o,P,X7) ) ) ).

% Free types (2)
tff(tfree_0,hypothesis,
    semiring_1(a) ).

tff(tfree_1,hypothesis,
    linorder(b) ).

% Conjectures (3)
tff(conj_0,hypothesis,
    tia = vEBT_Nodei(x11,x12,x13,x14) ).

tff(conj_1,hypothesis,
    ( ( x11 = 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)) )
    & ( x12 = aa(nat,nat,suc,aa(nat,nat,suc,va)) ) ) ).

tff(conj_2,conjecture,
    hoare_hoare_triple(option(nat),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,summary),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)),heap_Time_bind(product_unit,option(nat),refine_Imp_assert($true),aTP_Lamp_ao(product_unit,heap_Time_Heap(option(nat)))),aTP_Lamp_aq(option(nat),assn)) ).